Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Yanfang Xue; Chunlei Tang

doi:10.3934/cpaa.2018054

Previous Article
Sign-changing solutions for non-local elliptic equations with asymptotically linear term
CPAA Home
This Issue
Next Article
A nonlocal concave-convex problem with nonlocal mixed boundary data

May 2018, 17(3): 1121-1145. doi: 10.3934/cpaa.2018054

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Yanfang Xue ^1,2, and Chunlei Tang ^1,,

1.	School of Mathematics and Statistics, Southwest University, Chongqing, 400700, China
2.	School of Mathematics and Statistics, Xin-Yang Normal University, Xinyang, 464000, China

* Corresponding author: Chunlei Tang

Received January 2017 Revised November 2017 Published January 2018

Fund Project: The research is supported by National Natural Science Foundation of China(No. 11471267,11601438) and Chongqing Research Program of Basic Research and Frontier Technology(No.cstc2017jcyjAX0331).

In this paper, we are concerned with the existence of ground state solutions for the following quasilinear Schrödinger equation:

$-Δ u+V(x)u-Δ (u^2)u = K(x)|u|^{22^*-2}u+g(x,u), \ \ x∈ \mathbb{R}^N\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where $N≥ 3$, $V, \ g$ are asymptotically periodic functions in $x$. By combining variational methods and the concentration-compactness principle, we obtain a ground state solution for equation (1) under a new reformative condition which unify the asymptotic processes of $V, g $ at infinity.

Keywords: Asymptotically periodic, Sobolev critical exponent, ground state solution, Nehari manifold.

Mathematics Subject Classification: Primary: 35A15, 35B33, 35J60.

Citation: Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

References:

[1]	S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833. Google Scholar
[2]	H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. Google Scholar
[3]	M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. Google Scholar
[4]	M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. Google Scholar
[5]	Y. B. Deng, S. J. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504. Google Scholar
[6]	Y. B. Deng, S. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, Journal of Differential Equations, 260 (2016), 1228-1262. Google Scholar
[7]	J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644. Google Scholar
[8]	J.M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, Journal of Differential Equations, 248 (2010), 722-744. Google Scholar
[9]	X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032. Google Scholar
[10]	F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179. Google Scholar
[11]	Y. He and G. B. Li, Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Disctete and Continuous Dynamical Systems, 36 (2016), 731-762. Google Scholar
[12]	L. Jeanjean, T. J. Luo and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928. Google Scholar
[13]	G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. Google Scholar
[14]	H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905. Google Scholar
[15]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 223-283. Google Scholar
[16]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145. Google Scholar
[17]	J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976. Google Scholar
[18]	J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations with critical exponent, Comput. Math. Appl., 72 (2016), 1851-1864. Google Scholar
[19]	J. Q. Liu, X. Q. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations, 39 (2014), 2216-2239. Google Scholar
[20]	J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. Google Scholar
[21]	J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493. Google Scholar
[22]	J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448. Google Scholar
[23]	X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. Google Scholar
[24]	X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, Journal of Differential Equations, 254 (2013), 102-124. Google Scholar
[25]	X. Q. Liu, J. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calculus of Variations and Partial Differential Equations, 46 (2013), 641-669. Google Scholar
[26]	R. D. Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757. Google Scholar
[27]	A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N$, Journal of Differential Equations, 229 (2006), 570-587. Google Scholar
[28]	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. Google Scholar
[29]	D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233. Google Scholar
[30]	A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737. Google Scholar
[31]	H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858. Google Scholar
[32]	E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949. Google Scholar
[33]	E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. Google Scholar
[34]	X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728. Google Scholar
[35]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar
[36]	X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632. Google Scholar
[37]	H. Zhang, J. X. Xu and F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724. Google Scholar
[38]	H. Zhang, J. X. Xu and F. B. Zhang, Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38 (2015), 113-122. Google Scholar

show all references

References:

[1]	S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833. Google Scholar
[2]	H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. Google Scholar
[3]	M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. Google Scholar
[4]	M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. Google Scholar
[5]	Y. B. Deng, S. J. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504. Google Scholar
[6]	Y. B. Deng, S. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, Journal of Differential Equations, 260 (2016), 1228-1262. Google Scholar
[7]	J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644. Google Scholar
[8]	J.M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, Journal of Differential Equations, 248 (2010), 722-744. Google Scholar
[9]	X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032. Google Scholar
[10]	F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179. Google Scholar
[11]	Y. He and G. B. Li, Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Disctete and Continuous Dynamical Systems, 36 (2016), 731-762. Google Scholar
[12]	L. Jeanjean, T. J. Luo and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928. Google Scholar
[13]	G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. Google Scholar
[14]	H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905. Google Scholar
[15]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 223-283. Google Scholar
[16]	P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145. Google Scholar
[17]	J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976. Google Scholar
[18]	J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations with critical exponent, Comput. Math. Appl., 72 (2016), 1851-1864. Google Scholar
[19]	J. Q. Liu, X. Q. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations, 39 (2014), 2216-2239. Google Scholar
[20]	J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901. Google Scholar
[21]	J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493. Google Scholar
[22]	J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448. Google Scholar
[23]	X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. Google Scholar
[24]	X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, Journal of Differential Equations, 254 (2013), 102-124. Google Scholar
[25]	X. Q. Liu, J. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calculus of Variations and Partial Differential Equations, 46 (2013), 641-669. Google Scholar
[26]	R. D. Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757. Google Scholar
[27]	A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N$, Journal of Differential Equations, 229 (2006), 570-587. Google Scholar
[28]	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. Google Scholar
[29]	D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233. Google Scholar
[30]	A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737. Google Scholar
[31]	H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858. Google Scholar
[32]	E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949. Google Scholar
[33]	E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. Google Scholar
[34]	X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728. Google Scholar
[35]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar
[36]	X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632. Google Scholar
[37]	H. Zhang, J. X. Xu and F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724. Google Scholar
[38]	H. Zhang, J. X. Xu and F. B. Zhang, Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38 (2015), 113-122. Google Scholar

[1]	Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[2]	Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[3]	Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[4]	Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[5]	Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285
[6]	Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283
[7]	Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108
[8]	Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[9]	Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[10]	Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103
[11]	Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[12]	Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289
[13]	Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
[14]	Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[15]	Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[16]	A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419
[17]	Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[18]	T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
[19]	Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[20]	Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007

American Institute of Mathematical Sciences

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors