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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 $$\align -\Delta u+V(x)u+ \varphi u=|u|^{p-1}u &\quad\text{in }\Bbb R^3,\\ -\Delta\varphi=u^2 &\quad\text{in }\Bbb R^3. \endalign$$ 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:
35Q55NLS-like (nonlinear Schrödinger) equations
35B38Critical points in solutions of PDE
35B50Maximum principles (PDE)
49J40Variational methods including variational inequalities
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References:
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