MULTIPLICITY OF NONTRIVIAL SOLUTIONS TO PERTURBED SCHR¨ODINGER SYSTEM WITH MAGNETIC FIELDS
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초록
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.
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.
참고문헌 (22)
- A. Ambrosetti, "Semiclassical states of nonlinear Schr¨odinger equations", Arch. Ration. Mech. Anal, 140(3), pp.285~300 (1997)
- A. Ambrosetti, "Multiplicity results for some nonlinear Schr¨odinger equations with potentials", Arch. Ration. Mech. Anal, 159(3), pp.253~271 (2001)
- G. Arioli, "A semilinear Schr¨odinger equation in the presence of a magnetic field", Arch. Ration. Mech. Anal, 170(4), pp.277~295 (2003)
- T. Bartsch, "On mult-bump semi-classical bound states of nonlinear Schr¨odinger equations with electromagnetic fields", Adv. Differential Equations, 11(7), pp.781~812 (2006)
- V. Benci, "On critical point theory for indefinite functionals in the presence of symmetries", Trans. Amer. Math. Soc, 274(2), pp.533~572 (1982)
- D. Cao, "Existence and uniqueness of multi-bump bound states of nonlinear Schr¨odinger equations with electromagnetic fields", J. Differential Equations, 222(2), pp.381~424 (2006)
- S. Cingolani, "Semiclassical stationary states of nonlinear Schr¨odinger equation with an external magnetic field", J. Differential Equations, 188(1), pp.52~79 (2003)
- S. Cingolani, "Multiple positive solutions to nonlinear Schr¨odinger equation with competing potential functions", J. Differential Equations, 160(1), pp.118~138 (2000)
- S. Cingolani, "Multi-peaks periodic semiclassical states for a class of nonlinear Schr¨odinger equation", Proc. Roy. Soc. Edinburgh Sect. A, 128(6), pp.1249~1260 (1998)
- S. Cingolani, "Semiclassical states for NLS equations with magnetic potentials having polynomial growths", J. Math. Phys, 46(5), pp.053503~ (2005)
- M. Clapp, "Minimal nodal solutions of a Schr¨odinger equation with critical nonlinearity and symmetric potential", Differential Integral Equations, 16(8), pp.981~992 (2003)
- Y. H. Ding, "Solutions of perturbed Schr¨odinger equations with critical nonlinearity", Calc. Var. Partial Differential Equations, 30(2), pp.231~249 (2007)
- M. Esteban, "Stationary solutions of nonlinear Schr¨odinger equation with an external magnetic field, in PDE and Calculus of Variations, in honor of E. De Giorgi", pp.369~408 (1990), Brikh¨auser
- A. Floer, "Nonspreading wave packets for the cubic Schr¨odinger equation with a bounded potential", J. Funct. Anal, 69(3), pp.397~408 (1986)
- Y. G. Oh, "On positive multi-lump bound states of nonlinear Schr¨odinger equations under multiple well potential", Comm. Math. Phys, 131(2), pp.223~253 (1990)
- M. del Pino, "Semi-classical states for nonlinear Schr¨odinger equations", J. Funct. Anal, 149(1), pp.245~265 (1997)
- M. del Pino, "Multi-peak bound states for nonlinear Schr¨odinger equations", Ann. Inst. H. Poincar´e Anal. Non Lineaire, 15(2), pp.127~149 (1998)
- Z. Tang, "On the least energy solutions of nonlinear Schr¨odinger equations with electromagnetic fields", Comput. Math. Appl, 54(5), pp.627~637 (2007)
- Z. Tang, "Multi-bump bound states of nonlinear Schr¨odinger equations with electromagnetic fields and critical frequency", J. Differential Equations, 245(10), pp.2723~2748 (2008)
- F. Wang, "On an electromagnetic Schr¨odinger equation with critical growth", Nonlinear Anal, 69(11), pp.4088~4098 (2008)
- X. Wang, "On concentration of positive bound states of nonlinear Schr¨odinger equations", Comm. Math. Phys, 153(2), pp.229~244 (1993)
- M. Willem, "Minimax Theorems" (1996), Birkh¨auser