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

This paper is devoted to study the following equation $-\mathrm{\Delta }u+\lambda u={|u|}^{p-1}u\text{in}\mathrm{\Omega }$, with homogeneous Dirichlet or Neumann boundary conditions where $p>1$, $\lambda >0$, $\mathrm{\Omega }={\mathbb{R}}^{n-k}\times \omega$, $n\ge 2$, $k\ge 1$, 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 $\mathrm{\Omega }$. 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 $\mathrm{\Omega }$ for all $p>1$.