Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth

Abstract

This paper is concerned with the multiplicity of nontrivial solutions to the class of nonlocal boundary value problems of the Kirchhoff type, \begin{equation*} - \Big [M \Big (\int_{\Omega}|\nabla u|^{2} \ dx\ \Big ) \Big ]\Delta u = \lambda |u|^{q-2}u+|u|^{p-2}u \ \mbox{in} \ \Omega, \ \ \mbox{and} \ u=0 \ \mbox{on} \ \ \partial\Omega, \end{equation*} where $\Omega\subset\mathbb R^{N}$, for $N=1,$ 2, and 3, is a bounded smooth domain, $1 < q < 2 < p \leq 2^{*}=6$ in the case $N=3$ and $2^{*}=\infty$ in the case $N=1$ or $N=2$. Our approach is based on the genus theory introduced by Krasnoselskii [22].