March  2010, 9(2): 281-306. doi: 10.3934/cpaa.2010.9.281

Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations

1. 

Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy

2. 

Departamento de Matemática, Universidade Fededral da Paraíba, 58059-900, João Pessoa-PB, Brazil

3. 

Department of Mathematics and Statistics, Queen’s University Jeffery Hall, University Ave. Kingston, ON Canada, K7L 3N6, Canada

Received  January 2009 Revised  August 2009 Published  December 2009

In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension $N\ge 3$: we investigate the case of unbounded potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.
Citation: Daniele Cassani, João Marcos do Ó, Abbas Moameni. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Communications on Pure and Applied Analysis, 2010, 9 (2) : 281-306. doi: 10.3934/cpaa.2010.9.281
[1]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[2]

Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099

[3]

Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043

[4]

Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control and Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016

[5]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[6]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83

[7]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039

[8]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[9]

Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007

[10]

David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327

[11]

Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789

[12]

Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417

[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[14]

Yuanhong Wei, Yong Li, Xue Yang. On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1095-1106. doi: 10.3934/dcdss.2017059

[15]

Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435

[16]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[17]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[18]

Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209

[19]

Nghiem V. Nguyen, Zhi-Qiang Wang. Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1005-1021. doi: 10.3934/dcds.2016.36.1005

[20]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (16)

[Back to Top]