NODAL BOUND STATES WITH CLUSTERED SPIKES FOR NONLINEAR SCHRÖDINGER EQUATIONS

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doi:10.1016/S0252-9602(14)60133-8

Abstract:

We consider the following nonlinear Schr¨odinger equations −"2Δu + u = Q(x)|u|^p−2u in R^N, u ∈ H¹(R^N), where " is a small positive parameter, N ∈ 2, 2 < p < 1 for N = 2 and 2 < p < 2N /N−2 for N ≥3. We prove that this problem has sign-changing (nodal) semi-classical bound states with clustered spikes for sufficiently small " under some additional conditions on Q(x). Moreover, the number of this type of solutions will go to infinity as ε→ 0⁺.

Key words: nodal bound states, Lyapunov-Schmidt reduction, Schr¨odinger equations

CLC Number:

35J10

DAI Jin-Jun, HE Ji-Han, LI Bi-Wen. NODAL BOUND STATES WITH CLUSTERED SPIKES FOR NONLINEAR SCHRÖDINGER EQUATIONS[J].Acta mathematica scientia,Series B, 2014, 34(6): 1892-1906.

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[1] Ambrosetti A, Badiale M, Cingolani S. Semiclassical states of nonlinear Schr¨odinger equations. Arch Rational Mech Anal, 1997, 140: 285–300

[2] Ambrosetti A, Malchiodi A, Ni W. Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I Comm Math Phys, 2003, 235: 427–466

[3] Ba N, Deng Y, Peng S. Multi-peak bound states for Schr¨odinger equations with compactly supported or unbounded potentials. Ann Inst H Poincar´e Anal Non Lin´eaire, 2010, 27: 1205–1226

[4] Bartsch T, Peng S. Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II. J Differential Equations, 2010, 248: 2746–2767

[5] Bartsch T, Peng S. Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations I. Indiana Univ Math J, 2008, 57: 1599–1631

[6] Bartsch T,WillemM. Infinitely many radial solutions of a semilinear elliptic problem on RN. Arch Rational Mech Anal, 1993, 124: 261–276

[7] Cao D, Dancer N, Noussair E, Yan S. On the existence of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete Contin Dynam Systems, 1996, 2: 221–236

[8] Cao D, Noussair E. Multi-peak solutions for a singularly perturbed semilinear elliptic problem. J Differential Equations, 2000, 166: 266–289

[9] Cao D, Noussair E, Yan S. Existence and uniqueness results on single-peaked solutions of a semilinear problem. Ann Inst H Poincar´e Anal Non Lin´eaire, 1998, 15: 73–111

[10] Cao D, Noussair E, Yan S. Solutions with multiple peaks for nonlinear elliptic equations. Proc Roy Soc Edinburgh Sect A, 1999, 129: 235–264

[11] Cao D, Peng S. Multi-bump bound states of Schr¨odinger equations with a critical frequency. Math Ann, 2006, 336: 925–948

[12] Cao D, Peng S. Semi-classical bound states for Schr¨odinger equations with potentials vanishing or unbounded
at infinity. Comm Partial Differential Equations, 2009, 34: 1566–1591

[13] D’Aprile T, Pistoia A. Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical
nonlinear Schr¨odinger equation. Ann Inst H Poincar´e Anal Non Lin´eaire, 2009, 26: 1423–1451

[14] D’Aprile T, Pistoia A. Nodal solutions for some singularly perturbed Dirichlet problems. Trans Amer Math Soc, 2011, 363: 3601–3620

[15] D’Aprile T, Pistoia A. On the number of sign-changing solutions of a semiclassical nonlinear Schr´oinger equation. Adv Differ Equ, 2007, 12: 737–758

[16] D’Aprile T, Ruiz D. Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems. Math Z, 2011, 268: 605–634

[17] del Pino M, Felmer P. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc Var Partial Differential Equations, 1996, 4: 121–137

[18] del Pino M, Felmer P. Multi-peak bound states for nonlinear Schr¨odinger equations. Ann Inst H Poincar´e Anal Non Lin´eaire, 1998, 15: 127–149

[19] del Pino M, Felmer P, Wei J. Multi-peak solutions for some singular perturbation problems. Calc Var Partial Differ Equ, 2000, 10: 119–134

[20] Micheletti A, Pistoia A. On the multiplicity of nodal solutions to a singularly perturbed Neumann problem. Mediterranean J Math, 2008, 5: 285–294

[21] Floer A, Weinstein A. Nonspreading wave packets for the cubic Schr¨odinger equations with a bounded potential. J Funct Anal, 1986, 69: 397–408

[22] Gidas B, Ni W, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in RN. Adv in Math Suppl Stud, 1981, 7A: 369–402

[23] Gui C. Existence of multi-bump solutions for nonlinear Schr¨odinger equations via variational method. Comm Partial Differential Equations, 1996, 21: 787–820

[24] Gui C, Wei J. On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Canad J Math, 2000, 52: 522–538

[25] Kang X, Wei J. On interacting bumps of semi-classical states of nonlinear Schr¨odinger equations. Adv Differ Equ, 2000, 5: 899–928

[26] Lin F, NiW, Wei J. On the number of interior peak solutions for a singularly perturbed Neumann problem. Comm Pure Appl Math, 2007, 60: 252–281

[27] Liu F, Yang J. Nontrivial solutions for semilinear Schr¨odinger equations. Acta Math Sci, 2009, 29B(5): 1405–1420

[28] Noussair E, Wei J. On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems. Indiana Univ Math J, 1997, 46: 1255–1271

[29] Noussair E, Yan S. On positive multipeak solutions of a nonlinear elliptic problem. J London Math Soc, 2000, 62(2): 213–227

[30] Oh Y. Existence of semiclassical bound states of nonlinear Schr¨odinger equations with potentia of the class (V )a. Comm Partial Differ Equ, 1988, 13: 1499–1519

[31] Oh Y. On positive multi-lump bounded states of nonlinear Schr¨odinger equations under multiple well potential. Comm Math Phys, 1990, 131: 223–253

[32] Peng C, Yang J. Multiple solutions for Schr¨odinger equations with magnetic field. Acta Math Sci, 2009, 29B(5): 1323–1340

[33] Peng S, Wang Z. Segregated and synchronized vector solutions for nonlinear Schr¨odinger systems. Arch Rational Mech Anal, 2013, 208: 305–339

[34] Rey O. The role of the Green´’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J Funct Anal, 1990, 89: 1–52

[35] Tanaka K, Yan S. Many positive solutions for an elliptic problems in RN. preprint

[36] Wei J, Weth T. On the number of nodal solutions to a singularly perturbed Neumann problem. Manuscripta Math, 2005, 117: 333–344

[37] Wei J, Yan S. Infinitely many positive solutions for the nonlinear Schr¨odinger equations in RN. Calc Var Partial Differ Equ, 2010, 37: 423–439

[38] Wei J, Yan S. Infinitely many solutions for the prescribed scalar curvature problem on Sn. J Funct Anal, 2010, 258: 3048–3081

[39] Wei J, Yan S. An elliptic problem with critical growth and Lazer-Mckenna conjecture. Ann Sc Norm Super Pisa Cl Sci, 2010, 9(5): 423–457

[40] Wei J, Yan S. Infinitely many positive solutions for an elliptic problem with critical or supercritical growth. J Math Pures Appl, 2011, 96(9): 307–333

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