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

@article{ElMehdi2002,
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.},
author = {El Mehdi, Khalil, Pacella, Filomena},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Elliptic problem; Morse index; Blow-up analysis; elliptic problem; blow-up analysis},
language = {eng},
month = {6},
number = {2},
pages = {101-105},
publisher = {Accademia Nazionale dei Lincei},
title = {Morse index and blow-up points of solutions of some nonlinear problems},
url = {http://eudml.org/doc/252305},
volume = {13},
year = {2002},
}

TY - JOUR
AU - El Mehdi, Khalil
AU - Pacella, Filomena
TI - Morse index and blow-up points of solutions of some nonlinear problems
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/6//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 2
SP - 101
EP - 105
AB - 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.
LA - eng
KW - Elliptic problem; Morse index; Blow-up analysis; elliptic problem; blow-up analysis
UR - http://eudml.org/doc/252305
ER -