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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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참고문헌 (22)
참고문헌 신청
A. Ambrosetti / 1997 / Semiclassical states of nonlinear Schr¨odinger equations / Arch. Ration. Mech. Anal 140 (3) : 285 ~ 300
A. Ambrosetti / 2001 / Multiplicity results for some nonlinear Schr¨odinger equations with potentials / Arch. Ration. Mech. Anal 159 (3) : 253 ~ 271
A. Floer / 1986 / Nonspreading wave packets for the cubic Schr¨odinger equation with a bounded potential / J. Funct. Anal 69 (3) : 397 ~ 408
D. Cao / 2006 / Existence and uniqueness of multi-bump bound states of nonlinear Schr¨odinger equations with electromagnetic fields / J. Differential Equations 222 (2) : 381
F. Wang / 2008 / On an electromagnetic Schr¨odinger equation with critical growth / Nonlinear Anal 69 (11) : 4088 ~ 4098
A. Ambrosetti / 2001 / Multiplicity results for some nonlinear Schr¨odinger equations with potentials / Arch. Ration. Mech. Anal 159 (3) : 253 ~ 271
A. Floer / 1986 / Nonspreading wave packets for the cubic Schr¨odinger equation with a bounded potential / J. Funct. Anal 69 (3) : 397 ~ 408
D. Cao / 2006 / Existence and uniqueness of multi-bump bound states of nonlinear Schr¨odinger equations with electromagnetic fields / J. Differential Equations 222 (2) : 381
F. Wang / 2008 / On an electromagnetic Schr¨odinger equation with critical growth / Nonlinear Anal 69 (11) : 4088 ~ 4098
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