• Previous Article
    Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains
  • DCDS Home
  • This Issue
  • Next Article
    On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals
January  2015, 35(1): 427-440. doi: 10.3934/dcds.2015.35.427

Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities

1. 

School of Mathematical Sciences, Capital Normal University, Beijing 100037, China

2. 

Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China

Received  November 2013 Revised  March 2014 Published  August 2014

In this paper, we obtain the existence of infinitely many solutions for the following Schrödinger-Poisson system \begin{equation*} \begin{cases} -\Delta u+a(x)u+ \phi u=k(x)|u|^{q-2}u- h(x)|u|^{p-2}u,\quad &x\in \mathbb{R}^3,\\ -\Delta \phi=u^2,\ \lim_{|x|\to +\infty}\phi(x)=0, &x\in \mathbb{R}^3, \end{cases} \end{equation*} where $1 < q < 2 < p < +\infty$, $a(x)$, $k(x)$ and $h(x)$ are measurable functions satisfying suitable assumptions.
Citation: Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
References:
[1]

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

[2]

P. d'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system,, Nonlinear Anal. TMA, 74 (2011), 5705.  doi: 10.1016/j.na.2011.05.057.  Google Scholar

[3]

A. Azzollini, P. d'Avenia and A. Pomponio, On the Schroinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincare Anal. NonLineaire, 27 (2010), 779.  doi: 10.1016/j.anihpc.2009.11.012.  Google Scholar

[4]

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

[5]

T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549.  doi: 10.1142/S0219199701000494.  Google Scholar

[6]

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

[7]

K. Benmlih, Stationary solutions for a Schröinger-Poisson system in $\mathbbR^3$,, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, (2002), 65.   Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[9]

S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schrödinger Maxwell equations,, Nonlinear Anal. TMA, 71 (2009), 4927.  doi: 10.1016/j.na.2009.03.050.  Google Scholar

[10]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[11]

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), 893.  doi: 10.1017/S030821050000353X.  Google Scholar

[12]

D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems,, J. Funct. Anal., 199 (2003), 452.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[13]

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

[14]

H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations,, Nonlinear Anal. TMA, 67 (2007), 1445.  doi: 10.1016/j.na.2006.07.029.  Google Scholar

[15]

A. Kristály and D. Repovš, On the Schröinger-Maxwell system involving sublinear terms,, Nonlinear Anal. Real World Applications, 13 (2012), 213.  doi: 10.1016/j.nonrwa.2011.07.027.  Google Scholar

[16]

G. B. Li, S. J. Peng and S. S. Yan., Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system,, Commun. Contemp. Math., 12 (2010), 1069.  doi: 10.1142/S0219199710004068.  Google Scholar

[17]

P. L. Lions, Solutios of Hartree-Fock equations for Coulomb systems,, Commun. Math. Phys., 109 (1984), 33.  doi: 10.1007/BF01205672.  Google Scholar

[18]

S. B. Liu and S. J. Li, An elliptic equation with concave and convex nonlinearities,, Nonlinear Anal. TMA, 53 (2003), 723.  doi: 10.1016/S0362-546X(03)00020-8.  Google Scholar

[19]

D. Ruiz, The Schrödinger-Poisson equation under the effect of nonlinear local term,, J. Funct. Anal., 237 (2006), 655.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[20]

J. T. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 390 (2012), 514.  doi: 10.1016/j.jmaa.2012.01.057.  Google Scholar

[21]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[22]

E. Tonkes, A semilinear elliptic equation with concave and convex nonlinearities,, Topol. Methods Nonlinear Anal., 13 (1999), 251.   Google Scholar

[23]

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

[24]

Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,, Nonlinear Differ. Equ. Appl., 8 (2001), 15.  doi: 10.1007/PL00001436.  Google Scholar

[25]

M. B. Yang, Z. F. Shen and Y. H. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system,, Nonlinear Anal., 71 (2009), 730.  doi: 10.1016/j.na.2008.10.105.  Google Scholar

[26]

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

show all references

References:
[1]

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

[2]

P. d'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system,, Nonlinear Anal. TMA, 74 (2011), 5705.  doi: 10.1016/j.na.2011.05.057.  Google Scholar

[3]

A. Azzollini, P. d'Avenia and A. Pomponio, On the Schroinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincare Anal. NonLineaire, 27 (2010), 779.  doi: 10.1016/j.anihpc.2009.11.012.  Google Scholar

[4]

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

[5]

T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549.  doi: 10.1142/S0219199701000494.  Google Scholar

[6]

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

[7]

K. Benmlih, Stationary solutions for a Schröinger-Poisson system in $\mathbbR^3$,, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, (2002), 65.   Google Scholar

[8]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[9]

S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schrödinger Maxwell equations,, Nonlinear Anal. TMA, 71 (2009), 4927.  doi: 10.1016/j.na.2009.03.050.  Google Scholar

[10]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[11]

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), 893.  doi: 10.1017/S030821050000353X.  Google Scholar

[12]

D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems,, J. Funct. Anal., 199 (2003), 452.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[13]

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

[14]

H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations,, Nonlinear Anal. TMA, 67 (2007), 1445.  doi: 10.1016/j.na.2006.07.029.  Google Scholar

[15]

A. Kristály and D. Repovš, On the Schröinger-Maxwell system involving sublinear terms,, Nonlinear Anal. Real World Applications, 13 (2012), 213.  doi: 10.1016/j.nonrwa.2011.07.027.  Google Scholar

[16]

G. B. Li, S. J. Peng and S. S. Yan., Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system,, Commun. Contemp. Math., 12 (2010), 1069.  doi: 10.1142/S0219199710004068.  Google Scholar

[17]

P. L. Lions, Solutios of Hartree-Fock equations for Coulomb systems,, Commun. Math. Phys., 109 (1984), 33.  doi: 10.1007/BF01205672.  Google Scholar

[18]

S. B. Liu and S. J. Li, An elliptic equation with concave and convex nonlinearities,, Nonlinear Anal. TMA, 53 (2003), 723.  doi: 10.1016/S0362-546X(03)00020-8.  Google Scholar

[19]

D. Ruiz, The Schrödinger-Poisson equation under the effect of nonlinear local term,, J. Funct. Anal., 237 (2006), 655.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[20]

J. T. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 390 (2012), 514.  doi: 10.1016/j.jmaa.2012.01.057.  Google Scholar

[21]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[22]

E. Tonkes, A semilinear elliptic equation with concave and convex nonlinearities,, Topol. Methods Nonlinear Anal., 13 (1999), 251.   Google Scholar

[23]

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

[24]

Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,, Nonlinear Differ. Equ. Appl., 8 (2001), 15.  doi: 10.1007/PL00001436.  Google Scholar

[25]

M. B. Yang, Z. F. Shen and Y. H. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system,, Nonlinear Anal., 71 (2009), 730.  doi: 10.1016/j.na.2008.10.105.  Google Scholar

[26]

L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations,, J. Math. Anal. Appl., 346 (2008), 155.  doi: 10.1016/j.jmaa.2008.04.053.  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, 2020  doi: 10.3934/dcdss.2020447

[2]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[3]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[4]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[5]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[6]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[7]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[8]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[9]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[10]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[11]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[12]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[13]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[14]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[15]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[16]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[17]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[18]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[19]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[20]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]