Infinitely many solutions for Schrödinger–Kirchhoff-type equations involving indefinite potential

Abstract

In this paper, we study the multiplicity of solutions for the following Schrödinger–Kirchhoff-type equation \[ \begin{cases}-\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\triangle u+V(x)u=f(x,u)+g(x,u), \quad x\in \mathbb{R}^N,\\ u\in H^1(\mathbb{R}^N),\end{cases} \] where $N\geq 3$, $a,\,b>0$ are constants and the potential $V$ may be unbounded from below. Under some mild conditions on the nonlinearities $f$ and $g$, we obtain the existence of infinitely many solutions for this problem. Recent results from the literature are generalized and significantly improved.

Keywords

• schrödinger–kirchhoff-type equation
• symmetric mountain pass lemma
• variational method