Nodal solutions for Lane-Emden problems in almost-annular domains

Anna Lisa Amadori, Francesca Gladiali, Massimo Grossi

Published 2016-05-21Version 1

In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -\Delta u = |u|^{p-1} u \qquad & \text{in} \Omega, \\ u= 0 & \text{on} \partial\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in ${\mathbb R}^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1<p_2<..$ with $\lim\limits_{k\rightarrow+\infty}p_k=+\infty$ such that for any real number $p>1$ and $p\ne p_k$ there exist at least one solution with $m$ nodal zones. Keywords: semilinear elliptic equations, nodal solutions, supercritical problems.