American Institute of Mathematical Sciences

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July  2016, 15(4): 1309-1333. doi: 10.3934/cpaa.2016.15.1309

Soliton solutions for a quasilinear Schrödinger equation with critical exponent

Wentao Huang 1, and  Jianlin Xiang 2,

1. 

Department of Mathematics, Central China Normal University, Wuhan, 430079, China
2. 

Department of Mathematics, Wuhan University of Technology, Wuhan, 430070, China

Received  October 2015 Revised  January 2016 Published  April 2016

This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
Keywords: Ground state, quasilinear Schrödinger equation, positive solution, perturbation method., critical exponent.
Mathematics Subject Classification: Primary: 35J62; Secondary: 35B09, 35B3.
Citation: 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
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Func. Anal.}, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[3]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 248 (2010), 722.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma,, \emph{Phys. Fluids B}, 5 (1993), 3539.  doi: 10.1063/1.860828.  Google Scholar

[5]

F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves,, \emph{Phys. Rep.}, 189 (1990), 165.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math.}, 4 (1986), 279.   Google Scholar

[7]

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,, \emph{Phys. Rev. Lett.}, 70 (1993), 2082.  doi: 10.1103/PhysRevLett.70.2082.  Google Scholar

[8]

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

[9]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation,, \emph{Physica D}, 238 (2009), 38.  doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation,, \emph{Commun. Math. Phys.}, 189 (1997), 73.  doi: 10.1007/s002200050191.  Google Scholar

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4774153.  Google Scholar

[12]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$,, \emph{Commun. Math. Sci.}, 9 (2011), 859.  doi: 10.4310/CMS.2011.v9.n3.a9.  Google Scholar

[13]

Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 258 (2015), 115.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[14]

Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations,, \emph{J. Differential Equations}, 260 (2016), 1228.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

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Q. Han and F. Lin, Elliptic Partial Differential Equations,, Courant Lecture Notes in Mathematics, (1997).   Google Scholar

[16]

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, \emph{Z. Phys. B}, 37 (1980), 83.   Google Scholar

[17]

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

[18]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan}, 50 (1981), 3262.  doi: 10.1143/JPSJ.50.3262.  Google Scholar

[19]

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

[20]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I},, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 109.   Google Scholar

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P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II,, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 223.   Google Scholar

[22]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 24 (1999), 1399.  doi: 10.1080/03605309908821469.  Google Scholar

[23]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations}, 187 (2003), 473.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

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

[25]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 441.  doi: 10.1090/S0002-9939-02-06783-7 .  Google Scholar

[26]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method,, \emph{Proc. Amer. Math. Soc.}, 141 (2013), 253.  doi: 10.1090/S0002-9939-2012-11293-6 .  Google Scholar

[27]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method,, \emph{J. Differential Equations}, 254 (2013), 102.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[28]

X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 46 (2013), 641.  doi: 10.1007/s00526-012-0497-0.  Google Scholar

[29]

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

[30]

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

[31]

P. Pucci and J. Serrin, A general variational idnetity,, \emph{Indiana Univ. Math. J.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[32]

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

[33]

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

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E}, 50 (1994), 687.  doi: 10.1103/PhysRevE.50.R687.  Google Scholar

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations,, \emph{Nonlinear Anal. TMA.}, 80 (2013), 194.  doi: 10.1016/j.na.2012.10.005.  Google Scholar

[36]

E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 39 (2010), 1.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[37]

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4811394.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Func. Anal.}, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[3]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 248 (2010), 722.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma,, \emph{Phys. Fluids B}, 5 (1993), 3539.  doi: 10.1063/1.860828.  Google Scholar

[5]

F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves,, \emph{Phys. Rep.}, 189 (1990), 165.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math.}, 4 (1986), 279.   Google Scholar

[7]

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,, \emph{Phys. Rev. Lett.}, 70 (1993), 2082.  doi: 10.1103/PhysRevLett.70.2082.  Google Scholar

[8]

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

[9]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation,, \emph{Physica D}, 238 (2009), 38.  doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation,, \emph{Commun. Math. Phys.}, 189 (1997), 73.  doi: 10.1007/s002200050191.  Google Scholar

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4774153.  Google Scholar

[12]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$,, \emph{Commun. Math. Sci.}, 9 (2011), 859.  doi: 10.4310/CMS.2011.v9.n3.a9.  Google Scholar

[13]

Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 258 (2015), 115.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[14]

Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations,, \emph{J. Differential Equations}, 260 (2016), 1228.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[15]

Q. Han and F. Lin, Elliptic Partial Differential Equations,, Courant Lecture Notes in Mathematics, (1997).   Google Scholar

[16]

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, \emph{Z. Phys. B}, 37 (1980), 83.   Google Scholar

[17]

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

[18]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan}, 50 (1981), 3262.  doi: 10.1143/JPSJ.50.3262.  Google Scholar

[19]

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

[20]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I},, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 109.   Google Scholar

[21]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II,, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 223.   Google Scholar

[22]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 24 (1999), 1399.  doi: 10.1080/03605309908821469.  Google Scholar

[23]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations}, 187 (2003), 473.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

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

[25]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 441.  doi: 10.1090/S0002-9939-02-06783-7 .  Google Scholar

[26]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method,, \emph{Proc. Amer. Math. Soc.}, 141 (2013), 253.  doi: 10.1090/S0002-9939-2012-11293-6 .  Google Scholar

[27]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method,, \emph{J. Differential Equations}, 254 (2013), 102.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[28]

X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 46 (2013), 641.  doi: 10.1007/s00526-012-0497-0.  Google Scholar

[29]

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

[30]

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

[31]

P. Pucci and J. Serrin, A general variational idnetity,, \emph{Indiana Univ. Math. J.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[32]

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

[33]

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

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E}, 50 (1994), 687.  doi: 10.1103/PhysRevE.50.R687.  Google Scholar

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations,, \emph{Nonlinear Anal. TMA.}, 80 (2013), 194.  doi: 10.1016/j.na.2012.10.005.  Google Scholar

[36]

E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 39 (2010), 1.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[37]

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4811394.  Google Scholar

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