American Institute of Mathematical Sciences

Advanced
July  2011, 10(4): 1037-1054. doi: 10.3934/cpaa.2011.10.1037

Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents

Yimin Zhang 1, , Youjun Wang 2, and  Yaotian Shen 3,

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103, China
2. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084e/pjp, China
3. 

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received  April 2010 Revised  September 2010 Published  April 2011

By using a change of variable, the quasilinear Schrödinger equation is reduced to semilinear elliptic equation. Then, Mountain Pass theorem without $(PS)_c$ condition in a suitable Orlicz space is employed to prove the existence of positive standing wave solutions for a class of quasilinear Schrödinger equations involving critical Sobolev-Hardy exponents.
Keywords: quasilinear Schrödinger equations, Orlicz space., Critical Sobolev-Hardy exponents.
Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35Q5.
Citation: Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
References:
[1]

Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003.  Google Scholar

[2]

Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[3]

Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.  Google Scholar

[4]

Comm. Pure and Applied Anal., 8 (2009), 621-644. doi: 10.3934/cpaa.2009.8.621.  Google Scholar

[5]

J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[6]

Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[7]

Z. Phys. B, 37 (1980), 83-87. doi: 10.1007/BF01325508.  Google Scholar

[8]

Nonlinear Anal., 66 (2007), 241–252. doi: 10.1016/j.na.2005.11.028.  Google Scholar

[9]

Phys. Rep., 194 (1990), 117-238. doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[10]

J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/jpsj.50.3262.  Google Scholar

[11]

J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675.  Google Scholar

[12]

JETP Lett., 27 (1978), 517-520. doi: 0021-3640778/2710-0517.  Google Scholar

[13]

Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar

[14]

Proc. Amer. Math. Soc., 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[15]

J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/s0022-0396(02)0064-5.  Google Scholar

[16]

Phys. Rep, 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[17]

Springer-Verlag, 1989.  Google Scholar

[18]

Nonlinear Anal., 29 (1997), 773-781. doi: 10.1016/s0362-546x(96)00087-9.  Google Scholar

[19]

J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[20]

Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009.  Google Scholar

[21]

Calc. Var. Partial Differential Equations 14 (2002), 329-344. doi: 10.1007/s005260100105.  Google Scholar

[22]

Physica A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[23]

Progr. Theoret. Phys., 65 (1981), 172-189. doi: 10.1143/PTP.65.172.  Google Scholar

[24]

Z angew Math. Phy., 43 (1992), 272-291. doi: 10.1007/BF00946631.  Google Scholar

[25]

Nonlinear Anal., 71 (2009), 6157-6169. doi: 10.1016/j.na.2009.06.006.  Google Scholar

[26]

Applied Mathematics and Computation, 216 (2010), 849-856. doi: 10.1016/j.amc.2010.01.091.  Google Scholar

show all references

References:
[1]

Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003.  Google Scholar

[2]

Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[3]

Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.  Google Scholar

[4]

Comm. Pure and Applied Anal., 8 (2009), 621-644. doi: 10.3934/cpaa.2009.8.621.  Google Scholar

[5]

J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[6]

Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

[7]

Z. Phys. B, 37 (1980), 83-87. doi: 10.1007/BF01325508.  Google Scholar

[8]

Nonlinear Anal., 66 (2007), 241–252. doi: 10.1016/j.na.2005.11.028.  Google Scholar

[9]

Phys. Rep., 194 (1990), 117-238. doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[10]

J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/jpsj.50.3262.  Google Scholar

[11]

J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675.  Google Scholar

[12]

JETP Lett., 27 (1978), 517-520. doi: 0021-3640778/2710-0517.  Google Scholar

[13]

Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar

[14]

Proc. Amer. Math. Soc., 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[15]

J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/s0022-0396(02)0064-5.  Google Scholar

[16]

Phys. Rep, 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[17]

Springer-Verlag, 1989.  Google Scholar

[18]

Nonlinear Anal., 29 (1997), 773-781. doi: 10.1016/s0362-546x(96)00087-9.  Google Scholar

[19]

J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[20]

Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009.  Google Scholar

[21]

Calc. Var. Partial Differential Equations 14 (2002), 329-344. doi: 10.1007/s005260100105.  Google Scholar

[22]

Physica A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[23]

Progr. Theoret. Phys., 65 (1981), 172-189. doi: 10.1143/PTP.65.172.  Google Scholar

[24]

Z angew Math. Phy., 43 (1992), 272-291. doi: 10.1007/BF00946631.  Google Scholar

[25]

Nonlinear Anal., 71 (2009), 6157-6169. doi: 10.1016/j.na.2009.06.006.  Google Scholar

[26]

Applied Mathematics and Computation, 216 (2010), 849-856. doi: 10.1016/j.amc.2010.01.091.  Google Scholar

[1]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021022
[2]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[3]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216
[4]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[5]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115
[6]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038
[7]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024
[8]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016
[9]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067
[10]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[11]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002
[12]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013
[13]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392
[14]

Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100
[15]

Liviana Palmisano, Bertuel Tangue Ndawa. A phase transition for circle maps with a flat spot and different critical exponents. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021067
[16]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026
[17]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258
[18]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[19]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021043
[20]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences