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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
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show all references

References:
[1]

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

[2]

Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[3]

J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

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

[5]

Phys. Rep., 189 (1990), 165-223. doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

Exposition. Math., 4 (1986), 279-288.  Google Scholar

[7]

Phys. Rev. Lett., 70 (1993), 2082-2085. doi: 10.1103/PhysRevLett.70.2082.  Google Scholar

[8]

Nonlinear Anal. TMA., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[9]

Physica D, 238 (2009), 38-54. doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[10]

Commun. Math. Phys., 189 (1997), 73-105. doi: 10.1007/s002200050191.  Google Scholar

[11]

J. Math. Phys., 54 (2013), 011504, 27pp. doi: 10.1063/1.4774153.  Google Scholar

[12]

Commun. Math. Sci., 9 (2011), 859-878. doi: 10.4310/CMS.2011.v9.n3.a9.  Google Scholar

[13]

J. Differential Equations, 258 (2015), 115-147. doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[14]

J. Differential Equations, 260 (2016), 1228-1262. doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[15]

Courant Lecture Notes in Mathematics, vol. 1. New York University Courant Institute of Mathematical Sciences, New York, 1997.  Google Scholar

[16]

Z. Phys. B, 37 (1980), 83-87. Google Scholar

[17]

Phys. Rep., 194 (1990), 117-238. doi: doi:10.1016/0370-1573(90)90130-T.  Google Scholar

[18]

J. Phys. Soc. Japan, 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262.  Google Scholar

[19]

J. Math. Phys., 24 (1983), 2764-2769. doi: 10.1063/1.525675.  Google Scholar

[20]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  Google Scholar

[21]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[22]

Comm. Partial Differential Equations, 24 (1999), 1399-1418. doi: 10.1080/03605309908821469.  Google Scholar

[23]

J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.  Google Scholar

[25]

Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7 .  Google Scholar

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Proc. Amer. Math. Soc., 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6 .  Google Scholar

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J. Differential Equations, 254 (2013), 102-124. doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[28]

Calc. Var. Partial Differential Equations, 46 (2013), 641-669. doi: 10.1007/s00526-012-0497-0.  Google Scholar

[29]

J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[30]

Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[31]

Indiana Univ. Math. J., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[32]

Calc. Var. Partial Differential Equations, 14 (2002), 329-344. doi: 10.1007/s005260100105.  Google Scholar

[33]

Phys. A, 110 (1982), 41-80. doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[34]

Phys. Rev. E, 50 (1994), 687-689. doi: 10.1103/PhysRevE.50.R687.  Google Scholar

[35]

Nonlinear Anal. TMA., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.  Google Scholar

[36]

Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1.  Google Scholar

[37]

J. Math. Phys., 54 (2013), 071502, 19pp. doi: 10.1063/1.4811394.  Google Scholar

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