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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), 781-791. 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), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

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J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations:the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. 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), 621-644. doi: 10.3934/cpaa.2009.8.621.  Google Scholar

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A. Floer and A. Weisntein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

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N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. doi: 10.1090/S0002-9947-00-02560-5.  Google Scholar

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D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbbR^N$, Nonlinear Anal., 66 (2007), 241–252. 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), 117-238. doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

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[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), 2764-2769. doi: 10.1063/1.525675.  Google Scholar

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A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. 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), 879-901. 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), 441-448. 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), 473-493. 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), 1-86. 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), 773-781. doi: 10.1016/s0362-546x(96)00087-9.  Google Scholar

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A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbbR^N$, J. Differential Equations, 229 (2006), 570-587. 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), 937-957. 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 14 (2002), 329-344. 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), 41-80. 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), 172-189. 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), 272-291. 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), 6157-6169. 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), 849-856. 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), 781-791. 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), 213-226. 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), 3357-3372. 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), 621-644. 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), 397-408. 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), 5703-5743. 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), 83-87. 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), 241–252. 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), 117-238. 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), 3262-3267. 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), 2764-2769. 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), 517-520. 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), 879-901. 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), 441-448. 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), 473-493. 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), 1-86. 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), 773-781. 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), 570-587. 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), 937-957. 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 14 (2002), 329-344. 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), 41-80. 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), 172-189. 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), 272-291. 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), 6157-6169. 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), 849-856. doi: 10.1016/j.amc.2010.01.091.  Google Scholar

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