Existence and concentration of ground states of coupled nonlinear Schrödinger equations with bounded potentials

Abstract

A 2-coupled nonlinear Schrödinger equations with bounded varying potentials and strongly attractive interactions is considered. When the attractive interaction is strong enough, the existence of a ground state for sufficiently small Planck constant is proved. As the Planck constant approaches zero, it is proved that one of the components concentrates at a minimum point of the ground state energy function which is defined in Section 4.

Keywords

Concentration / Nehari’s manifold / Critical point theory / Concentration-compactness principle

References

[1] Ambrosetti A., Colorado E.. Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc., 2007, 75(1): 67-82

[2] Ambrosetti A., Colorado E.. Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris, 2006, 342(7): 453-458

[3] Agmon S.. Bounds on exponential decay of eigenfunctions of Schrödinger operators, 1985, Berlin: Springer-Verlag 1-39

[4] Alikakos N., Kowalczyk M.. Critical point of a singular perturbation problem via reduced energy and local linking. J. Diff. Eqs., 1999, 159: 403-426

[5] Ambrosetti A., Malchiodi A., Ni W.-M.. Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part I. Comm. Math. Phys., 2003, 235: 427-466

[6] Ambrosetti A., Malchiodi A., Secchi S.. Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Rat. Mech. Anal., 2001, 159: 253-271

[7] Badiale M., D’Aprile T.. Concentration around a sphere for a singularly perturbed Schrödinger equation. Nonlinear Anal., 2002, 49: 947-985

[8] Brezis H., Kato T.. Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures et Appl., 1979, 58: 137-151

[9] Berestycki H., Lions P. L.. Nonlinear scalar field equations, I Existence of a ground state; II Existence of infinitely many solutions. Arch. Rat. Mech. Anal., 1982, 82: 313-345

[10] Busca J., Sirakov B.. Symmetry results for semilinear elliptic systems in the whole space. J. Diff. Eqs., 2000, 163: 41-56

[11] Bartsch T., Wang Z.-Q.. Note on ground states of nonlinear Schrödinger systems. J. Partial Diff. Eqs., 2006, 19: 200-207

[12] Bartsch T., Wang Z.-Q.. Existence and multiple results for some superlinear elliptic problems on ℝ^N. Comm. in Partial Diff. Eqs., 1995, 20: 1725-1741

[13] Bartsch T., Willem M.. Infinitely many radial solutions of a semilinear elliptic problem on ℝ^N. Arch. Rat. Mech. Anal., 1993, 124: 261-276

[14] Chen J., Li S.. Existence and multiplicity of nontrivial solutions for an elliptic equation on ℝ^N with indefinite linear part. Manuscript Math., 2003, 111: 221-239

[15] Cingolani S., Lazzo M.. Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Diff. Eqs., 2000, 160: 118-138

[16] Conti M., Terracini S., Verzini G.. Nehari’s problem and competing species systems. Ann. I. H. Poincaré-AN, 2002, 19(6): 871-888

[17] Ding Y., Lin F.. Semiclassical satates of Hamiltonian system of Schrödinger equations with subcritical and critical nonlinearities. J. Partial Diff. Eqs., 2006, 19: 232-255

[18] Del Pino M., Felmer P.. Local mountain passes for Semilinear elliptic problems in unbounded domian. Calc. Var. Partial Diff. Eqs., 1996, 4: 121-137

[19] Del Pino M., Felmer P.. Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal., 1997, 149: 245-265

[20] Del Pino M., Felmer P.. Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann., 2002, 324(2): 1-32

[21] Dancer E., Wei J.. On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem. J. Diff. Eqs., 1999, 157: 82-101

[22] Floer A., Weinstein A.. Nonspreading wave packets for the cubic Schrödinger equation with the bounded potential. J. Funct. Anal., 1986, 69: 397-408

[23] Grossi M.. On the number of single-peak solutions of the nonlinear Schrödinger equation. Ann. I. H. Poincaré Nonlineaire, 2002, 19: 261-280

[24] Jeanjean L., Tanaka K.. Singularly perturbed problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Diff. Eqs., 2004, 21(3): 287-318

[25] Kowalczyk M.. Multiple spike layers in the shadow Gierer-Meinhardt systems: Existence of equilibria and the quasi-invariant manifold. Duke Math. J., 1999, 98(1): 59-111

[26] Lieb E. H.. The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys., 1977, 53: 185-194

[27] Lions P. L.. The concentration-compactness principle in the calculus of variations; The locally compact case, part 1 and part 2. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1984, 1: 109-145

[28] Lin C. S., Ni W. M., Takagi I.. Large amplitude stationary solutions to a chemotaxis system. J. Diff. Eqs., 1988, 72: 1-27

[29] Lin T. C., Wei J.. Ground state of N coupled nonlinear Schrödinger equations in ℝⁿ, n ≤ 3. Comm. Math. Phys., 2005, 255: 629-653

[30] Lin T. C., Wei J.. Spikes in two copuled nonlinear Schrödinger equations. Ann. I. H. Poincaré-AN, 2005, 22: 403-439

[31] Lin T. C., Wei J.. Spikes on two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Diff. Eqs., 2006, 229: 538-569

[32] Ni, W. M., Diffusion, cross-diffusion, and their spike-layer steady states, Notices of the A. M. S., 1998, 9–18.

[33] Ni W. M., Takagi I.. On the Neumann problem for some semilinear elliptic equations and systems of activator-inhabitor type. Trans. Amer. Math. Soc., 1986, 297: 351-368

[34] Ni W. M., Takagi I.. On the shape of least-energy solutions to a semilinear problem. Comm. Pure Appl. Math., 1991, XLIV: 819-851

[35] Ni W. M., Takagi I.. Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J., 1993, 70: 247-281

[36] Oh Y. G.. On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys., 1990, 131: 223-253

[37] Pomponio A.. Singularly perturbed Neumann problems with potentials. Topol. Methods Nonlinear Anal., 2004, 23: 301-322

[38] Rabinowitz P. H.. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys., 1992, 43: 270-291

[39] Ramos M., Yang J.. Spike-layered solutions for an elliptic system with Neumann boundary condition. Trans. Amer. Math. Soc., 2004, 357: 3265-3284

[40] Shaker A. W.. On symmetry in elliptic systems. Appl. Anal., 1991, 41: 1-9

[41] Strauss W. A.. Existence of solitary waves in higher dimensions. Comm. Math. Phys., 1977, 55: 149-162

[42] Takagi I.. Point-condensation for a reaction-diffusion system. J. Diff. Eqs., 1986, 61: 208-249

[43] Trudinger N. S.. On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm. Pure Appl. Math., 1967, XX: 721-747

[44] Wang X.. On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys., 1993, 153: 229-244

[45] Wang X., Zeng B.. On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal., 1997, 28(3): 633-655

[46] Willem M.. Minimax theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, 1996, Boston: Birkhäuser

[47] Wei J., Winter M.. Spikes for the Gierer-Mainhardt system on two dimension: the strong coupling case. J. Diff. Eqs., 2002, 178: 478-518