New solutions for the Lane-Emden problem on planar domains

TL;DR

This paper investigates the Lane-Emden problem on planar domains, revealing how large exponents influence solution existence, multiplicity, and concentration phenomena, with new sign-changing solutions discovered.

Contribution

It introduces new sign-changing solutions exhibiting concentration phenomena for large exponents, highlighting the impact of domain geometry on solution behavior.

Findings

Existence of multiple solutions depending on domain geometry

Discovery of sign-changing solutions with concentration phenomena

Solutions become highly localized as the exponent increases

Abstract

We consider the Lane-Emden problem on planar domains. When the exponent 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 the exponent is sufficiently large. In this paper, we focus on this topic and fine new sign-changing solutions that exhibit an unexpected concentration phenomenon as the exponent approaches infinity.