Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 47, pp. 1-10.

Title: Positive solutions of Schrodinger-Poisson systems with Hardy
       potential and indefinite nonlinearity
       
Authors: Yongyi Lan (Jimei Univ., Xiamen, China)
         Biyun Tang (Jimei Univ., Xiamen, China)
         Xian Hu    (Jimei Univ., Xiamen, China)

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.
 
Submitted April 6, 2020. Published May 21, 2020.
Math Subject Classifications: 35J20, 35J70.
Key Words: Hardy potential; variational methods; indefinite nonlinearity;
           positive solution.