Boundary clustered layers near the higher critical exponents

Abstract

We consider the supercritical problem {equation*} -\Delta u=|u| ^{p-2}u\text{\in}\Omega,\quad u=0\text{\on}\partial\Omega, {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 $L^{q}(\mathbb{R}^{N-k})$, $1\leq k\leq N-3.$ 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.