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 and 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, J. Func. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[3]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030.

[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, Phys. Fluids B, 5 (1993), 3539-3550. doi: 10.1063/1.860828.

[5]

F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223. doi: 10.1016/0370-1573(90)90093-H.

[6]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.

[7]

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085. doi: 10.1103/PhysRevLett.70.2082.

[8]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.

[9]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation, Physica D, 238 (2009), 38-54. doi: 10.1016/j.physd.2008.08.010.

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105. doi: 10.1007/s002200050191.

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504, 27pp. doi: 10.1063/1.4774153.

[12]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$, Commun. Math. Sci., 9 (2011), 859-878. doi: 10.4310/CMS.2011.v9.n3.a9.

[13]

Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147. doi: 10.1016/j.jde.2014.09.006.

[14]

Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262. doi: 10.1016/j.jde.2015.09.021.

[15]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1. New York University Courant Institute of Mathematical Sciences, New York, 1997.

[16]

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.

[17]

A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238. doi: doi:10.1016/0370-1573(90)90130-T.

[18]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262.

[19]

E. Laedke, K. 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.

[20]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I}, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

[21]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

[22]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24 (1999), 1399-1418. doi: 10.1080/03605309908821469.

[23]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.

[24]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.

[25]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7 .

[26]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6 .

[27]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124. doi: 10.1016/j.jde.2012.09.006.

[28]

X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669. doi: 10.1007/s00526-012-0497-0.

[29]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.

[30]

V.G. Makhankov and V.K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.

[31]

P. Pucci and J. Serrin, A general variational idnetity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.

[32]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. doi: 10.1007/s005260100105.

[33]

G.R.W. Quispel and H.W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. doi: 10.1103/PhysRevE.50.R687.

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.

[36]

E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1.

[37]

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 071502, 19pp. doi: 10.1063/1.4811394.

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[3]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030.

[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, Phys. Fluids B, 5 (1993), 3539-3550. doi: 10.1063/1.860828.

[5]

F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223. doi: 10.1016/0370-1573(90)90093-H.

[6]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.

[7]

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085. doi: 10.1103/PhysRevLett.70.2082.

[8]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.

[9]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation, Physica D, 238 (2009), 38-54. doi: 10.1016/j.physd.2008.08.010.

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105. doi: 10.1007/s002200050191.

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504, 27pp. doi: 10.1063/1.4774153.

[12]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$, Commun. Math. Sci., 9 (2011), 859-878. doi: 10.4310/CMS.2011.v9.n3.a9.

[13]

Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147. doi: 10.1016/j.jde.2014.09.006.

[14]

Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262. doi: 10.1016/j.jde.2015.09.021.

[15]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1. New York University Courant Institute of Mathematical Sciences, New York, 1997.

[16]

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.

[17]

A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238. doi: doi:10.1016/0370-1573(90)90130-T.

[18]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262.

[19]

E. Laedke, K. 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.

[20]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I}, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

[21]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

[22]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24 (1999), 1399-1418. doi: 10.1080/03605309908821469.

[23]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.

[24]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.

[25]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7 .

[26]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6 .

[27]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124. doi: 10.1016/j.jde.2012.09.006.

[28]

X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669. doi: 10.1007/s00526-012-0497-0.

[29]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.

[30]

V.G. Makhankov and V.K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.

[31]

P. Pucci and J. Serrin, A general variational idnetity, Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.

[32]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. doi: 10.1007/s005260100105.

[33]

G.R.W. Quispel and H.W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. doi: 10.1103/PhysRevE.50.R687.

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.

[36]

E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1.

[37]

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 071502, 19pp. doi: 10.1063/1.4811394.

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