# American Institute of Mathematical Sciences

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

## Soliton solutions for a quasilinear Schrödinger equation with critical exponent

 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.
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:
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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

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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