## Differential and Integral Equations

### 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].

#### Article information

Source
Differential Integral Equations, Volume 25, Number 9/10 (2012), 853-868.

Dates
First available in Project Euclid: 20 December 2012