Boundary clustered layers near the higher critical exponents

Abstract

We consider the supercritical problem {equation*} -Δu=|u| ^{p-2}u\text{\in}Ω,\quad u=0\text{\on}\partialΩ, {equation*} where $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^N$ and $p$ smaller than the critical exponent $2_{N,k}^{\ast }:=\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^1({\mathbb{R}}^{N-k})$ in $L^q({\mathbb{R}}^{N-k})$, $1\le k\le N-3.$ We show that in some suitable domains $\Omega$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\partial \Omega$ as $p$ approaches $2_{N,k}^{\ast }$ from below. Key words:Nonlinear elliptic boundary value problem; critical and supercritical exponents; existence of positive and sign changing solutions.

References in corpus (1)

• Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

Cited by in corpus (5)

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• Supercritical problems in domains with thin toroidal holes