January 2019, 18(1): 493-517. doi: 10.3934/cpaa.2019025

Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity

1. 

Department of Mathematics, Nanchang University, Nanchang, 330031 Jiangxi, China

2. 

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

3. 

School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 Yunnan, China

* Corresponding author

Received  March 2018 Revised  May 2018 Published  August 2018

Fund Project: This work was supported by National Natural Science Foundation of China (Grant No. 11461043, 11571370 and 11601525), and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009) and the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), Yunnan Local Colleges Applied Basic Research Projects (2017FH001-011) and Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2016B037)

In this paper, we study the following quasilinear Schrödinger equation
$\begin{equation*}-Δ u+V(x)u-Δ(u^2)u = g(u),\,\, x∈\mathbb{R}^N,\end{equation*}$
where
$ N>4, 2^* = \frac{2N}{N-2}, V: \mathbb{R}^N \to \mathbb{R}$
satisfies suitable assumptions. Unlike
$ g∈ \mathcal{C}^1(\mathbb{R},\mathbb{R})$
, we only need to assume that
$ g∈ \mathcal{C}(\mathbb{R},\mathbb{R})$
. By using a change of variable, we obtain the existence of ground state solutions with general critical growth. Our results extend some known results.
Citation: Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025
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V. Ambrosio and G. M. Figueiredo, Ground state solutions for a fractional Schrödinger equation with critical growth, Asymptotic Anal., 105 (2017), 159-191. doi: 10.3233/ASY-171438.

[2]

F. G. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[4]

J. H. ChenX. H. Tang and B. T. Cheng, Non-Nehari manifold method for a class of generalized quasilinear Schrödinger equations, Appl. Math. Lett., 74 (2017), 20-26. doi: 10.1016/j.aml.2017.04.032.

[5]

J. H. ChenX. H. Tang and B. T. Cheng, Ground states for a class of generalized quasilinear Schrödinger equations in $ \mathbb{R}^N$, Mediterr. J. Math., 14 (2017), 190. doi: 10.1007/s00009-017-0990-y.

[6]

J. H. ChenX. H. Tang and B. T. Cheng, Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition, Appl. Math. Lett., 79 (2018), 27-33. doi: 10.1016/j.aml.2017.11.007.

[7]

J. H. Chen, X. H. Tang and B. T. Cheng, Ground state solutions for a class of quasilinear Schrödinger equations via Pohažaev manifold, Submitted. doi: 10.3934/cpaa.2018054.

[8]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst. A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096.

[9]

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.

[10]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation, Phys. D, 238 (2009), 38-54. doi: 10.1016/j.physd.2008.08.010.

[11]

Y. Deng and W. Huang, Ground state solutions for a quasilinear Elliptic equations with critical growth, Discrete Contin. Dyn. Syst. A, 37 (2017), 4213-4230. doi: 10.3934/dcds.2017179.

[12]

Y. DengS. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504. doi: 10.1063/1.4774153.

[13]

Y. DengS. Peng and S. Yan, Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147. doi: 10.1016/j.jde.2014.09.006.

[14]

Y. DengS. 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.

[15]

J. M. do ÓO. Miyagaki and S. 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.

[16]

J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations, 38 (2010), 275-315. doi: 10.1007/s00526-009-0286-6.

[17]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin: Springer, 1983. doi: 10.1007/978-3-642-61798-0.

[18]

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

[19]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $ \mathbb{R}^N$, Proc. R. Soc. Edinburgh Sect A., 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[20]

L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $ \mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614. doi: 10.1051/cocv:2002068.

[21]

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

[22]

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

[23]

J. Q. LiuY. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.

[24]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc., 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7.

[25]

X. Q. LiuJ. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669. doi: 10.1007/s00526-012-0497-0.

[26]

X. Q. LiuJ. 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.

[27]

Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448. doi: 10.1016/j.jmaa.2013.10.066.

[28]

V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.

[29]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $ \mathbb{R}^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.

[30]

A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009.

[31]

M. PoppenbergK. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equation, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. doi: 10.1007/s005260100105.

[32]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. Theory Methods Appl., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.

[33]

E. A. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with crirical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1.

[34]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Schröinger Poisson problems with general potentials, Discrete Contin. Dyn. Syst. A, 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214.

[35]

X. H. TangX. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., (2018), 1-15.

[36]

Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations, Nonlinear Differ. Equ. Appl., 19 (2012), 19-47. doi: 10.1007/s00030-011-0116-3.

[37]

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

[38]

H. Ye and G. Li, Concentrating solition solutions for quasilinear Schrödinger equations involving critical Sobolev exponents, Discrete Contin. Dyn. Syst. A, 36 (2016), 731-762. doi: 10.3934/dcds.2016.36.731.

[39]

J. ZhangW. Zhang and X. H. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst. A, 37 (2017), 4565-4583. doi: 10.3934/dcds.2017195.

[40]

J. Zhang and W. Zou, The critical case for a Berestycki-Lions theorem, Sci. China Math., 14 (2014), 541-554. doi: 10.1007/s11425-013-4687-9.

[41]

X. Zhu and D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (1989), 307-328. doi: 10.1016/S0252-9602(18)30356-4.

show all references

References:
[1]

V. Ambrosio and G. M. Figueiredo, Ground state solutions for a fractional Schrödinger equation with critical growth, Asymptotic Anal., 105 (2017), 159-191. doi: 10.3233/ASY-171438.

[2]

F. G. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[4]

J. H. ChenX. H. Tang and B. T. Cheng, Non-Nehari manifold method for a class of generalized quasilinear Schrödinger equations, Appl. Math. Lett., 74 (2017), 20-26. doi: 10.1016/j.aml.2017.04.032.

[5]

J. H. ChenX. H. Tang and B. T. Cheng, Ground states for a class of generalized quasilinear Schrödinger equations in $ \mathbb{R}^N$, Mediterr. J. Math., 14 (2017), 190. doi: 10.1007/s00009-017-0990-y.

[6]

J. H. ChenX. H. Tang and B. T. Cheng, Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition, Appl. Math. Lett., 79 (2018), 27-33. doi: 10.1016/j.aml.2017.11.007.

[7]

J. H. Chen, X. H. Tang and B. T. Cheng, Ground state solutions for a class of quasilinear Schrödinger equations via Pohažaev manifold, Submitted. doi: 10.3934/cpaa.2018054.

[8]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst. A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096.

[9]

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.

[10]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation, Phys. D, 238 (2009), 38-54. doi: 10.1016/j.physd.2008.08.010.

[11]

Y. Deng and W. Huang, Ground state solutions for a quasilinear Elliptic equations with critical growth, Discrete Contin. Dyn. Syst. A, 37 (2017), 4213-4230. doi: 10.3934/dcds.2017179.

[12]

Y. DengS. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504. doi: 10.1063/1.4774153.

[13]

Y. DengS. Peng and S. Yan, Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147. doi: 10.1016/j.jde.2014.09.006.

[14]

Y. DengS. 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.

[15]

J. M. do ÓO. Miyagaki and S. 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.

[16]

J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations, 38 (2010), 275-315. doi: 10.1007/s00526-009-0286-6.

[17]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin: Springer, 1983. doi: 10.1007/978-3-642-61798-0.

[18]

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

[19]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $ \mathbb{R}^N$, Proc. R. Soc. Edinburgh Sect A., 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[20]

L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $ \mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614. doi: 10.1051/cocv:2002068.

[21]

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

[22]

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

[23]

J. Q. LiuY. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. doi: 10.1081/PDE-120037335.

[24]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc., 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7.

[25]

X. Q. LiuJ. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669. doi: 10.1007/s00526-012-0497-0.

[26]

X. Q. LiuJ. 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.

[27]

Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448. doi: 10.1016/j.jmaa.2013.10.066.

[28]

V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6.

[29]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $ \mathbb{R}^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001.

[30]

A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009.

[31]

M. PoppenbergK. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equation, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. doi: 10.1007/s005260100105.

[32]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. Theory Methods Appl., 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005.

[33]

E. A. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with crirical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1.

[34]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Schröinger Poisson problems with general potentials, Discrete Contin. Dyn. Syst. A, 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214.

[35]

X. H. TangX. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., (2018), 1-15.

[36]

Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations, Nonlinear Differ. Equ. Appl., 19 (2012), 19-47. doi: 10.1007/s00030-011-0116-3.

[37]

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

[38]

H. Ye and G. Li, Concentrating solition solutions for quasilinear Schrödinger equations involving critical Sobolev exponents, Discrete Contin. Dyn. Syst. A, 36 (2016), 731-762. doi: 10.3934/dcds.2016.36.731.

[39]

J. ZhangW. Zhang and X. H. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst. A, 37 (2017), 4565-4583. doi: 10.3934/dcds.2017195.

[40]

J. Zhang and W. Zou, The critical case for a Berestycki-Lions theorem, Sci. China Math., 14 (2014), 541-554. doi: 10.1007/s11425-013-4687-9.

[41]

X. Zhu and D. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci., 9 (1989), 307-328. doi: 10.1016/S0252-9602(18)30356-4.

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