Abstract
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 R N , 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].
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