A study on the nonexistence of stable solutions for nonlinear elliptic equations in strips

Abstract

In this paper, we investigate the nonexistence of solutions of certain nonlinear elliptic equations, focusing on solutions that are stable or stable outside a compact set, potentially unbounded, and sign-changing. Our primary methods include integral estimates, Pohozaev-type identity and the monotonicity formula. Our classification approaches as a sharp result, specifically, in the subcritical case (i.e, \(1< p < \frac{n+4}{n-4}\)), we establish the existence of a mountain pass solution with a Morse index of 1 in the subspace of \(H^2 \cap H_0^1(\Omega )\) that exhibits cylindrical symmetry.