Geodesic
Positive ground state solutions to Schrödinger-Poisson systems with a negative non-local term
Electronic journal of differential equations, Tome 2015 (2015)
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.
Classification : 35J47, 35J50, 35J99
Keywords: Schrödinger-Poisson system, ground state solution, variational methods 
@article{EJDE_2015__2015__a70,
     author = {Gao,  Yan-Ping and Yu,  Sheng-Long and Tang,  Chun-Lei},
     title = {Positive ground state solutions to {Schr\"odinger-Poisson} systems with a negative non-local term},
     journal = {Electronic journal of differential equations},
     year = {2015},
     volume = {2015},
     zbl = {1322.35026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a70/}
}
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Gao,  Yan-Ping; Yu,  Sheng-Long; Tang,  Chun-Lei. Positive ground state solutions to Schrödinger-Poisson systems with a negative non-local term. Electronic journal of differential equations, Tome 2015 (2015). http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a70/