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MULTIPLICITY OF NONTRIVIAL SOLUTIONS TO PERTURBED SCHRÖDINGER SYSTEM WITH MAGNETIC FIELDS

  • Zhang, Huixing (Department of Mathematics China University of Mining and Technology) ;
  • Liu, Wenbin (Department of Mathematics China University of Mining and Technology)
  • Received : 2011.07.17
  • Published : 2012.11.30

Abstract

We are concerned with the multiplicity of semiclassical solutions of the following Schr$\ddot{o}$dinger system involving critical nonlinearity and magnetic fields $$\{-({\varepsilon}{\nabla}+iA(x))^2u+V(x)u=H_u(u,v)+K(x)|u|^{2*-2}u,\;x{\in}\mathbb{R}^N,\\-({\varepsilon}{\nabla}+iB(x))^2v+V(x)v=H_v(u,v)+K(x)|v|^{2*-2}v,\;x{\in}\mathbb{R}^N,$$ where $2^*=2N/(N-2)$ is the Sobolev critical exponent and $i$ is the imaginary unit. Under proper conditions, we prove the existence and multiplicity of the nontrivial solutions to the perturbed system.

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