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

In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -Δu = |u|^{p-1} u \qquad & \text{in} Ω, \\ u= 0 & \text{on} \partialΩ, \end{array} \right. $$ where $Ω$ 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. 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.