Concentrating  solutions for an anisotropic elliptic problem withlarge exponent

Liping Wang; Dong Ye

doi:10.3934/dcds.2015.35.3771

Article Contents

2015, Volume 35, Issue 8: 3771-3797. Doi: 10.3934/dcds.2015.35.3771

This issue Previous Article Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems Next Article Global attractor for weakly damped gKdV equations in higher sobolev spaces

Concentrating solutions for an anisotropic elliptic problem with large exponent

Liping Wang^1, and
Dong Ye^2,

1.
Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
2.
IECL, UMR 7502, Département de Mathématiques, Université de Lorraine, Bât. A, Ile de Saulcy, 57045 Metz Cedex 1

Received: August 31, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

We consider the following anisotropic boundary value problem $$\nabla (a(x)\nabla u) + a(x)u^p = 0, \;\; u>0 \ \ \mbox{in} \ \Omega, \quad u = 0 \ \ \mbox{on} \ \partial\Omega,$$ where $\Omega \subset \mathbb{R}^2$ is a bounded smooth domain, $p$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of anisotropic coefficient $a(x)$ on the existence of concentrating solutions. We show that at a given strict local maximum point of $a(x)$, there exist arbitrarily many concentrating solutions.

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Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J40.

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