A Note on the Existence of a Positive Solution for a Non-autonomous Schrödinger–Poisson System

Abstract

We consider the system $\left\{\begin{array}{cl}{{-\Delta u + V(x)u + K(x)\phi}(x)u = a(x){|u|^{p-1}}u, \quad x \in \mathbb{R}^{3}} \\{-\Delta \phi = K(x)u^{2},}\qquad\qquad\qquad\qquad\qquad\qquad\quad{x\in \mathbb{R}^{3}}\end{array} \right.$ where 3 < p < 5 and the potentials $K(x), a(x) ]\rm {and} V(x)$ has finite limits as $|x|\rightarrow + \infty .$ By imposing some conditions on the decay rate of the potentials we obtain the existence of a ground state solution. In the proof we apply variational methods.

Mathematics Subject Classification (2010). 35J20, 35J60, 35B38.