Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in <inline-formula><tex-math id="M1">\begin{document} \mathbb R^N \end{document}</tex-math></inline-formula><inline-formula><tex-math id="M2">\begin{document} ^\diamondsuit \end{document}</tex-math></inline-formula>

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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 and Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3801-3805. doi: 10.1143/JPSJ.50.3801.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[33]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[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.
[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.
[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.
[37]	M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn., 50 (1981), 3801-3805. doi: 10.1143/JPSJ.50.3801.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[33]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[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.
[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.
[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.
[37]	M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.
[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.
[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.

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