Vector Solutions with Prescribed Component-Wise Nodes for a Schr?dinger System
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摘要
For the Schr?dinger system ■where k ≥ 2 and N = 2,3, we prove that for any λ_j> 0 and β_(jj)> 0 and any positive integers p_j, j = 1,2,···,k, there exists b > 0 such that if β_(ij)= β_(ji)≤ b for all i ≠ j then there exists a radial solution(u_1,u_2,···,u_k) with ujhaving exactly p_j-1 zeroes. Moreover,there exists a positive constant C_0 such that if β_(ij)= β_(ji)≤ b(i ≠ j) then any solution obtained satisfies ■Therefore, the solutions exhibit a trend of phase separations as β_(ij)→-∞ for i ≠ j.
For the Schr?dinger system ■where k ≥ 2 and N = 2,3, we prove that for any λ_j> 0 and β_(jj)> 0 and any positive integers p_j, j = 1,2,···,k, there exists b > 0 such that if β_(ij)= β_(ji)≤ b for all i ≠ j then there exists a radial solution(u_1,u_2,···,u_k) with ujhaving exactly p_j-1 zeroes. Moreover,there exists a positive constant C_0 such that if β_(ij)= β_(ji)≤ b(i ≠ j) then any solution obtained satisfies ■Therefore, the solutions exhibit a trend of phase separations as β_(ij)→-∞ for i ≠ j.
For the Schr?dinger system ■where k ≥ 2 and N = 2,3, we prove that for any λ_j> 0 and β_(jj)> 0 and any positive integers p_j, j = 1,2,···,k, there exists b > 0 such that if β_(ij)= β_(ji)≤ b for all i ≠ j then there exists a radial solution(u_1,u_2,···,u_k) with ujhaving exactly p_j-1 zeroes. Moreover,there exists a positive constant C_0 such that if β_(ij)= β_(ji)≤ b(i ≠ j) then any solution obtained satisfies ■Therefore, the solutions exhibit a trend of phase separations as β_(ij)→-∞ for i ≠ j.
引文
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[19] H. Liu and Z. Liu, Positive solutions of a nonlinear Schr ?dinger system with nonconstant potentials, Discrete Contin. Dyn. Syst., 36(2016), 1431–1464.
[20] H. Liu and Z. Liu, Multiple positive solutions of elliptic systems in exterior domains, Commun. Contemp. Math., 20(6)(2018), 1750063(32 pages), DOI:10.1142/S0219199717500638.
[21] H. Liu, Z. Liu and J. Chang, Existence and uniqueness of positive solutions of nonlinear Schr ?dinger systems, Proc. Royal Soc. Edinburgh, 145A(2015), 365–390.
[22] J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schr ?dinger systems, Calc. Var. Partial Differential Equations, 52(2015), 565–586.
[23] J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schr?dinger equations with critical growth, J. Differential Equations, 261(2016), 7194–7236.
[24] Z. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Advanced Nonlinear Studies, 4(2004), 563–574.
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[26] Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schr?dinger system,Advanced Nonlinear Studies, 10(2010), 175–193.
[27] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387(1997), 880–883.
[28] E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schr?dinger systems, J. Euro. Math. Soc., 10(2008), 41–71.
[29] Z. Nehari, Characteristic values associated with a class of nonlinear second-order differential equations, Acta Math., 105(1960), 141–175.
[30] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform H?lder bounds for nonlinear Schr ?dinger systems with strong competition, Commun. Pure Appl. Math., 63(2010), 267–302.
[31] S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrodinger systems, Arch. Rational Mech. Anal., 208(2013), 305–339.
[32] Ch. Rüegg et al., Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423(2003), 62–65.
[33] Y. Sato and Z.-Q. Wang, Least energy solutions for nonlinear Schr?dinger systems with mixed attractive and repulsive couplings, Advanced Nonlinear Studies, 15(2015), 1–22.
[34] Y. Sato and Z.-Q. Wang, Multiple positive solutions for Schr?dinger systems with mixed couplings, Calc. Var. Partial Differential Equations, 54(2015), 1373–1392.
[35] Y. Sato and Z.-Q. Wang, On the least energy sign-changing solutions for a nonlinear elliptic system, Discrete Contin. Dyn. Syst., 35(2015), 2151–2164.
[36] B. Sirakov, Least energy solitary waves for a system of nonlinear Schr?dinger equations in Rn, Commun. Math. Phys., 271(2007), 199–221.
[37] N. Soave, On existence and phase separation of solitary waves for nonlinear Schr?dinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differential Equations, 53(2015), 689–718.
[38] S. Terracini and G. Verzini, Multipulse phase in k-mixtures of Bose-Einstein condenstates,Arch. Rational Mech. Anal., 194(2009), 717–741.
[39] R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schr?dinger systems,Topol. Methods Nonl. Anal., 37(2011), 203–223.
[40] E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81(1998),5718–5721.
[41] Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal.Math., 122(2014), 69–85.
[42] J. Wei and T. Weth, Nonradial symmetric bound states for s system of two coupled Schr ?dinger equations, Rend. Lincei Mat. Appl., 18(2007), 279–293.
[43] J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schr ?dinger equations, Arch. Rational Mech. Anal., 190(2008), 83–106.
[2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schr?dinger equations, C. R. Math. Acad. Sci. Paris, 342(2006), 453–458.
[3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schr?dinger equations, J. Lond. Math. Soc., 75(2007), 67–82.
[4] T. Bartsch, E. N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37(2010), 345–361.
[5] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schr?dinger systems, J. Partial Differential Equations, 19(2006), 200–207.
[6] T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schr?dinger system, J. Fixed Point Theory Appl., 2(2007), 353–367.
[7] T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on RN, Arch. Rational Mech. Anal., 124(1993), 261–276.
[8] J. Byeon, Y. Sato and Z.-Q. Wang, Pattern formation via mixed attractive and repulsive interactions for nonlinear Schr ?dinger systems, J. Math. Pures Appl.,(9)(2016), 477–511.
[9] D. Cao and X. Zhu, On the existence and nodal character of solutions of semilinear elliptic equations, Acta Math. Sci.(English Ed.), 8(1988), 345–359.
[10] S. M. Chang, C. S. Lin, T. C. Lin and W. W. Lin, Segregated nodal domains of twodimensional multispecies Bose-Einstein condensates, Physica D:Nonlinear Phenomena, 196(2004), 341–361.
[11] M. Conti, S. Terracini and G. Verzini, Nehari’s problem and competing species systems, Ann.Inst. H. PoincaréAnal. Non Linéaire, 19(2002), 871–888.
[12] E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schr ?dinger system, Ann. Inst. H. PoincaréAnal. Non Linéaire, 27(2010), 953–969.
[13] D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schr?dinger systems,Ann. Inst. H. PoincaréAnal. Non Linéaire, 25(2008), 149–161.
[14] B. D. Esry, C. H. Greene, J. P. Jr. Burke and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78(1997), 3594–3597.
[15] T. C. Lin and J. Wei, Ground state of N coupled nonlinear Schr ?dinger equations in Rn, n≤3,Commun. Math. Phys., 255(2005), 629–653.
[16] T. C. Lin, and J. Wei, Spikes in two coupled nonlinear Schr?dinger equations, Ann. Inst. H.PoincaréAnal. Non Linéaire, 22(2005), 403–439.
[17] T. C. Lin and J. Wei, Solitary and self-similar solutions of two-component systems of nonlinear Schr ?dinger equations, Physica D:Nonlinear Phenomena, 220(2006), 99–115.
[18] H. Liu and Z. Liu, Ground states of a nonlinear Schr?dinger system with nonconstant potentials, Science China:Mathematics, 58(2015), 257–278.
[19] H. Liu and Z. Liu, Positive solutions of a nonlinear Schr ?dinger system with nonconstant potentials, Discrete Contin. Dyn. Syst., 36(2016), 1431–1464.
[20] H. Liu and Z. Liu, Multiple positive solutions of elliptic systems in exterior domains, Commun. Contemp. Math., 20(6)(2018), 1750063(32 pages), DOI:10.1142/S0219199717500638.
[21] H. Liu, Z. Liu and J. Chang, Existence and uniqueness of positive solutions of nonlinear Schr ?dinger systems, Proc. Royal Soc. Edinburgh, 145A(2015), 365–390.
[22] J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schr ?dinger systems, Calc. Var. Partial Differential Equations, 52(2015), 565–586.
[23] J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schr?dinger equations with critical growth, J. Differential Equations, 261(2016), 7194–7236.
[24] Z. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Advanced Nonlinear Studies, 4(2004), 563–574.
[25] Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schr?dinger systems, Commun.Math. Phys., 282(2008), 721–731.
[26] Z. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schr?dinger system,Advanced Nonlinear Studies, 10(2010), 175–193.
[27] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387(1997), 880–883.
[28] E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schr?dinger systems, J. Euro. Math. Soc., 10(2008), 41–71.
[29] Z. Nehari, Characteristic values associated with a class of nonlinear second-order differential equations, Acta Math., 105(1960), 141–175.
[30] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform H?lder bounds for nonlinear Schr ?dinger systems with strong competition, Commun. Pure Appl. Math., 63(2010), 267–302.
[31] S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrodinger systems, Arch. Rational Mech. Anal., 208(2013), 305–339.
[32] Ch. Rüegg et al., Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl3, Nature, 423(2003), 62–65.
[33] Y. Sato and Z.-Q. Wang, Least energy solutions for nonlinear Schr?dinger systems with mixed attractive and repulsive couplings, Advanced Nonlinear Studies, 15(2015), 1–22.
[34] Y. Sato and Z.-Q. Wang, Multiple positive solutions for Schr?dinger systems with mixed couplings, Calc. Var. Partial Differential Equations, 54(2015), 1373–1392.
[35] Y. Sato and Z.-Q. Wang, On the least energy sign-changing solutions for a nonlinear elliptic system, Discrete Contin. Dyn. Syst., 35(2015), 2151–2164.
[36] B. Sirakov, Least energy solitary waves for a system of nonlinear Schr?dinger equations in Rn, Commun. Math. Phys., 271(2007), 199–221.
[37] N. Soave, On existence and phase separation of solitary waves for nonlinear Schr?dinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differential Equations, 53(2015), 689–718.
[38] S. Terracini and G. Verzini, Multipulse phase in k-mixtures of Bose-Einstein condenstates,Arch. Rational Mech. Anal., 194(2009), 717–741.
[39] R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schr?dinger systems,Topol. Methods Nonl. Anal., 37(2011), 203–223.
[40] E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81(1998),5718–5721.
[41] Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal.Math., 122(2014), 69–85.
[42] J. Wei and T. Weth, Nonradial symmetric bound states for s system of two coupled Schr ?dinger equations, Rend. Lincei Mat. Appl., 18(2007), 279–293.
[43] J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schr ?dinger equations, Arch. Rational Mech. Anal., 190(2008), 83–106.