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

Abstract

We consider the semilinear Lane–Emden problem $$ \label{problemAbstract}\left{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega\ u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right.\tag{$\mathcal E\_p$} $$ 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 \eqref{problemAbstract}, 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 $\mathbb R^2$.