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
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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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