Morse index and blow-up points of solutions of some nonlinear problems

El Mehdi, Khalil ; Pacella, Filomena

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 2, pp. 101-105.

Abstract (VO)

In this Note we consider the following problem $$ \begin{cases} - \triangle u = N(N-2) u^{p_{\epsilon}} - \lambda u \, \text{in } \Omega \\ u > 0 \text{in } \Omega \\ u = 0 \text{on } \partial \Omega. \end{cases} $$ where $\Omega$ is a bounded smooth starshaped domain in $\mathbb{R}^{N}$, $N \ge 3$, $p_{\epsilon} = \frac{N+2}{N-2} - \epsilon$, $\epsilon > 0$, and $\lambda \ge 0$. We prove that if $u_{\epsilon}$ is a solution of Morse index $m > 0$ than $u_{\epsilon}$ cannot have more than $m$ maximum points in $\Omega$ for $\epsilon$ sufficiently small. Moreover if $\Omega$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $\epsilon$ sufficiently small.

Riassunto

Si consideri il seguente problema $$ \begin{cases} - \triangle u = N(N-2) u^{p_{\epsilon}} - \lambda u \, \text{in } \Omega \\ u > 0 \text{in } \Omega \\ u = 0 \text{on } \partial \Omega. \end{cases} $$ dove $\Omega$ è un dominio regolare limitato e stellato in $\mathbb{R}^{N}$, $N \ge 3$, $p_{\epsilon} = \frac{N+2}{N-2} - \epsilon$, $\epsilon > 0$, e $\lambda \ge 0$. Si dimostra che se $u_{\epsilon}$ è una soluzione di indice di Morse m > 0 allora $u_{\epsilon}$ non può avere più di $m$ punti di massimo in $\Omega$, se $\epsilon$ è sufficientemente piccolo. Inoltre, se $\Omega$ è convesso si dimostra che ogni soluzione di indice di Morse $1$ ha un unico punto critico e gli insiemi di livello sono stellati, se $\epsilon$ è sufficientemente piccolo.

Bibliographie

[1] A. Bahri - Y.Y. Li - O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Cal. Var. Partial Differential Equatios, 3, 1995, 67-93. | DOI | MR | Zbl

[2] B. Gidas - J. Spruck, A priori bounds for a positive solutions of nonlinear elliptic equation. Comm. Partial Differential Equation, 6, 1981, 883-901. | DOI | MR | Zbl

[3] D. Gilbarg - N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin-New York 1983. | MR | Zbl

[4] M. Grossi - R. Molle, On the shape of the solutions of some semilinear elliptic problems. Preprint 2001. | DOI | MR | Zbl

[5] Y.Y. Li, Prescribing scalar curvature on $S^{n}$ and related topics, Part I. J. Differential Equations, 120, 1995, 319-410. | DOI | MR | Zbl

[6] S.I. Pohozaev, Eigenfunctions of $\triangle u + \lambda f(u) = 0$. Soviet. Math. Dokhl., 6, 1965, 1408-1411. | Zbl

[7] O. Rey, The question of interior blow-up for an elliptic neumann problem: the critical case. Preprint Ecole Polytechnique 2001. | Zbl

[8] R. Schoen, Courses at Stanford University (1988) and New-York University (1989). Unpublished.