Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

Abstract

Abstract In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R 3 , u ∈ H 1 ( R 3 ) , $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ where a , b > 0 $a,b>0$ are constants, and the potential V ( x ) $V(x)$ is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.

Keywords

• Kirchhoff-type equation
• Variational methods
• Palais–Smale condition
• Local linking
• Morse theory