Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system

Abstract

In this paper we study the existence of ground states solutions for non-autonomous Schrödinger–Bopp–Podolsky system \begin{equation*} \begin{cases} -\Delta u + u +\lambda K(x)\phi u = b(x)|u|^{p-2}u & \text{in} \ \mathbb{R}^{3}, \\ -\Delta \phi + a^2\Delta^2\phi = 4\pi K(x) u^{2} & \text{in}\ \mathbb{R}^{3}, \end{cases} \end{equation*} where $\lambda>0, 20$ and ${\lim_{\vert{x}\vert \to +\infty}}b(x)=b_\infty >0$ and satisfying suitable assumptions, but not requiring any symmetry property on them. We show that the existence of a positive solution depends on the parameters $\lambda$ and $p$. We also establish the existence of ground state solutions for the case $3.18\approx\frac{1+\sqrt{73}}{3}<p\le{4}$.

Keywords

• non-autonomous schrödinger–bopp–podolsky system
• variational methods
• pohožaev identity
• nehari manifold