Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations. (English) Zbl 1311.35286
Summary: In this paper, by using variational methods and critical point theory, we study the existence of semiclassical solutions for the following nonlinear Schrödinger-Maxwell equations
\[
\begin{cases}-\varepsilon^2\Delta u+V(x)u+\phi u=f(x,u),\quad & \text{in }\mathbb R^3,\\-\Delta\phi=4\pi u^2,\quad &\text{in }\mathbb R^3,\end{cases}
\]
where \(\varepsilon>0\) and \(V(x)\geq 0\) for all \(x\in\mathbb R^3\). Under some weaker assumptions on \(V\) and \(f\), we prove that the system has at least one nontrivial solution for sufficient small \(\varepsilon>0\).
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 78A30 | Electro- and magnetostatics |
| 81V70 | Many-body theory; quantum Hall effect |
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