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A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition
Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential
1. | School of Mathematics and Computational Science, Hunan First Normal University, Changsha, 410205 Hunan, China |
2. | School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China |
$ -{\rm div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u=f(x,u) \mbox{ in }\mathbb{R}^{N},$ |
$ g(u):\mathbb{R}\to\mathbb{R}^{+} $ |
$ |u| $ |
$ V(x) $ |
$ f(x,u) $ |
References:
[1] |
T. Bartsch and Z. Wang,
Existence and multiple results for some superlinear elliptic problems on $ \mathbb{R}^{N} $, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[2] |
A. V. Borovskii and A. L. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1993), 562-573. Google Scholar |
[3] |
H. S. 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. Google Scholar |
[4] |
X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082-2085. Google Scholar |
[5] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. TMA, 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[6] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations,
J. Math. Phys. , 55 (2014), 051501.
doi: 10.1063/1.4874108. |
[7] |
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. |
[8] |
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. |
[9] |
R. W. Hasse,
A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[10] |
S. Kurihura,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3801. |
[11] |
E. W. Laedke, K. H. 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. |
[12] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅰ, Proc. Amer. Math. Soc., 131 (2002), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[13] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[14] |
A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. Google Scholar |
[15] |
V. G. Makhankov and V. K. Fedyanin,
Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[16] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[17] |
M. Porkolab and M. V. Goldman,
Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.
doi: 10.1063/1.861553. |
[18] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Ser. in Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[19] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. Google Scholar |
[20] |
Y. Shen and Y. Wang,
Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[21] |
H. Shi and H. Chen,
Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858.
doi: 10.1016/j.camwa.2016.01.007. |
[22] |
H. Shi and H. Chen,
Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity, Applied Mathematics Letters, 61 (2016), 137-142.
doi: 10.1016/j.aml.2016.06.004. |
[23] |
H. Shi and H. Chen,
Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations, J. Math. Anal. Appl., 452 (2017), 578-594.
doi: 10.1016/j.jmaa.2017.03.020. |
[24] |
E. 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. |
[25] |
E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949. Google Scholar |
[26] |
M. Willem, Minimax Thorem, Birkhäuser, Berlin, 1996. Google Scholar |
[27] |
X. Wu,
Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.
doi: 10.1016/j.jde.2014.01.026. |
[28] |
M. B. Yang,
Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Anal., 75 (2012), 5362-5373.
doi: 10.1016/j.na.2012.04.054. |
[29] |
J. Zhang, X. H. Tang and W. Zhang,
Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl., 420 (2014), 1762-1775.
doi: 10.1016/j.jmaa.2014.06.055. |
show all references
References:
[1] |
T. Bartsch and Z. Wang,
Existence and multiple results for some superlinear elliptic problems on $ \mathbb{R}^{N} $, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[2] |
A. V. Borovskii and A. L. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1993), 562-573. Google Scholar |
[3] |
H. S. 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. Google Scholar |
[4] |
X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082-2085. Google Scholar |
[5] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. TMA, 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[6] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations,
J. Math. Phys. , 55 (2014), 051501.
doi: 10.1063/1.4874108. |
[7] |
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. |
[8] |
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. |
[9] |
R. W. Hasse,
A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[10] |
S. Kurihura,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3801. |
[11] |
E. W. Laedke, K. H. 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. |
[12] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅰ, Proc. Amer. Math. Soc., 131 (2002), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[13] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[14] |
A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. Google Scholar |
[15] |
V. G. Makhankov and V. K. Fedyanin,
Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[16] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[17] |
M. Porkolab and M. V. Goldman,
Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.
doi: 10.1063/1.861553. |
[18] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Ser. in Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[19] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. Google Scholar |
[20] |
Y. Shen and Y. Wang,
Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[21] |
H. Shi and H. Chen,
Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858.
doi: 10.1016/j.camwa.2016.01.007. |
[22] |
H. Shi and H. Chen,
Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity, Applied Mathematics Letters, 61 (2016), 137-142.
doi: 10.1016/j.aml.2016.06.004. |
[23] |
H. Shi and H. Chen,
Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations, J. Math. Anal. Appl., 452 (2017), 578-594.
doi: 10.1016/j.jmaa.2017.03.020. |
[24] |
E. 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. |
[25] |
E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949. Google Scholar |
[26] |
M. Willem, Minimax Thorem, Birkhäuser, Berlin, 1996. Google Scholar |
[27] |
X. Wu,
Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.
doi: 10.1016/j.jde.2014.01.026. |
[28] |
M. B. Yang,
Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Anal., 75 (2012), 5362-5373.
doi: 10.1016/j.na.2012.04.054. |
[29] |
J. Zhang, X. H. Tang and W. Zhang,
Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl., 420 (2014), 1762-1775.
doi: 10.1016/j.jmaa.2014.06.055. |
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