非线性Schr（？）dinger方程组基态解的存在性
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摘要
考虑如下的Schr(o|¨)dinger方程组:其中:λ_j,μ_j＞0.j=1,2,(?)＞0.n=2,3.
本文主要运用Morse理论来找上述方程组的完全非平凡径向基态解.当λ_1,λ_2给定了适当的限制后,我们给出了[Sirakov,Comm.Math.Phys,271(2007),199-221]中结果的改进.
Consider the following Schr(o|¨)dinger system:whoreλ_j,μ_j＞0, j = 1.2.β＞0. n = 2.3.
In this paper, we study the existence of nontrivial radial ground states of the upper system by the Morse theory. A result from [Sirakov. Comm. Math. Phys, 271(2007). 199-221] is improved when proper restraint onλ_1 andλ_2 is given.
本文主要运用Morse理论来找上述方程组的完全非平凡径向基态解.当λ_1,λ_2给定了适当的限制后,我们给出了[Sirakov,Comm.Math.Phys,271(2007),199-221]中结果的改进.
Consider the following Schr(o|¨)dinger system:whoreλ_j,μ_j＞0, j = 1.2.β＞0. n = 2.3.
In this paper, we study the existence of nontrivial radial ground states of the upper system by the Morse theory. A result from [Sirakov. Comm. Math. Phys, 271(2007). 199-221] is improved when proper restraint onλ_1 andλ_2 is given.
引文
[1] N. Akhmediev and A. Ankiewiez. Partially coherent solitons on a finite background. Phys. Rev. Lett. 82(1999). 2661-2664.
[2] B.Gidas. W.-M.Ni, L.Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R~n. in:Mathematical Analysis and Applications. Part A, in: Adv. Math. Suppl.
[3] D. N. Christodoulides. T. H. Coskun, M. Mitchell and M. Segev. Theory of incoherent self-focusing in biased photorefractive media, Phys. Rev. Lett., 78(1997). 646-649.
[4] M.K.kwong, Uniqueness of positive solutions of △u - u + u~p = 0 in R~n, Arch. Rational Mech. Anal. 105(1989). 243-266.
[5] T.Bartsch and Z.-Q Wang, Note on ground states of nonlinear Schrodinger systems, J. partial Differential Equations, 19(2006). 200-207.
[6] B. D. Esry. C. H. Greene. J. P. Burke Jr.. J. L. Bohn. Hartree-Fock theory for double condensates. Phys. Rev. Lett. 78(1997), 3594-3597.
[7] B. Sirakov. Least energy solitary waves for a system of nonlinear Schrodinger equations in R~n, Comm. Math. Phys., 271(2007), 199-221.
[8] G. M. Genkin, Modification of superfluidity in a resonantly strongly driven Bose-Einstein condensate, Phys. Rev. A. 65(2002), No.035604.
[9] T.-C. Lin and J. Wei. Ground state of N coupled nonlinear Schrodinger equations in R~n, n ≤ 3, Comm. Math. Phys. 255(2005), 629-653.
[10] T. Bartsch. Z.-Q. Wang and J. C. Wei. Bound states for a coupled Schrodinger system, J. Fixed Point Theory Appl, 2(2007), 353-367.
[11] K. C. Chang, Infinite dimensional Morse theory and multiple solution problems. Progress in Nonlinear Differential Equations and Their Applications, 6. Birkh(a|¨)user. Boston. 1993.
[12] T. -C. Lin and J. C. Wei, Spikes in two coupled nonlinear Schr(o|¨)dinger equations, Ann. Inst. H. Poincar(?) Anal. Non Lin(?)aire, 22(2005). 403-439.
[13] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Euqations and Their Applications, 24, Birkh(a|¨)user, Boston, 1996.
[14] F. T. Hioe. Solitary waves for N coupled nonlinear Schr(o|¨)dinger equations, Phys. Rev. Lett, 82(1999), 1152-1155.
[15] F. T. Hioe and T. S. Saltcr. Special set and solutions of coupled nonlinear Schr(o|¨)dinger equations, J. Phys. A: Math. Cen, 35(2002), 8912-8928.
[16] T. Kanna and M. Lakshmanan. Exact soliton solutions, shape changing collisions. and partially coherent solitons in coupled nonlinear Schr(o|¨)dinger equations. Phys. Rev. Lett., 86(2001), 5043-5046.
[17] M. Mitchell, Z. Chen. M. Shih. and M. Segev. Self-Trapping of partially spatially incoherent light. Phys. Rev. Lett., 77(1996), 490-493.
[18] E. Timmcrmans, Phase separation of Bose-Einstcin condensates, Phys. Rev. Lett, 81(1998), 5718-5721.
[19] Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schr(o|¨)dinger system. Preprint
[2] B.Gidas. W.-M.Ni, L.Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R~n. in:Mathematical Analysis and Applications. Part A, in: Adv. Math. Suppl.
[3] D. N. Christodoulides. T. H. Coskun, M. Mitchell and M. Segev. Theory of incoherent self-focusing in biased photorefractive media, Phys. Rev. Lett., 78(1997). 646-649.
[4] M.K.kwong, Uniqueness of positive solutions of △u - u + u~p = 0 in R~n, Arch. Rational Mech. Anal. 105(1989). 243-266.
[5] T.Bartsch and Z.-Q Wang, Note on ground states of nonlinear Schrodinger systems, J. partial Differential Equations, 19(2006). 200-207.
[6] B. D. Esry. C. H. Greene. J. P. Burke Jr.. J. L. Bohn. Hartree-Fock theory for double condensates. Phys. Rev. Lett. 78(1997), 3594-3597.
[7] B. Sirakov. Least energy solitary waves for a system of nonlinear Schrodinger equations in R~n, Comm. Math. Phys., 271(2007), 199-221.
[8] G. M. Genkin, Modification of superfluidity in a resonantly strongly driven Bose-Einstein condensate, Phys. Rev. A. 65(2002), No.035604.
[9] T.-C. Lin and J. Wei. Ground state of N coupled nonlinear Schrodinger equations in R~n, n ≤ 3, Comm. Math. Phys. 255(2005), 629-653.
[10] T. Bartsch. Z.-Q. Wang and J. C. Wei. Bound states for a coupled Schrodinger system, J. Fixed Point Theory Appl, 2(2007), 353-367.
[11] K. C. Chang, Infinite dimensional Morse theory and multiple solution problems. Progress in Nonlinear Differential Equations and Their Applications, 6. Birkh(a|¨)user. Boston. 1993.
[12] T. -C. Lin and J. C. Wei, Spikes in two coupled nonlinear Schr(o|¨)dinger equations, Ann. Inst. H. Poincar(?) Anal. Non Lin(?)aire, 22(2005). 403-439.
[13] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Euqations and Their Applications, 24, Birkh(a|¨)user, Boston, 1996.
[14] F. T. Hioe. Solitary waves for N coupled nonlinear Schr(o|¨)dinger equations, Phys. Rev. Lett, 82(1999), 1152-1155.
[15] F. T. Hioe and T. S. Saltcr. Special set and solutions of coupled nonlinear Schr(o|¨)dinger equations, J. Phys. A: Math. Cen, 35(2002), 8912-8928.
[16] T. Kanna and M. Lakshmanan. Exact soliton solutions, shape changing collisions. and partially coherent solitons in coupled nonlinear Schr(o|¨)dinger equations. Phys. Rev. Lett., 86(2001), 5043-5046.
[17] M. Mitchell, Z. Chen. M. Shih. and M. Segev. Self-Trapping of partially spatially incoherent light. Phys. Rev. Lett., 77(1996), 490-493.
[18] E. Timmcrmans, Phase separation of Bose-Einstcin condensates, Phys. Rev. Lett, 81(1998), 5718-5721.
[19] Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schr(o|¨)dinger system. Preprint