×

zbMATH — the first resource for mathematics

Nonradial symmetric bound states for a system of coupled Schrödinger equations. (English) Zbl 1229.35019
Summary: We consider bound state solutions of the coupled elliptic system \[ \begin{aligned} \Delta u - u + u^{3} + \beta v^{2} u = 0 \quad &{\text{in}} \quad \mathbb{R}^{N},\\ \Delta v - v + v^{3} + \beta u^{2} v = 0 \quad &{\text{in}} \quad \mathbb{R}^{N},\\ u>0, \quad v>0,\quad u, v &\in H^{1} (\mathbb{R}^{N}),\end{aligned} \] where \(N=2,3\).
It is known [T. C. Lin and J. C. Wei, Commun. Math. Phys. 255, No. 3, 629–653 (2005; Zbl 1119.35087)] that when \(\beta < 0\), there are no ground states, i.e., no least energy solutions. We show that, for certain finite subgroups of \(O (N)\) acting on \(H^{1} (\mathbb{R}^{N})\), least energy solutions can be found within the associated subspaces of symmetric functions. For \(\beta \leq -1\), these solutions are nonradial. From this we deduce, for every \(\beta \leq -1\), the existence of infinitely many nonradial bound states of the system.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
92C40 Biochemistry, molecular biology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. AKHMEDIEV - A. ANKIEWICZ, Partially coherent solitons on a finite background . Phys. Rev. Lett. 82 (1999), 2661-2664.
[2] A. AMBROSETTI - E. COLORADO, Bound and ground states of coupled nonlinear Schrödinger equations . C. R. Math. Acad. Sci. Paris 342 (2006), 453-458. · Zbl 1094.35112 · doi:10.1016/j.crma.2006.01.024
[3] A. AMBROSETTI - E. COLORADO, Standing waves of some coupled nonlinear Schrödinger equations . J. London Math. Soc. 75 (2007), 67-82. · Zbl 1130.34014 · doi:10.1112/jlms/jdl020
[4] T. BARTSCH - Z.-Q. WANG - J. C. WEI, Bound states for a coupled Schrödinger system . Preprint. · Zbl 1153.35390 · doi:10.1007/s11784-007-0033-6
[5] P. CHOSSAT - R. LAUTERBACH - I. MELBOURNE, Steady-state bifurcation with O(3) - symmetry . Arch. Ration. Mech. Anal. 113 (1990), 313-376. · Zbl 0722.58031 · doi:10.1007/BF00374697
[6] D. N. CHRISTODOULIDES - T. H. COSKUN - M. MITCHELL - M. SEGEV, Theory of incoherent self-focusing in biased photorefractive media . Phys. Rev. Lett. 78 (1997), 646-649.
[7] C. V. COFFMAN, Uniqueness of the ground state solution for \Delta u - u+u3 = 0 and a variational characterization of other solutions . Arch. Ration. Mech. Anal. 46 (1972), 81-95. · Zbl 0249.35029 · doi:10.1007/BF00250684
[8] 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.
[9] B. GIDAS - W. M. NI - L. NIRENBERG, Symmetry and related properties via the maximum principle . Comm. Math. Phys. 68 (1979), 209-243. · Zbl 0425.35020 · doi:10.1007/BF01221125
[10] F. T. HIOE, Solitary waves for N coupled nonlinear Schrödinger equations . Phys. Rev. Lett. 82 (1999), 1152-1155.
[11] F. T. HIOE - T. S. SALTER, Special set and solutions of coupled nonlinear Schrödinger equations . J. Phys. A 35 (2002), 8913-8928; Corrigendum, ibid. 37 (2004), 7821. · Zbl 1040.35115 · doi:10.1088/0305-4470/35/42/303
[12] T. KANNA - M. LAKSHMANAN, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations . Phys. Rev. Lett. 86 (2001), 5043-5046.
[13] T. C. LIN - J. C. WEI, Ground state of N coupled nonlinear Schrödinger equations in Rn, n \leq 3. Comm. Math. Phys. 255 (2005), 629-653. · Zbl 1119.35087 · doi:10.1007/s00220-005-1313-x
[14] T. C. LIN - J. C. WEI, Spikes in two coupled nonlinear Schrodinger equations . Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403-439. · Zbl 1080.35143 · doi:10.1016/j.anihpc.2004.03.004 · numdam:AIHPC_2005__22_4_403_0 · eudml:78662
[15] T. C. LIN - J. C. WEI, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials . J. Differential Equations 229 (2006), 538-569. · Zbl 1105.35117 · doi:10.1016/j.jde.2005.12.011
[16] T. C. LIN - J. C. WEI, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations . Phys. D 220 (2006), 99-115. · Zbl 1105.35116 · doi:10.1016/j.physd.2006.07.009
[17] P.-L. LIONS, The concentration-compactness principle in the calculus of variations. The locally compact case . Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109-145 and 223-283. · Zbl 0704.49004 · numdam:AIHPC_1984__1_4_223_0 · eudml:78074
[18] L. A. MAIA - E. MONTEFUSCO - B. PELLACCI, Positive solutions for a weakly coupled nonlinear Schrödinger system . J. Differential Equations 229 (2006), 743-767. · Zbl 1104.35053 · doi:10.1016/j.jde.2006.07.002
[19] M. MITCHELL - Z. CHEN - M. SHIH - M. SEGEV, Self-trapping of partially spatially incoherent light . Phys. Rev. Lett. 77 (1996), 490-493.
[20] M. MITCHELL - M. SEGEV, Self-trapping of incoherent white light . Nature 387 (1997), 880-882.
[21] W.-M. NI - I. TAKAGI, Locating the peaks of least energy solutions to a semilinear Neumann problem . Duke Math. J. 70 (1993), 247-281. · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4
[22] B. SIRAKOV, Least energy solitary waves for a system of nonlinear Schrödinger equations in n R . Comm. Math. Phys. 271 (2007), 199-221. J. C. WEI - T. WETH · Zbl 1147.35098 · doi:10.1007/s00220-006-0179-x
[23] M. STRUWE, Variational Methods . 2nd ed., Springer, Berlin, 1996.
[24] E. TIMMERMANS, Phase separation of Bose-Einstein condensates . Phys. Rev. Lett. 81 (1998), 5718-5721.
[25] W. C. TROY, Symmetry properties in systems of semilinear elliptic equations . J. Differential Equations 42 (1981), 400-413. · Zbl 0486.35032 · doi:10.1016/0022-0396(81)90113-3
[26] T. WETH, Energy bounds for entire nodal solutions of autonomous elliptic equations via the moving plane method . Calc. Var. 27 (2006), 421-437. · Zbl 1151.35365 · doi:10.1007/s00526-006-0015-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.