A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

Dancer, E.N.; Wei, Juncheng; Weth, Tobias

Dancer, E.N. ; Wei, Juncheng ; Weth, Tobias

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 953-969 / Harvested from Numdam

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Résumé

We study the set of solutions of the nonlinear elliptic system $\begin{matrix} {\begin{matrix} - Δ u + λ_{1} u = μ_{1} u^{3} + β v^{2} u in Ω, \\ - Δ v + λ_{2} v = μ_{2} v^{3} + β u^{2} v in Ω, \\ u, v > 0 in Ω, u = v = 0 on \partial Ω, \end{matrix} & (P) \end{matrix}$ in a smooth bounded domain $Ω \subset ℝ^{N}$ , $N ⩽ 3$ , with coupling parameter $β \in ℝ$ . This system arises in the study of Bose–Einstein double condensates. We show that the value $β = - \sqrt{μ_{1} μ_{2}}$ is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for $β > - \sqrt{μ_{1} μ_{2}}$ , solutions of (P) are a priori bounded. In contrast, when $λ_{1} = λ_{2}$ , $μ_{1} = μ_{2}$ , (P) admits an unbounded sequence of solutions if $β ⩽ - \sqrt{μ_{1} μ_{2}}$ .

Publié le : 2010-01-01

MR 2629888 | Zbl 1191.35121 | 2 citations dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2010.01.009

@article{AIHPC_2010__27_3_953_0,
     author = {Dancer, E.N. and Wei, Juncheng and Weth, Tobias},
     title = {A priori bounds versus multiple existence of positive solutions for a nonlinear Schr\"odinger system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {953-969},
     doi = {10.1016/j.anihpc.2010.01.009},
     mrnumber = {2629888},
     zbl = {1191.35121},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_3_953_0}
}

Dancer, E.N.; Wei, Juncheng; Weth, Tobias. A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 953-969. doi : 10.1016/j.anihpc.2010.01.009. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_3_953_0/

Bibliographie

[1] A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris 342 (2006), 453-458 | MR 2214594 | Zbl 1094.35112

[2] T. Bartsch, Z.-Q. Wang, J.C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl. 2 no. 2 (2007), 353-367 | MR 2372993 | Zbl 1153.35390

[3] H. Berestycki, L.A. Caffarelli, L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains. I, Duke Math. J. 81 (1996), 467-494 | MR 1395408 | Zbl 0860.35004

[4] H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), 59-78 | MR 1321809 | Zbl 0816.35030

[5] D.G. De Figueiredo, B. Sirakov, Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann. 333 (2005), 231-260 | MR 2195114 | Zbl 1165.35360

[6] D.G. De Figueiredo, J.F. Yang, A priori bounds for positive solutions of a non-variational elliptic system, Comm. Partial Differential Equations 26 (2001), 2305-2321 | MR 1876419 | Zbl 0997.35015

[7] B.D. Esry, C.H. Greene, J.P. Burke, J.L. Bohn, Hartree–Fock theory for double condensates, Phys. Rev. Lett. 78 (1997), 3594-3597

[8] B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, Nonlinear Partial Differential Equations in Engineering and Applied Science, Proc. Conf., Univ. Rhode Island, Kingston, RI, 1979, Lect. Notes Pure Appl. Math. vol. 54, Dekker, New York (1980), 255-273

[9] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883-901 | MR 619749 | Zbl 0462.35041

[10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (2001) | MR 473443 | Zbl 0691.35001

[11] S. Gupta, Z. Hadzibabic, M.W. Zwierlein, C.A. Stan, K. Dieckmann, C.H. Schunck, E.G.M. Van Kempen, B.J. Verhaar, W. Ketterle, Radio-frequency spectroscopy of ultracold fermions, Science 300 (2003), 1723-1726

[12] D.S. Hall, R. Matthews, J.R. Ensher, C.E. Wieman, E.A. Cornell, Dynamics of component separation in a binary mixture of Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998), 1539-1542

[13] C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, C.E. Wieman, Production of two overlapping Bose–Einstein condensates by sympathetic cooling, Phys. Rev. Lett. 78 (1997), 586-589

[14] G.G. Laptev, Absence of global positive solutions of systems of semilinear elliptic inequalities in cones, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 107-124 | MR 1817251 | Zbl 1013.35041

[15] G.G. Laptev, On the nonexistence of solutions of elliptic differential inequalities in conic domains, Mat. Zametki 71 (2002), 855-866 | MR 1933106

[16] T.-C. Lin, J.-C. Wei, Ground state of N coupled nonlinear Schrödinger equations in $R^{n}$ , $n ⩽ 3$ , Comm. Math. Phys. 255 (2005), 629-653 | MR 2135447 | Zbl 1119.35087

[17] T.-C. Lin, J.-C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403-439 | Numdam | MR 2145720 | Zbl 1080.35143

[18] T.C. Lin, J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations 229 (2006), 538-569 | MR 2263567 | Zbl 1105.35117

[19] E. Mitidieri, S. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1-384 | MR 1879326 | Zbl 1074.35500

[20] L.A. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2006), 743-767 | MR 2263573 | Zbl 1104.35053

[21] P. Poláčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Duke Math. J. 139 (2007), 555-579 | MR 2350853 | Zbl 1146.35038

[22] P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004), 49-81 | MR 2092996 | Zbl 1113.35062

[23] W. Reichel, H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations 161 (2000), 219-243 | MR 1740363 | Zbl 0962.35054

[24] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations, Comm. Math. Phys. 271 (2007), 199-221 | MR 2283958 | Zbl 1147.35098

[25] M.A.S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations 8 (1995), 1245-1258 | MR 1325555 | Zbl 0823.35064

[26] E. Timmermans, Phase separation of Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998), 5718-5721

[27] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergeb. Math. Grenzgeb. (3) vol. 34, Springer-Verlag, Berlin (1996) | MR 1411681 | Zbl 0864.49001

[28] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. vol. 24, Birkhäuser Boston Inc., Boston, MA (1996) | MR 1400007 | Zbl 0856.49001

[29] J.C. Wei, T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equations, Rend. Lincei Mat. Appl. 18 (2007), 279-293 | MR 2318821 | Zbl 1229.35019

[30] J.C. Wei, T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal. 190 no. 1 (2008), 83-106 | MR 2434901 | Zbl 1161.35051

[31] H. Zou, A priori estimates for a semilinear elliptic system without variational structure and their application, Math. Ann. 323 (2002), 713-735 | MR 1924277 | Zbl 1005.35024

[32] H. Zou, A priori estimates and existence for quasi-linear elliptic equations, Calc. Var. 33 (2008), 417-437 | MR 2438741 | Zbl 1169.35336