On the Existence of Solutions for a Class of Schrödinger–Kirchhoff-Type Equations with Sign-Changing Potential

Abstract

In this paper, we consider the following Kirchhoff problem $\left\{\begin{array}{c}-\left(a+b{\displaystyle {\int }_{R^3}{\left|\nabla u\right|}^2\text{d}x}\right)\mathrm{\Delta }u+\lambda V\left(x\right)u={\left|u\right|}^{p-2}u,\text{\hspace{0.17em}in\hspace{0.17em}}{R}^3\hfill \\ u\in H^1\left(R^3\right)\hfill \end{array}\right.$ where $a,b>0$ are constants, $\lambda$ is a positive parameter, and $4<p<6$ . Under suitable assumptions on $V\left(x\right)$ , the existence of nontrivial solution is obtained via variational methods. The potential $V\left(x\right)$ is allowed to be sign-changing.