Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents

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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

Abstract
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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 and 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 $\mathbb{R}^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbb{R}^N2$, Nonlinear Anal., 66 (2007), 241–252. doi: 10.1016/j.na.2005.11.028.
[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.
[10]	S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/jpsj.50.3262.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, 1989.
[18]	O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N2$ with critical growth, Nonlinear Anal., 29 (1997), 773-781. doi: 10.1016/s0362-546x(96)00087-9.
[19]	A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N2$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.
[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.
[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.
[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.
[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.
[24]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z angew Math. Phy., 43 (1992), 272-291. doi: 10.1007/BF00946631.
[25]	Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbb{R}^N2$, Nonlinear Anal., 71 (2009), 6157-6169. doi: 10.1016/j.na.2009.06.006.
[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.

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References:

[1]	C. O. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbb{R}^N2$, Nonlinear Anal., 66 (2007), 241–252. doi: 10.1016/j.na.2005.11.028.
[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.
[10]	S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/jpsj.50.3262.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, 1989.
[18]	O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N2$ with critical growth, Nonlinear Anal., 29 (1997), 773-781. doi: 10.1016/s0362-546x(96)00087-9.
[19]	A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N2$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.
[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.
[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.
[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.
[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.
[24]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z angew Math. Phy., 43 (1992), 272-291. doi: 10.1007/BF00946631.
[25]	Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbb{R}^N2$, Nonlinear Anal., 71 (2009), 6157-6169. doi: 10.1016/j.na.2009.06.006.
[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.

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