Bull. Korean Math. Soc. 2012; 49(6): 1311-1326
Printed November 30, 2012
https://doi.org/10.4134/BKMS.2012.49.6.1311
Copyright © The Korean Mathematical Society.
Huixing Zhang and Wenbin Liu
China University of Mining and Technology, China University of Mining and Technology
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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