Positive ground state solutions to Schrodinger-Poisson systems with a negative non-local term

Abstract

In this article, we study the Schrodinger-Poisson system $$\displaylines{ -\Delta u+u-\lambda K(x)\phi(x)u=a(x)|u|^{p-1}u, \quad x\in\mathbb{R}^3, \cr -\Delta\phi=K(x)u^{2},\quad x\in\mathbb{R}^3, }$$ with $p\in(1,5)$. Assume that $a:\mathbb{R}^3\to \mathbb{R^{+}}$ and $K:\mathbb{R}^3\to \mathbb{R^{+}}$ are nonnegative functions and satisfy suitable assumptions, but not requiring any symmetry property on them, we prove the existence of a positive ground state solution resolved by the variational methods.

Keywords

• Schrodinger-Poisson system
• ground state solution
• variational methods