Boundary clustered layers near the higher critical exponents

We consider the supercritical problem {equation*} -\Delta u=|u| ^{p-2}u$\text{in}$\Omega,\quad u=0$\text{on}$\partial\Omega, {equation*} where $\mathrm{\Omega }$ 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 $L^q({\mathbb{R}}^{N-k})$, $1\leq k\leq N-3.$Weshowthatinsomesuitabledomains$\Omega$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\mathrm{\partial }\mathrm{\Omega }$ 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.