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http://dx.doi.org/10.4134/BKMS.2012.49.6.1311

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.49, no.6, 2012 , pp. 1311-1326 More about this Journal
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.
Keywords
perturbed Schr$\ddot{o}$dinger system; critical nonlinearity; variational methods; magnetic fields;
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