Signed and sign-changing solutions of Kirchhoff type problems

Abstract

We consider the following nonlinear Kirchhoff type problem of the form

$$\begin{aligned} \left\{ \begin{array}{ll} -(a+b\int _{\Omega }|\nabla u|^{2})\triangle u = \mu g(x,u)+f(x,u), &{}\quad \hbox {in}\ \ \Omega ,\\ u=0, &{}\quad \hbox {on}\ \ \partial \Omega ,\end{array}\right. \end{aligned}$$

where $\Omega \subset \mathbb {R}^{3}$ is a bounded domain with smooth boundary $\partial \Omega$ and $a>0$, $b\ge 0$. The nonlinearity $\mu g(x,u)+f(x,u)$ may involve a combination of concave and convex terms. Under some suitable conditions on $f,g\in C(\overline{\Omega }\times \mathbb {R},\mathbb {R})$ and $\mu \in \mathbb {R}$, we prove the existence of infinitely many high-energy solutions using Fountain theorem. In particular, using the method of invariant sets of descending flow, we prove the existence of at least one sign-changing solutions.