Stable Solutions of $-\Delta u+\lambda u=|u|^{p-1}u $ in Strips

Abstract

This paper is devoted to study the following equation $-\Delta u+\lambda u= |u|^{p-1}u \;\;\text{in}\;\Omega $ , with homogeneous Dirichlet or Neumann boundary conditions where $p>1$ , $\lambda >0$ , $\Omega =\mathbb{R}^{n-k}\times \omega $ , $n\geq 2$ , $k\geq 1$ , and $\omega $ is a smoothly bounded domain of $\mathbb{R}^{k}$ . We prove Liouville-type theorems for $C^{2}$ solutions which are stable or stable outside a compact set of $\Omega $ . We first provide an integral estimate using stability argument which combined with Pohozaev-type identity allow to obtain nonexistence results for $p\in [p_{s}(n), p_{s}(n-k)]$ , where $p_{s}(n)$ is the classical Sobolev exponent in dimension $n$ . Also, we establish monotonicity formula to prove the nonexistence of nontrivial solution which is stable or stable outside a compact set of $\Omega $ for all $p>1$ .