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

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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 and 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.
[2]	A. V. Borovskii and A. L. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1993), 562-573.
[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.
[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.
[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.
[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.
[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.
[26]	M. Willem, Minimax Thorem, Birkhäuser, Berlin, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	M. Willem, Minimax Thorem, Birkhäuser, Berlin, 1996.
[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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