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Ground state solutions for the nonlinear Schrödinger-Maxwell equations. (English) Zbl 1147.35091

Summary: We study the nonlinear Schrödinger-Maxwell equations
\[ \begin{aligned} -\Delta u+V(x)u+ \varphi u=|u|^{p-1}u &\quad\text{in }\mathbb R^3,\\ -\Delta\varphi=u^2 &\quad\text{in }\mathbb R^3. \end{aligned} \]
If \(V\) is a positive constant, we prove the existence of a ground state solution \((u,\varphi)\) for \(2<p<5\). The non-constant potential case is treated for \(3<p<5\), and \(V\) possibly unbounded below. Existence and nonexistence results are proved also when the nonlinearity exhibits a critical growth.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35B50 Maximum principles in context of PDEs
49J40 Variational inequalities
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