Existence of multiple sign-changing solutions for an asymptotically linear elliptic problem and the topology of the configuration space of the domain

Abstract

In the case $f \in C(\mathbb R,\mathbb R)$ is asymptotically linear, we give a lower estimate of number of sign-changing solutions to the problem $$-d^2 \Delta u + u =f(u)\;\;\text{in $\Omega$,}\quad u=0 \;\;\text{on $\partial\Omega$,} $$ where $\Omega$ is a bounded domain in $\\mathbb R^N$ ($N \geq 2$) and $d>0$ is an appropriate small number.