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The Nehari manifold for fractional systems involving critical nonlinearities
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math.}, 4 (1986), 279.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
Q. Han and F. Lin, Elliptic Partial Differential Equations,, Courant Lecture Notes in Mathematics, (1997).
|
[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. |
[18] |
S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan}, 50 (1981), 3262.
doi: 10.1143/JPSJ.50.3262. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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 . |
[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 . |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math.}, 4 (1986), 279.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
Q. Han and F. Lin, Elliptic Partial Differential Equations,, Courant Lecture Notes in Mathematics, (1997).
|
[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. |
[18] |
S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan}, 50 (1981), 3262.
doi: 10.1143/JPSJ.50.3262. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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 . |
[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 . |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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