Nodal solutions for a nonhomogeneous elliptic equation with symmetry

Abstract

We consider the semilinear problem $-\Delta u + \lambda u =|u|^{p-2}u + f(u)$ in $\Omega$, $u=0$ on $\partial \Omega$ where $\Omega \subset {\mathbb R}^N$ is a bounded smooth domain, $2< p< 2^*=2N/(N-2)$ and $f(t)$ behaves like $t^{p-1-\varepsilon}$ at infinity. We show that if $\Omega$ is invariant by a nontrivial orthogonal involution then, for $\lambda> 0$ sufficiently large, the equivariant topology of $\Omega$ is related with the number of solutions which change sign exactly once. The results are proved by using equivariant Lusternik-Schnirelmann theory.