Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials

Abstract

This paper is dedicated to studying the following Schrödinger-Poisson problem \begin{document} $\left\{ \begin{array}{ll} -\triangle u+V(x)u+φ u=f(u), \ \ \ \ x∈ \mathbb{R}^{3},\\ -\triangle φ=u^2, \ \ \ \ x∈ \mathbb{R}^{3}, \end{array} \right. $ \end{document} where $V(x)$ is weakly differentiable and $f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$. By introducing some new tricks, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on $V$ and $f$. Our results generalize and improve the ones in [D. Ruiz, J. Funct. Anal. 237 (2006) 655-674], [J.J. Sun, S.W. Ma, J. Differential Equations 260 (2016) 2119-2149] and some related literature.