• Previous Article
    Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions
  • ERA Home
  • This Issue
  • Next Article
    Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition
March  2020, 28(1): 383-404. doi: 10.3934/era.2020022

Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system

1. 

College of Mathematics and Physics, Fujian Jiangxia University, Fuzhou 350108, China

2. 

College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Qishan Campus, Fuzhou 350117, China

* Corresponding author: Jianqing Chen

Received  December 2019 Published  March 2020

Fund Project: Lirong Huang is supported by NSF of Fujian (No. 2017J01549); Jianqing Chen is supported by NNSF of China (No. 11871152, 11671085)

This paper is concerned with the following Schrödinger-Poisson system
$ \begin{equation*} (P_\mu): -\Delta u +u + K(x)\phi u = |u|^{p-1}u + \mu h(x)u, \ -\Delta \phi = K(x) u^2, \ x\in\mathbb{R}^3, \end{equation*} $
where
$ p\in (3,5) $
,
$ K(x) $
and
$ h(x) $
are nonnegative functions, and
$ \mu $
is a positive parameter. Let
$ \mu_1 > 0 $
be an isolated first eigenvalue of the eigenvalue problem
$ -\Delta u + u = \mu h(x)u $
,
$ u\in H^1(\mathbb{R}^3) $
. As
$ 0<\mu\leq\mu_1 $
, we prove that
$ (P_{\mu}) $
has at least one nonnegative bound state with positive energy. As
$ \mu > \mu_1 $
, there is
$ \delta > 0 $
such that for any
$ \mu\in (\mu_1, \mu_1 + \delta) $
,
$ (P_\mu) $
has a nonnegative ground state
$ u_{0,\mu} $
with negative energy, and
$ u_{0,\mu^{(n)}}\to 0 $
in
$ H^1(\mathbb{R}^3) $
as
$ \mu^{(n)}\downarrow \mu_1 $
. Besides,
$ (P_\mu) $
has another nonnegative bound state
$ u_{2,\mu} $
with positive energy, and
$ u_{2,\mu^{(n)}}\to u_{\mu_1} $
in
$ H^1(\mathbb{R}^3) $
as
$ \mu^{(n)}\downarrow \mu_1 $
, where
$ u_{\mu_1} $
is a bound state of
$ (P_{\mu_1}) $
.
Citation: Lirong Huang, Jianqing Chen. Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system. Electronic Research Archive, 2020, 28 (1) : 383-404. doi: 10.3934/era.2020022
References:
[1]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), no. 4,439–475. doi: 10.1007/BF01206962.  Google Scholar

[2]

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

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), no. 3,391–404. doi: 10.1142/S021919970800282X.  Google Scholar

[5]

J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984.  Google Scholar

[6]

A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), no. 7, 1746–1765. doi: 10.1016/j.jde.2010.07.007.  Google Scholar

[7]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), no. 1, 90–108. doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar

[8]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), no. 2,283–293. doi: 10.12775/TMNA.1998.019.  Google Scholar

[9]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), no. 4,409–420. doi: 10.1142/S0129055X02001168.  Google Scholar

[10]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), no. 3,486–490. doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[11]

G. Cerami and G. Vaira, Positive solution for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), no. 3,521–543. doi: 10.1016/j.jde.2009.06.017.  Google Scholar

[12]

J. Chen, Z. Wang and X. Zhang, Standing waves for nonlinear Schrödinger-Poisson equation with high frequency, Topol. Methods Nonlinear Anal., 45 (2015), no. 2,601–614. doi: 10.12775/TMNA.2015.028.  Google Scholar

[13]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), no. 2-3,417–423.  Google Scholar

[14]

D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $R^N$, Calc. Var. Partial Differential Equations, 13 (2001), no. 2,159–189. doi: 10.1007/PL00009927.  Google Scholar

[15]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), no. 5,893–906. doi: 10.1017/S030821050000353X.  Google Scholar

[16]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), no. 3,307–322. doi: 10.1515/ans-2004-0305.  Google Scholar

[17]

T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), no. 1,321–342. doi: 10.1137/S0036141004442793.  Google Scholar

[18]

P. D'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), no. 2,177–192. doi: 10.1515/ans-2002-0205.  Google Scholar

[19]

P. D'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal., 74 (2011), no. 16, 5705–5721. doi: 10.1016/j.na.2011.05.057.  Google Scholar

[20]

L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), no. 8, 2463–2483. doi: 10.1016/j.jde.2013.06.022.  Google Scholar

[21]

Y. Jiang and H.-S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), no. 3,582–608. doi: 10.1016/j.jde.2011.05.006.  Google Scholar

[22]

G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), no. 5, 053505, 19 pp. doi: 10.1063/1.3585657.  Google Scholar

[23]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), no. 2,655–674. doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[24]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Model. Methods Appl. Sci., 15 (2005), no. 1,141–164. doi: 10.1142/S0218202505003939.  Google Scholar

[25]

J. Sun, H. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), no. 5, 3365–3380. doi: 10.1016/j.jde.2011.12.007.  Google Scholar

[26]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), no. 2,263–297. doi: 10.1007/s11587-011-0109-x.  Google Scholar

[27]

J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Calc. Var. Partial Differential Equations, 48 (2013), no. 1-2,243–273. doi: 10.1007/s00526-012-0548-6.  Google Scholar

[28]

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

[29]

Z. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), no. 4,809–816. doi: 10.3934/dcds.2007.18.809.  Google Scholar

[30]

Z. Wang and H.-S. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), no. 3,545–573. doi: 10.4171/JEMS/160.  Google Scholar

[31]

L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), no. 6, 2150–2164. doi: 10.1016/j.na.2008.02.116.  Google Scholar

show all references

References:
[1]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), no. 4,439–475. doi: 10.1007/BF01206962.  Google Scholar

[2]

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

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), no. 3,391–404. doi: 10.1142/S021919970800282X.  Google Scholar

[5]

J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984.  Google Scholar

[6]

A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), no. 7, 1746–1765. doi: 10.1016/j.jde.2010.07.007.  Google Scholar

[7]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), no. 1, 90–108. doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar

[8]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), no. 2,283–293. doi: 10.12775/TMNA.1998.019.  Google Scholar

[9]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), no. 4,409–420. doi: 10.1142/S0129055X02001168.  Google Scholar

[10]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), no. 3,486–490. doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[11]

G. Cerami and G. Vaira, Positive solution for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), no. 3,521–543. doi: 10.1016/j.jde.2009.06.017.  Google Scholar

[12]

J. Chen, Z. Wang and X. Zhang, Standing waves for nonlinear Schrödinger-Poisson equation with high frequency, Topol. Methods Nonlinear Anal., 45 (2015), no. 2,601–614. doi: 10.12775/TMNA.2015.028.  Google Scholar

[13]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), no. 2-3,417–423.  Google Scholar

[14]

D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $R^N$, Calc. Var. Partial Differential Equations, 13 (2001), no. 2,159–189. doi: 10.1007/PL00009927.  Google Scholar

[15]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), no. 5,893–906. doi: 10.1017/S030821050000353X.  Google Scholar

[16]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), no. 3,307–322. doi: 10.1515/ans-2004-0305.  Google Scholar

[17]

T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), no. 1,321–342. doi: 10.1137/S0036141004442793.  Google Scholar

[18]

P. D'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), no. 2,177–192. doi: 10.1515/ans-2002-0205.  Google Scholar

[19]

P. D'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal., 74 (2011), no. 16, 5705–5721. doi: 10.1016/j.na.2011.05.057.  Google Scholar

[20]

L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), no. 8, 2463–2483. doi: 10.1016/j.jde.2013.06.022.  Google Scholar

[21]

Y. Jiang and H.-S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), no. 3,582–608. doi: 10.1016/j.jde.2011.05.006.  Google Scholar

[22]

G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), no. 5, 053505, 19 pp. doi: 10.1063/1.3585657.  Google Scholar

[23]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), no. 2,655–674. doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[24]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Model. Methods Appl. Sci., 15 (2005), no. 1,141–164. doi: 10.1142/S0218202505003939.  Google Scholar

[25]

J. Sun, H. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), no. 5, 3365–3380. doi: 10.1016/j.jde.2011.12.007.  Google Scholar

[26]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), no. 2,263–297. doi: 10.1007/s11587-011-0109-x.  Google Scholar

[27]

J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Calc. Var. Partial Differential Equations, 48 (2013), no. 1-2,243–273. doi: 10.1007/s00526-012-0548-6.  Google Scholar

[28]

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

[29]

Z. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), no. 4,809–816. doi: 10.3934/dcds.2007.18.809.  Google Scholar

[30]

Z. Wang and H.-S. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), no. 3,545–573. doi: 10.4171/JEMS/160.  Google Scholar

[31]

L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), no. 6, 2150–2164. doi: 10.1016/j.na.2008.02.116.  Google Scholar

[1]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[2]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[3]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[4]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[5]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[6]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[7]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[8]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[9]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[10]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[11]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931

[12]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[13]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[14]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[15]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[16]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[17]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[18]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[19]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[20]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

 Impact Factor: 0.263

Metrics

  • PDF downloads (106)
  • HTML views (248)
  • Cited by (0)

Other articles
by authors

[Back to Top]