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

Résumé

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.