A Nondegeneracy Result For A Nonlinear Elliptic Equation

A nondegeneracy result for a nonlinear elliptic equation, Massimo Grossi∗ Abstract Let Ω be a smooth bounded domain of IRN , with N ≥ 5. In this paper we prove, for ε > 0 small, the nondegeneracy of the solution of the problem  N +2  −∆u = u N −2 + εu in Ω (0.1) u>0 in Ω,  u=0 on ∂Ω, under a nondegeneracy condition on the critical points of the Robin function. Our proof uses different techniques with respect to other known papers on this topic. 1 Introduction Let Ω be a smooth bounded domain of IRN , N ≥ 3. For ε > 0 let us consider the following problem  N +2  −∆u = u N −2 + εu u>0  u=0 in Ω in Ω, on ∂Ω, (1.1) Let λ1 the first eigenvalue of −∆ in H01 (Ω). In a famous paper Brezis and Nirenberg (see [4]) proved that there exists a constant λ∗ ≥ 0 such that if ε ∈ (λ∗ , λ1 ) then (1.1) admits at least one solution (actually in [4] was proved that λ∗ = 0 if N ≥ 4). This problem was widely studied in the last years. Among the many results on this subject, we would like to recall the papers of Han (see [12]) and Rey (see [13] and [14]) where the authors studied the asymptotic behaviour of solutions of (1.1) verifying R lim ε→0  R Ω |∇uε |2 dx 2N  NN−2 = S. (1.2) |uε | N −2 dx Ω where S is the best constant in Sobolev inequality. This solutions are usually called one-bump solutions since it is possible to show that ∗ Dipartimento di Matematica, Universit` a di Roma ”La Sapienza” P.le Aldo Moro, 2 - 00185 Roma. Supported by M.U.R.S.T., project “Variational methods and nonlinear differential equations”. 1 |∇uε |2 * C(N )δP (1.3) weakly in the sense of the measures. In (1.3) C(N ) denotes a positive constant depending only on N and δP is the Dirac-function centered at P . The point P is often called the concentration point of uε