• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1311-1326
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• 2012

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.