Geodesic
A Note on the Ground State Solutions for the Nonlinear Schrödinger-Maxwell Equations
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 93-104.

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In this paper we study the nonlinear Schrödinger-Maxwell equations $$\begin{cases} - \Delta u + V(x) u + \phi u= |u|^{p-1} u \quad \text{in} \,\, \mathbb{R}^{3}, \\ - \Delta \phi = u^{2} \text{in} \,\, \mathbb{R}^{3}.\end{cases}$$ If $V$ is a positive constant, we prove the existence of a ground state solution $(u,\phi)$ for $2 p 5$. The non-constant potential case is treated for $3 p 5$, and $V$ possibly unbounded below. 
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Azzollini, A.; Pomponio, A. A Note on the Ground State Solutions for the Nonlinear Schrödinger-Maxwell Equations. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 93-104. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a4/

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