Schrodinger-Poisson systems with singular potential and critical exponent

Abstract

In this article we study the Schrodinger-Poisson system $$\displaylines{ -\Delta u +V(|x|)u+\lambda\phi u = f(u), \quad x\in\mathbb{R}^3, \cr -\Delta \phi =u^2, \quad x\in\mathbb{R}^3, }$$ where V is a singular potential with the parameter $\alpha$ and the nonlinearity f satisfies critical growth. By applying a generalized version of Lions-type theorem and the Nehari manifold theory, we establish the existence of the nonnegative ground state solution when $\lambda=0$. By the perturbation method, we obtain a nontrivial solution to above system when $\lambda\neq 0$.

Keywords

• schrodinger-poisson system
• lions-type theorem
• singular potential
• ground state solution
• critical exponent