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Applicable Analysis
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Volume 100, 2021 - Issue 14
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Articles

Concentrating ground state solutions for quasilinear Schrödinger equations with steep potential well

Pages 3065-3082 | Received 12 Sep 2019, Accepted 12 Dec 2019, Published online: 27 Dec 2019

References

  • Bartsch T, Pankov A, Wang Z-Q. Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math. 2001;3:549–569.
  • Bartsch T, Wang Z-Q. Existence and multiplicity results for superlinear elliptic problems on RN. Commun Partial Differ Equ. 1995;20:1725–1741.
  • Bellazzini J, Jeanjean L. On dipolar quantum gases in the unstable regime. SIAM J Math Anal. 2017;48:2028–2058.
  • Carles R, Markowich P, Sparber C. On the Gross-Pitaevskii equation for trapped dipolar quantum gases. Nonlinearity. 2008;21:2569–2590.
  • Lushnikov P. Collapse of Bose-Einstein condensates with dipole-dipole interactions. Phys Rev A. 2002;66:051601(R).
  • Lushnikov P. Collapse and stable self-trapping for Bose-Einstein condensates with 1/rb type attractive interatomic interaction potential. Phys Rev A. 2010;82:023615.
  • Bartsch T, Tang Z. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete Contin Dyn Syst. 2013;33:7–26.
  • Jia H, Luo X. Existence and concentrating behavior of solutions for Kirchhoff type equations with steep potential well. J Math Anal Appl. 2018;467:893–915.
  • Stuart C, Zhou H. Global branch of solutions for nonlinear Schrödinger equations with deepening potential well. Proc London Math Soc. 2006;92:655–681.
  • Wang Z, Zhou H. Positive solutions for nonlinear Schrödinger equations with deepening potential well. J Eur Math Soc. 2009;11:545–573.
  • Kurihara S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Japan. 1981;50:3262–3267.
  • Bartsch T, Willem M. Infinitely many radial solutions of a semilinear elliptic problem on RN. Arch Ration Mech Anal. 1993;124:261–276.
  • Berestycki H, Lions PL. Nonlinear scalar field euqations I: existence of a ground state. Arch Ration Mech Anal. 1983;82:313–346.
  • Cerami G, Passaseo D, Solomini S. Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients. Comm Pure Appl Math. 2013;66:372–413.
  • Rabinowitz PH. On a class of nonlinear Schrödinger equations. Z Angew Math Phys. 1992;43:272–291.
  • Strauss WA. Existence of solitary waves in higher dimensions. Comm Math Phys. 1977;55:149–162.
  • Liu J, Wang Z-Q. Soliton solutions for quasilinear Schrödinger equations. II. Proc Am Math Soc. 2003;131:441–448.
  • Poppenberg M, Schnitt K, Wang Z-Q. On the existence of solitary solutions to quasilinear Schrödinger equations. Calc Var Partial Differ Equ. 2002;14:329–344.
  • Liu J, Wang Y, Wang Z-Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differ Equ. 2004;29:879–901.
  • Liu J, Wang Y, Wang Z-Q. Soliton solutions for quasilinear Schrödinger equations. Π. J Differ Equ. 2003;187:473–493.
  • Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J Func Anal. 1973;14:349–381.
  • Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 2004;56:213–226.
  • Bezerra do Ó JM, Miyagaki OH, Soares SHM. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differ Equ. 2010;248:722–744.
  • Fang X-D, Szulkin A. Multiple solutions for a quasilinear Schrödinger equation. J Differ Equ. 2013;254:2015–2032.
  • Lins HF, Silva EAB. Quasilinear asymptotically periodic elliptic equations with critical growth. Nonlinear Anal. 2009;71:2890–2905.
  • Moameni A. Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN. J Differ Equ. 2006;229:570–587.
  • Silva EAB, Vieira GF. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc Var Partial Differ Equ. 2010;39:1–33.
  • Wu X. Multiple solutions for quasilinear Schrödinger equations with a parameter. J Differ Equ. 2014;256:2619–2632.
  • Zhang J, Tang X, Zhang W. Infinitely many solutions of quasilinear Schrodinger equation with sign-changing potential. J Math Anal Appl. 2014;420:1762–1775.
  • Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal. 2013;80:194–201.
  • Alves CO, Wang Y, Shen Y. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differ Equ. 2015;259:318–343.
  • Aires JFL, Souto MAS. Equation with positive coefficient in the quasilinear term and vanishing potential. Topol Methods Nonlinear Anal. 2015;46:813–833.
  • Wang YJ. A class of quasilinear Schrödinger equations with critical or supercritical exponents. Comput Math Appl. 2015;70:562–572.
  • Severo UB, Gloss E, da Silva ED. On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms. J Differ Equ. 2017;263:3550–3580.
  • Ambrosetti A, Wang Z-Q. Positive solutions to a class of quasilinear elliptic equations on R. Discrete Contin Dyn Syst. 2003;9:55–68.
  • Brüll L, Lange H. Stationary, oscillatory and solitary waves type solutions of singular nonlinear Schrödinger equations. Math Mech Appl Sci. 1986;8:559–575.
  • Lange H, Poppenberg M, Teisniann H. Nash-Moser methods for the solution of quasilinear Schrödinger equations. Comm Partial Differ Equ. 1999;24:1399–1418.
  • Deng Y, Huang W. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete Contin Dyn Syst. 2017;37:4213–4230.
  • Ekeland I. Convexity methods in Hamiltonian mechanics. Berlin: Springer; 1990.
  • Willem M. Minimax Theorems. Boston: Birkhauser; 1996.

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