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

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