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]

C. O. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbbR^2$ involving critical growth,, Nonlinear Anal., 56 (2004), 781. doi: 10.1016/j.na.2003.06.003. Google Scholar

[2]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar

[3]

J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations:the critical exponential case,, Nonlinear Anal., 67 (2007), 3357. doi: 10.1016/j.na.2006.10.018. Google Scholar

[4]

J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities,, Comm. Pure and Applied Anal., 8 (2009), 621. doi: 10.3934/cpaa.2009.8.621. Google Scholar

[5]

A. Floer and A. Weisntein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar

[6]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents,, Trans. Amer. Math. Soc., 352 (2000), 5703. doi: 10.1090/S0002-9947-00-02560-5. Google Scholar

[7]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, Z. Phys. B, 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar

[8]

D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbbR^N$,, Nonlinear Anal., 66 (2007). doi: 10.1016/j.na.2005.11.028. Google Scholar

[9]

A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons,, Phys. Rep., 194 (1990), 117. doi: 10.1016/0370-1573(90)90130-T. Google Scholar

[10]

S. Kurihura, Large-amplitude quasi-solitons in superfluid films,, J. Phys. Soc. Japan, 50 (1981), 3262. doi: 10.1143/jpsj.50.3262. Google Scholar

[11]

E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions,, J. Math. Phys., 24 (1983), 2764. doi: 10.1063/1.525675. Google Scholar

[12]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves,, JETP Lett., 27 (1978), 517. doi: 0021-3640778/2710-0517. Google Scholar

[13]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar

[14]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, Proc. Amer. Math. Soc., 131 (2002), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar

[15]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/s0022-0396(02)0064-5. Google Scholar

[16]

V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory,, Phys. Rep, 104 (1984), 1. doi: 10.1016/0370-1573(84)90106-6. Google Scholar

[17]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). Google Scholar

[18]

O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: 10.1016/s0362-546x(96)00087-9. Google Scholar

[19]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbbR^N$,, J. Differential Equations, 229 (2006), 570. doi: 10.1016/j.jde.2006.07.001. Google Scholar

[20]

A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations,, Nonlinearity, 19 (2006), 937. doi: 10.1088/0951-7715/19/4/009. Google Scholar

[21]

M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. Partial Differential Equations \textbf{14} (2002), 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar

[22]

G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain,, Physica A, 110 (1982), 41. doi: 10.1016/0378-4371(82)90104-2. Google Scholar

[23]

S. Takeno and S. Homma, Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations,, Progr. Theoret. Phys., 65 (1981), 172. doi: 10.1143/PTP.65.172. Google Scholar

[24]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z angew Math. Phy., 43 (1992), 272. doi: 10.1007/BF00946631. Google Scholar

[25]

Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbbR^N$,, Nonlinear Anal., 71 (2009), 6157. doi: 10.1016/j.na.2009.06.006. Google Scholar

[26]

Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schröinger equations involving critical exponent,, Applied Mathematics and Computation, 216 (2010), 849. doi: 10.1016/j.amc.2010.01.091. Google Scholar

show all references

References:
[1]

C. O. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbbR^2$ involving critical growth,, Nonlinear Anal., 56 (2004), 781. doi: 10.1016/j.na.2003.06.003. Google Scholar

[2]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar

[3]

J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations:the critical exponential case,, Nonlinear Anal., 67 (2007), 3357. doi: 10.1016/j.na.2006.10.018. Google Scholar

[4]

J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities,, Comm. Pure and Applied Anal., 8 (2009), 621. doi: 10.3934/cpaa.2009.8.621. Google Scholar

[5]

A. Floer and A. Weisntein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar

[6]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents,, Trans. Amer. Math. Soc., 352 (2000), 5703. doi: 10.1090/S0002-9947-00-02560-5. Google Scholar

[7]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, Z. Phys. B, 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar

[8]

D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbbR^N$,, Nonlinear Anal., 66 (2007). doi: 10.1016/j.na.2005.11.028. Google Scholar

[9]

A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons,, Phys. Rep., 194 (1990), 117. doi: 10.1016/0370-1573(90)90130-T. Google Scholar

[10]

S. Kurihura, Large-amplitude quasi-solitons in superfluid films,, J. Phys. Soc. Japan, 50 (1981), 3262. doi: 10.1143/jpsj.50.3262. Google Scholar

[11]

E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions,, J. Math. Phys., 24 (1983), 2764. doi: 10.1063/1.525675. Google Scholar

[12]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves,, JETP Lett., 27 (1978), 517. doi: 0021-3640778/2710-0517. Google Scholar

[13]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar

[14]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, Proc. Amer. Math. Soc., 131 (2002), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar

[15]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/s0022-0396(02)0064-5. Google Scholar

[16]

V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory,, Phys. Rep, 104 (1984), 1. doi: 10.1016/0370-1573(84)90106-6. Google Scholar

[17]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). Google Scholar

[18]

O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth,, Nonlinear Anal., 29 (1997), 773. doi: 10.1016/s0362-546x(96)00087-9. Google Scholar

[19]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbbR^N$,, J. Differential Equations, 229 (2006), 570. doi: 10.1016/j.jde.2006.07.001. Google Scholar

[20]

A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations,, Nonlinearity, 19 (2006), 937. doi: 10.1088/0951-7715/19/4/009. Google Scholar

[21]

M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. Partial Differential Equations \textbf{14} (2002), 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar

[22]

G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain,, Physica A, 110 (1982), 41. doi: 10.1016/0378-4371(82)90104-2. Google Scholar

[23]

S. Takeno and S. Homma, Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations,, Progr. Theoret. Phys., 65 (1981), 172. doi: 10.1143/PTP.65.172. Google Scholar

[24]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z angew Math. Phy., 43 (1992), 272. doi: 10.1007/BF00946631. Google Scholar

[25]

Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbbR^N$,, Nonlinear Anal., 71 (2009), 6157. doi: 10.1016/j.na.2009.06.006. Google Scholar

[26]

Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schröinger equations involving critical exponent,, Applied Mathematics and Computation, 216 (2010), 849. doi: 10.1016/j.amc.2010.01.091. Google Scholar

[1]

Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731
[2]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[3]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[4]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[5]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[6]

Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259
[7]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[8]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
[9]

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 & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435
[10]

Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[11]

Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103
[12]

Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025
[13]

Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016
[14]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[15]

Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[16]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273
[17]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025
[18]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054
[19]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[20]

Yaotian Shen, Youjun Wang. A class of generalized quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2016, 15 (3) : 853-870. doi: 10.3934/cpaa.2016.15.853

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences