New solutions for the Lane-Emden problem in planar domains

Abstract

We consider the Lane-Emden problem$-\Delta u=|u{|}^{p-1}u\text{in}\Omega ,u=0\text{on}\partial \Omega ,$ where $\Omega \subset R^2$ is a smooth bounded domain. When the exponent p is large, the existence and multiplicity of solutions strongly depend on the geometric properties of the domain, which also deeply affect their qualitative behavior. Remarkably, a wide variety of solutions, both positive and sign-changing, have been found when p is sufficiently large. In this paper, we focus on this topic and find new sign-changing solutions that exhibit an unexpected concentration phenomenon as p approaches +∞.