Prescribing scalar and boundary mean curvature on the three dimensional half sphere

Zindine Djadli  Andrea Malchiodi  Mohameden Ould Ahmedou

April 16, 2001

Abstract We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques, we prove some existence and compactness results.

1 Introduction

In this paper we study some equation arising in differential geometry, when the metric of a Riemannian manifold is conformally deformed. Precisely, is given a manifold with boundary (M, g) of dimension greater or equal than three; transforming the metric g into g′ = v^{\frac{4}{n−2}} g, where v is a smooth positive function, the scalar curvatures R_{g}, R_{g′} and the mean curvatures of the boundary h_{g}, h_{g′}, with respect to g and g′ respectively, are related by the formulas

\begin{cases}−4\frac{n−1}{n−2}Δ_{g}v + R_{g}v = R_{g′}v^{\frac{n+2}{n−2}} , & in M; \\ \frac{2}{n−2} \frac{∂v}{∂ν} + h_{g}v = h_{g′}v^{\frac{n}{n−2}} , & on ∂M, \end{cases} (1)

see e.g. [5]. In the above equation, ν denotes the outward unit vector perpendicular to ∂M, with respect to the metric g.

A problem arises naturally when looking at equation (1): assigned two functions K : M → ℝ and H : ∂M → ℝ, does exists a metric g′ conformally equivalent to g such that R_{g′} ≡ K and h_{g′} ≡ H? From equation (1), the problem is equivalent to finding a smooth positive solution v of the equation

\begin{cases}−4\frac{n−1}{n−2}Δ_{g}v + R_{g}v = Kv^{\frac{n+2}{n−2}} , & in M; \\ \frac{2}{n−2} \frac{∂v}{∂ν} + h_{g}v = Hv^{\frac{n}{n−2}} , & on ∂M. \end{cases} (2)

The requirement about the positivity of v is necessary for the metric g′ to be Riemannian. For the two-dimensional case, there are analogous equations involving exponential nonlinearities.

We are mainly interested in the so-called positive case, see [21], when the quadratic part of the Euler functional associated to (2) is positive definite.

A first criterion for existence of solutions of (2), and also a proof of regularity, was given by P. Cherrier, [17], who discovered the second equation in (1). He proved that if the energy of some test function is below an explicit threshold, then problem (2) admits a solution as a mountain pass critical point. Combining this criterion with the Positive Mass Theorem by R. Schoen and S.T. Yau [35], J. Escobar proved some existence results in the interesting particular case of constant K and H, see [21], [22]. He proved that almost every compact manifold with boundary can be conformally deformed so that its scalar curvature is 1 and the boundary is minimal, i.e. the mean curvature is zero. He also gives some results when H is a constant close to zero. More recently Z.C. Han and Y.Y. Li, see [25], [26], extended most of the results of J. Escobar to the case in which K ≡ 1 and H is any constant. They also prove a compactness results in the locally conformally flat manifolds with umbilic boundary.

We consider here the case of the standard half sphere S_{+}^{n} = {x ∈ ℝ^{n+1} : ∥x∥ = 1, x_{n+1} > 0} endowed with its standard metric g_{0}, and in particular the case n = 3; the functions K and H are now non constant and K will always assumed to be positive. We are thus reduced to find positive solutions v of the problem

\begin{cases}−4\frac{n−1}{n−2} Δv + n(n − 1) v = K(x) v^{\frac{n+2}{n−2}} , & in S_{+}^{n}; \\ \frac{2}{n−2} \frac{∂v}{∂ν} = H(x′) v^{\frac{n}{n−2}} , & on ∂S_{+}^{n}. \end{cases} (3)

Problem (3) is in some sense related to the well-known Scalar Curvature Problem on S^{n}

−4\frac{n − 1}{n − 2} Δv + n(n − 1) v = K(x) v^{\frac{n+2}{n−2}} , in S^{n}, (4)

to which much work has been devoted, see [3], [7], [8], [9], [11], [12], [14], [15], [16], [27], [29], [30], [36] and references therein. As for (4), also for problem (3) there are topological obstructions for existence of solutions, based on Kazdan-Warner type conditions, see [10] and also the proof of Proposition 8.7. Hence it is not expectable to solve problem (3) for all the functions K and H, and it is natural to impose some conditions on them.

We would like to point out the following features of the scalar curvature problem on S^{n} in lower dimensions, referring to the above-mentioned papers. For n = 2, non-converging Palais Smale sequences are characterized by the presence of just one bubble. Under generic assumptions on K, it turns out that when n = 3, 4, solutions of (4), or of some subcritical approximation, possess only isolated simple blow ups, see Section 4 for the definition. When n = 3 there is indeed just one blow up point, while for n = 4 blow ups may as well occur at more points.

We now discuss problem (3). For n = 2 (about the corresponding equation with exponential nonlinearities), P.L. Li and J.Q. Liu proved in [28] that compactness is lost along one bubble only, as in the case of the problem on the sphere. The only difference is that blow up can only occur at the boundary of S_{+}^{2}. For the case H ≡ 0 and positive K, they prove existence results which are in some sense reminiscent of those of [11], see also [14], [15].

In [31] Y.Y. Li considered the case of n = 3 and H ≡ 0. Under generic assumptions on K, he proved that blow ups are isolated simple and at only the boundary. He also stated that, as for (4) when n = 3, blow ups may occur at most one point. Actually the last statement is not true: we correct this here and we prove that (also for non constant H) blow ups, which are always isolated simple at the boundary, can be multiple, see Section 6. Hence the situation could be considered similar to that of (4) for n = 4. This fact, in the case of H ≡ 0, could be roughly explained as follows. Reflecting both K and v evenly to the whole S^{3}, one could study symmetric solutions of (4). The blow up analysis for the three dimensional case strongly relies on the differentiability of K at blow up points. This implies that blow ups of symmetric functions outside ∂S_{+}^{3}, which are multiple, are ruled out. This argument does not apply when blow up points are on ∂S_{+}^{3}, since the symmetric extension of K is not regular there. See Remark 6.7 for a more quantitative explanation of this fact.

For the case of any n, some results are proved in [4] when K and H are close to some constants; here we are extending some of those results for n = 3 without the close to constant conditions, see also Remark 7.2. In the paper [13] the case K ≡ 0 and H close to a positive constant is considered, for n ≥ 3. In the forthcoming paper [19] we will extend some of those results to the non-perturbative case.

Our first result is the following.

Theorem 1.1 Assume n = 3, let K : \overline{S_{+}^{3}} → ℝ be a positive C^{1} function and let H : ∂S_{+}^{3} → ℝ be of class C^{2}. Let φ : ∂S_{+}^{3} → ℝ be the function defined by

φ(x′) = 4π \sqrt{\frac{6}{K(x′)}} (\frac{π}{2} − arctan (H(x′) \sqrt{\frac{6}{K(x′)}})) , x′ ∈ ∂S_{+}^{3}. (5)

Suppose that for some point q ∈ ∂S_{+}^{3} the following condition holds

φ(q) = min_{∂S_{+}^{3}} φ; \frac{∂K}{∂ν} (q) < 0. (6)

Then there exists a positive solution of problem (3).

The proof of Theorem 1.1 relies on the study of the following subcritical approximation of equation (3)

\begin{cases}−4\frac{n−1}{n−2} Δu + n(n − 1) u = K u^{p}, & in S_{+}^{n}; \\ \frac{2}{n−2} \frac{∂u}{∂ν} = H u^{\frac{p+1}{2}} , & on ∂S_{+}^{n}. \end{cases} (7)

Here the exponent p is converging to \frac{n+2}{n−2} from below. As mentioned before, blow ups of equation (7) can occur only at the boundary of S_{+}^{3}. Nevertheless, if v_{p} denotes a mountain pass solution of (7) for p < \frac{n+2}{n−2}, condition (6) implies that \{v_{p}\}_{p} is uniformly bounded for p → \frac{n+2}{n−2}, and hence converges to a solution of (3). The function φ(x′) represents the blow up energy at a point x′ ∈ ∂S_{+}^{3} and plays a crucial role in the blow up analysis. Indeed, see Section 6, blow ups can occur only at critical points of φ. We note that when H is a constant function, critical points of φ coincide with critical points of K|_{∂S_{+}^{3}}, see also [31].

Under generic assumptions on K and H, ((K, H) ∈ A in the notation below), it is possible to give a complete description of the behavior of general solutions of (7) when p converges to \frac{n+2}{n−2}, and to deduce existence and compactness results for equation (3). The blow up analysis provides necessary conditions on these solutions, while the Implicit Function Theorem gives sufficient conditions for existence of solutions highly concentrating at some points of ∂S_{+}^{3}. In this way one can compute the total Leray Schauder degree of the solutions of (3) in the space C^{2,α}(\overline{S_{+}^{3}}), for some α ∈ (0, 1). Such a method has been used in [36] and [30] for problem (4) in dimensions 3 and 4 respectively.

To state our next result we need to introduce some notation, which considerably simplifies in the case H ≡ 0, see Remark 1.3. Given K ∈ C^{2}(\overline{S_{+}^{3}}) and H ∈ C^{2}(∂S_{+}^{3}), let φ ∈ C^{2}(∂S_{+}^{3}) be defined by formula (5), and set

F = \{q ∈ ∂S_{+}^{3} : ∇φ(q) = 0\}; F^{+(−)} = \{q ∈ ∂S_{+}^{3} : ∇′φ(q) = 0, \frac{∂K}{∂ν} (q) > 0 (< 0)\};

M_{K,H} = \{v ∈ C^{2,α}(\overline{S_{+}^{3}}) : v satisfies(3)\} .

Here ∇′ denotes the gradient of functions defined on ∂S_{+}^{n}.

For q ∈ ∂S_{+}^{3}, let π_{q} : S_{+}^{3} → ℝ_{+}^{3} denote the stereographic projection with pole −q. In π_{q}-stereographic coordinates, we define the function G_{q} : S_{+}^{3} → ℝ by

G_{q}(x) = (\frac{1 + |x|^{2}}{2} )^{\frac{1}{2}} \frac{1}{|x|}, x ∈ ℝ_{+}^{3}. (8)

The function G_{q} is the Green’s function for the conformal laplacian −8Δ + 6 on S^{3} with pole q. Define also ψ : ∂S_{+}^{3} → ℝ by

ψ(x′) = 1 + H(x′)\sqrt{\frac{6}{K(x′)}} (arctan (H(x′)\sqrt{\frac{6}{K(x′)}}) − \frac{π}{2} ) = 1 − \frac{H(x′)φ(x′)}{4π}. (9)

To each {q^{1}, …, q^{N}} ⊆ F \ F^{−}, N ≥ 1, we associate an N × N symmetric matrix M = M(q^{1}, …, q^{N}) defined by

\begin{cases}M_{jj} = \frac{∂K}{∂ν} (q^{j}) \frac{ψ(q^{j})}{K(q^{j})^{\frac{3}{2}}} , & j ∈ \{1, …, N\} \\ M_{lj} = −4\sqrt{2} \frac{G_{q^{l}} (q^{j})}{K(q^{l})^{\frac{1}{4}} K(q^{j})^{\frac{1}{4}}} , & l, j ∈ \{1, …, N\}, l ≠ j. \end{cases} (10)

Let ρ = ρ(q^{1}, …, q^{N}) denote the least eigenvalue of M. It has been first pointed out by A. Bahri, [6], see also [8], that when the interaction between different bubbles is of the same order as the self interaction, the function ρ for a matrix as in (10) plays a fundamental role in the theory of the critical points at infinity. For problem (3), such kind of phenomenon appears when n = 3.

Define the set A to be

A = \{(K, H) ∈ C^{2}(\overline{S_{+}^{3}}) × C^{2}(∂S_{+}^{3}) : K > 0, φ is a Morse function on ∂S_{+}^{3},

\frac{∂K}{∂ν} ≠ 0 on F, and ρ = ρ(q^{1}, …, q^{N}) ≠ 0, ∀ q^{1}, …, q^{N} ∈ F^{+}\}.

Let us observe that the condition (K, H) ∈ A is generic. We introduce an integer valued continuous function Index : A → ℕ by the following formula

Index(K, H) = −1 + ∑_{j=1}^{ℓ} ∑_{1≤i_{1}<i_{2}≤···≤i_{j}≤ℓ^{ρ(q^{i_{1}} ,...q^{i_{j}} )>0,}}(−1)^{j+∑_{l=1}^{j}(2−m(φ,q^{j}))},

where m(φ, q^{i_{l}}) denotes the Morse index of φ at q^{i_{l}}, and ℓ = card |F^{+}|. Now we are able to state our next result, about existence and compactness of solutions of (3).

Theorem 1.2 Let n = 3 and suppose (K, H) ∈ A. Then for all α ∈ (0, 1), there exists some constant R depending only on min\overline{S_{+}^{3}} K, ∥K∥_{C^{1}(\overline{S_{+}^{3}})}, ∥H∥_{C^{2}(S_{+}^{3})}, min\{|ρ(q^{1}, …, q^{N})| : q^{1}, …, q^{N} ∈ F^{+}, N ≥ 2\} and α such that

\frac{1}{R} ≤ v ≤ R, ∥v∥_{C^{2,α}(\overline{S_{+}^{3}})} ≤ R, (11)

for all positive solutions v of equation (3). Moreover (3) possesses a solution provided Index(K, H) ≠ 0.

Notice that only the least eigenvalue of M(q^{1}, …, q^{N}) plays a role in counting the total degree of solutions of (3) and in the compactness. For instance, considering a continuous family of functions (K_{t}, H_{t}), the total degree changes when the least eigenvalue of M_{t}(q^{1}, …, q^{N}) crosses zero, while it remains unchanged when other eigenvalues cross zero.

Remark 1.3 (a) When the function H is identically equal to zero, the functions φ(x′) and ψ(x′) assume the simpler form

φ(x′) = 2π^{2}\sqrt{\frac{6}{K(x′)}}; ψ(x′) = 1.

In particular minima of φ coincide with maxima of K restricted to the boundary and viceversa. (b) We note that equation (3) for n = 3 is invariant under the rescaling

K → γK;  H → γ^{\frac{1}{2}} H;  u → γ^{− \frac{1}{4}} u,

where γ is any positive number. The hypotheses involving K and H in Theorems 1.1 and 1.2 are both invariant under such a rescaling.

(c) While Theorem 1.2 is related to some known results for equation (4), Theorem 1.1 has no counterpart in the problem on the whole sphere. The existence argument is strictly related to the presence of the boundary.

The authors have been recently informed about some related results obtained in [23].

The paper is organized as follows. In Section 2 we collect some useful technical tools, while in Section 3 we compute the blow up energies, depending on the values of K and H at the blow up point. In section 4 we recall some known facts about blow up analysis of equations (3) and (4), and in section 5 we specialize to the case of boundary blow ups. Then in Section 6 we prove that blow ups are isolated simple, see Definition 4.5, and occur only at the boundary of ∂S_{+}^{3}. Finally, Sections 7 and 8 are devoted to the proofs of Theorems 1.1 and 1.2 respectively.

Acknowledgments

This work was first began while Z. D. enjoyed the hospitality of SISSA at Trieste. He would like to thank A. Ambrosetti for the invitation. Part of it was accomplished when Z. D. was visiting the Mathematical Sciences Research Institute (MSRI) at Berkeley. His research at MSRI is supported in part by NSF grant DMS-9701755. A. M. is supported by a Fulbright fellowship for the academic year 2000-2001 and by M.U.R.S.T., under the project Variational Methods and Nonlinear Differential Equations. M.O.A.’s research is supported by a S.I.S.S.A. postdoctoral fellowship.

2 Some preliminaries

We will use the notation x for variables belonging to the half sphere S_{+}^{n}, or to the half space ℝ_{+}^{n}, which is defined by ℝ_{+}^{n} := {x ∈ ℝ^{n} : x_{n} > 0}; variables in both the boundaries will be denoted in general by x′.

Solutions of problem (3) are critical points of the Euler functional J_{K,H} : H^{1}(S_{+}^{n}) → ℝ defined in the following way

J_{K,H}(v) = \frac{1}{2} ∫_{S_{+}^{n}} (\frac{4(n − 1)}{n − 2}|∇v|^{2} + n(n − 1) v^{2}) − \frac{1}{2^{∗}} ∫_{S_{+}^{n}} K(x) |v|^{2^{∗}} − (n − 2) ∫_{∂S_{+}^{n}} H(x′) |v|^{2 \frac{n−1}{n−2}} ,

v ∈ H^{1}(S_{+}^{n}), (12)

where 2^{∗} = \frac{2n}{n−2}. It will be convenient to perform some stereographic projection in order to reduce the above problem to ℝ_{+}^{n}. Let D^{1,2}(ℝ_{+}^{n}) denote the completion of C_{c}^{∞} (\overline{ℝ_{+}^{n}}) with respect to the Dirichlet norm. The stereographic projection π_{q} through a point q ∈ ∂S_{+}^{n} induces an isometry ι : H^{1}(S_{+}^{n}) → D^{1,2}(ℝ_{+}^{n}) according to the following formula

(ι v)(x) = (\frac{2}{1 + |x|^{2}} )^{\frac{n−2}{2}} v(π_{q}^{−1}(x)), v ∈ H^{1}(S_{+}^{n}), x ∈ ℝ_{+}^{n}. (13)

In particular, one can check that the following relations hold true, for every v ∈ H^{1}(S_{+}^{n})

∫_{S_{+}^{n}} (\frac{4(n − 1)}{n − 2}|∇v|^{2} + n(n − 1) v^{2}) = ∫_{ℝ_{+}^{n}} \frac{4(n − 1)}{n − 2}|∇(ι v)|^{2};

∫_{S_{+}^{n}} |v|^{2^{∗}} = ∫_{ℝ_{+}^{n}} |ι v|^{2^{∗}}; ∫_{∂S_{+}^{n}} |v|^{2 \frac{n−1}{n−2}} = ∫_{∂ℝ_{+}^{n}} |ι v|^{2 \frac{n−1}{n−2}} .

By means of these equations, the functional J_{K,H} transforms into I_{K,H} : D^{1,2}(ℝ_{+}^{n}) → ℝ given by

I_{K,H}(u) = \frac{1}{2} ∫_{ℝ_{+}^{n}} \frac{4(n − 1)}{n − 2}|∇u|^{2} − \frac{1}{2^{∗}} ∫_{ℝ_{+}^{n}} K(x) |u|^{2^{∗}} − (n − 2) ∫_{∂ℝ_{+}^{n}} H(x′) |u|^{2 \frac{n−1}{n−2}} ,

meaning that J_{K,H}(v) = I_{K,H}(ι v) for every v ∈ H^{1}(S_{+}^{n}). Here we are identifying the functions K and H and their compositions with the stereographic projection π_{q}. This fact will be assumed as understood in the sequel.

Critical points of the functional I_{K,H} are solutions of the following problem

\begin{cases}−4\frac{n−1}{n−2} Δu = K(x) u^{\frac{n+2}{n−2}} , & in ℝ_{+}^{n}; \\ −\frac{2}{n−2} \frac{∂u}{∂x_{n}} = H(x′) u^{\frac{n}{n−2}} , & on ∂ℝ_{+}^{n}. \end{cases} (14)

Using similar arguments, one finds that the counterpart in ℝ_{+}^{n} of equation (7) is given by

\begin{cases}−4\frac{n−1}{n−2} Δu = W (x)^{τ} K(x) u^{p}, & in ℝ_{+}^{n}; \\ −\frac{2}{n−2} \frac{∂u}{∂x_{n}} = W (x′)^{\frac{τ}{2}} H(x′) u^{\frac{p+1}{2}} , & on ∂ℝ_{+}^{n}, \end{cases} (15)

where W (x) = (\frac{2}{1+|x|^{2}} )^{\frac{n−2}{2}} , and where τ = \frac{n+2}{n−2} − p. The terms W (x)^{τ} and W (x′)^{\frac{τ}{2}} in the equation

above are corrections due to the non conformality of equation (7) when p ≠ \frac{n+2}{n−2}.

As a typical feature of non compact variational problems like (14), it is fundamental to analyze the associated problems at infinity. Solutions of such problems describe the asymptotic profile of nonconverging Palais Smale sequences. In the specific case of (14), these problems at infinity are of two kinds, namely

−4\frac{n − 1}{n − 2} Δu = K(\overline{x}) u^{\frac{n+2}{n−2}} , in ℝ^{n}, (16)

for some fixed \overline{x} ∈ \overline{ℝ_{+}^{n}}, and

\begin{cases}−4\frac{n−1}{n−2} Δu = K(\overline{x}′) u^{\frac{n+2}{n−2}} , & in ℝ_{+}^{n}; \\ −\frac{2}{n−2} \frac{∂u}{∂x_{n}} = H(\overline{x}′) u^{\frac{n}{n−2}} , & on ∂ℝ_{+}^{n}, \end{cases} (17)

for some \overline{x}′ ∈ ∂ℝ_{+}^{n}. Roughly, problem (16) corresponds to the case in which the functions are mostly concentrated in the interior of ℝ_{+}^{n}, while problem (17) corresponds to the case in which the functions are concentrated near the boundary. We note that solutions of problem (17) are critical points of the functional

I_{\overline{K},\overline{H}}(u) = \frac{1}{2} ∫_{ℝ_{+}^{n}} \frac{4(n − 1)}{n − 2}|∇u|^{2} − \frac{1}{2^{∗}} \overline{K} ∫_{ℝ_{+}^{n}} |u|^{2^{∗}} − (n − 2)\overline{H} ∫_{∂ℝ_{+}^{n}} |u|^{2 \frac{n−1}{n−2}} ,

where \overline{K} = K(\overline{x}′) and \overline{H} = H(\overline{x}′).

Positive solutions of problems (16) and (17) have been completely classified in [24] and [33], see also [20]; we recall the results in the following Lemma.

Lemma 2.1 The positive solutions u of problems (16) and (17) are, modulo translations in ℝ^{n} and ℝ^{n−1}, respectively of the form

U_{λ}(x) = (\frac{λ}{1 + λ^{2}k|x|^{2}} )^{\frac{n−2}{2}} ; λ > 0, (18)

where k = \frac{K(\overline{x})}{4n(n−1)}, and

\overline{U}_{λ}(x) = (\frac{λ}{1 + λ^{2}k (|x′|^{2} + (x_{n} + t_{λ})^{2})})^{\frac{n−2}{2}} ; λ > 0, (19)

where k = \frac{K(\overline{x}′)}{4n(n−1)} and where t_{λ} is given by 2 k t_{λ} λ = H(\overline{x}′).

It will be convenient to consider the expression ∫_{ℝ_{+}^{n}} x_{n}\overline{U} 1^{\frac{2n}{n−2}} with n = 3. Using the formula

∫_{0}^{∞} \frac{r^{α}}{(1 + r^{2})^{β}} = \frac{Γ (\frac{α+1}{2}) Γ (β − \frac{α+1}{2})}{2Γ (β)},

and integrating first in the variable x′ and then in the variable x_{3}, one obtains

∫_{ℝ_{+}^{3}} x_{3}\overline{U}_{1}^{6} = \frac{144π}{K(\overline{x}′)^{2}} [1 + H(\overline{x}′)\sqrt{\frac{6}{K(\overline{x}′)}} (arctan (H(\overline{x}′)\sqrt{\frac{6}{K(\overline{x}′)}} − \frac{π}{2} ))] = \frac{144π}{K^{2}(\overline{x}′)} ψ(\overline{x}′). (20)

The projection to ℝ_{+}^{n} is not the only transformation we will perform. In the next section we will use a conformal transformation of S_{+}^{n} onto some suitable spherical cap Σ_{θ}. In a similar way as before, this transformation induces an isometry between H^{1}(S^{n}) and H^{1}(Σ_{θ}). We will not write the explicit formulas for this transformation, as in (13).

3 Study of the blow up energies

In this section we compute the energies of the solutions of problems (16) and (17) (i.e. of the functions given in (18) and (19)), highlighting the dependence on the values of K (and H) at \overline{x} (resp. at \overline{x}′). It is well known that these energies are strongly related to those of non converging Palais Smale sequences of the functionals J_{K,H} and I_{K,H}.

In order to simplify the computations it is convenient, using a suitable stereographic transformations, to reduce problems (16) and (17) to S^{n} and to some spherical cap respectively, see [26]. In the following ω_{d} denotes the volume of the unit d-dimensional sphere in ℝ^{d+1}.

Given θ ∈ (0, π), we define the spherical cap Σ_{θ} in the following way

Σ_{θ} = {x ∈ S^{n} : x_{n+1} ≥ cos θ}.

One can find with elementary computations that the mean curvature of ∂Σ_{θ} with respect to Σ_{θ} endowed with the standard metric g_{0} is given by

h_{g_{0}}(∂Σ_{θ}, Σ_{θ})(x′) = h_{θ} := \frac{cos θ}{sin θ} , for all x ∈ ∂Σ_{θ}.

We set for brevity \overline{K} = K(\overline{x}) (or K(\overline{x}′)) and \overline{H} = H(\overline{x}′). We want to choose an appropriate θ in such a way that some solution of the problem

\begin{cases}−4\frac{n−1}{n−2}Δv + n(n − 1)v = \overline{K}v^{\frac{n+2}{n−2}} & in Σ_{θ}, \\ \frac{2}{n−2} \frac{∂v}{∂ν} + h_{θ}v = \overline{H} v^{\frac{n}{n−2}} & in ∂Σ_{θ}. \end{cases} (21)

can be chosen to be a constant v_{θ}. In this way the problem transforms into

\begin{cases}n(n − 1)v_{θ} = \overline{K}vθ^{\frac{n+2}{n−2}} & in Σ_{θ}, \\ h_{θ}v_{θ} = \overline{H} vθ^{\frac{n}{n−2}} & in ∂Σ_{θ}. \end{cases} (22)

From equation (22) it follows that θ must satisfy the relation

\overline{H} sin θ = (\frac{\overline{K}}{n(n − 1)})^{\frac{1}{2}} cos θ, (23)

and that v_{θ} solves

\overline{K} vθ^{\frac{4}{n−2}} = n(n − 1). (24)

As far as the interior blow up is concerned, we look for a constant function v̂ on the whole sphere. Since v̂ solves the equation

−4\frac{n − 1}{n − 2}Δv̂ + n(n − 1)v̂ = \overline{K}v̂^{\frac{n+2}{n−2}} , in S^{n}, (25)

and is constant, it must also satisfy \overline{K} v̂^{\frac{4}{n−2}} = n(n − 1). (26) Boundary blow up energy We now compute the energy of a boundary blow up. Let J_{\overline{K},\overline{H}} be the Euler functional as in (12) corresponding to K ≡ K(\overline{x}′) and H ≡ H(\overline{x}′); we have

J_{\overline{K},\overline{H}}(v_{θ}) = \frac{1}{2}n(n − 1) ∫_{Σ_{θ}} v_{θ}^{2} + \frac{(n − 1)}{tan θ} ∫_{∂Σ_{θ}} v_{θ}^{2} − \frac{\overline{K}}{2^{∗}} ∫_{Σ_{θ}} v_{θ}^{2^{∗}} − \overline{H} (n − 2) ∫_{∂Σ_{θ}} v_{θ}^{2 \frac{n−1}{n−2}} .

Taking into account the fact that v_{θ} is a critical point of J_{\overline{K},\overline{H}}, and in particular that J′_{\overline{K},\overline{H}}(v_{θ})[v_{θ}] = 0, it turns out that

n(n − 1) ∫_{Σ_{θ}} v_{θ}^{2} + 2\frac{(n − 1)}{tan θ} ∫_{∂Σ_{θ}} v_{θ}^{2} − \overline{K} ∫_{Σ_{θ}} v_{θ}^{2^{∗}} − 2\overline{H}(n − 1) ∫_{∂Σ_{θ}} v_{θ}^{2 \frac{n−1}{n−2}} = 0.

Hence it follows that

J_{\overline{K},\overline{H}}(v_{θ}) = (\frac{1}{2} − \frac{1}{2^{∗}} ) \overline{K} ∫_{Σ_{θ}} v_{θ}^{2^{∗}} + \overline{H} ∫_{∂Σ_{θ}} v_{θ}^{2 \frac{n−1}{n−2}} .

Setting

F (θ) = ∫_{0}^{θ} sin^{n−1} s ds; θ ∈ [0, π],

one immediately checks that

|Σ_{θ}| = ω_{n−1} · F (θ); |∂Σ_{θ}| = \frac{d}{dθ} |Σ_{θ}| = ω_{n−1} sin^{n−1} θ.

So, taking into account formula (24), J_{\overline{K},\overline{H}}(v_{θ}) can be written as

J_{\overline{K},\overline{H}}(v_{θ}) = \frac{ω_{n−1}}{n} \overline{K}F (θ) · (\frac{n(n − 1)}{\overline{K}})^{\frac{n}{2}} + \overline{H} ω_{n−1} sin^{n−1} θ · (\frac{n(n − 1)}{\overline{K}})^{\frac{n−1}{2}} . (27)

Finally, using equation (23) we deduce

J_{\overline{K},\overline{H}}(v_{θ}) = ω_{n−1} (\frac{n(n − 1)}{\overline{K}})^{\frac{n}{2} −1} [(n − 1) F (θ) + cos θ sin^{n−2} θ] . (28)

Interior blow up energy Solutions of (25) are critical points of the functional J\overline{K} : H^{1}(S^{n}) → ℝ defined by

J\overline{K}(v) = \frac{1}{2} ∫_{S^{n}} (\frac{4(n − 1)}{n − 2}|∇v|^{2} + n(n − 1)v^{2}) − \frac{\overline{K}}{2^{∗}} ∫_{S^{n}} |v|^{2^{∗}}, v ∈ H^{1}(S^{n}).

If v̂ is the constant given by formula (26), then its energy is

J\overline{K}(v̂) = \frac{1}{2}n(n − 1) ∫_{S^{n}} v̂^{2} − \frac{1}{2^{∗}} ∫_{S^{n}} \overline{K}v̂^{2^{∗}}.

Since v̂ is a critical point of J\overline{K} it turns out that the following relation must be also satisfied

n(n − 1) ∫_{S^{n}} v̂^{2} − ∫_{S^{n}} \overline{K}v̂^{2^{∗}} = 0.

Hence it follows that

J\overline{K}(v̂) = (\frac{1}{2} − \frac{1}{2^{∗}} ) ∫_{S^{n}} \overline{K}v̂^{2^{∗}} = \frac{ω_{n}}{n} (n(n − 1))^{\frac{n}{2}} (\overline{K})^{− \frac{n−2}{2}} . (29)

Comparison of energies We conclude this section by proving that, for the same value of \overline{K}, the interior blow up energy is always greater than the boundary blow up energy, namely we show that J_{\overline{K},\overline{H}}(v_{θ}) < J\overline{K}(v̂). (30) From equation (23) and from the obvious relation

ω_{n} = ω_{n−1} F (π),

one deduces that J\overline{K}(v̂) = ω_{n−1} [(n − 1) tan^{2−n} θ F (π)] \overline{H}^{2−n}. (31) Taking into account (23) and (27), showing inequality (30) is equivalent to prove

G(θ) := F (θ) + \frac{1}{n − 1} sin^{n−2} θ cos θ < F (π). (32)

Since it is clearly G(π) = F (π), we are done if we prove that G′(θ) > 0 for all θ ∈ (0, π). There holds

G′(θ) = sin^{n−1} θ + \frac{1}{n − 1} ((n − 2) sin^{n−3} θ cos^{2} θ − sin^{n−1} θ) .

With straightforward computations one finally finds that

G′(θ) = \frac{n − 2}{n − 1} sin^{n−3} θ · (sin^{2} θ + cos^{2} θ) ,

hence equation (32) is proved.

We also note that in the case when \overline{H} = 0, the boundary blow up energy is exactly one half of the interior blow up energy, see formulas (28) and (29). Remark 3.1 We are particularly interested in the boundary blow up case for n = 3. In this situation, using elementary trigonometric formulas, the explicit expression of J_{\overline{K},\overline{H}}(v_{θ}) becomes

J_{\overline{K},\overline{H}}(v_{θ}) = 4π \sqrt{\frac{6}{\overline{K}}} (\frac{π}{2} − arctan (\overline{H} \sqrt{\frac{6}{\overline{K}}} )) .

The above function will play a crucial role in the blow up analysis performed later.

4 Blow up analysis: definitions and preliminary results

In this section we recall the definition of isolated and isolated simple blow up due to R. Schoen, [34]; we also collect some useful tools and known results.

For a smooth bounded domain Ω ⊆ ℝ^{n} set Ω_{+} = Ω ∩ {x_{n} > 0}, ∂_{1}Ω = Ω ∩ ∂ℝ_{+}^{n} and ∂_{2}Ω = ∂Ω ∩ ℝ_{+}^{n}, hence ∂Ω_{+} = \overline{∂_{1}Ω} ∪ ∂_{2}Ω. We also assume that ∂Ω and ∂ℝ_{+}^{n} intersect transversally, so that ∂Ω ∩ ∂ℝ_{+}^{n} is a smooth manifold of dimension n − 2. Let ν denote the unit exterior normal to Ω, and let ν′ denote the exterior unit normal of ∂_{1}Ω in ∂ℝ_{+}^{n}. Given w : ∂ℝ_{+}^{n} → ℝ, the expression ∇′w stands for the gradient in ℝ^{n−1}. If w is defined on \overline{Ω_{+}}, the same symbol will be used for the gradient of the restriction of w to ∂_{1}Ω. In the following B_{σ}(x) denotes the open ball in ℝ^{n} of radius σ centered at x; we just write B_{σ} if x = 0.

We will consider equation (14) restricted to Ω, or equation (15) when the exponent p is converging to \frac{n+2}{n−2}. For this reason we will not keep the functions K and H fixed, but we will allow them to vary; more precisely, we consider positive solutions u_{i} of the sequence of problems

\begin{cases}−Δu_{i} = \frac{n−2}{4(n−1)}K_{i}(x)u_{i}^{p_{i}}, & in Ω_{+}; \\ −\frac{∂u_{i}}{∂x_{n}} = \frac{n−2}{2}H_{i}(x′)ui^{\frac{p_{i}+1}{2}} , & on ∂_{1}Ω. \end{cases} (33)

We are interested in the case where the supremum of the functions u_{i} is tending to infinity, trying to give a precise characterization of the blow up phenomenon, as in [34] and [29]. A typical ingredient of blow up analysis of scalar curvature equations is a Pohozahev type identity, which we provide in the next Lemma.

Lemma 4.1 Let p ≥ 1, let Ω ⊆ ℝ^{n} be as above, and let K ∈ C^{1}(\overline{Ω_{+}}), H ∈ C^{1}(∂_{1}Ω). Assume u ∈ C^{2}(\overline{Ω_{+}}) is a positive solution of

\begin{cases}−Δu = \frac{n−2}{4(n−1)}K(x) u^{p},  in Ω; \\ −\frac{∂u}{∂x_{n}} = \frac{n−2}{2}H(x′) u^{\frac{p+1}{2}} ,  on ∂_{1}Ω. \end{cases} (34)

Then there holds

\frac{n − 2}{4(n − 1)} (\frac{n − 2}{2} − \frac{n}{p + 1}) ∫_{Ω_{+}} Ku^{p+1} + \frac{n − 2}{2} (\frac{n − 2}{2} − \frac{2(n − 1)}{p + 3}) ∫_{∂_{1}Ω} H u^{\frac{p+3}{2}}

(35)  = \frac{n − 2}{4(n − 1)} \frac{1}{p + 1} ∫_{Ω_{+}}(x · ∇K) u^{p+1} + \frac{n − 2}{p + 3} ∫_{∂_{1}Ω}(x′ · ∇′H) u^{p+1}

+ ∫_{∂_{2}Ω} B − \frac{n − 2}{4(n − 1)} ∫_{∂_{2}Ω} K u^{p+1} x · ν − \frac{n − 2}{p + 3} ∫_{∂(∂_{1}Ω)} H u^{\frac{p+3}{2}} x′ · ν′

where

B = B(x, u, ∇u) = \frac{∂u}{∂ν}x · ∇u + \frac{n − 2}{2}u\frac{∂u}{∂ν} − \frac{|∇u|^{2}}{2} x · ν.

Proof. Multiply the first equation in (34) by ∑_{j=1}^{n} x_{j} \frac{∂u}{∂x_{j}} and integrate by parts: we obtain

\frac{n − 2}{4(n − 1)} (\frac{n − 2}{2} − \frac{n}{p + 1}) ∫_{Ω_{+}} Ku^{p+1} − \frac{n − 2}{4(n − 1)} \frac{1}{p + 1} ∫_{Ω_{+}}(x · ∇K) u^{p+1}
= ∫_{∂_{2}Ω} B − \frac{n − 2}{4(n − 1)} ∫_{∂_{2}Ω} K u^{p+1} x · ν.

Integrating by parts on ∂_{1}Ω, we deduce

∫_{∂_{1}Ω} H u^{\frac{p+1}{2}} x′ · ∇′u = −2 \frac{n − 1}{p + 1} ∫_{∂_{1}Ω} u^{\frac{p+3}{2}} H − \frac{2}{p + 1} ∫_{∂_{1}Ω} u^{\frac{p+3}{2}} x′ · ∇′H + ∫_{∂(∂_{1}Ω)} x′ · ν′ H u^{\frac{p+3}{2}} ,

Using the second equation in (34), we easily reach the conclusion.

We have also the following Proposition, which proof is elementary.

Proposition 4.2 Suppose the function h : (B_{σ})_{+} \ {0} → ℝ is of the form

h(x) = a |x|^{2−n} + b(x),

with a > 0 and b(x) of class C^{1} on \overline{(B_{σ})_{+}}. Then there holds

lim_{σ→0} ∫_{∂_{2}B_{σ}} B(x, h, ∇h) = −\frac{(n − 2)^{2}}{4} ω_{n−1} a b(0).

Let Ω ⊆ ℝ^{n} be as above, let 1 < p_{i} ≤ \frac{n+2}{n−2}, p_{i} → \frac{n+2}{n−2}, and let τ_{i} = \frac{n+2}{n−2} − p_{i}, so that τ_{i} → 0. Let \{K_{i}\}_{i} ⊆ C^{1}(\overline{Ω_{+}}), \{H_{i}\}_{i} ⊆ C^{1}(\overline{∂_{1}Ω}) satisfy for some constant A_{1} > 0

(36)  \frac{1}{A_{1}} ≤ K_{i}(x) ≤ A_{1}, −A_{1} ≤ H_{i}(x′) ≤ A_{1};  for all x ∈ \overline{Ω_{+}}, all x′ ∈ \overline{∂_{1}Ω}, and all i.

For every i, let also u_{i} ∈ C^{2}(\overline{Ω_{+}}) be a positive solution of problem (33).

Definition 4.3 The point \overline{x} ∈ Ω_{+} ∪ ∂_{1}Ω is called a blow up point for \{u_{i}\}_{i} if there exists a sequence of points x_{i} ∈ Ω_{+} ∪ ∂_{1}Ω tending to \overline{x} such that u_{i}(x_{i}) → +∞.

Definition 4.4 Let \overline{x} ∈ Ω_{+} ∪ ∂_{1}Ω, and let \{x_{i}\} be a sequence of local maxima of u_{i} such that x_{i} → \overline{x} and u_{i}(x_{i}) → +∞. The point \overline{x} is called an isolated blow up point if there exist 0 < \overline{r} < dist(\overline{x}, ∂_{2}Ω) and \overline{C} > 0 such that

u_{i}(x) ≤ \overline{C} |x − x_{i}|^{− \frac{2}{p_{i}−1}} , |x − x_{i}| ≤ \overline{r}, x ∈ Ω_{+}.

If \overline{x} is a blow up point for \{u_{i}\}_{i} we will write for brevity x_{i} → \overline{x} meaning that \{x_{i}\}_{i} is a sequence of points as in Definition 4.4. It is possible to prove, using Proposition 5.1 and Lemma 4.6 below, that the points x_{i} having the properties in Definition 4.4 are uniquely determined, provided the functions K_{i} and H_{i} in (33) are uniformly bounded in C^{1} and C^{2} norm respectively, see [25].

If x_{i} → \overline{x} is a simple blow up for \{u_{i}\}_{i} and if \overline{r} is given by Definition 4.4 we define

\overline{u}_{i}(r) = \frac{1}{|∂B_{r}(x_{i}) ∩ Ω_{+}|} ∫_{∂B_{r}(x_{i})∩Ω_{+}} u_{i}, r ∈ (0, \overline{r}), (37)

and

ũ_{i}(r) = r^{\frac{2}{p_{i}−1}} \overline{u}_{i}(r), r ∈ (0, \overline{r}).

Definition 4.5 The isolated blow up point x_{i} → \overline{x} is called isolated simple if there exists ϱ ∈ (0, \overline{r}) such that for large i there holds

(38)  ũ_{i} has precisely one critical point in (0, ϱ).

If \overline{x} is a blow up point, we will call it interior blow up point if \overline{x} ∈ Ω_{+}, or boundary blow up point if \overline{x} ∈ ∂_{1}Ω.

Another fundamental tool for the blow up analysis is the Harnack inequality; we recall the following version from [25], Appendix A.

Lemma 4.6 (Harnack-type inequality) Let {K_{i}} ∈ L^{∞}(Ω_{+}) and {H_{i}} ∈ L^{∞}(∂_{1}Ω) satisfy (36). Assume also that \{u_{i}\}_{i} satisfy (33) with p_{i} ≥ p_{0} > 1, and let x_{i} → \overline{x} be an isolated blow up point. Then for every 0 < r < \frac{1}{4} \overline{r} the following Harnack-type inequality holds

sup_{x∈(B_{2r})_{+}(x_{i})\(B_{r/2})_{+}(x_{i})} u_{i}(x) ≤ C inf_{x∈(B_{2r})_{+}(x_{i})\(B_{r/2})_{+}(x_{i})} u_{i}(x),

where C is a positive constant depending only on n, A_{1} and \overline{C}.

For the blow up analysis of the first equation in (33) we mainly refer to [29], where the following proposition regarding the interior blow up points is proved.

Proposition 4.7 Assume Ω ⊆ ℝ^{3} and that \{K_{i}\}_{i} is uniformly bounded in C^{1}(\overline{Ω}). Assume that p_{i} ≤ \frac{n+2}{n−2}, p_{i} → \frac{n+2}{n−2}, and \{u_{i}\}_{i} are solutions of

−Δu_{i} = \frac{n − 2}{4(n − 1)} K_{i} u_{i}^{p_{i}}, u_{i} > 0 in Ω.

Then, if \overline{x} ∈ Ω is a blow up point for u_{i}, it is also an isolated simple blow up point. Moreover there exists an harmonic function b : B_{ϱ/2}(\overline{x}) → ℝ such that, passing to a subsequence

u_{i}(x_{i}) u_{i}(x) → a |x − \overline{x}|^{2−n} + b(x), in C_{loc}^{2}(B_{ϱ/2} \ \{\overline{x}\}), (39)

where a = (4n(n − 1))^{\frac{n−2}{2}} (lim_{i} K_{i}(x_{i}))^{\frac{2−n}{2}} , and where ϱ is given in Definition 4.5.

5 Behavior of isolated simple blow ups

In this section we perform the study of isolated simple blow ups of equation (33). The situation of interior blow up has been treated in [29], hence we are reduced to consider the case in which the blow up point \overline{x} is in ∂_{1}Ω. We will refer sometimes to the paper [25], where it is studied equation (33) when K_{i} and H_{i} are converging to constant functions.

Proposition 5.1 Assume \{K_{i}\}_{i} ⊆ C^{1}(\overline{Ω_{+}}) and \{H_{i}\}_{i} ⊆ C^{2}(\overline{∂_{1}Ω}) satisfy (36) for some A_{1} > 0, and satisfy also the condition

∥∇K_{i}∥_{C(\overline{Ω_{+}})} ≤ A_{2}, ∥∇′H_{i}∥_{C^{1}(\overline{∂_{1}Ω})} ≤ A_{2}, (40)

for some A_{2} > 0. For every i, let u_{i} be a positive solution of (33), and let x_{i} → \overline{x}′ ∈ ∂_{1}Ω be an isolated blow up point for {u_{i}}_{i}. Let also x′_{i} denote the projection of x_{i} onto ∂_{1}Ω. Then, given R_{i} → +∞ and ε_{i} → 0^{+}, after passing to a subsequence of {u_{i}}_{i} (still denoted with {u_{i}}_{i}) we have

\begin{cases}r_{i} := u_{i}(x_{i})^{− \frac{p_{i}−1}{2}} R_{i} → 0 as i → +∞, \\ ∥u_{i}(x_{i})^{−1}u_{i}(u_{i}(x_{i})^{− \frac{p_{i}−1}{2}} x + x_{i}) − (\frac{λ}{(1+kλ^{2}(|x′|^{2}+|x_{n}+t|^{2})})^{\frac{n−2}{2}} ∥_{C^{2}((u_{i}(x_{i})^{\frac{p_{i}−1}{2}} (\overline{Ω_{+}}−x_{i}))∩\overline{B_{2R_{i}}} )} ≤ ε_{i}, \end{cases}

where k = (4n(n − 1))^{−1} K_{i}(x′_{i}), while λ, t satisfy 2kλt = H_{i}(x′_{i}) and

λ = \begin{cases}1 + kλ^{2}t^{2} & if H_{i}(x′_{i}) ≥ 0, \\ 1 & if H_{i}(x′_{i}) < 0. \end{cases} (41)

Proof. Consider the functions

w_{i}(x) = u_{i}(x_{i})^{−1} u_{i} (u_{i}(x_{i})^{− \frac{p_{i}−1}{2}} x + x_{i}) , x ∈ u_{i}(x_{i})^{\frac{p_{i}−1}{2}} (Ω_{+} − x_{i}).

It follows immediately that w_{i}(0) = 1 for all i and that 0 is a local maximum point for w_{i}. Moreover from the assumption of isolated blow up we have

w_{i}(x) ≤ C |x|^{− \frac{p_{i}−1}{2}} , x ∈ u_{i}(x_{i})^{\frac{p_{i}−1}{2}} (Ω_{+} − x_{i}) ∩ \{|x| < u_{i}(x_{i})^{\frac{p_{i}−1}{2}} \overline{r}\},

where \overline{r} is given in Definition 4.4.

The function w_{i} is a solution of the problem

\begin{cases}−Δw_{i}(x) = \frac{n−2}{4(n−1)}K_{i} (u_{i}(x_{i})^{\frac{p_{i}−1}{2}} x + x_{i}) w_{i}(x)^{p_{i}}, & in u_{i}(x_{i})^{\frac{p_{i}−1}{2}} (Ω_{+} − x_{i}); \\ −\frac{∂w_{i}}{∂x_{n}} (x) = \frac{n−2}{2}H_{i} (u_{i}(x_{i})^{\frac{p_{i}−1}{2}} x + x_{i}) u_{i}(x)^{\frac{p_{i}+1}{2}} , & on u_{i}(x_{i})^{\frac{p_{i}−1}{2}} (∂_{1}Ω − x_{i}). \end{cases}

Denoting by x_{i,n} the n-th component of x_{i} and setting T_{i} = u_{i}(x_{i})^{\frac{p_{i}−1}{2}} x_{i,n}, two cases may occur, namely

T_{i} → +∞,  or  T_{i} → T ∈ ℝ.

In the latter one, we can use (36), (40) and the results in [1] to prove that the functions w_{i} converge up to subsequence, and then one can conclude as in [25], Proposition 1.4.

Hence it is sufficient to exclude the first case. In order to do this, define the functions

ξ_{i}(x) = xi,n^{\frac{2}{p_{i}−1}} u_{i}(x_{i} + x_{i,n} x), x ∈ T_{i} (Ω_{+} − x_{i}).

First, letting Ω_{i} = T_{i} (Ω_{+} − x_{i}), it is clear that Ω_{i} are relatively open sets which invade the half space ℝ_{1}^{n} := {x ∈ ℝ^{n} : x_{n} > −1}. Then, since we are supposing by contradiction that T_{i} → +∞, 0 is an interior blow up point for the functions ξ_{i}, so from Proposition 4.7 it follows that 0 is an isolated simple blow up point. Using Lemma 4.6 and the inequality

ξ_{i}(x) ≤ C |x|^{− \frac{p_{i}−1}{2}} , x ∈ Ti^{\frac{p_{i}−1}{2}} (Ω_{+} − x_{i}) ∩ \{|x| < Ti^{\frac{p_{i}−1}{2}} \overline{r}\},

the convergence in (39) can be extended to the whole ℝ_{1}^{n} \ {0}. Namely one has

ξ_{i}(0) ξ_{i}(x) → h(x) in C_{loc}^{2}(ℝ_{1}^{n} \ {0}),

where h(x) is a non-negative harmonic function in ℝ_{1}^{n} \ {0} singular at 0 and satisfying

\frac{∂h}{∂x_{n}} = 0, on ∂ℝ_{1}^{n}. (42)

By equation (42) and by the Schwartz’s Reflection Principle, the function h possesses an harmonic extension to the set ℝ^{n} \ {0, ˜0}, where ˜0 is the symmetric point of 0 with respect to the plane ∂ℝ_{1}^{n}. By uniqueness of harmonic extensions, this must coincide with the symmetric prolongation of h through ∂ℝ_{1}^{n}. Hence the positivity of h implies that h(x) = a|x|^{2−n} + A + o(|x|) for x close to 0 , where a, A > 0. Reasoning as in Proposition 3.1 of [29], one can reach a contradiction.

Next, we establish the counterpart of Proposition 4.7 for blow up points in ∂_{1}Ω.

Proposition 5.2 Let Ω = B_{2}, suppose \{K_{i}\}_{i} ⊆ C^{1}(\overline{Ω_{+}}), \{H_{i}\}_{i} ⊆ C^{2}(\overline{∂_{1}Ω}) satisfy conditions (36) and (40) for some A_{1}, A_{2} > 0. Suppose that for every i, u_{i} satisfies (33) and that x_{i} → 0 is an isolated simple blow up with

|x − x_{i}|^{\frac{2}{p_{i}−1}} u_{i}(x) ≤ A_{3}, for all x ∈ Ω_{+}.

Then there exists some positive constant C = C(A_{1}, A_{2}, A_{3}, n, ρ) such that u_{i}(x) ≤ Cu_{i}(x_{i})^{−1}|x − x_{i}|^{2−n} for all x ∈ (B_{1}(x_{i}))_{+}. (43) Furthermore, there exists b : \overline{(B_{1})_{+}} satisfying

\begin{cases}−Δb = 0 & in (B_{1})_{+}; \\ −\frac{∂b}{∂y_{n}} = 0 & on B_{1} ∩ ∂ℝ_{+}^{n}, \end{cases}

such that

u_{i}(x_{i})u_{i}(x) → a |x|^{2−n} + b, in C_{loc}^{2} (\overline{(B_{1})_{+}} \ \{0\}) .

The coefficient a is given by

(44) a = \begin{cases}(4n(n − 1))^{\frac{n−2}{2}} (lim_{i} K_{i}(x′_{i}))^{\frac{2−n}{2}} & if lim_{i} H_{i}(x′_{i}) < 0; \\ lim_{i} (\frac{K_{i}(x′_{i})}{4n(n−1)} + \frac{H_{i}(x′_{i})^{2}}{4} )^{\frac{2−n}{2}} & if lim_{i} H_{i}(x′_{i}) ≥ 0, \end{cases}

where x′_{i} is the projection of x_{i} onto ∂_{1}Ω. Before proving Proposition 5.2, we need some preliminary Lemmas. Lemma 5.3 Under the hypotheses of Proposition 5.2, except for condition (40), there exist δ_{i} > 0, δ_{i} = O(R_{i}^{−2+o(1)}) such that

u_{i}(x) ≤ C u_{i}(x_{i})^{−λ_{i}} |x − x_{i}|^{2−n+δ_{i}}, for R_{i} u_{i}(x_{i})^{− \frac{p_{i}−1}{2}} ≤ |x − x_{i}| ≤ 1,

where λ_{i} = (n − 2 − δ_{i}) \frac{p_{i}−1}{2} − 1. Proof. It follows from [25], pages 511-513. Lemma 5.4 Under the hypotheses of Proposition 5.2 there holds

τ_{i} = O(u_{i}(x_{i})^{− \frac{2}{n−2} +o(1)}), as i → +∞

and therefore

u_{i}(x_{i})^{τ_{i}} = 1 + o(1), as i → +∞.

Proof. Let B(x, u, ∇u) be the function defined in Lemma 4.1. By Lemma 5.3, Proposition 5.1 and standard elliptic theories we have

∫_{∂_{2}B_{1}} B(x, u_{i}, ∇u_{i}) = O (u_{i}(x_{i})^{−2+o(1)}); ∫_{∂_{2}B_{1}} K_{i} |u_{i}|^{p_{i}+1} = O (u_{i}(x_{i})^{−p_{i}−1+o(1)});

∫_{∂(∂_{1}B_{1})} H_{i} ui^{\frac{p_{i}+3}{2}} x′ · ν′ = O (u_{i}(x_{i})^{− \frac{p_{i}+3}{2} +o(1)}) .

Furthermore, since the gradients of K_{i} and H_{i} are uniformly bounded, one can deduce from Lemma 5.3, Proposition 5.1 and a rescaling argument that

∫_{(B_{1})_{+}} u_{i}^{p_{1}+1} x · ∇K_{i} = O (u_{i}(x_{i})^{− \frac{2}{n−2} +o(1)}); ∫_{∂_{1}B_{1}} ui^{\frac{p_{i}+3}{2}} x′ · ∇′H_{i} = O (u_{i}(x_{i})^{− \frac{2}{n−2} +o(1)}) .

On the other hand, using again Lemma 5.3 and Proposition 5.1 we have also

\frac{n − 2}{4(n − 1)} (\frac{n}{p_{i} + 1} − \frac{n − 2}{2}) ∫_{(B_{1})_{+}} K_{i} u_{i}^{p_{i}+1} + \frac{n − 2}{2} (\frac{n − 2}{2} − 2 \frac{n − 1}{p_{i} + 3}) ∫_{∂_{1}B_{1}} ui^{\frac{p_{i}+3}{2}} H_{i}

= τ_{i} \frac{(n − 2)^{3}}{16(n − 1)} (\frac{1}{n}K_{i}(x′_{i}) ∫_{ℝ_{+}^{n}} \overline{U}_{λ}^{2^{∗}} + H_{i}(x′_{i}) ∫_{∂ℝ_{+}^{n}} \overline{U}_{λ}^{2 \frac{n−1}{n−2}} + o(1)) ,  i → +∞.

Here the function \overline{U}_{λ} is given by formula (19) with K(\overline{x}′) (resp. H(\overline{x}′)) replaced by K_{i}(x′_{i}) (resp. H_{i}(x′_{i})); we note that the values of the above integrals do not depend on the parameter λ.

Using the relation I′_{K_{i}(x′_{i}),H_{i}(x′_{i})}(\overline{U}_{λ})[\overline{U}_{λ}] = 0, it follows that

\frac{1}{n}K_{i}(x′_{i}) ∫_{ℝ_{+}^{n}} \overline{U}_{λ}^{2^{∗}} + H_{i}(x′_{i}) ∫_{∂ℝ_{+}^{n}} \overline{U}_{λ}^{2 \frac{n−1}{n−2}} = I_{lim_{i} K_{i}(x′_{i}),lim_{i} H_{i}(x′_{i})}(\overline{U}_{λ}) + o(1) > α > 0,

where α is a positive constant depending only on n, A_{1} and A_{2}. Then the conclusion follows from equation (35) and the above estimates.

Lemma 5.5 Under the same assumptions of Proposition 5.2 there holds

lim_{i} u_{i}(x_{i}) ∫_{∂_{2}B_{1}} \frac{∂u_{i}}{∂ν} < 0.

Proof. Using the divergence formula and (33), we can write

u_{i}(x_{i}) ∫_{∂_{2}B_{1}} \frac{∂u_{i}}{∂ν} = u_{i}(x_{i}) (\frac{n − 2}{4(n − 1)} ∫_{(B_{1})_{+}} K_{i}u_{i}^{p_{i}} − \frac{n − 2}{2} ∫_{∂_{1}B_{1}} H_{i}ui^{\frac{p_{i}+1}{2}} ) . (45)

From Lemma 5.3 we deduce that

∫_{(B_{1})_{+}\(B_{r_{i}} (x_{i}))_{+}} u_{i}^{p_{i}} ≤ C ∫_{(B_{1})_{+}\(B_{r_{i}})_{+}} (u_{i}(x_{i})^{−λ_{i}} |y − x_{i}|^{2−n+δ_{i}})^{p_{i}}
≤ C R_{i}^{n−p_{i}(n−2−δ_{i})} u_{i}(x_{i})^{−1+O(τ_{i})} = o(1) u_{i}(x_{i})^{−1}.

In the same way we have that

∫_{∂_{1}B_{1}\∂_{1}B_{r_{i}} (x_{i})} ui^{\frac{p_{i}+1}{2}} = o(1) u_{i}(x_{i})^{−1}.

Hence, using Proposition 5.1 and a rescaling argument, choosing ε_{i} → 0 sufficiently fast, formula (45) can be written as

u_{i}(x_{i}) ∫_{∂_{2}B_{1}} \frac{∂u_{i}}{∂ν} = \frac{n − 2}{4(n − 1)} lim_{i} K_{i}(x′_{i}) ∫_{(B_{R_{i}})_{+}} \overline{U} λ^{\frac{n+2}{n−2}} − \frac{n − 2}{2} lim_{i} H_{i}(x′_{i}) ∫_{∂_{1}B_{R_{i}}} \overline{U} λ^{\frac{n}{n−2}} + o(1),

where λ satisfies equation (41). Again from the divergence theorem, we have

lim_{i} (\frac{n − 2}{4(n − 1)}K_{i}(x′_{i}) ∫_{(B_{R_{i}})_{+}} \overline{U} λ^{\frac{n+2}{n−2}} − \frac{n − 2}{2}H_{i}(x′_{i}) ∫_{∂_{1}B_{R_{i}}} \overline{U} λ^{\frac{n}{n−2}} ) = lim_{R→+∞} ∫_{∂_{2}B_{R}} \frac{∂\overline{U}_{λ}}{∂ν} < 0.

This concludes the proof.

Proof of Proposition 5.2 Inequality (43) for |x − x_{i}| ≤ r_{i} is an immediate consequence of Lemma 5.3 and Lemma 5.4. We now prove it for r_{i} ≤ |x − x_{i}| ≤ 1. Let \overline{u}_{i} be given by formula (37) and set ξ_{i}(x) = \overline{u}_{i}(1) u_{i}(x). It is easy to see that ξ_{i} satisfies

\begin{cases}−Δ ξ_{i} = \frac{n−2}{4(n−1)} \overline{u}_{i}(1)^{p_{i}−1} K_{i}(x) ξ_{i}^{p_{i}}, & in (B_{2})_{+}; \\ −\frac{∂ξ_{i}}{∂x_{n}} = \frac{n−2}{2} \overline{u}_{i}(1)^{\frac{p_{i}−1}{2}} H_{i}(x′) ξi^{\frac{p_{i}+1}{2}} , & on ∂_{1}B_{2}. \end{cases}

Reasoning as in the proof of Proposition 5.1 it follows that, passing to a subsequence, {ξ_{i}}_{i} converges in C_{loc}^{2}(\overline{(B_{2})_{+}} \ 0) to some positive function h ∈ C_{loc}^{2}(\overline{(B_{2})_{+}} \ 0) satisfying

\begin{cases}−Δh = 0, & in (B_{2})_{+}; \\ −\frac{∂h}{∂x_{n}} = 0, & on ∂_{1}B_{2}. \end{cases}

Moreover, it follows from condition (38) that h must be singular at 0. Reflecting the function h evenly to B_{2} and reasoning as above, we deduce that h must be of the form

h(x) = a_{1} |x|^{2−n} + b(x), x ∈ \overline{(B_{2})_{+}} \ 0,

where a_{1} > 0, and b ∈ C^{2}(\overline{(B_{2})_{+}}) satisfies

Δb(x) = 0, x ∈ (B_{2})_{+};  \frac{∂b}{∂x_{n}} (x′) = 0,  x′ ∈ ∂_{1}B_{2}.

Now we can prove (43) for |x − x_{i}| = 1, namely

\overline{u}_{i}(1) ≤ C u_{i}(x_{i})^{−1}. (46)

To do this we observe that, by the harmonicity of b(y) we have

∫_{∂_{2}B_{1}} \frac{∂b}{∂ν} = ∫_{∂_{1}B_{1}} \frac{∂b}{∂x_{n}} + ∫_{(B_{1})_{+}}(Δb) = 0,

and so we deduce that

lim_{i} \overline{u}_{1}(1)^{−1} ∫_{∂_{2}B_{1}} \frac{∂u_{i}}{∂ν} = ∫_{∂_{2}B_{1}} \frac{∂h}{∂ν} = a_{1} ∫_{∂_{2}B_{1}} \frac{∂|x|^{2−n}}{∂ν} < 0.

Hence formula (46) follows from Lemma 5.5. The inequality for a general x with r_{i} ≤ |x − x_{i}| ≤ 1 follows from a rescaling argument, as in [29] page 340. The value of the constant a in (44) can be computed multiplying the first equation in (33) by u_{i}, integrating by parts, and using Proposition 5.1.

We collect now a couple of technical lemmas which will be needed later.

Lemma 5.6 Suppose that the hypotheses of Proposition 5.2 hold true. Then we have the following estimates

∫_{(B_{r_{i}} (x_{i}))_{+}} |x − x_{i}|^{s} u_{i}(x)^{p_{i}+1} = O (u_{i}(x_{i})^{− \frac{2s}{n−2}} ) , 0 < s < 2;

∫_{(B_{1}(x_{i}))_{+}\(B_{r_{i}} (x_{i}))_{+}} |x − x_{i}|^{s} u_{i}(x)^{p_{i}+1} = o (u_{i}(x_{i})^{− \frac{2s}{n−2}} ) , 0 < s < 2.

∫_{∂_{1}B_{r_{i}} (x_{i})} |x′ − x′_{i}|^{s} u_{i}(x′)^{\frac{p_{i}+3}{2}} = O (u_{i}(x_{i})^{− \frac{2s}{n−2}} ) , 0 < s < 2;

∫_{∂_{1}B_{1}(x_{i})\∂_{1}B_{r_{i}} (x_{i})} |x′ − x′_{i}|^{s} u_{i}(x′)^{\frac{p_{i}+3}{2}} = o (u_{i}(x′_{i})^{− \frac{2s}{n−2}} ) , 0 < s < 2;

∫_{∂_{1}B_{1}(x_{i})} |x′ − x′_{i}|^{s} u_{i}(x′)^{p_{i}+1} = O (u_{i}(x_{i})^{− \frac{4}{n−2}} log u_{i}(x_{i})) , s = 2.

Proof. The proof is a simple consequence of Proposition 5.1 and Proposition 5.2.

Lemma 5.7 Suppose that n = 3 and that the hypotheses of Proposition 5.2 hold true. Then we have

τ_{i} = O(u_{i}(x_{i})^{−2}).

Proof. It is sufficient to use (35) and Lemma 5.6.

6 Blow up points are isolated simple and at the boundary

In this section we show that blow up points of equation (33) are isolated simple and that the case of interior blow up can be ruled out. We will use the same terminology about blow ups for describing both functions on S_{+}^{n} or functions defined on some domain Ω_{+} ⊆ ℝ_{+}^{n}, having in mind the natural transformation (13) induced by the stereographic projection.

Proposition 6.1 Let n = 3, let Ω be as above, suppose that {K_{i}} and {H_{i}} satisfy conditions (36) and (40), that for every i u_{i} is a positive solution of (33) and that \overline{x}′ ∈ ∂_{1}Ω is an isolated blow up point for \{u_{i}\}. Then \overline{x}′ is also an isolated simple blow up point.

Proof. The proof follows that of Proposition 3.1 in [25], combined with some argument in [29] page 353, and Lemma 5.6. We omit the details.

Proposition 6.2 Let Ω, \{K_{i}\}, \{H_{i}\}, \{u_{i}\} be as in Proposition 6.1, and let \overline{x}′ be an isolated simple blow up point for \{u_{i}\}. Suppose also that \{K_{i}\} are uniformly bounded in C^{2}(\overline{Ω_{+}}). Let φ_{i} : ∂_{1}B_{1} → ℝ be the sequence of functions defined by

φ_{i}(x′) = 4π \sqrt{\frac{6}{K_{i}(x′)}} (\frac{π}{2} − arctan (H_{i}(x′) \sqrt{\frac{6}{K_{i}(x′)}})) .

Let also x′_{i} denote the projection of x_{i} onto ∂_{1}B_{1}. Then there holds

|∇′φ_{i}(x′_{i})| ≤ O(u_{i}(x_{i})^{−2}), as i → +∞.

Proof. Choose a test function η ∈ C^{∞}(B_{1}) which satisfies

η(x) = 1, x ∈ B_{1/4}; η(x) = 0, x ∈ B_{1} \ B_{1/2}.

Multiplying equation (33) by η \frac{∂u_{i}}{∂x_{j}}, j = 1, 2, we obtain

∫_{(B_{1})_{+}}(−Δu_{i})η \frac{∂u_{i}}{∂x_{j}} = \frac{1}{8} ∫_{(B_{1})_{+}} K_{i} u_{i}^{p_{i}}η \frac{∂u_{i}}{∂x_{j}} .

Integrating by parts we deduce

∫_{(B_{1})_{+}} K_{i} u_{i}^{p_{i}}η \frac{∂u_{i}}{∂x_{j}} = −\frac{1}{p_{i} + 1} ∫_{(B_{1})_{+}} u_{i}^{p_{i}+1} (η \frac{∂K_{i}}{∂x_{j}} + K_{i} \frac{∂η}{∂x_{j}} ) .

and also

∫_{(B_{1})_{+}}(−Δu_{i}) η \frac{∂u_{i}}{∂x_{j}} = \frac{1}{p_{i} + 3} ∫_{∂_{1}B_{1}} ui^{\frac{p_{i}+3}{2}} (η \frac{∂H_{i}}{∂x_{j}} + H_{i} \frac{∂η}{∂x_{j}} ) − \frac{1}{2} ∫_{(B_{1})_{+}} |∇u_{i}|^{2} \frac{∂η}{∂x_{j}}

+ ∫_{(B_{1})_{+}} ∇u_{i} \frac{∂u_{i}}{∂x_{j}} ∇η.

From the above equations, Proposition 5.2, and the fact that ∇η has support in \overline{(B_{1/2})_{+}} \ (B_{1/4})_{+}, we obtain

\frac{1}{p_{i} + 1} ∫_{(B_{1})_{+}} u_{i}^{p_{i}+1} η \frac{∂K_{i}}{∂x_{j}} + \frac{1}{p_{i} + 3} ∫_{∂_{1}B_{1}} ui^{\frac{p_{i}+3}{2}} η \frac{∂H_{i}}{∂x_{j}} = O(u_{i}(x_{i})^{−2}).

Using the uniform bounds on the second derivatives of K_{i} and H_{i}, and taking into account Lemmas 5.6 and 5.7 we deduce

\frac{1}{6} \frac{∂K_{i}}{∂x_{j}} (x′_{i}) ∫_{(B_{1})_{+}} u_{i}^{p_{i}+1} η + \frac{1}{8} \frac{∂H_{i}}{∂x_{j}} (x′_{i}) ∫_{∂_{1}B_{1}} ui^{\frac{p_{i}+3}{2}} η = O(u_{i}(x_{i})^{−2}). (47)

Let \overline{U}_{λ} be the function given in formula (19) with K(\overline{x}′) replaced by K_{i}(x′_{i}) and H(\overline{x}′) replaced by H_{i}(x′_{i}). Using Proposition 5.1 and Lemma 5.7, equation (47) becomes

\frac{1}{6} \frac{∂K_{i}}{∂x_{j}} (x′_{i}) ∫_{ℝ_{+}^{n}} \overline{U}_{λ}^{6} + \frac{1}{8} \frac{∂H_{i}}{∂x_{j}} (x′_{i}) ∫_{∂ℝ_{+}^{n}} \overline{U}_{λ}^{4} = O(u_{i}(x_{i})^{−2}). (48)

By Remark 3.1, it turns out that

φ_{i}(x′_{i}) = I_{K_{i}(x′_{i}),H_{i}(x′_{i})}(\overline{U}_{λ}) = 4 ∫_{ℝ_{+}^{n}} |∇\overline{U}_{λ}|^{2} − \frac{1}{6} K_{i}(x′_{i}) ∫_{ℝ_{+}^{n}} \overline{U}_{λ}^{6} − H_{i}(x′_{i}) ∫_{∂ℝ_{+}^{n}} \overline{U}_{λ}^{4}.

Differentiating with respect to x′_{i}, and taking into account that I′_{K_{i}(x′_{i}),H_{i}(x′_{i})}(\overline{U}_{λ}) = 0, we deduce that

\frac{∂φ_{i}}{∂y′_{j}} (x′_{i}) = \frac{∂I_{K_{i}(·),H_{i}(·)}}{∂y′_{j}} |_{x′_{i}}(\overline{U}_{λ}) + I_{K_{i}(·),H_{i}(·)} (\frac{∂\overline{U}_{λ(·)}(x, ·)}{∂y′_{j}}|_{x′_{i}})

= \frac{1}{6} \frac{∂K_{i}}{∂x_{j}} (x′_{i}) ∫_{ℝ_{+}^{n}} \overline{U}_{λ}^{6} + \frac{1}{8} \frac{∂H_{i}}{∂x_{j}} (x′_{i}) ∫_{∂ℝ_{+}^{n}} \overline{U}_{λ}^{4}.

In the above formula the boundary point y′_{i} is considered as a parameter on which I_{K_{i},H_{i}} and \overline{U}_{λ} depend, through the functions K_{i} and H_{i}. The conclusion then follows from equation (48) and the last expression.

Remark 6.3 We note that if {K_{i}} is just bounded in C^{1} norm, the above proof yields anyway ∇′φ_{i}(x′_{i}) → 0 as i → +∞.

Now the local blow up analysis will be applied to equation (7) on the whole half sphere. We begin with the following Proposition which can be proved as in [25] pages 499-502, with minor modifications.

Proposition 6.4 Assume K ∈ C^{1}(\overline{S_{+}^{n}}) and H ∈ C^{2}(∂S_{+}^{n}) satisfy

\frac{1}{A_{1}} ≤ K(x) ≤ A_{1}, ∀x ∈ S_{+}^{n}; ∥K∥_{C^{1}(\overline{S_{+}^{n}})} ≤ A_{2}; (49)

−A_{1} ≤ H(x′) ≤ A_{1}, ∀x′ ∈ ∂S_{+}^{n}; ∥K∥_{C^{2}(∂S_{+}^{n})} ≤ A_{2}. (50)

Then, given any R ≥ 1 and any 0 < ε < 1, there exist positive constants δ_{0}, C_{0}, C_{1}, depending only on n, ε, R, A_{1} and A_{2} such that, for all τ ≤ δ_{0}, and for all the solutions v of equation (33) with sup_{S_{+}^{n}} v ≥ C_{0}, the following properties hold true. There exist {q^{1}, …, q^{N}} ⊆ S_{+}^{n}, with N ≥ 1, such that i) each q^{j} is a local maximum for v and

\overline{B_{\overline{r}_{j}}(q^{j})} ∩ \overline{B_{\overline{r}_{l}}(q^{l})} = ∅, for j ≠ l,

where \overline{r}_{j} = R v(q^{j})^{− \frac{p−1}{2}} ;

ii) either dist(q^{j}, ∂S_{+}^{n}) > \overline{r}_{j} and

∥v(q^{j})^{−1} v (v(q^{j})^{− \frac{p−1}{2}} x) − (\frac{1}{1 + k_{j} |x|^{2}} )^{\frac{n−2}{2}} ∥_{C^{2}(B_{2R})} < ε, (a)

or dist(q^{j}, ∂S_{+}^{n}) < \overline{r}_{j} and

∥v(q_{i})^{−1} v (v(q^{j})^{− \frac{p−1}{2}} x) − (\frac{λ_{j}}{1 + k_{j}λ_{j}^{2}(|x′|^{2} + |x_{n} + t_{j}|^{2})})^{\frac{n−2}{2}} ∥_{C^{2}((B_{2R})_{+})} < ε, (b)

In the above two formulas it is k_{j} = (4n(n − 1))^{−1}K(q^{j}), while λ_{j} and t_{j} satisfy 2k_{j}λ_{j}t_{j} = H(q^{j}), with

λ_{j} = \begin{cases}1 + k_{j}λ_{j}^{2}t_{j}^{2}, & if H(q^{j}) ≥ 0; \\ 1 & if H(q^{j}) < 0. \end{cases}

The function v is identified with its image through the map ι, the projection being suitably chosen depending on the point q^{j}.

iii) |q^{j} − q^{l}|^{\frac{2}{p−1}} v(q^{l}) ≥ C_{0}, for j < l, while v(q) ≤ C_{1} dist (q, \{q^{1}, …, q^{N}\})^{− \frac{2}{p−1}} for all q ∈ S_{+}^{n}.

Properties (a) and (b) in assertion ii) above distinguish, roughly, the cases of interior and boundary blow ups. Property iii) implies that, if the mutual distance of the points {q^{j}} is bounded from below along some sequence of solutions, then blow ups are isolated. This fact is the content of the next Proposition.

Proposition 6.5 Suppose that K ∈ C^{1}(\overline{S_{+}^{n}}) and H ∈ C^{2}(∂S_{+}^{n}) satisfy conditions (49) and (50) respectively. Then, given any R ≥ 1, and any 0 < ε < 1, there exist positive constants δ_{0}, δ_{1} and C_{0} such that, for all τ ≤ δ_{0}, and for all the solutions v of equation (7) with sup_{S_{+}^{n}} v ≥ C_{0} the following property holds true. If {q^{1}, …, q^{N}} ⊆ S_{+}^{n} are the points given by Proposition 6.4, then there holds

min_{l≠j} |q^{l} − q^{j}| ≥ δ_{1}.

Proof. The proof is very similar to that of Proposition 1.2 in [25], and is based on the use of formula (35) and on a rescaling argument. The only difference is that K and H here are non constant, but one can use conditions (49), (50), Proposition 5.1 and Lemma 5.6.

Proposition 6.6 Suppose that \{K_{i}\}_{i} ⊆ C^{1}(\overline{S_{+}^{3}}) and \{H_{i}\}_{i} ⊆ C^{2}(∂S_{+}^{3}) satisfy assumptions (49) and (50) uniformly in i. Suppose that {v_{i}}_{i} is a sequence of positive solutions of (7); then there are no interior blow-ups for {v_{i}}_{i}.

Proof. By Propositions 4.7, 6.1, 6.4, 6.5 we know that both interior and boundary blow ups are isolated simple and hence isolated. As a consequence, by Definition 4.4, the number of blow up points is bounded above by a constant depending on A_{1} and A_{2} only.

Suppose by contradiction that \overline{x} ∈ S_{+}^{3} is an interior blow-up point for v_{i}. Then, it follows from the Harnack inequality, the fact that there is just a finite number of blow-up points and Propositions 4.7, 5.2 that for some finite set \{q^{1}, …, q^{N}\} ⊆ \overline{ℝ_{+}^{3}}, with q^{1} ∈ ℝ_{+}^{3}, and some harmonic function b : ℝ_{+}^{3} → ℝ, the following holds

u_{i}(x_{i}) u_{i}(x) → a_{1} |x − q^{1}|^{−1} + ∑_{j=2}^{N} a_{j}|x − q^{j}|^{−1} + b(x), in C_{loc}^{2}(\overline{R}_{+}^{3} \ \{q^{1}, …, q^{N}\}).

Here a_{j} > 0 for all j, u_{i} = ι v_{i}, and x_{i} is the local maximum point of u_{i} converging to q^{1} with u_{i}(x_{i}) → +∞; for simplicity we can suppose that the pole of the stereographic projection is not a blow up point for v_{i}.

It follows from the Liouville Theorem and from \frac{∂b}{∂x_{3}} = 0 on ∂ℝ_{+}^{3} that b is constant on ℝ_{+}^{3}; reasoning as above we deduce

h(x) = a_{0} |x − \overline{x}|^{−1} + \overline{b} + O(|x − q^{1}|), for x close to q^{1},

where \overline{b} > 0. Let σ > 0 be such that B_{σ}(q^{1}) ⊆ ℝ_{+}^{3} \ \{q^{2}, …, q^{N}\}; as for (35), the function u_{i} satisfies

\frac{1}{8} (\frac{1}{2} − \frac{3}{p_{i} + 1}) ∫_{B_{σ}(q^{1})} K_{i}u_{i}^{p_{i}+1} − \frac{1}{8(p_{i} + 1)} ∫_{B_{σ}(q^{1})} x · ∇K_{i} u_{i}^{p_{i}+1} + \frac{1}{8} ∫_{∂B_{σ}(q^{1})} K_{i} u_{i}^{p_{i}+1} x · ν
(51)  = ∫_{∂B_{σ}(q^{1})} B(x, u_{i}, ∇u_{i}).

The estimates of the above terms are completely analogous to the corresponding ones in boundary blow up analysis, see [29]. Hence, using also Proposition 4.2, one deduces that

(\frac{1}{2} − \frac{3}{p_{i} + 1}) ∫_{B_{σ}(q^{1})} K_{i}u_{i}^{p_{i}+1} = \frac{τ_{i}}{4} (lim_{i} K_{i}(x_{i}) ∫_{ℝ^{3}} U_{1}^{6} + o(1));

∫_{B_{σ}(q^{1})} x · ∇K_{i}(x) u_{i}^{p_{i}+1} = ∫_{B_{σ}(q^{1})} x · ∇K_{i}(x_{i}) u_{i}^{p_{i}+1} + ∫_{B_{σ}(q^{1})} x · ∇(K_{i}(x) − K_{i}(x_{i})) u_{i}^{p_{i}+1} = o (u_{i}(x_{i})^{−2});

∫_{∂B_{σ}(q^{1})} K_{i} u_{i}^{p_{i}+1} = o (u_{i}(x_{i})^{−2}); ∫_{∂B_{σ}(q^{1})} B(x, u_{i}, ∇u_{i}) = −π u_{i}(x_{i})^{−2}a_{1} \overline{b} + o (u_{i}(x_{i})^{−2}) ,

for σ small. Using the last estimates and (51) we reach a contradiction. This concludes the proof.

Remark 6.7 As anticipated in the Introduction, the fact that there are no interior blow ups for equation (7) is strongly related to the fact that there are no multiple blow ups for the scalar curvature equation on S^{3}. The proof relies on the above estimate

∫_{B_{σ}} x · ∇K_{i}ui^{\frac{p+1}{2}} = o (u_{i}(x_{i})^{−2}) .

In our case, the corresponding term ∫_{(B_{σ})_{+}} x · ∇K_{i}ui^{\frac{p+1}{2}} may be of order u_{i}(x_{i})^{−2} (indeed this always happens if (K_{i}, H_{i}) ≡ (K, H) ∈ A, see the proof of Theorem 1.2) and the above proof breaks down; this is the reason of the possible presence of multiple blow ups. See Proposition 8.5.

We can summarize the above results with the following proposition.

Proposition 6.8 Suppose that {K_{i}}_{i} ⊆ C^{1}(S_{+}^{3}) and that {H_{i}}_{i} ⊆ C^{2}(∂S_{+}^{3}) satisfy (49) and (50) uniformly in i. Suppose that {v_{i}}_{i} is a sequence of positive solutions of

\begin{cases}−8Δv_{i} + 6v_{i} = K_{i}(x)v_{i}^{p_{i}}, & in S_{+}^{3}; \\ 2\frac{∂v_{i}}{∂ν} = H_{i}(x′)vi^{\frac{p_{i}+1}{2}} , & on ∂S_{+}^{3}. \end{cases}

with sup_{i} v_{i} = +∞. Then the functions {v_{i}} blow up at a finite number of points of ∂S_{+}^{3}. These blow ups are isolated simple and their distance is bounded below by a positive constant depending on min_{i} K_{i}, the C^{1} bounds of {K_{i}}_{i} and the C^{2} bounds of {H_{i}}_{i}. As a consequence the number of blow ups is bounded above by a constant depending only on these numbers. If (K_{i}, H_{i}) = (K, H) for some fixed functions K and H, then the blow up points are critical for φ.

7 Proof of Theorem 1.1

Consider the subcritical approximation (7) of equation (3), with p < 5. From the discussion in Section 2, we are reduced to find solutions of the equivalent problem (15) in the half-space. We can choose as pole of the projection the point −q, where q is a global minimum point of φ, as in the statement of Theorem 1.1. In this way the image of q under the projection is the origin in ℝ^{3}.

Solutions of (15) can be found as critical points of the Euler functional I_{τ} : D^{1,2}(ℝ_{+}^{3}) → ℝ defined as

I_{τ} (u) = \frac{1}{2} ∫_{ℝ_{+}^{3}} |∇u|^{2} − \frac{1}{6 − τ} ∫_{ℝ_{+}^{3}} W^{τ} (x)K(x)|u|^{6−τ} − \frac{4}{4 − τ} ∫_{∂ℝ_{+}^{n}} W^{\frac{τ}{2}} (x′)H(x′)|u|^{4−τ} , u ∈ D^{1,2}(ℝ_{+}^{3}).

Let also J_{τ} denote the corresponding functional on H^{1}(S_{+}^{3}). Since the standard half sphere is of positive type (see the Introduction), it is clear that the functional I_{τ} possesses a mountain pass structure; we denote by \overline{I}_{τ} the mountain pass level of I_{τ}. When τ = 0, namely when the problem is purely critical, the functional I_{τ} is simply I_{K,H}, see the notation in Section 2. We first give an estimate from above of \overline{I}_{τ}.

Lemma 7.1 There exist δ_{0} and τ_{0}, depending on K and H, such that

\overline{I}_{τ} ≤ φ(q) − δ_{0}, for all τ ∈ (0, τ_{0}).

Proof. For λ > 0, let \overline{U}_{λ} be the function defined in formula (19), with k = \frac{K(0)}{4n(n−1)} and with t satisfying 2kλt = H(0). Using a rescaling, it is easy to prove that

(52) ∫_{ℝ_{+}^{3}} |∇\overline{U}_{λ}|^{2} = ∫_{ℝ_{+}^{3}} |∇\overline{U}_{1}|^{2};  ∫_{∂ℝ_{+}^{3}} H(x′)\overline{U}_{λ}^{4} = H(0) ∫_{∂ℝ_{+}^{3}} \overline{U}_{1}^{4} + O (λ^{−2} log λ) ,  for λ large,

and

(53)  ∫_{ℝ_{+}^{3}} K(x)\overline{U}_{λ}^{6} = K(0) ∫_{ℝ_{+}^{3}} \overline{U}_{1}^{6} + λ^{−1} \frac{∂K}{∂x_{3}} (0) ∫_{ℝ_{+}^{3}} x_{3}\overline{U}_{1}^{6} + o(λ^{−1}),  for λ large.

Using equations (52), (53) and some simple computations one finds

sup_{t≥0} I_{K,H}(t\overline{U}_{λ}) = φ(q) − \frac{1}{6}λ^{−1} \frac{∂K}{∂x_{3}} (0) ∫_{ℝ_{+}^{3}} x_{3}|\overline{U}_{1}|^{2^{∗}} + o(λ^{−1}), for λ large.

We note that the condition \frac{∂K}{∂ν} (q) < 0 implies \frac{∂K}{∂x_{3}} (0) > 0 in ℝ_{+}^{3}. Hence, choosing λ_{0} sufficiently large, we find the existence of δ_{0}, depending on K and H, such that

sup_{t≥0} I_{K,H}(t\overline{U}_{λ_{0}}) ≤ φ(q) − 2δ_{0}.

By continuity, choosing τ_{0} > 0 sufficiently small we deduce that

\overline{I}_{τ} ≤ sup_{t≥0} I_{τ} (t\overline{U}_{λ_{0}}) ≤ φ(q) − δ_{0}, τ ∈ (0, τ_{0}).

The proof is thereby completed.

Proof of Theorem 1.1 concluded. For τ > 0 small, let v_{τ} be the mountain pass solution of (7). We claim that the functions {v_{τ} }_{τ} remain bounded in L^{∞}(S_{+}^{3}) as τ → 0. In fact, supposing by contradiction that {v_{τ} }_{τ} blows up, by Proposition 6.8 blow ups of {v_{τ} }_{τ} occur at the boundary of S_{+}^{3} only; let q^{1}, …, q^{N} be the blow up points. It follows from Proposition 5.1 and Lemma 5.6 that

lim_{τ→0} J_{τ} (v_{τ}) = ∑_{j=1}^{N} φ(q^{j}). (54)

On the other hand, since φ(q) = min_{∂S_{+}^{3}} φ, and since φ > 0, Lemma 7.1 contradicts equation (54).

Hence, the functions {v_{τ} } converge to a solution v of (7). We note that the function v is non-zero and strictly positive: this follows from the fact that v_{τ} is uniformly away from zero in H^{1}(S_{+}^{3}), or also from the Harnack inequality. This concludes the proof.

Remark 7.2 Using the Mountain Pass scheme and the standard analysis of Palais Smale sequences for the functional I_{K,H}, one can prove some existence results of problem (3) for any n with more restrictive hypotheses. Let φ̃(x′), x′ ∈ ∂S_{+}^{n}, denote the blow up energy corresponding to K(x′) and H(x′), computable for example by formula (28) (note that for n = 3 φ̃ is nothing but φ). The assumptions on K and H are the following. There exists q ∈ ∂S_{+}^{n} with

φ̃(q) = min\{φ̃(x′) : x′ ∈ ∂S_{+}^{n}\};  \frac{∂K}{∂ν} (q) < 0;  φ̃(q) ≤ \frac{ω_{n}}{n} (n(n − 1))^{\frac{n}{2}} (sup_{S_{+}^{n}} K)^{− \frac{n−2}{2}} . (55)

By formula (29), the last inequality asserts that the interior blow up has energy larger that the boundary blow up. We also note that, by (30), this assumption is non empty.

When n = 3, the first two conditions in (55) coincide with (6), and the last condition can be completely removed.

Remark 7.3 For the case H ≡ 0, Theorem 1.1 could be proved also using the observations in Remark 7.2 and the minimization technique in [27].

In fact, if sup_{S_{+}^{3}} K ≤ 4 sup_{∂S_{+}^{3}} K, condition (55) is satisfied, see Remark 1.3 (a).

On the other hand, if sup_{S_{+}^{3}} K > sup_{∂S_{+}^{3}} K, we can reflect K evenly on all S^{3} and look for symmetric solutions of (4), see the discussion in the Introduction. Then, using the condition sup_{S_{+}^{3}} K > 4 sup_{∂S_{+}^{3}} K, one can reason as in the proof of Theorem 4 in [27], ruling out concentration of mountain pass Palais Smale sequences outside ∂S_{+}^{3}.

8 Proof of Theorem 1.2

In this section we prove Theorem 1.2. We start by giving some further characterizations of blow up point for solutions of (33). We recall the definition of the matrix M_{lj} given in formula (10) and its least eigenvalue ρ.

Proposition 8.1 Let K ∈ C^{1}(\overline{S_{+}^{3}}) be a positive function, and let H ∈ C^{2}(∂S_{+}^{3}). Then there exists some number δ^{∗} > 0, depending only on min\overline{S_{+}^{3}} K, ∥K∥_{C^{1}(\overline{S_{+}^{3}})} and ∥H∥_{C^{2}(∂S_{+}^{3})}, with the following properties.

Let \{p_{i}\} be such that p_{i} ≤ 5, p_{i} → 5, let K_{i} → K in C^{1}(\overline{S_{+}^{3}}), H_{i} → H in C^{2}(∂S_{+}^{3}), and let v_{i} > 0 satisfy

\begin{cases}−4\frac{n−1}{n−2}Δv_{i} = K_{i}(x) v_{i}^{p_{i}} & in ∂S_{+}^{3}, \\ \frac{2}{n−2} \frac{∂v_{i}}{∂ν} = H_{i}(x′)vi^{\frac{p_{i}+1}{2}} & on ∂S_{+}^{3}. \end{cases} (56)

with max\overline{S_{+}^{3}} v_{i} → +∞ as i → +∞. Then, after passing to a subsequence, the following properties hold true

i) {v_{i}} has only isolated simple blow up points (q^{1}, …, q^{N}) ∈ F \F^{−} (N ≥ 1), with |q^{j} −q^{l}| ≥ δ^{∗} ∀j ≠ l, and ρ(q^{1}, …, q^{N}) ≥ 0. Furthermore q^{1}, …, q^{N} ∈ F^{+} if N ≥ 2.

ii) Setting

μ_{j} = 2 (\frac{K(q^{j})}{6} + H^{+}(q^{j})^{2})^{− \frac{1}{2}} K(q^{j})^{\frac{1}{4}} lim_{i} \frac{v_{i}(q_{i}^{1})}{v_{i}(q_{i}^{l})}

λ_{j} = \frac{1}{16π} (\frac{K(q^{j})}{6} + H^{+}(q^{j})^{2}) \frac{φ(q^{j})}{K(q^{j})^{\frac{1}{2}}} lim_{i} τ_{i} v_{i}(q_{i}^{j})^{2},

where H^{+} is the positive part of H and q_{i}^{j} → q^{j} is the local maximum of v_{i}, there holds

μ_{j} ∈ (0, +∞),  λ_{j} ∈ [0, +∞),  ∀j = 1, …, N.

iii) When N = 1

λ_{1} = \frac{∂K}{∂ν} (q^{1}) \frac{ψ(q^{1})}{K(q^{1})^{\frac{3}{2}}} ;

when N ≥ 2

∑_{l=1}^{N} M_{lj}μ_{l} = λ_{j} μ_{j}, ∀j = 1, …, N. (57)

iv) λ_{j} ∈ (0, +∞) ∀j = 1, …, N if and only if ρ(q^{1}, …, q^{N}) > 0.

Proof. Assertion ii) follows from Proposition 5.2 and Lemmas 4.6, 5.7. From another part, it follows from Propositions 6.1, 6.5 and Remark 6.3 that v_{i} has only isolated simple blow up points q^{1}, …, q^{N} ∈ F (N ≥ 1) with |q^{j} − q^{l}| ≥ δ^{∗} (j ≠ l) for a fixed δ^{∗} > 0 depending only on the above quantities.

Let q_{i}^{1} → q^{1} be the local maximum of v_{i} for which v_{i}(q_{i}^{1}) → +∞; performing a stereographic projection through the point −q^{1}, equation (56) is transformed into

\begin{cases}−Δu_{i}(x) = K_{i}(y) W (x)^{τ_{i}} u_{i}^{p_{i}} & in ℝ_{+}^{3}, \\ −\frac{2}{n−2} \frac{∂u_{i}}{∂x_{n}} = H_{i}(x′)W (x′)^{\frac{τ_{i}}{2}} ui^{\frac{p_{i}+1}{2}} & on ∂ℝ_{+}^{3}. \end{cases}

By our choice of the projection, it is clear that 0 is also an isolated simple blow up point for {u_{i}}. We can also suppose that none of the points {q^{1}, …, q^{N}} is mapped to +∞ by the stereographic projection, and we still denote their images by q^{1}, …, q^{N} (in particular we have q^{1} = 0). It follows from Proposition 5.2 that

u_{i}(q_{i}^{1}) u_{i}(x) → h_{1}(x) := a(q^{1}) |x|^{−1} + b_{1}(x) in C_{loc}^{2}(\overline{ℝ_{+}^{3}} \ \{q^{1}, …, q^{N}\}), (58)

where a(q^{1}) is the coefficient in (44) with lim_{i} K_{i}(x_{i}) replaced by K(q^{1}) and lim_{i} H_{i}(x_{i}) replaced by K(q^{1}). The function b_{1}(x) is harmonic in \overline{ℝ_{+}^{3}} \ \{q^{2}, …, q^{N}\}, and we have still used the notation q_{i}^{j} for the local maxima of u_{i} converging to q^{j}.

Coming back to v_{i} on S_{+}^{3} we have

lim_{i} v_{i}(q_{i}^{1}) v_{i}(x) = \frac{1}{2}a(q^{1}) G_{q^{1}}(x) + b̃_{1}(x), in C_{loc}^{2}(\overline{S_{+}^{3}} \ \{q^{2}, …, q^{N}\}),

where b̃_{1} is some regular function on \overline{S_{+}^{3}} \ \{q^{2}, …, q^{N}\} satisfying (−8Δ + 6) b̃_{1} = 0 with \frac{∂b̃_{1}}{∂ν} = 0 on ∂S_{+}^{3}.

If N = 1, then b̃_{1} = 0 by the maximum principle, while for N ≥ 2, taking into account the contribution of all the poles, we deduce that

lim_{i} v_{i}(q_{i}^{j}) v_{i}(x) = \frac{1}{\sqrt{2}}a(q^{1}) G_{q^{1}}(x) + \frac{1}{\sqrt{2}} ∑_{l≠1} a(q^{l}) G_{q^{l}}(x) lim_{i} \frac{v_{i}(q_{i}^{1})}{v_{i}(q_{i}^{l})}.

In fact, subtracting all the poles from the limit function, we obtain a regular function r : S_{+}^{3} → ℝ such that (−8Δ+6)r = 0 and \frac{∂r}{∂ν} = 0 on ∂S_{+}^{3}, so it must be r ≡ 0. In the above formula, G_{q}(x) is the function defined in the Introduction, and the convergence is in C_{loc}^{2}(\overline{S_{+}^{3}} \ \{q^{1}, …, q^{N}\}).

Using the last expression, we can compute the value of b_{1}(0) in (58), which is

b_{1}(0) = \sqrt{2} a(q^{1}) ∑_{l≠1} a(q^{l}) G_{q^{l}}(x) lim_{i} \frac{v_{i}(q_{i}^{1})}{v_{i}(q_{i}^{l})}. (59)

Hence, using (59) and Proposition 4.2, we deduce that

lim_{σ→0} ∫_{∂_{2}B_{σ}} B(x, h_{1}, ∇h_{1}) = −\sqrt{2} π a(q^{1}) ∑_{l≠1} a(q^{l}) G_{q^{l}}(q^{1}) lim_{i} \frac{v_{i}(q_{i}^{1})}{v_{i}(q_{i}^{l})}.

From another part, it follows from Lemma 4.1, Proposition 5.1, Lemma 5.7 and some computations as in Lemma 5.5 that

∫_{∂_{2}B_{σ}} B(x, h_{1}, ∇h_{1}) = \frac{1}{16}φ(q^{1}) lim_{i} τ_{i}v_{i}(q_{i}^{1})^{2} − \frac{1}{48} lim_{i} u_{i}(q_{i}^{1})^{2} ∫_{B_{σ}} x · ∇K_{i} u_{i}^{p_{i}+1}.

By Proposition 5.1, Lemma 5.7 and equation (20) it follows that

\frac{1}{48} lim_{i} u_{i}(q_{i}^{1})^{2} ∫_{B_{σ}} x · ∇K_{i} u_{i}^{p_{i}+1} = 3π ψ(q^{1}) (1 + 6\frac{H^{+}(q^{1})^{2}}{K(q^{1})} )^{−1} \frac{∂K}{∂x_{3}} (q^{1}),

where ψ is defined in (9). The tangent map of the stereographic projection π, calculated in q^{1}, is \frac{1}{2} Id,

hence it turns out that \frac{∂(K∘π^{−1})}{∂x_{3}}(q^{1}) = −2\frac{∂K}{∂ν} (q^{1}). Then, always identifying K with K ∘ π^{−1}, from the last two formulas we obtain

\frac{1}{16}φ(q^{1}) lim_{i} τ_{i}v_{i}(q_{i}^{1})^{2} = −\sqrt{2} π a(q^{1}) ∑_{l≠1} a(q^{l}) G_{q^{l}}(q^{1}) lim_{i} \frac{v_{i}(q_{i}^{1})}{v_{i}(q_{i}^{l})} + \frac{1}{4}π \frac{∂K}{∂ν} (q^{1}) a(q^{1})^{2} \frac{ψ(q^{1})}{K(q^{1})}.

Finally, using the expression of {μ_{l}} and λ_{1} we get

\frac{ψ(q^{1})}{K(q^{1})^{\frac{3}{2}}} \frac{∂K}{∂ν} (q^{1}) μ_{1} − 4\sqrt{2} \frac{G_{q^{l}}(q^{1})}{K(q^{1})^{\frac{1}{4}} K(q^{l})^{\frac{1}{4}}} μ_{l} = λ_{1} μ_{1}.

Of course a similar formula holds for every q^{j} with j ≠ 1. We have thus established (57) and completed the proof of iii).

From the last formula it follows that q^{j} ∈ F \ F^{−}, ∀j = 1, …, N, and when N ≥ 2, q^{j} ∈ F^{+}. Furthermore, since M_{jj} ≥ 0 for every j, and M_{lj} < 0 for l ≠ j, it follows from linear algebra and the variational characterization of the least eigenvalue that there exists some y = (y_{1}, …, y_{N}) ≠ 0, y_{l} ≥ 0 ∀l, such that ∑_{j=1}^{N} M_{lj} y_{j} = ρ y_{l}.

Multiplying (57) by y_{j} and summing over j, we have

ρ ∑_{l} μ_{l} y_{l} = ∑_{l,j} M_{lj} y_{j} μ_{l} = ∑_{j} λ_{j} μ_{j} y_{j} ≥ 0.

It follows that ρ ≥ 0, so we have verified part i). Part iv) follows from i)-iii).

As a consequence of i), iii) and iv) in Proposition 8.1 we deduce the following corollary. It asserts that, in the nondegenerate situation (K, H) ∈ A, one can obtain a-priori estimates for positive solutions of the subcritical problem (7), provided p is sufficiently close to \frac{n+2}{n−2}.

Corollary 8.2 Suppose (K, H) ∈ A. Then there exist constants τ_{∗} and C_{∗}, depending only on min\overline{S_{+}^{3}} K,

∥K∥_{C^{1}(\overline{S_{+}^{3}})} and ∥H∥_{C^{2}(∂S_{+}^{3})} and min\{|ρ(q^{1}, …, q^{N})| : q^{1}, …, q^{N} ∈ F, N ≥ 1\} such that

max_{\overline{S_{+}^{3}}} v ≤ \frac{C_{∗}}{\sqrt{τ}}

for every positive solution v of (7) with 0 < τ = \frac{n+2}{n−2} − p < τ_{∗}.

For \overline{x}′ ∈ ∂S_{+}^{3} and γ > 0 large, let δ_{\overline{x}′,γ} : S_{+}^{3} → ℝ be the function defined in the following way

δ_{\overline{x}′,γ}(x) = (\frac{24}{K(\overline{x}′)})^{\frac{1}{4}} (\frac{γ}{γ^{2} + 1 + (1 − γ^{2}) cos d(x̃, x)})^{\frac{1}{2}} ,

where x̃ = (x̃′, x̃_{4}) ∈ S^{3} is given by

\begin{matrix} & x̃′ \\ | & x̃′ & | \end{matrix} = \overline{x}′;  x̃_{4} = −\sqrt{24} \frac{γ}{γ^{2} − 1} \frac{H(\overline{x}′)}{\sqrt{K(\overline{x}′)}},

and where d(x̃, x) denotes the geodesic distance in S^{3}.

For all the choices of \overline{x}′ and γ, δ_{\overline{x}′,γ} satisfies the equation (−8Δ + 6)δ_{\overline{x}′,γ} = K(\overline{x}′)δ_{\overline{x}′,γ}^{5} with the

boundary condition 2\frac{∂δ_{\overline{x}′,γ}}{∂ν} = H(\overline{x}′)δ_{\overline{x}′,γ}^{3}. The functions \{δ_{\overline{x}′,γ}\}_{\overline{x}′,γ}, restricted to the half sphere S_{+}^{3}, are nothing but the pre-images under the map ι of the family \{\overline{U}_{λ}\}_{λ} defined in (19), or of some of their translations in ℝ^{n−1}.

For τ = p − \frac{n+2}{n−2}, τ > 0, let J_{τ} denote the Euler functional corresponding to problem (7), namely

J_{τ} (v) = 4 ∫_{S_{+}^{3}} |∇v|^{2} + 3 ∫_{S_{+}^{3}} v^{2} − \frac{1}{6 − τ} ∫_{S_{+}^{3}} K(x) |v|^{6−τ} − \frac{4}{4 − \frac{τ}{2}} ∫_{∂S_{+}^{3}} H(x′) |v|^{4− \frac{τ}{2}} , v ∈ H^{1}(S_{+}^{3}).

Let q^{1}, …, q^{N} ∈ F^{+} be critical points of φ with ρ(q^{1}, …, q^{N}) > 0. For ε small, define the set V_{ε} = V_{ε}(τ, q^{1}, …, q^{N}) ⊆ H^{1}(S_{+}^{3}) as

V_{ε} = \{∑_{i=1}^{N} δ_{a_{i},γ_{i}} : (γ, a) ∈ ℝ_{+}^{N} × (∂S_{+}^{3})^{N}, |a_{i} − q^{i}| < ε, ε < τ γ_{i} < \frac{1}{ε}, i = 1, …, N\} .

We also define U_{ε} = U_{ε}(τ, q^{1}, …, q^{N}) to be the ε-tubular neighborhood of V_{ε}, namely

U_{ε} = {v + z : v ∈ V_{ε}, z ∈ (T_{v}V_{ε})^{⊥}, ∥z∥ < ε} ,

where (T_{v}V_{ε})^{⊥} denotes the subspace of H^{1}(S_{+}^{3}) orthogonal to T_{v}V_{ε}.

For R > 0, set

O_{R} = \{v ∈ C^{2,α}(\overline{S_{+}^{3}}) | \frac{1}{R} ≤ v ≤ R, ∥v∥_{C^{2,α}(\overline{S_{+}^{3}})} ≤ R\} .

Using the last definitions and standard regularity results, Proposition 8.1 can be reformulated as follows.

Proposition 8.3 Let (K, H) ∈ A and let α ∈]0, 1[. Then there exist a small positive constant ε, and a large positive constant R such that, when τ > 0 is sufficiently small, there holds

v ∈ O_{R} ∪ {U_{ε}(τ, q^{1}, …, q^{N}) : q^{1}, …, q^{N} ∈ F^{+}, ρ(q^{1}, … q^{N}) > 0, N ≥ 1}

for all v ∈ H^{1}(S_{+}^{3}) satisfying v ≥ 0 a.e. and J′_{τ} (v) = 0.

Using blow up analysis, we gave necessary conditions on blowing up solutions of (7) when p tends to \frac{n+2}{n−2} from below. Now we are going to show that if (K, H) ∈ A, one can construct solutions highly concentrating at any N points q^{1}, …, q^{N} ∈ F^{+} provided ρ(q^{1}, …, q^{N}) > 0, see Proposition 8.5 below. The main tool is Implicit Function Theorem. Since the procedure is well-known, see [30], [36], we just give a general idea of the proof omitting some details.

We begin with the following technical Lemma, which proof is a consequence of standard estimates, see [6].

Lemma 8.4 Let ε and δ̃ be fixed positive numbers, let a_{i} ∈ ∂S_{+}^{3}, i = 1, 2 be such that d(a_{1}, a_{2}) ≥ δ̃, and let γ_{i} ∈ (0, +∞) be such that ε < τγ_{i} < \frac{1}{ε}, i = 1, 2. Then there exist a positive constant C such that for τ sufficiently small, the following estimates hold

∥δ_{a_{i},γ_{i}}^{4−τ}δ_{a_{j},γ_{j}}∥_{L^{\frac{6}{5}} (S_{+}^{3})} ≤ C τ, i ≠ j; ∥δ_{a_{i},γ_{i}}^{5−τ} − δ_{a_{i},γ_{i}}^{5}∥_{L^{\frac{6}{5}} (S_{+}^{3})} ≤ C τ| log τ|;
∥d(·, a_{i})δ_{a_{i},γ_{i}}^{5}∥_{L^{\frac{6}{5}} (S_{+}^{3})} ≤ C τ; ∥δ_{a_{i},γ_{i}}^{3−τ}δ_{a_{j},γ_{j}}∥_{L^{\frac{4}{3}} (∂S_{+}^{3})} ≤ C τ, i ≠ j;
∥δ_{a_{i},γ_{i}}^{3−τ} − δ_{a_{i},γ_{i}}^{3}∥_{L^{\frac{4}{3}} (∂S_{+}^{3})} ≤ C τ| log τ|; ∥d(·, a_{i})δ_{a_{i},γ_{i}}^{3}∥_{L^{\frac{4}{3}} (∂S_{+}^{3})} ≤ C τ.

Following the original arguments in [6], [30], and using the estimates in Lemma 8.4, one can prove that for τ sufficiently small

∥J′_{τ} (v)∥ ≤ O(τ | log τ|), for τ small and v ∈ V_{ε}.

Moreover, from Proposition 3.2 in [26] and standard computations, it follows that, for τ small, I′′_{τ}(u) is invertible in (T_{v}V_{ε})^{⊥}, uniformly with respect to τ and v ∈ V_{ε}. Hence by the local inversion theorem, see [2], there exists ε > 0 small (independent of τ) with the following property. For any v ∈ V_{ε}, there exists a unique w(v, τ) such that

w(v, τ) ∈ (T_{v}V_{ε})^{⊥}; J′_{τ} (v + w(v, τ)) ∈ T_{v}V_{ε}. (60)

Furthermore, the norm of w(v, τ) can be estimated as

∥w(v, τ)∥ ≤ C∥J′_{τ} (v)∥ ≤ C′ τ | log τ|, (61)

where C and C′ are fixed constants. As a consequence of the above discussion and of some computations, one finds

J_{τ} (v + w(v, τ)) = J_{τ} (v) + J′_{τ} (v)[w(v, τ)] + O (∥w(v, τ)∥^{2}) = J_{τ} (v) + O (τ | log τ|)^{2}
(62)  = ∑_{i=1}^{N} φ(a_{i}) − \frac{1}{6} ∑_{i=1}^{N} (γ_{i}^{− \frac{τ}{2}} − 1) K(a_{i}) ∫_{S_{+}^{3}} δ_{a_{i},γ_{i}}^{6} − ∑_{i=1}^{N} (γ_{i}^{− \frac{τ}{4}} − 1) H(a_{i}) ∫_{∂S_{+}^{3}} δ_{a_{i},γ_{i}}^{4}

+4 π\sqrt{6} ∑_{i=1}^{N} \frac{∂K}{∂ν} (a_{i})\frac{ψ(a_{i})}{K(a_{i})^{\frac{3}{2}}} \frac{1}{γ_{i}} − 64 π\sqrt{3} ∑_{l≠j} \frac{1}{\sqrt{γ_{j}}} \frac{1}{\sqrt{γ_{j}}} \frac{G_{a_{j}}(a_{l})}{K(a_{j})^{\frac{1}{4}} K(a_{j})^{\frac{1}{4}}} + o(τ),

as τ → 0. By means of equation (60), the manifold

Ṽ_{ε} = {v + w(v, τ), : v ∈ V_{ε}}

is a natural constraint for J_{τ}, namely a point u which is critical for J_{τ} |Ṽ_{ε} is also critical for J_{τ}. In order to find critical points of J_{τ} |Ṽ_{ε}, we differentiate J_{τ} (v + w(v, τ)) with respect to the parameters a_{i}, γ_{i}. Using standard estimates we obtain

\frac{∂}{∂a_{i}} J_{τ} (v + w(v, τ)) = \frac{∂φ}{∂a_{i}} + o(1), v ∈ V_{ε}, τ → 0; (63)

\frac{∂}{∂γ_{j}} J_{τ} (v + w(v, τ)) = \frac{1}{12} \frac{τ}{γ_{j}} K(a_{j}) ∫_{S_{+}^{3}} δ_{a_{j},γ_{j}}^{6} + \frac{1}{4} \frac{τ}{γ_{j}} H(a_{j}) ∫_{∂S_{+}^{3}} δ_{a_{j},γ_{j}}^{4}

−4 π\sqrt{6} ∑_{i=1}^{N} \frac{∂K}{∂ν} (a_{j})\frac{ψ(a_{j})}{K(a_{j})^{\frac{3}{2}}} \frac{1}{γ_{j}^{2}} + 32 π\sqrt{3} ∑_{l≠j} \frac{1}{γj^{\frac{3}{2}}} \frac{1}{γl^{\frac{1}{2}}} \frac{G_{a_{j}}(a_{l})}{K(a_{j})^{\frac{1}{4}} K(a_{j})^{\frac{1}{4}}} (64)

+o(τ^{2}), v ∈ V_{ε}, τ → 0.

Let us point out that the coefficients of \frac{1}{γ_{j}^{2}} and of _{\frac{1}{γ \frac{3}{2} ; j}} _{\frac{1}{γ \frac{1}{2} ; l}} in formula (64) coincide, when \{a_{j}\} ≡ \{q^{j}\},

with a constant multiple of the numbers M_{jj} and M_{lj} given in (10). As a consequence, since we are assuming that the least eigenvalue ρ of (M_{lj}) is positive, the above coefficients form a positive definite and invertible matrix. Using this condition, equation (63) and the fact that φ is Morse, one can prove that

deg_{H^{1}(S_{+}^{3})} (J′_{τ} |Ṽ_{ε}, Ṽ_{ε}, 0) = (−1)^{∑_{j=1}^{N}(2−m(φ,q^{j}))}. (65)

By the invertibility of J′′_{τ} in the normal direction to V_{ε}, and by the fact that the functions δ_{a_{i},γ_{i}} have Morse index 1, it follows from (65) that

deg_{H^{1}(S_{+}^{3})} (J′_{τ} , U_{ε}, 0) = (−1)^{N+∑_{j=1}^{N}(2−m(φ,q^{j}))}. (66)

Since the above degree is always different from zero, J_{τ} has at least one critical point in U_{ε}; moreover it is standard to prove that critical points of J_{τ} in U_{ε} are non-negative functions when τ is sufficiently small. From [1] and [17] then it follows that these solutions are also regular and strictly positive.

We collect the above discussion in the following Proposition.

Proposition 8.5 Let (K, H) ∈ A, and let ε > 0 be small enough. Then, if q^{1}, …, q^{N} ∈ F^{+} with ρ(q^{1}, …, q^{N}) > 0, and if τ > 0 is sufficiently small, the functional J_{τ} possesses a critical point in U_{ε}(τ, q^{1}, …, q^{N}). Moreover, formula (66) holds true and all the critical points of J_{τ} are strictly positive functions on S^{n}.

We need now the following lemma which, combined with Corollary 8.2, will be useful to obtain a-priori estimates for the computation of some degree formula, see Proposition 8.7 below.

Lemma 8.6 Suppose (K, H) ∈ A. Then there exists an homotopy (K_{t}, H_{t}) : [0, 1] → A with the following properties.

(j) H_{t} = tH for all t ∈ [0, 1]; moreover 4π\sqrt{\frac{6}{K_{0}}} ≡ φ and K_{1} = K.

(jj) Setting

φ_{t}(x′) = 4π \sqrt{\frac{6}{K_{t}(x′)}} (\frac{π}{2} − arctan (H_{t}(x′) \sqrt{\frac{6}{K_{t}(x′)}})) , x′ ∈ ∂S_{+}^{3},

then there holds φ_{t} ≡ φ for all t ∈ [0, 1].

(jjj) Let F_{t}, F_{t}^{±}, ρ_{t} denote the counterparts of F, F^{±}, ρ corresponding to the functions (K_{t}, H_{t}). Then F_{t} ≡ F and F_{t}^{±} ≡ F^{±} for all t ∈ [0, 1]. Moreover there exists a positive constant \overline{C}, depending only on min\overline{S_{+}^{3}} K, ∥K∥_{C^{2}(\overline{S_{+}^{3}})}, ∥H∥_{C^{2}(∂S_{+}^{3})} and min\{|ρ(q^{1}, …, q^{N})| : q^{1}, …, q^{N} ∈ F, N ≥ 1\} such that

(67)  min_{\overline{S_{+}^{3}}} K_{t} ≥ \frac{1}{\overline{C}} ,  ∥K∥_{C^{2}(\overline{S_{+}^{3}})} ≤ \overline{C},  for all t ∈ [0, 1];

(68)  min\{|ρ(q^{1}, …, q^{N})| : q^{1}, …, q^{N} ∈ F_{t}, N ≥ 2\} ≥ \frac{1}{\overline{C}} ,  for all t ∈ [0, 1].

Proof. First we note that, for a fixed valued of H(x′) we have

(69)  lim_{K(x′)→+∞} φ(x′) = 0;  lim_{K(x′)→0} φ(x′) = \begin{cases}+∞ & if H(x′) ≤ 0, \\ \frac{4π}{H(x′)} & if H(x′) > 0. \end{cases}

Moreover, using some simple computations, one finds

\frac{∂φ(x′)}{∂K(x′)} = −4π \frac{\sqrt{6}}{K(x′)^{\frac{3}{2}}} [\frac{π}{2} − arctan (H(x′)\sqrt{\frac{6}{K(x′)}}) − \frac{H(x′)\sqrt{6K(x′)}}{K(x′) + 6H(x′)^{2}} ] < 0. (70)

As a consequence of (69), (70) and the implicit function theorem, one finds a unique positive function K_{t}(x′), x′ ∈ ∂S_{+}^{3}, for which

4π \sqrt{\frac{6}{K_{t}(x′)}} (\frac{π}{2} − arctan (tH(x′) \sqrt{\frac{6}{K_{t}(x′)}})) = φ(x′), for all x′ ∈ ∂S_{+}^{3}.

We point out that, since φ is of class C^{2}, also K_{t} is of class C^{2} on ∂S_{+}^{3}. With such a choice of K_{t}, properties (j) and (jj) are clearly satisfied.

We are now going prove (jjj), finding a suitable extension of K_{t} in the interior of S_{+}^{3}. Note that by (jj), F_{t} coincides with F for all t. For q^{j} ∈ F, choose \frac{∂K_{t}}{∂ν} (q^{j}) satisfying

\frac{∂K_{t}}{∂ν} (q^{j}) = \frac{K_{t}}{K} (q^{j}) \frac{4π − H(q^{j})φ(q^{j})}{4π − tH(q^{j})φ(q^{j})} \frac{∂K}{∂ν} (q^{j}), q^{j} ∈ F. (71)

Let (M_{t})_{lj} be the counterpart of the matrix M_{lj} defined in (10) for the functions (K_{t}, H_{t}). It is clear from (9) and (71) that

(M_{t})_{lj} = \frac{K(q^{l})^{\frac{1}{4}} K(q^{j})^{\frac{1}{4}}}{K_{t}(q^{l})^{\frac{1}{4}} K_{t}(q^{j})^{\frac{1}{4}}} M_{lj}, q^{j} ∈ F.

As a consequence, from the multi-linearity of the determinant one deduces that

det M_{t}(q^{1}, …, q^{N}) = ∏_{j=1}^{N} \frac{K_{t}(q^{j})^{\frac{1}{2}}}{K(q^{j})^{\frac{1}{2}}} det M(q^{1}, …, q^{N}),

and hence it follows that ρ_{t}(q^{1}, …, q^{N}) ≠ 0 whenever ρ(q^{1}, …, q^{N}) ≠ 0. This implies that (K_{t}, H_{t}) ∈ A for all t, that F_{t}^{±} ≡ F^{±} for all t, and that (68) is satisfied. Then it is easy to extend K_{t} in the interior of S_{+}^{3} so that also (67) holds true. This concludes the proof.

Consider the following problem in S_{+}^{3}

\begin{cases}−8Δv + 6v = f_{1} & in S_{+}^{3}; \\ 2\frac{∂v}{∂ν} = f_{2} & on ∂S_{+}^{3}. \end{cases} (72)

It is standard, see e.g. [1], that if f_{1} ∈ C^{α}(\overline{S_{+}^{3}}) and if f_{2} ∈ C^{1,α}(∂S_{+}^{3}) for some α ∈ (0, 1), then there exists a solution v ∈ C^{2,α} of (72). We denote by Ξ the operator which associates to (f_{1}, f_{2}) the solution v of (72), and we extend the definition of Ξ also to the case of weak solutions of (72).

When (K, H) ∈ A and the number τ is bounded from below, we have compactness result for positive solutions of (7) and we can compute their total degree. We recall the above definition of the set O_{R}.

Proposition 8.7 Suppose (K, H) ∈ A. Then there exist constants τ_{0}, C_{0} and δ_{0}, depending only on min_{S_{+}^{3}} K, ∥K∥_{C^{1}(\overline{S_{+}^{3}})}, ∥H∥_{C^{2}(∂S_{+}^{3})} and min\{|ρ(q^{1}, …, q^{N})| : q^{1}, …, q^{N} ∈ F, N ≥ 1\} with the following properties

i) {v ∈ H^{1}(S_{+}^{3}) : v ≥ 0 a.e., J′_{τ_{0}}(v) = 0} ⊆ O_{C_{0}};

ii) for C, δ > 0 set O_{C,δ} = {u ∈ H^{1}(S_{+}^{3}) : ∃v ∈ O_{C} such that ∥u − v∥_{H^{1}(S_{+}^{3})} < δ}: then J′_{τ_{0}} ≠ 0 on ∂O_{C_{0},δ_{0}}, and

deg_{H^{1}(S_{+}^{3})}(u − Ξ(K |u|^{4−τ_{0}}u, H |u|^{2− \frac{τ_{0}}{2}} u), O_{C_{0},δ_{0}}, 0) = −1. (73)

Proof. Let K̃ : \overline{S_{+}^{3}} → ℝ be the function defined in the following way, using stereographic coordinates

K̃(x) = 2 + \frac{|x|^{2} − 1}{x_{1}^{2} + x_{2}^{2} + (x_{3} + 1)^{2}} , x ∈ \overline{ℝ_{+}^{3}}.

We point out that K̃ is smooth and strictly positive on \overline{S_{+}^{3}} and satisfies

x · ∇K̃(x) ≥ 0, for all x ∈ \overline{ℝ_{+}^{3}}. (74)

As a consequence, by equation (35), there is no solution of (3) with (K, H) = (K̃, 0).

Consider the following homotopy from [0, 2] into C^{2}(\overline{S_{+}^{3}}) × C^{2}(∂S_{+}^{3})

s → (K_{s}, H_{s}) = \begin{cases}((1 − s)K̃ + sK_{0}, 0) & for s ∈ [0, 1]; \\ (K_{s−1}, H_{s−1}) & for s ∈ [1, 2], \end{cases}

where K_{t} and H_{t} are the functions given in Lemma 8.6. This homotopy connects (K̃, 0) to (K, H), when the parameter goes from 0 to 2.

Define J_{s,τ} : H^{1}(S_{+}^{3}) → ℝ to be the Euler functional corresponding to (7) for (K, H) = (K_{s}, H_{s}). We claim that for C_{0} sufficiently large and τ_{0} sufficiently small there holds

{v ∈ H^{1}(S_{+}^{3}) : v ≥ 0 a.e., J′_{s,τ_{0}}(v) = 0} ⊆ O_{C_{0}}, for all s ∈ [0, 2]. (75)

Of course, by the above discussion, all these weak solutions are of class C^{2,α} and positive.

For s ∈ [0, 1], upper bounds follow from standard blow up arguments and from non-existence results for the problems

−Δu = u^{p}  in ℝ^{n},  u > 0, (76)

and

\begin{cases}−Δu = u^{p} & in ℝ_{+}^{n}, u > 0; \\ \frac{∂u}{∂x_{n}} = 0 & on ∂ℝ_{+}^{n}. \end{cases} (77)

The non-existence result for (76) has been proved in [24] while that for (77) is a consequence of the former.

For s ∈ [1, 2], upper bounds follow from Corollary 8.2 and Lemma 8.6, choosing τ_{0} sufficiently small. Once upper bounds are achieved, lower bounds follow from the Harnack inequality, see Lemma 4.6.

This proves (75) and hence property i) in the statement.

Using (75), it is standard to prove that J′_{s,τ_{0}} ≠ 0 on ∂O_{C_{0},δ_{0}} for δ_{0} sufficiently small and for all s ∈ [0, 2]; this simply follows by testing J′_{s,τ_{0}} on the positive parts of the solutions.

Therefore, for the homotopy property of the degree, we only need to establish (73) for (K, H) = (K̃, 0). In this case the formula follows from Propositions 8.1, 8.3 and 8.5, since there are no solutions of (3) with (K, H) = (K̃, 0) and since K̃|_{∂S_{+}^{3}} possesses just one critical point with \frac{∂K̃}{∂ν} > 0. This concludes the proof.

Remark 8.8 In the case in which H ≥ 0 on ∂S_{+}^{3}, the a-priori estimates in the previous proof could be obtained from the non existence results for (76) and for the problem

\begin{cases}−Δu = u^{p} & in ℝ_{+}^{n}, u > 0; \\ \frac{∂u}{∂ν} = a u^{q} & on ∂ℝ_{+}^{n}, \end{cases} (78)

where 1 < p < \frac{n+2}{n−2}, 1 < q < \frac{n}{n−2} and a ≤ 0, see [18] and [32]. As far as our knowledge, existence or non-existence of solutions is not known for a > 0 and general subcritical exponents p and q.

Proof of Theorem 1.2 The norm inequality in (11) follows from Propositions 5.1, 6.8 and some standard estimates. Suppose by contradiction that the second inequality in (11) is not true; then there exist a sequence of solutions {v_{i}} of (3) blowing up at q^{1}, …, q^{N} ∈ ∂S_{+}^{3}, and these blow ups are isolated simple. Taking into account that (K, H) ∈ A and that λ_{j} = 0 for all j (since τ_{i} = 0 for all i), we get a contradiction from Proposition 8.1 iv). Hence (11) is proved.

Using Proposition 8.3, (11) and the homotopy invariance of the Leray-Schauder degree, we have

(79) deg_{C^{2,α}(\overline{S_{+}^{3}})}(u − Ξ(K |u|^{4}u, H |u|^{2}u)), O_{R}, 0) = deg_{C^{2,α}(\overline{S_{+}^{3}})}(u − Ξ(K |u|^{4−τ} u, H |u|^{2− \frac{τ}{2}} u), O_{R}, 0),

for τ sufficiently small. By Propositions 8.3 and 8.5, for suitable value of ε, and for τ small, we know that the non-negative solutions of J′_{τ} = 0 are either in O_{R} or in some U_{ε}(τ, q^{1}, …, q^{N}); viceversa for all q^{1}, …, q^{N} ∈ F_{+} with ρ(q^{1}, …, q^{k}) > 0, there are (positive) solutions of J′_{τ} = 0 in U_{ε}, and degree of J′_{τ} on U_{ε} is given by (66).

Let C_{0} >> R, τ_{0} and δ_{0} be given by Proposition 8.7; take also δ_{1} << δ_{0}. By Proposition 8.5, (73) and by the excision property of the degree, we have

(80) deg_{H^{1}(S_{+}^{3})}(u − Ξ(K |u|^{4−τ_{0}}u, H |u|^{2− \frac{τ_{0}}{2}} u), O_{R,δ_{1}}, 0) = Index(K, H).

As in the proof of Proposition 8.7, one can check that there are no critical points of J_{τ_{0}} in \overline{O_{R,δ_{1}}} \ O_{R}, hence Theorem B.2 of [29] applies and yields

deg_{H^{1}(S_{+}^{3})}(u − Ξ(K |u|^{4−τ_{0}}u, H |u|^{2− \frac{τ_{0}}{2}} u), O_{R,δ_{1}}, 0) = deg_{C^{2,α}(\overline{S_{+}^{3}})}(u − Ξ(K |u|^{4−τ_{0}}u, H |u|^{2− \frac{τ_{0}}{2}} u), O_{R}, 0). (81)

Then the conclusion follows from (79), (80) and (81). The proof of Theorem 1.2 is thereby completed.

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Z.D. : Mathematical Sciences Research Institute (MSRI) - 1000 Centennial Drive - Berkeley, CA 94720-5070 and Université de Cergy-Pontoise - Département de Mathématique - Site de Saint-Martin - 2 Avenue Adolphe Chauvin - F 95302 Cergy-Pontoise Cedex, France A.M. : Rutgers University - Department of Mathematics - 110 Frelinghuysen Road, Busch Campus, Hill Center - 08854-8019 Piscataway, NJ - USA M.O.A. : Scuola Internazionale Superiore di Studi avanzati (SISSA) - Via Beirut, 2-4 - 34014 Trieste - Italy

e-mail : Zindine.Djadli@math.u-cergy.fr, malchiod@math.rutgers.edu, ahmedou@sissa.it