SEMICLASSICAL SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER-MAXWELL EQUATIONS WITH CRITICAL NONLINEARITY

Abstract

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 \[\left\{\begin{array}{lll} -\varepsilon^{2}\triangle u+V(x)u+\phi u=K(x)|u|^4u+f(x,u), &in \, \mathbb R^{3}, \\ -\triangle\phi=4\pi u^{2}, & in \, \mathbb R^{3}, \end{array} \right.\] where $\varepsilon\gt 0$, $V(x)\geq0$ and $K(x)\gt 0$ for all $x\in\mathbb R^{3}$, under some more assumptions on $V$, $K$ and $f$, we prove that the system has at least one nontrivial solution for sufficient small $\varepsilon\gt 0$. Our approach is much more straightforward.