A Note on the Ground State Solutions for the Nonlinear Schrödinger-Maxwell Equations

Azzollini, A. ; Pomponio, A.

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 93-104

Résumé

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.