On a geometric equation involving the Sobolev trace critical exponent

Journal of Inequalities and Applications, Aug 2013

In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of R n , n ≥ 4 . Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem. MSC: 58E05, 35J65, 53C21, 35B40.

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On a geometric equation involving the Sobolev trace critical exponent

Journal of Inequalities and Applications On a geometric equation involving the Sobolev trace critical exponent Mohamed Ali Al-Ghamdi 0 Hichem Chtioui 0 Khadija Sharaf 0 0 Department of Mathematics, King Abdulaziz University , Jeddah , Saudi Arabia In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of Rn, n ≥ 4. Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem. MSC: 58E05; 35J65; 53C21; 35B40 boundary mean curvature; variational problem; loss of compactness; β-flatness condition; critical point at infinity 1 Introduction and hg is the mean curvature of g on Sn–. which solves the following nonlinear boundary value equation: u =  in Bn, n ⎩ ∂∂uν + n– u = n– Hu n– on Sn–, ∇H(y) = . (–)ind(H,y) = . H(x) = H() + where bi = bi(y) ∈ R \ {}, ∀i = , . . . , n – , in=– bi =  and [sβ=] |∇sR(x)||x|s–β = o() as x tends to zero. Here ∇s denotes all possible derivatives of order s and [β] is an integer part of β. Let we have already mentioned, another obstruction to solving the problem, the so-called Kazdan-Warner obstruction. There have been many papers on the problem and related ones, please see [–] and the references therein. One group of existence results has been obtained under hypotheses involving the Laplacian H at the critical points y of H; see [, ] for n = , and [–] for n ≥ . For example, in [] and [], it is assumed that H is a Morse function and The result has been extended to any dimension n ≥  in []. Roughly, it is assumed that there exists β, n –  < β < n – , such that in some geodesic normal coordinate system centered at y, we have K = y ∈ Sn–, ∇H(y) =  , K+ = y ∈ K, y∈K+ (–)n––˜i(y) = ; see []. Let us observe that a condition like (.) appeared first in [] concerning the scalar curvature problem; see also []. In this work we restrict our attention to problem (.) under condition that H is a C-function satisfying (f)β condition with n –  ≤ β < n – . This leads to an interesting new phenomenon, that is, the presence of multiple blow-up points. In fact, when looking to the possible formations of blow-up points, it comes out that the strong interaction of the bubbles in the case where n –  < β < n –  forces all blow-up points to be single, while in the case where β = n – , we have a balance phenomenon, that is, any interaction of two bubbles is of the same order with respect to the self-interaction. We denote by the operator which associates to H the solution v of (.), and we extend the definition of to the case of weak solutions of (.). Let Kn– = y ∈ K, β = β(y) = n –  . ⎧ – kn=– bk(yij ) ⎪⎪⎨ Mjj = nn–– c˜ H(ylj )n– , j ∈ {, . . . , N }, ⎪⎪⎩ Mlj = – n– c [H(yGil )(Hyil(y,yijij))] n– , l, j ∈ {, . . . , N }, l = j, Here G(q, ·) denotes Green’s function for the operator with point q. Let ρ = ρ(yi , . . . , yiN ) be the least eigenvalue of M. We assume the following: We now introduce the following set: We then have the following theorem. (–)n––˜i(y) + y∈K+\Kn– (yi ,...,yiN )∈Cn+– (–)N–+ jN= n–+˜i(yij) = , then (.) has at least one solution. Moreover, for generic H, we have S ≥  – (–)n––˜i(y) – y∈K+\Kn– (yi ,...,yiN )∈Cn+– (–)N–+ jN= n–+˜i(yij) , where S denotes the set of solutions of (.). tion of a suitable pseudo-gradient, for which the Palais-Smale condition is satisfied along the decreasing flow lines as long as these flow lines do not enter the neighborhood of some specific critical points of H. A similar Morse lemma at infinity has been established for problem (.) under the hypothesis that the function H is of class C and the order of flatness at critical points of H is β ∈ ]n – , n – [; see []. The rest of this paper is organized as follows. In Section , we set up the variational problem and we recall the expansion of the gradient of the associated Euler-Lagrange functional near infinity. In Section , we construct a suitable pseudo-gradient and we characterize the critical points at infinity. Lastly, in Section , we prove our main result. 2 General framework and some known facts 2.1 Variational problem First, we recall the functional setting and the variational problem and its main features. Problem (.) has a variational structure. The Euler-Lagrange functional is J(u) = defined on H(Bn) equipped with the norm u  = Bn |∇u| dvg + where dvg and dσg denote the Riemannian measure on Bn and Sn– induced by the metric g. We denote by the unit sphere of H(Bn), and we set + = {u ∈ /u ≥ }. u |x | +(xxn + ) , |x|x||++(xxnn+–) , Bn |∇u| + u = In the sequel, we identify the function H a (...truncated)


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Mohamed Al-Ghamdi, Hichem Chtioui, Khadija Sharaf. On a geometric equation involving the Sobolev trace critical exponent, Journal of Inequalities and Applications, 2013, pp. 405, 2013,