Mediterranean Journal of Mathematics

, Volume 13, Issue 5, pp 3469–3481 | Cite as

Ground-State Solutions for Asymptotically Cubic Schrödinger–Maxwell Equations



In this paper, using variational methods and critical point theory, we study the existence of ground-state solutions for the following nonlinear Schrödinger–Maxwell equations
$$\left\{\begin{array}{l@{\quad}l} -\triangle u + V(x)u + \phi u = f(x, u), & {\rm in}\, \mathbb{R}^{3},\\ -\triangle\phi = 4\pi u^{2}, & {\rm in} \, \mathbb{R}^{3},\end{array}\right. $$
where f is asymptotically cubic, V 1-periodic in each of \({x_1, x_2, x_3}\) and \({\underline{V}:= {\rm inf}_{x\in\mathbb{R}^3}V(x) > 0}\). Under some more assumptions on V and f, we develop a direct and simple method to find ground-state solutions for \({(\mathrm{NSM})}\). The main idea is to find a minimizing (PS) sequence for the energy functional outside the Nehari manifold \({\mathcal{N}}\) using the diagonal method. This seems to be the first result for \({(\mathrm{NSM})}\) satisfying the assumptions (V) and (N).


Schrödinger–Maxwell equations ground-state solution Nehari manifold asymptotically cubic 

Mathematics Subject Classification

35J20 35J25 35J60 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsGuangxi Normal UniversityGuilinPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

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