Positive solutions of Schrodinger-Poisson systems with Hardy potential and indefinite nonlinearity

Abstract

In this article, we study the nonlinear Schrodinger-Poisson system $$\displaylines{ -\Delta u+u-\mu\frac{u}{|x|^2}+l(x) \phi u=k(x)|u|^{p-2}u \quad x\in\mathbb{R}^3, \cr -\Delta\phi=l(x)u^2 \quad x\in\mathbb{R}^3, }$$ where $k\in C(\mathbb{R}^3)$ and 4<p<6, k changes sign in $\mathbb{R}^3$ and $\limsup_{|x|\to\infty}k(x)=k_{\infty}<0$. We prove that Schrodinger-Poisson systems with Hardy potential and indefinite nonlinearity have at least one positive solution, using variational methods.

Keywords

• hardy potential
• variational methods
• indefinite nonlinearity
• positive solution