## Differential and Integral Equations

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

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

Giovany M. Figueiredo and João R. Santos Junior

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012371

**Mathematical Reviews number (MathSciNet)**

MR2985683

**Zentralblatt MATH identifier**

1274.35087

**Subjects**

Primary: 45M20: Positive solutions 35J25: Boundary value problems for second-order elliptic equations 34B18: Positive solutions of nonlinear boundary value problems 34C11: Growth, boundedness 34K12: Growth, boundedness, comparison of solutions

#### Citation

Figueiredo, Giovany M.; Santos Junior, João R. Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differential Integral Equations 25 (2012), no. 9/10, 853--868. https://projecteuclid.org/euclid.die/1356012371