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

Abstract

In this paper we prove an existence result to the problem $$\{\begin{array}{ll}-\Delta u=|u{|}^{p-1}u& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}$$ 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 $\underset{k\to +\infty }{\lim }{p}_k=+\infty$ 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.

12 pages

Cited by in corpus (2)

• Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem
• Nonradial sign changing solutions to Lane Emden equation