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September  2019, 18(5): 2299-2324. doi: 10.3934/cpaa.2019104

Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent

Yong-Yong Li 1, , Yan-Fang Xue 2, and  Chun-Lei Tang 1,,

1. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China
2. 

College of Mathematics and Information Sciences, Xin-Yang Normal University, Xinyang, 464000, China

* Corresponding author

Received  October 2017 Revised  October 2017 Published  April 2019

Fund Project: The research is supported by National Natural Science Foundation of China (No.11471267).

In this paper, we study the modified Schr
$ \ddot{\mbox{o}} $
dinger-Poisson system:
$ \begin{align*} \begin{cases} -\Delta u+V(x)u-K(x)\phi|u|^8u-\Delta(u^2)u = g(x,u),\ \ \ \ &\mbox{in}\ \mathbb{R}^3,\\ -\Delta\phi = K(x)|u|^{10},\ \ \ \ &\mbox{in}\ \mathbb{R}^3, \end{cases} \end{align*} $
where
$ V,K,g $
 are asymptotically periodic functions of
$ x $
. Based on variational methods and the dual approach, we prove the existence of ground state solution by using the Nehari manifold method, the Mountain Pass theorem and the concentration-compactness principle.
Keywords: Modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system, asymptotically periodic, critical nonlocal term, Nehari manifold, ground state solution.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
References:
[1]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schr$\ddot{\mbox o}$dinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.

[2]

C. O. Alves, M. A. S. Souto and S. H. M. Soares, Schr$\ddot{\mbox o}$dinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 377 (2011), 584-592.  doi: 10.1016/j.jmaa.2010.11.031.

[3]

G. Bao, Infinitely many small solutions for a sublinear Schr$\ddot{\mbox o}$dinger-Poisson system with sign-changing potential, Comput. Math. Appl., 71 (2016), 2082-2088.  doi: 10.1016/j.camwa.2016.04.006.

[4]

V. Benci and D. Fortunato, An eigenvalue problem for the Schr$\ddot{\mbox o}$dinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[5]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[6]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schr$\ddot{\mbox o}$dinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.

[7]

S.-J. Chen and C.-L. Tang, High energy solutions for the superlinear Schr$\ddot{\mbox o}$dinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.  doi: 10.1016/j.na.2009.03.050.

[8]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schr$\ddot{\mbox o}$dinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.

[9]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schr$\ddot{\mbox o}$dinger-Maxwell equations, Proc. Roy. Soc. Edinb. Set. A, 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.

[10]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[11]

X. Feng and Y. Zhang, Existence of non-trivial solution for a class of modified Schr$\ddot{\mbox o}$dinger-Poisson equations via perturbation method, J. Math. Anal. Appl., 442 (2016), 637-684.  doi: 10.1016/j.jmaa.2016.05.002.

[12]

Y.-P. Gao, S.-L. Yu and C.-L. Tang, Positive ground state solutions to Schr$\ddot{\mbox o}$dinger-Poisson systems with a negative non-local term, E. J. Differential Equations, 118 (2015), 1-11. 

[13]

R. W. Hasse, A general method for the solution of nonlinear soliton and Kink Schr$\ddot{\mbox o}$dinger equations, Z. Phys. B., 37 (1980), 83-87.  doi: 10.1007/BF01325508.

[14]

L. R. Huang, E. M. Rocha and J. Q. Chen, On the Schr$\ddot{\mbox o}$dinger-Poisson system with a general indefinite nonlinearity, Nonlinear Anal. Real World Appl., 28 (2016), 1-19.  doi: 10.1016/j.nonrwa.2015.09.001.

[15]

H. Liu, Positive solutions of an asymptotically periodic Schr$\ddot{\mbox o}$dinger-Poisson system with critical exponent, Nonlinear Anal. Real World Appl., 32 (2016), 198-212.  doi: 10.1016/j.nonrwa.2016.04.007.

[16]

H. Liu and H. Chen, Multiple solutions for a nonlinear Schr$\ddot{\mbox o}$dinger-Poisson system with sign-changing potential, Comput. Math. Appl., 71 (2016), 1405-1416.  doi: 10.1016/j.camwa.2016.02.010.

[17]

W. Liu and L. Gan, Existence of solutions for modified Schr$\ddot{\mbox o}$dinger-Poisson system with critical nonlinearity in $\mathbb{R}^3$, Taiwan. J. Math., 20 (2016), 411-429.  doi: 10.11650/tjm.20.2016.6144.

[18]

Z. Liu and Y. Huang, Multiple solutions of asymptotically linear Schr$\ddot{\mbox o}$dinger-Poisson system with radial potentials vanishing at infinity, J. Math. Anal. Appl., 411 (2014), 693-706.  doi: 10.1016/j.jmaa.2013.10.023.

[19]

J. Liu, J.-F. Liao and C.-L. Tang, A positive ground state solution for a class of asymptotically periodic Schr$\ddot{\mbox o}$dinger equations, Comput. Math. Appl., 71 (2016), 965-976.  doi: 10.1016/j.camwa.2016.01.004.

[20]

F.-Y. Li, Y.-H. Li and J.-P. Shi, Existence of positive solutions to Schr$\ddot{\mbox o}$dinger-Poisson type systems with critical exponent, Commun. Contempt. Math., 16 (2014), 1450036. doi: 10.1142/S0219199714500369.

[21]

X. Liu, J. Liu and Z.-Q. W, Ground states for quasilinear Schr$\ddot{\mbox o}$dinger equations with critical growth, Calc. Var. Partial Differ. Equ., 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[22]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.

[23]

M.-M. Li and C.-L. Tang, Multiple positive solutions for Schr$\ddot{\mbox o}$dinger-Poisson system in $\mathbb{R}^3$ involving concave-convex nonlinearities with critical exponent, Commun. Pure Appl. Anal., 16 (2017), 1587-1602.  doi: 10.3934/cpaa.2017076.

[24]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schr$\ddot{\mbox o}$dinger equations, Ⅱ, J. Differential Equations, 187 (2003), 473-793.  doi: 10.1016/S0022-0396(02)00064-5.

[25]

Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schr$\ddot{\mbox o}$dinger-Poisson system, Ann. Mat. Pur. Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8.

[26]

F. Li and Q. Zhang, Existence of positive solutions to the Schr$\ddot{\mbox o}$dinger-Poisson system without compactness conditions, J. Math. Anal. Appl., 401 (2013), 754-762.  doi: 10.1016/j.jmaa.2013.01.002.

[27]

C. Mercuri, Positive solutions of nonlinear Schr$\ddot{\mbox o}$dinger-Poisson systems with radial potential vanishing at infinity, Rend. Lincei Mat. Appl., 19 (2008), 211-227.  doi: 10.4171/RLM/520.

[28]

J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for a class of modified Schr$\ddot{\mbox o}$dinger-Poisson systems, J. Math. Anal. Appl., 408 (2013), 713-724.  doi: 10.1016/j.jmaa.2013.06.011.

[29]

D. Ruiz, The Schr$\ddot{\mbox o}$dinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.

[30]

J. Sun, Infinitely many solutions for a class of sublinear Schr$\ddot{\mbox o}$dinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.  doi: 10.1016/j.jmaa.2012.01.057.

[31]

J. Sun and S. Ma, Ground state solutions for some Schr$\ddot{\mbox o}$dinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057.

[32]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schr$\ddot{\mbox o}$dinger equations with critical growth, Calc. Var. Partial Differ. Equ., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[33]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schr$\ddot{\mbox o}$dinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.

[34]

G. Vaira, Ground states for Schr$\ddot{\mbox o}$dinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.

[35]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996., Basel (1996). doi: 10.1007/978-1-4612-4146-1.

[36]

M.-H. Yang and Z.-Q. Han, Existence and multiplicity results for the nonlinear Schr$\ddot{\mbox o}$dinger-Poisson systems, Nonlinear Anal. Real World Appl., 13 (2012), 1093-1101.  doi: 10.1016/j.nonrwa.2011.07.008.

[37]

Y. Ye and C.-L. Tang, Existence and multiplicity of solutions for Schr$\ddot{\mbox o}$dinger-Poisson equations with sign-changing potential, Calc. Var. Partial Differ. Equ., 53 (2015), 383-411.  doi: 10.1007/s00526-014-0753-6.

[38]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schr$\ddot{\mbox o}$dinger-Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.

[39]

L. Zhao and F. Zhao, On the existence of solutions for the Schr$\ddot{\mbox o}$dinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.

show all references

References:
[1]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schr$\ddot{\mbox o}$dinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.

[2]

C. O. Alves, M. A. S. Souto and S. H. M. Soares, Schr$\ddot{\mbox o}$dinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 377 (2011), 584-592.  doi: 10.1016/j.jmaa.2010.11.031.

[3]

G. Bao, Infinitely many small solutions for a sublinear Schr$\ddot{\mbox o}$dinger-Poisson system with sign-changing potential, Comput. Math. Appl., 71 (2016), 2082-2088.  doi: 10.1016/j.camwa.2016.04.006.

[4]

V. Benci and D. Fortunato, An eigenvalue problem for the Schr$\ddot{\mbox o}$dinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[5]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[6]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schr$\ddot{\mbox o}$dinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.

[7]

S.-J. Chen and C.-L. Tang, High energy solutions for the superlinear Schr$\ddot{\mbox o}$dinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.  doi: 10.1016/j.na.2009.03.050.

[8]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schr$\ddot{\mbox o}$dinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.

[9]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schr$\ddot{\mbox o}$dinger-Maxwell equations, Proc. Roy. Soc. Edinb. Set. A, 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.

[10]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[11]

X. Feng and Y. Zhang, Existence of non-trivial solution for a class of modified Schr$\ddot{\mbox o}$dinger-Poisson equations via perturbation method, J. Math. Anal. Appl., 442 (2016), 637-684.  doi: 10.1016/j.jmaa.2016.05.002.

[12]

Y.-P. Gao, S.-L. Yu and C.-L. Tang, Positive ground state solutions to Schr$\ddot{\mbox o}$dinger-Poisson systems with a negative non-local term, E. J. Differential Equations, 118 (2015), 1-11. 

[13]

R. W. Hasse, A general method for the solution of nonlinear soliton and Kink Schr$\ddot{\mbox o}$dinger equations, Z. Phys. B., 37 (1980), 83-87.  doi: 10.1007/BF01325508.

[14]

L. R. Huang, E. M. Rocha and J. Q. Chen, On the Schr$\ddot{\mbox o}$dinger-Poisson system with a general indefinite nonlinearity, Nonlinear Anal. Real World Appl., 28 (2016), 1-19.  doi: 10.1016/j.nonrwa.2015.09.001.

[15]

H. Liu, Positive solutions of an asymptotically periodic Schr$\ddot{\mbox o}$dinger-Poisson system with critical exponent, Nonlinear Anal. Real World Appl., 32 (2016), 198-212.  doi: 10.1016/j.nonrwa.2016.04.007.

[16]

H. Liu and H. Chen, Multiple solutions for a nonlinear Schr$\ddot{\mbox o}$dinger-Poisson system with sign-changing potential, Comput. Math. Appl., 71 (2016), 1405-1416.  doi: 10.1016/j.camwa.2016.02.010.

[17]

W. Liu and L. Gan, Existence of solutions for modified Schr$\ddot{\mbox o}$dinger-Poisson system with critical nonlinearity in $\mathbb{R}^3$, Taiwan. J. Math., 20 (2016), 411-429.  doi: 10.11650/tjm.20.2016.6144.

[18]

Z. Liu and Y. Huang, Multiple solutions of asymptotically linear Schr$\ddot{\mbox o}$dinger-Poisson system with radial potentials vanishing at infinity, J. Math. Anal. Appl., 411 (2014), 693-706.  doi: 10.1016/j.jmaa.2013.10.023.

[19]

J. Liu, J.-F. Liao and C.-L. Tang, A positive ground state solution for a class of asymptotically periodic Schr$\ddot{\mbox o}$dinger equations, Comput. Math. Appl., 71 (2016), 965-976.  doi: 10.1016/j.camwa.2016.01.004.

[20]

F.-Y. Li, Y.-H. Li and J.-P. Shi, Existence of positive solutions to Schr$\ddot{\mbox o}$dinger-Poisson type systems with critical exponent, Commun. Contempt. Math., 16 (2014), 1450036. doi: 10.1142/S0219199714500369.

[21]

X. Liu, J. Liu and Z.-Q. W, Ground states for quasilinear Schr$\ddot{\mbox o}$dinger equations with critical growth, Calc. Var. Partial Differ. Equ., 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[22]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.  doi: 10.1016/j.na.2009.01.171.

[23]

M.-M. Li and C.-L. Tang, Multiple positive solutions for Schr$\ddot{\mbox o}$dinger-Poisson system in $\mathbb{R}^3$ involving concave-convex nonlinearities with critical exponent, Commun. Pure Appl. Anal., 16 (2017), 1587-1602.  doi: 10.3934/cpaa.2017076.

[24]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schr$\ddot{\mbox o}$dinger equations, Ⅱ, J. Differential Equations, 187 (2003), 473-793.  doi: 10.1016/S0022-0396(02)00064-5.

[25]

Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schr$\ddot{\mbox o}$dinger-Poisson system, Ann. Mat. Pur. Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8.

[26]

F. Li and Q. Zhang, Existence of positive solutions to the Schr$\ddot{\mbox o}$dinger-Poisson system without compactness conditions, J. Math. Anal. Appl., 401 (2013), 754-762.  doi: 10.1016/j.jmaa.2013.01.002.

[27]

C. Mercuri, Positive solutions of nonlinear Schr$\ddot{\mbox o}$dinger-Poisson systems with radial potential vanishing at infinity, Rend. Lincei Mat. Appl., 19 (2008), 211-227.  doi: 10.4171/RLM/520.

[28]

J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for a class of modified Schr$\ddot{\mbox o}$dinger-Poisson systems, J. Math. Anal. Appl., 408 (2013), 713-724.  doi: 10.1016/j.jmaa.2013.06.011.

[29]

D. Ruiz, The Schr$\ddot{\mbox o}$dinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.

[30]

J. Sun, Infinitely many solutions for a class of sublinear Schr$\ddot{\mbox o}$dinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.  doi: 10.1016/j.jmaa.2012.01.057.

[31]

J. Sun and S. Ma, Ground state solutions for some Schr$\ddot{\mbox o}$dinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057.

[32]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schr$\ddot{\mbox o}$dinger equations with critical growth, Calc. Var. Partial Differ. Equ., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[33]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schr$\ddot{\mbox o}$dinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.

[34]

G. Vaira, Ground states for Schr$\ddot{\mbox o}$dinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.

[35]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996., Basel (1996). doi: 10.1007/978-1-4612-4146-1.

[36]

M.-H. Yang and Z.-Q. Han, Existence and multiplicity results for the nonlinear Schr$\ddot{\mbox o}$dinger-Poisson systems, Nonlinear Anal. Real World Appl., 13 (2012), 1093-1101.  doi: 10.1016/j.nonrwa.2011.07.008.

[37]

Y. Ye and C.-L. Tang, Existence and multiplicity of solutions for Schr$\ddot{\mbox o}$dinger-Poisson equations with sign-changing potential, Calc. Var. Partial Differ. Equ., 53 (2015), 383-411.  doi: 10.1007/s00526-014-0753-6.

[38]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schr$\ddot{\mbox o}$dinger-Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.

[39]

L. Zhao and F. Zhao, On the existence of solutions for the Schr$\ddot{\mbox o}$dinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.

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