American Institute of Mathematical Sciences

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January  2018, 17(1): 53-66. doi: 10.3934/cpaa.2018004

Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential

Hongxia Shi 1, and  Haibo Chen 2,,

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

* Corresponding author

Received  November 2016 Revised  July 2017 Published  September 2017

Fund Project: This research was supported by National Natural Science Foundation of China 11671403 and by the Mathematics and Interdisciplinary Sciences project of CSU.

We investigate a class of generalized quasilinear Schrödinger equations
$ -{\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},$
where
$ g(u):\mathbb{R}\to\mathbb{R}^{+} $
 is a nondecreasing function with respect to
$ |u| $
, the potential
$ V(x) $
 and the primitive of the nonlinearity
$ f(x,u) $
 are allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions. The proof is based on a change of variable as well as symmetric Mountain Pass Theorem.
Keywords: Quasilinear Schrödinger equations, sign-changing potential, change of variable, Mountain Pass Theorem.
Mathematics Subject Classification: 35J20, 35J60.
Citation: Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure & Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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