Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Abstract

We consider the semilinear Lane–Emden problem

where $p>1$ and $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of (${\mathcal{E}}_p$), as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in ${\mathbb{R}}^2$.