Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth

Abstract

In this paper, we consider a Neumann problem of Kirchhoff type equation \begin{equation*} \begin{cases} \displaystyle-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u= Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle\frac{\partial u}{\partial v}=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^3$ is a bounded domain with a smooth boundary, $a,b>0$, $10$ is a real parameter, $Q(x)$ and $P(x)$ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions.

Keywords

• kirchhoff type equation
• neumann problem
• critical growth
• variation methods
• nontrivial solution