含指数增长非线性项的Chern-Simons-Schrdinger方程组解的存在性（英文）
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摘要
本文研究了带指数增长的非线性项的非线性Chern-Simons-Schrdinger方程组.利用山路引理的方法,得到该方程组解的存在性.
In this paper, the nonlinear Chern-Simons-Schrdinger systems with exponential nonlinearities are studied. By mountain pass theorem, the existence of a solution to these systems is obtained.
In this paper, the nonlinear Chern-Simons-Schrdinger systems with exponential nonlinearities are studied. By mountain pass theorem, the existence of a solution to these systems is obtained.
引文
[1]Do J M,Medeiros E,Severo U.A nonhomogeneous elliptic problem involving critical growth in dimension two[J].J.Math.Anal.Appl.,2008,345:286-304.
[2]Dunne V.Self-dual Chern-Simons theories[M].Berlin:Springer,1995.
[3]Jackiw R,Pi S Y.Classical and quantal nonrelativistic Chern-Simons theory[J].Phys.Rev.D,1990,42(10):3500-3513.
[4]Jackiw R,Pi S-Y.Self-dual Chern-Simons solitons[J].Progr.Theoret.Phys.Suppl.,1992,107:1-40.
[5]Berge L,De Bouard A,Saut J-C.Blowing up time-dependent solutions of the planar,Chern-Simons gauged nonlinear Schrdinger equation[J].Nonlinearity,1995,8:235-253.
[6]Huh H.Blow-up solutions of the Chern-Simons-Schrdinger equations[J].Nonlinearity,2009,22(5):967-974.
[7]Liu B,Smith P,Tataru D.Local wellposedness of Chern-Simons-Schrdinger[J].Int.Math.Res.Notices,doi:10.1093/imrn/rnt161.
[8]Byeon J,Huh H,Seok J.Standing waves of nonlinear Schrdinger equations with the gauge field[J].J.Funct.Anal.,2012,263:1575-1608.
[9]Cunha P L,DAvenia P,Pomponio A,Siciliano G.A multiplicity result for Chern-SimonsSchrdinger equation with a general nonlinearity[J].Nonl.Diff.Equ.Appl.No DEA,2015,22(6):1831-1850.
[10]Huh H.Standing waves of the Schrdinger equation coupled with the Chern-Simons gauge field[J].J.Math.Phys.,2012,53(6):3500.
[11]Huh H.Nonexistence results of semilinear elliptic equations coupled the the Chern-Simons gauge field[J].Abstr.Appl.Anal.,2013,2:168-184.
[12]Pomponio A,Ruiz D.A variational analysis of a gauged nonlinear Schrdinger equation[J].J.Eur.Math.Soc.,2015,17:1463-1486.
[13]Jiang Y,Pomponio A,Ruiz D.Standing waves for a gauged nonlinear Schrdinger equation with a vortex point[J].Commun.Contemp.Math.,2015,18(4),DOI:10.1142/S0219199715500741.
[14]Pomponio A,Ruiz D.Boundary concentration of a gauged nonlinear Schrdinger equation on large balls[J].Calc.Vari.PDEs,2015,53(1):289-316.
[15]Wan Y,Tan J.Standing waves for the Chern-Simons-Schrdinger systems without(AR)condition[J].J.Math.Anal.Appl.,2014,415(1):422-434.
[16]Yuan J.Multiple normalized solutions of Chern-Simons-Schrdinger system[J].Nonl.Diff.Equ.Appl.,2015,22(6):1801-1816.
[17]Wan Y,Tan J.The existence of nontrivial solutions to Chern-Simons-Schrdinger systems[J].Disc.Conti.Dyn.Sys.-Ser.A,2017,37(5):2765-2786.
[18]Kondratev V,Shubin M.Discreteness of spectrum for the Schrdinger operators on manifolds of bounded geometry[J].Oper.Theory Adv.Appl.,1999,110:185-226.
[19]Liu Z,Wang Z Q.Schrdinger equations with concave and convex nonlinearities[J].Z.Angew.Math.Phys.,2005,56(4):609-629.
[20]Ambrosetti A,Rabinowitz P H.Dual varitional methods in critical point theory and applications[J].J.Funct.Anal.,1973,14:349-381.
[21]Cao D M.Nontrivial solution of semilinear elliptic equation with critical exponent in R2[J].Comm.Part.Diff.Equ.,1992,17:407-435.
[22]Ruf B.A sharp Trudinger-Moser type inequality for unbounded domains in R2[J].J.Funct.Anal.,2005,219(2):340-367.
[2]Dunne V.Self-dual Chern-Simons theories[M].Berlin:Springer,1995.
[3]Jackiw R,Pi S Y.Classical and quantal nonrelativistic Chern-Simons theory[J].Phys.Rev.D,1990,42(10):3500-3513.
[4]Jackiw R,Pi S-Y.Self-dual Chern-Simons solitons[J].Progr.Theoret.Phys.Suppl.,1992,107:1-40.
[5]Berge L,De Bouard A,Saut J-C.Blowing up time-dependent solutions of the planar,Chern-Simons gauged nonlinear Schrdinger equation[J].Nonlinearity,1995,8:235-253.
[6]Huh H.Blow-up solutions of the Chern-Simons-Schrdinger equations[J].Nonlinearity,2009,22(5):967-974.
[7]Liu B,Smith P,Tataru D.Local wellposedness of Chern-Simons-Schrdinger[J].Int.Math.Res.Notices,doi:10.1093/imrn/rnt161.
[8]Byeon J,Huh H,Seok J.Standing waves of nonlinear Schrdinger equations with the gauge field[J].J.Funct.Anal.,2012,263:1575-1608.
[9]Cunha P L,DAvenia P,Pomponio A,Siciliano G.A multiplicity result for Chern-SimonsSchrdinger equation with a general nonlinearity[J].Nonl.Diff.Equ.Appl.No DEA,2015,22(6):1831-1850.
[10]Huh H.Standing waves of the Schrdinger equation coupled with the Chern-Simons gauge field[J].J.Math.Phys.,2012,53(6):3500.
[11]Huh H.Nonexistence results of semilinear elliptic equations coupled the the Chern-Simons gauge field[J].Abstr.Appl.Anal.,2013,2:168-184.
[12]Pomponio A,Ruiz D.A variational analysis of a gauged nonlinear Schrdinger equation[J].J.Eur.Math.Soc.,2015,17:1463-1486.
[13]Jiang Y,Pomponio A,Ruiz D.Standing waves for a gauged nonlinear Schrdinger equation with a vortex point[J].Commun.Contemp.Math.,2015,18(4),DOI:10.1142/S0219199715500741.
[14]Pomponio A,Ruiz D.Boundary concentration of a gauged nonlinear Schrdinger equation on large balls[J].Calc.Vari.PDEs,2015,53(1):289-316.
[15]Wan Y,Tan J.Standing waves for the Chern-Simons-Schrdinger systems without(AR)condition[J].J.Math.Anal.Appl.,2014,415(1):422-434.
[16]Yuan J.Multiple normalized solutions of Chern-Simons-Schrdinger system[J].Nonl.Diff.Equ.Appl.,2015,22(6):1801-1816.
[17]Wan Y,Tan J.The existence of nontrivial solutions to Chern-Simons-Schrdinger systems[J].Disc.Conti.Dyn.Sys.-Ser.A,2017,37(5):2765-2786.
[18]Kondratev V,Shubin M.Discreteness of spectrum for the Schrdinger operators on manifolds of bounded geometry[J].Oper.Theory Adv.Appl.,1999,110:185-226.
[19]Liu Z,Wang Z Q.Schrdinger equations with concave and convex nonlinearities[J].Z.Angew.Math.Phys.,2005,56(4):609-629.
[20]Ambrosetti A,Rabinowitz P H.Dual varitional methods in critical point theory and applications[J].J.Funct.Anal.,1973,14:349-381.
[21]Cao D M.Nontrivial solution of semilinear elliptic equation with critical exponent in R2[J].Comm.Part.Diff.Equ.,1992,17:407-435.
[22]Ruf B.A sharp Trudinger-Moser type inequality for unbounded domains in R2[J].J.Funct.Anal.,2005,219(2):340-367.