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

Abstract

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 $p_1 < p_2 < \cdots$ 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. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as $p\to 1$.