Existence of positive solutions for Kirchhoff problems

Abstract

We study problems for the Kirchhoff equation $$\displaylines{ -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =\nu u^3+ Q(x)u^{q},\quad \text{in }\Omega, \cr u=0, \quad \text{on }\partial\Omega, }$$ where $\Omega\subset \mathbb{R}^3$ is a bounded domain, $a,b\geq0$ and $a+b>0$, $\nu>0$, $30$ in $\Omega$. By the mountain pass lemma, the existence of positive solutions is obtained. Particularly, we give a condition of Q to ensure the existence of solutions for the case of q=5.

Keywords

• Kirchhoff type equation
• resonance
• positive solution
• mountain pass lemma