Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein– Gordon–Maxwell equations on R 3

  • Shang-Jie Chen
  • Chun-Lei TangEmail author


In this paper we study the following nonhomogeneous Schrödinger–Maxwell equations
$$\left\{\begin{array}{ll} {-\triangle u+V(x)u+ \phi u=f(x,u)+h(x),} \quad {\rm in}\,\,\,{\mathbf{R}}^3,\\ {-\triangle \phi=u^2, \qquad\qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,{\mathbf{R}}^3,} \end{array} \right.$$
where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V, f and h, the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory. Similar results for the nonhomogeneous Klein–Gordon–Maxwell equations
$$\left\{\begin{array}{ll} {-\triangle u+[m^2-(\omega+\phi)^2]u=|u|^{q-2}u+h(x), \quad {\rm in} \,\,\,{\mathbf{R}}^3,}\\ {-\triangle \phi+ \phi u^2=-\omega u^2, \qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,\,{\mathbf{R}}^3,} \end{array} \right.$$
are also obtained when 2 < q < 6.

Mathematics Subject Classification (2000)



Schrödinger–Maxwell equations Klein–Gordon–Maxwell equations Nonhomogeneous Superlinear Ekeland’s variational principle Mountain Pass Theorem Variational methods 


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© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China

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