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Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines
Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent
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 |
$ \ddot{\mbox{o}} $ |
$ \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*} $ |
$ V,K,g $ |
$ x $ |
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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