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

El Mehdi, Khalil ; Pacella, Filomena

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 13 (2002), p. 101-105 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

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.

Résumé

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.

Détail

Publié le : 2002-06-01

MR 1949483 | Zbl 1221.35145

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