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January  2017, 37(1): 605-625. doi: 10.3934/dcds.2017025

Bound state solutions of Schrödinger-Poisson system with critical exponent

Xu Zhang , Shiwang Ma , and  Qilin Xie

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received  March 2016 Revised  July 2016 Published  November 2016

Fund Project: Research supported by the National Natural Science Foundation of China (No. 11571187).

In this paper, we consider the following Schrödinger-Poisson problem
$\tag{P}\label{0.1} \begin{cases}- Δ u+V(x)u+K(x)φ u=|u|^{2^*-2}u, &x∈ \mathbb{R}^3,\\-Δ φ=K(x)u^2,&x∈ \mathbb{R}^3,\end{cases}$
where
$2^*=6 $
 is the critical exponent in
$\mathbb R^3$
,
$ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$
and
$V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$
are given nonnegative functions. When
$|V|_{\frac{3}{2}}+|K|_{\frac{1}{2}}$
is suitable small, we prove that problem (P) has at least one bound state solution via a linking theorem.
Keywords: Schrödinger-Poisson system, critical exponent, bound state solutions.
Mathematics Subject Classification: Primary:35J20;Secondary:35J6.
Citation: Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.

[2]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.

[3]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.

[4]

V. Benci and D. Fortunato, Some compact embedding theorems for weighted Sobolev spaces, Boll. Un. Mat. Ital. B (5), 13 (1976), 832-843. 

[5]

V. Benci and C. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbb R^3$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.

[6]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. 

[7]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.

[8]

H. Brezis, Analyse Fonctionnelle, Masson, Parias, 1983.

[9]

G. M. Coclite, A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79 (2002), 21-30. 

[10]

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

[11]

T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.  doi: 10.1137/S0036141004442793.

[12]

X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth J. Math. Phys. 53 (2012), 023702, 19pp. doi: 10.1063/1.3683156.

[13]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I: Necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720.  doi: 10.1142/S0218202509003589.

[14]

I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part II: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.  doi: 10.1142/S0218202509003656.

[15]

Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential. Equations., 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.

[16]

G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system J. Maht. phys. 52 (2011), 053505, 19pp. doi: 10.1063/1.3585657.

[17]

P. L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283. 

[18]

Z. S. Liu and S. J. 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.

[19]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.  doi: 10.1142/S0218202505003939.

[20]

D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27 (2011), 253-271.  doi: 10.4171/RMI/635.

[21]

A. Salvatore, Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Adv. Nonlinear strud., 6 (2006), 157-169.  doi: 10.1515/ans-2006-0203.

[22]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[23]

J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.  doi: 10.1088/0951-7715/26/5/1377.

[24]

M. Willem, Analyse Harmomque Réelle, Hermann, Parias, 1995.

[25]

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

[26]

Z. P. Wang and H. S. Zhou, Positive solutions for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3 $, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.

[27]

J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinea Anal., 75 (2012), 6391-6401.  doi: 10.1016/j.na.2012.07.008.

[28]

J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.

[29]

L. G. Zhao, H. Liu and F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential. Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.

[30]

L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.

[31]

L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anual. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.

show all references

References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.

[2]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.

[3]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.

[4]

V. Benci and D. Fortunato, Some compact embedding theorems for weighted Sobolev spaces, Boll. Un. Mat. Ital. B (5), 13 (1976), 832-843. 

[5]

V. Benci and C. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbb R^3$, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.

[6]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. 

[7]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.

[8]

H. Brezis, Analyse Fonctionnelle, Masson, Parias, 1983.

[9]

G. M. Coclite, A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79 (2002), 21-30. 

[10]

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

[11]

T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.  doi: 10.1137/S0036141004442793.

[12]

X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth J. Math. Phys. 53 (2012), 023702, 19pp. doi: 10.1063/1.3683156.

[13]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I: Necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720.  doi: 10.1142/S0218202509003589.

[14]

I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part II: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.  doi: 10.1142/S0218202509003656.

[15]

Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential. Equations., 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.

[16]

G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system J. Maht. phys. 52 (2011), 053505, 19pp. doi: 10.1063/1.3585657.

[17]

P. L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283. 

[18]

Z. S. Liu and S. J. 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.

[19]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.  doi: 10.1142/S0218202505003939.

[20]

D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential, Rev. Mat. Iberoam., 27 (2011), 253-271.  doi: 10.4171/RMI/635.

[21]

A. Salvatore, Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Adv. Nonlinear strud., 6 (2006), 157-169.  doi: 10.1515/ans-2006-0203.

[22]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[23]

J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.  doi: 10.1088/0951-7715/26/5/1377.

[24]

M. Willem, Analyse Harmomque Réelle, Hermann, Parias, 1995.

[25]

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

[26]

Z. P. Wang and H. S. Zhou, Positive solutions for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3 $, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.

[27]

J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinea Anal., 75 (2012), 6391-6401.  doi: 10.1016/j.na.2012.07.008.

[28]

J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.

[29]

L. G. Zhao, H. Liu and F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential. Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.

[30]

L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.

[31]

L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anual. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.

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