Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki–Lions type conditions

Abstract

In this paper, we discuss the following Kirchhoff-type problem with convolution nonlinearity

$$\begin{aligned} -\left( 1+ b\int _{ \mathbb {R}^{3}}|\nabla u|^{2} dx \right) \triangle u+ V(x)u=(I_{\alpha }*F(u))f( u),~x\in \mathbb {R}^{3},~u\in H^{1}(\mathbb {R}^{3}), \end{aligned}$$

where $b>0$, $I_{\alpha }:\mathbb {R}^{3}\rightarrow \mathbb {R}$, with $\alpha \in (0,3)$, is the Riesz potential, V is differentiable, $f\in \mathbb {C}(\mathbb {R},\mathbb {R})$ and $F(t)=\int ^{t}_{0}f(s)ds$. Let f satisfies some relatively weak conditions in the absence of the usual Ambrosetti-Rabinowitz or monotonicity conditions. We get two classes of ground state solutions under the general “Berestycki–Lions conditions” on the nonlinearity f and we also give a minimax characterization of the ground state energy.