径向对称位势下Klein-Gordon-Maxwell方程解的存在性
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摘要
研究了非线性Klein-Gordon-Maxwell方程问题■,其中参数ω>0,λ>0。当V(x)为径向对称位势并且方程的非线性项f(u)只在零点附近有定义时,可以通过变分法证明该方程解的存在性,并得到方程的解关于参数λ的依赖性。
We have studied the nonlinear Klein-Gordon-Maxwell equation ■, where the parameter ω>0, λ>0. When V(x) is a radial symmetric potential and the nonlinear term of the equation f(u) is defined only near the zero point, the existence of solution to the equation can be proved by the calculus of variation, and the dependence on the parameter λ of the solution to the equation can be obtained.
We have studied the nonlinear Klein-Gordon-Maxwell equation ■, where the parameter ω>0, λ>0. When V(x) is a radial symmetric potential and the nonlinear term of the equation f(u) is defined only near the zero point, the existence of solution to the equation can be proved by the calculus of variation, and the dependence on the parameter λ of the solution to the equation can be obtained.
引文
[1] BENCI V, FORTUNATO D. The nonlinear Klein-Gordon equation coupled with the Maxwell equations[J]. Nonlinear Analysis, 2001, 47(9): 6065- 6072.
[2] BENCI V, FORTUNATO D. Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations[J]. Reviews in Mathematical Physics, 2002, 14(4): 409- 420.
[3] D’APRILE T, MUGNAI D. Non-existence results for the coupled Klein-Gordon-Maxwell equations[J]. Advanced Nonlinear Studies, 2004, 4(3): 307- 322.
[4] CASSANI D. Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations[J]. Nonlinear Analysis, 2004, 58(7): 733- 747.
[5] CARRIAO P C, CUNHA P L, MIYAGAKI O H. Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials[J]. Nonlinear Analysis, 2012, 75(10): 4068- 4078.
[6] JING Y T, LIU Z L. Elliptic systems with a partially sublinear local term[J]. Journal of Mathematical Study, 2015, 48(3): 290- 305.
[7] 陆文端. 微分方程中的变分方法[M]. 北京: 北京科技出版社, 2003. LU W D . The variational method in differential equations [M]. Beijing: Beijing Science and Technology Press, 2003. (in Chinese)
[8] COSTAD G, WANG Z Q. Multiplicity results for a class of superlinear elliptic problems[J]. Proceedings of the American Mathematical Society, 2005, 133(3): 787- 794.
[9] D’ APRILE T, MUGNAI D. Solitary waves for nonlinear Klein- Gordon- Maxwell and Schr?dinger- Maxwell equations[J]. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2004, 134(5): 893- 906.
[10] BADIALEM, SERRA E. Semilinear elliptic equations for beginners[M]. London: Springer-Verlag London Limited, 2011.
[11] NI W M. A nonlinear Dirichlet problem on the unit ball and its applications[J]. Indiana University Mathematics Journal, 1982, 31(6): 801- 807.
[2] BENCI V, FORTUNATO D. Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations[J]. Reviews in Mathematical Physics, 2002, 14(4): 409- 420.
[3] D’APRILE T, MUGNAI D. Non-existence results for the coupled Klein-Gordon-Maxwell equations[J]. Advanced Nonlinear Studies, 2004, 4(3): 307- 322.
[4] CASSANI D. Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations[J]. Nonlinear Analysis, 2004, 58(7): 733- 747.
[5] CARRIAO P C, CUNHA P L, MIYAGAKI O H. Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials[J]. Nonlinear Analysis, 2012, 75(10): 4068- 4078.
[6] JING Y T, LIU Z L. Elliptic systems with a partially sublinear local term[J]. Journal of Mathematical Study, 2015, 48(3): 290- 305.
[7] 陆文端. 微分方程中的变分方法[M]. 北京: 北京科技出版社, 2003. LU W D . The variational method in differential equations [M]. Beijing: Beijing Science and Technology Press, 2003. (in Chinese)
[8] COSTAD G, WANG Z Q. Multiplicity results for a class of superlinear elliptic problems[J]. Proceedings of the American Mathematical Society, 2005, 133(3): 787- 794.
[9] D’ APRILE T, MUGNAI D. Solitary waves for nonlinear Klein- Gordon- Maxwell and Schr?dinger- Maxwell equations[J]. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2004, 134(5): 893- 906.
[10] BADIALEM, SERRA E. Semilinear elliptic equations for beginners[M]. London: Springer-Verlag London Limited, 2011.
[11] NI W M. A nonlinear Dirichlet problem on the unit ball and its applications[J]. Indiana University Mathematics Journal, 1982, 31(6): 801- 807.