Infinitely many solutions for Kirchhoff-type problems depending on a parameter

Abstract

In this article, we study a Kirchhoff type problem with a positive parameter $\lambda$, $$\displaylines{ -K\Big( \int_{\Omega }|\nabla u|^{2}dx\Big) \Delta u=\lambda f(x,u) , \quad \text{in } \Omega , \cr u=0, \quad \text{on } \partial \Omega , }$$ where $K:[0,+\infty )\to \mathbb{R} $ is a continuous function and $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a $L^{1}$-Caratheodory function. Under suitable assumptions on K(t) and f(x,u), we obtain the existence of infinitely many solutions depending on the real parameter $\lambda$. Unlike most other papers, we do not require any symmetric condition on the nonlinear term $f(x,u)$. Our proof is based on variational methods.

Keywords

• Infinitely many solutions
• Kirchhoff type problem
• variational method