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

A. Azzollini; A. Pomponio — 2009

Bollettino dell'Unione Matematica Italiana

In this paper we study the nonlinear Schrödinger-Maxwell equations $\begin{cases}-\Delta u+V(x)u+\phi u=|u|^{p-1}u&\text{in}\,\,\mathbb{R}^{3},\\ -\Delta\phi=u^{2}&\text{in}\,\,\mathbb{R}^{3}.\end{cases}$ 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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