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]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[2]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[3]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[4]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[5]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[6]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[7]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[8]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[9]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[10]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[11]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264
[12]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[13]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[14]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[15]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[16]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250
[17]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[18]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274
[19]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[20]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences