Multiplicity of nontrivial solutions to perturbed Schr\"{o}dinger system with magnetic fields
Bull. Korean Math. Soc. 2012 Vol. 49, No. 6, 1311-1326
Printed November 30, 2012
Huixing Zhang and Wenbin Liu
China University of Mining and Technology, China University of Mining and Technology
Abstract : We are concerned with the multiplicity of semiclassical solutions of the following Schr\"{o}dinger system involving critical nonlinearity and magnetic fields \begin{align*} \left\{ \begin{array}{ll} -(\varepsilon\nabla+i A(x))^{2}u+V(x)u=H_u(u,v)+K(x)|u|^{2^{\ast}-2}u,\ x\in{\mathbb{R}^{N}},\\ -(\varepsilon\nabla+i B(x))^{2}v+V(x)v=H_v(u,v)+K(x)|v|^{2^{\ast}-2}v,\ x\in{\mathbb{R}^{N}}, \end{array} \right. \end{align*} where $2^{\ast}=2N/(N-2)$ 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\"{o}dinger system, critical nonlinearity, variational methods, magnetic fields
MSC numbers : 35B33, 35J60, 35J65
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