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September  2021, 14(9): 3285-3303. doi: 10.3934/dcdss.2020281

Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $

Ziqing Yuan 1,2, and  Jianshe Yu 1,,

1. 

Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China
2. 

Department of Mathematics, Shaoyang University, Shaoyang, Hunan, 422000, China

* Corresponding author: Jianshe Yu

Received  July 2019 Revised  October 2019 Published  September 2021 Early access  March 2020

Fund Project: Research are supported by the National Natural Science Foundation of China (Grant No. 11901126), the Hunan Province Natural Science Foundation of China (Grant No. 2017JJ3222), and the China Postdoctoral Science Foundation funded project (Grant No. 2018M643039)

This work concerns with the existence and multiplicity of positive solutions for the following quasilinear Schrödinger equation
$ -\Delta u+V(x)u-\Delta(u^2)u = a(\epsilon x)g(u), \; \; \; \; x\in\mathbb R^N, $
where
$ V(x)>0 $
,
$ u>0 $
,
$ a $
 and
$ g $
 are continuous functions and
$ a $
 has
$ m $
 maximum points. With the change of variables we show that this equation has at least
$ m $
 nontrivial solutions by using variational methods, the Ekeland's variational principle, and some properties of the Nehari manifold. Some recent results are improved.
Keywords: Quasilinear Schrödinger equation, Variational method, Dual approach, Positive solution.
Mathematics Subject Classification: 35J85; 47J30; 49J52.
Citation: Ziqing Yuan, Jianshe Yu. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281
References:
[1]

C. O. Alves, G. M. Figueiredo and J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456.  doi: 10.12775/TMNA.2014.055.  Google Scholar

[2]

C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb R^N$, J. Differential Equations, 246 (2009), 1288-1311.  doi: 10.1016/j.jde.2008.08.004.  Google Scholar

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A. Borovskiik and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1983), 562-573.   Google Scholar

[5]

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, Phys. Fluids B, 5 (1993), 3539-3550.   Google Scholar

[6]

J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[7]

K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

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D. Cassani, J. M. Bezerra do Ó and A. Moameni, Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 9 (2010), 281-306.  doi: 10.3934/cpaa.2010.9.281.  Google Scholar

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D. M. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb R^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.  Google Scholar

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D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463.  doi: 10.1017/S0308210500022836.  Google Scholar

[11]

P. C. Carrião, R. Lehrer and O. H. Miyagaki, Existence of solutions to a class of asymptotically linear Schrödinger equations in $\mathbb R^N$ via the Pohozaev manifold, J. Math. Anal. Appl., 428 (2015), 165-183.  doi: 10.1016/j.jmaa.2015.02.060.  Google Scholar

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M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[13]

F. Esposito, A. Farina and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983.  doi: 10.1016/j.jde.2018.04.030.  Google Scholar

[14]

X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.  doi: 10.1016/j.jde.2012.11.017.  Google Scholar

[15]

E. Gloss, Existence and concentration of positive solution for a quasilinear equation in $\mathbb R^N$, J. Math. Anal. Appl., 371 (2010), 465-484.  doi: 10.1016/j.jmaa.2010.05.033.  Google Scholar

[16]

X. He, A. Qian and W. Zou, Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.  Google Scholar

[17]

T. S. Hsu, H. L. Lin and C. C. Hu, Multiple positive solutions of quasilinear elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 388 (2012), 500-512.  doi: 10.1016/j.jmaa.2011.11.010.  Google Scholar

[18]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3801-3805.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[19]

J. Q. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equation, II, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations, I. Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[21]

J. Liu, S. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.  doi: 10.1016/j.jmaa.2012.05.063.  Google Scholar

[22]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[23]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[24]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[25]

V. G. Makhankov and V. K. Fedyanin, Nonlinear 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

[26]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$, Commun. Pure Appl. Anal., 7 (2008), 89-105.  doi: 10.3934/cpaa.2008.7.89.  Google Scholar

[27]

K. Mahiout and C. O. Alves, Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces, Complex Var. Elliptic Equ., 62 (2016), 767-785.  doi: 10.1080/17476933.2016.1243669.  Google Scholar

[28]

O. H. Miyagaki and S. I. Moreira, Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign, J. Math. Anal. Appl., 421 (2015), 643-655.  doi: 10.1016/j.jmaa.2014.06.074.  Google Scholar

[29]

Y. Miyamoto and Y. Naito, Singular extremal solutions for supercritical elliptic equations in a ball, J. Differential Equations, 265 (2018), 2842-2885.  doi: 10.1016/j.jde.2018.04.055.  Google Scholar

[30]

Q. H. Miyagaki and S. I. Moreira, Nonnegative solution for quasilinear Schrödinger equations involving supercritical exponent with nonlinearities indefinitie in sign, J. Math. Anal. Appl., 421 (2015), 643-655.  doi: 10.1016/j.jmaa.2014.06.074.  Google Scholar

[31]

M. Poppenbery, 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

[32]

M. D. Pina and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[33]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[34]

E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.  Google Scholar

[35]

W. Wang, X. Yang and F. K. Zhao, Existence and concentration of ground states to a quasilinear problem with competing potentials, Nonlinear Anal., 102 (2014), 120-132.  doi: 10.1016/j.na.2014.01.025.  Google Scholar

[36]

Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comput., 216 (2010), 849-856.  doi: 10.1016/j.amc.2010.01.091.  Google Scholar

[37]

M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[38]

X. Wu and K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasilinear elliptic equations on $\mathbb R^N$, Nonlinear Anal. Real World Appl., 16 (2014), 48-64.   Google Scholar

[39]

J. Zhang, X. 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]

C. O. Alves, G. M. Figueiredo and J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435-456.  doi: 10.12775/TMNA.2014.055.  Google Scholar

[2]

C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $\mathbb R^N$, J. Differential Equations, 246 (2009), 1288-1311.  doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[3]

C. O. Alves and C. Torres Ledesma, Existence and multiplicity of solutions for a non-linear Schrödinger equation with non-local regional diffusion, J. Math. Phys., 58 (2017), 111507, 19pp. doi: 10.1063/1.5011724.  Google Scholar

[4]

A. Borovskiik and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1983), 562-573.   Google Scholar

[5]

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, Phys. Fluids B, 5 (1993), 3539-3550.   Google Scholar

[6]

J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[7]

K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[8]

D. Cassani, J. M. Bezerra do Ó and A. Moameni, Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 9 (2010), 281-306.  doi: 10.3934/cpaa.2010.9.281.  Google Scholar

[9]

D. M. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb R^N$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.  Google Scholar

[10]

D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 443-463.  doi: 10.1017/S0308210500022836.  Google Scholar

[11]

P. C. Carrião, R. Lehrer and O. H. Miyagaki, Existence of solutions to a class of asymptotically linear Schrödinger equations in $\mathbb R^N$ via the Pohozaev manifold, J. Math. Anal. Appl., 428 (2015), 165-183.  doi: 10.1016/j.jmaa.2015.02.060.  Google Scholar

[12]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[13]

F. Esposito, A. Farina and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983.  doi: 10.1016/j.jde.2018.04.030.  Google Scholar

[14]

X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.  doi: 10.1016/j.jde.2012.11.017.  Google Scholar

[15]

E. Gloss, Existence and concentration of positive solution for a quasilinear equation in $\mathbb R^N$, J. Math. Anal. Appl., 371 (2010), 465-484.  doi: 10.1016/j.jmaa.2010.05.033.  Google Scholar

[16]

X. He, A. Qian and W. Zou, Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.  Google Scholar

[17]

T. S. Hsu, H. L. Lin and C. C. Hu, Multiple positive solutions of quasilinear elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 388 (2012), 500-512.  doi: 10.1016/j.jmaa.2011.11.010.  Google Scholar

[18]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3801-3805.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[19]

J. Q. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equation, II, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations, I. Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[21]

J. Liu, S. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $\mathbb R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.  doi: 10.1016/j.jmaa.2012.05.063.  Google Scholar

[22]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[23]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[24]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[25]

V. G. Makhankov and V. K. Fedyanin, Nonlinear 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

[26]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$, Commun. Pure Appl. Anal., 7 (2008), 89-105.  doi: 10.3934/cpaa.2008.7.89.  Google Scholar

[27]

K. Mahiout and C. O. Alves, Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces, Complex Var. Elliptic Equ., 62 (2016), 767-785.  doi: 10.1080/17476933.2016.1243669.  Google Scholar

[28]

O. H. Miyagaki and S. I. Moreira, Nonnegative solution for quasilinear Schrödinger equations that include supercritical exponents with nonlinearities that are indefinite in sign, J. Math. Anal. Appl., 421 (2015), 643-655.  doi: 10.1016/j.jmaa.2014.06.074.  Google Scholar

[29]

Y. Miyamoto and Y. Naito, Singular extremal solutions for supercritical elliptic equations in a ball, J. Differential Equations, 265 (2018), 2842-2885.  doi: 10.1016/j.jde.2018.04.055.  Google Scholar

[30]

Q. H. Miyagaki and S. I. Moreira, Nonnegative solution for quasilinear Schrödinger equations involving supercritical exponent with nonlinearities indefinitie in sign, J. Math. Anal. Appl., 421 (2015), 643-655.  doi: 10.1016/j.jmaa.2014.06.074.  Google Scholar

[31]

M. Poppenbery, 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

[32]

M. D. Pina and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[33]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[34]

E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.  Google Scholar

[35]

W. Wang, X. Yang and F. K. Zhao, Existence and concentration of ground states to a quasilinear problem with competing potentials, Nonlinear Anal., 102 (2014), 120-132.  doi: 10.1016/j.na.2014.01.025.  Google Scholar

[36]

Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comput., 216 (2010), 849-856.  doi: 10.1016/j.amc.2010.01.091.  Google Scholar

[37]

M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[38]

X. Wu and K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasilinear elliptic equations on $\mathbb R^N$, Nonlinear Anal. Real World Appl., 16 (2014), 48-64.   Google Scholar

[39]

J. Zhang, X. 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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