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

• Volume: 13, Issue: 2, page 101-105
• ISSN: 1120-6330

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## Abstract

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

## How to cite

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El Mehdi, Khalil, and Pacella, Filomena. "Morse index and blow-up points of solutions of some nonlinear problems." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.2 (2002): 101-105. <http://eudml.org/doc/252305>.

@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 -

## References

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3. Gilbarg, D. - Trudinger, N.S., Elliptic Partial Differential Equations of Second Order. Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin-New York1983. Zbl0562.35001MR737190
4. Grossi, M. - Molle, R., On the shape of the solutions of some semilinear elliptic problems. Preprint 2001. Zbl1136.35364MR1958020DOI10.1142/S0219199703000914
5. Li, Y.Y., Prescribing scalar curvature on ${S}^{n}$ and related topics, Part I. J. Differential Equations, 120, 1995, 319-410. Zbl0827.53039MR1347349DOI10.1006/jdeq.1995.1115
6. Pohozaev, S.I., Eigenfunctions of $\mathrm{△}u+\lambda f\left(u\right)=0$. Soviet. Math. Dokhl., 6, 1965, 1408-1411. Zbl0141.30202
7. Rey, O., The question of interior blow-up for an elliptic neumann problem: the critical case. Preprint Ecole Polytechnique 2001. Zbl1066.35033
8. Schoen, R., Courses at Stanford University (1988) and New-York University (1989). Unpublished.

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