On the existence of a solution for elliptic system related to the Maxwell–Schrödinger equations

In this paper we study the existence and the multiplicity of standing wave solutions for the nonlinear Schrödinger equation coupled with the Maxwell equations. It seems difficult to obtain a boundedness of a Palais–Smale sequence for the functional associated with this system. We overcome this difficulty by Jeanjean’s result [L. Jeanjean, On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on RNRN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787–809], which generalizes Struwe’s argument [M. Struwe, Variational Methods, 2nd ed., Springer, 1996; M. Struwe, The existence of a surface of constant mean curvature with free boundaries, Acta Math. 160 (1988) 19–64] and Zou’s result [W. Zou, Variant fountain theorem and their applications, Manuscripta Math. 104 (2001) 343–358].