Ground state solution for a class of Kirchhoff-type equation with general convolution nonlinearity

Abstract

In this paper, we consider the following class of Kirchhoff-type equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -(a+b\mathop \int \limits _{{\mathbb {R}}^N}|\nabla u|^2\text {d}x)\Delta u+V(x)u=(I_{\alpha }*F(u))f(u),&{} \text {in}\,\,{\mathbb {R}}^N,\\ u\,\in \,H^1({\mathbb {R}}^N), \end{array}\right. } \end{aligned}$$

where $a>0$, $b\ge 0$, $N\ge 3$, $\alpha \,\in \,(N-2,N)$, $V:{\mathbb {R}}^N \rightarrow {\mathbb {R}}$ is a potential function and $I_{\alpha }$ is a Riesz potential of order $\alpha \,\in \,(N-2,N)$. Under certain assumptions on V(x) and f(u), we prove that the equation has ground state solutions by variational methods.