Bubble concentration on spheres for supercritical elliptic problems

TL;DR

This paper proves the existence of solutions to a supercritical elliptic PDE that concentrate and blow up on spheres within an annular domain as a parameter approaches zero, using a reduction method.

Contribution

It introduces a novel application of finite dimensional reduction to find solutions concentrating on spheres for supercritical elliptic problems.

Findings

Existence of positive solutions concentrating on spheres.

Existence of sign-changing solutions with blow-up behavior.

Application of Ljapunov-Schmidt reduction in a nonhomogeneous setting.

Abstract

We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and $p_\eps={(m+1)+2\over(m+1)-2}-\eps$, $\eps>0.$ We prove the existence of positive and sign changing solutions of $(P_\eps)$ concentrating and blowing-up, as $\eps\to0$, on $(m-1)-$dimensional spheres. Using a reduction method (see Ruf-Srikanth (2010) J. Eur. Math. Soc. and Pacella-Srikanth (2012) arXiv:1210.0782)we transform problem $(P_\eps)$ into a nonhomogeneous problem in an annulus $\mathcal D\subset \rr^{m+1}$ which can be solved by a Ljapunov-Schmidt finite dimensional reduction.