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.
Categories: math.AP