November  2018, 38(11): 5461-5504. doi: 10.3934/dcds.2018241

Positive solutions for a nonlinear Schrödinger-Poisson system

1. 

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

2. 

School of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, China

* Corresponding author: Chunhua Wang

Received  September 2017 Published  August 2018

Fund Project: The authors would like to thank professor Shuangjie Peng for helpful and valuable discussions. This paper was partially supported by NSFC (No. 11671162, No. 11601194) and CCNU18CXTD04.

In this paper, we study the following nonlinear Schrödinger-Poisson system
$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$
where
$K(x)$
is a positive and continuous potential and
$f(u)$
is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some
$\epsilon_{0}>0$
such that for
$0<\epsilon<\epsilon_{0}$
, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [3] and a result of Li, Peng and Wang in [26].
Citation: Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
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]

W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with nonsymmetric potential, Calc. Var. Partial Differential Equ., 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.

[4]

W. AoJ. Wei and J. Zeng, An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.  doi: 10.1016/j.jfa.2013.06.016.

[5]

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

[6]

A. Azzollini and A. Pomponio, Ground state solutions for the non-linear Schrödinger-Maxwell equations with a singular potential, arXiv: 0706.1679[math.AP].

[7]

A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $ \mathbb{R}^{N}$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.  doi: 10.4171/RMI/92.

[8]

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.

[9]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[10]

R. BenguriaH. Brézis and E. Lieb, The Thomas-Fermi-Von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.  doi: 10.1007/BF01942059.

[11]

D. Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal., 58 (2004), 733-747.  doi: 10.1016/j.na.2003.05.001.

[12]

I. Catto and P. Loins, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Ⅰ. A necessary and sufficient condition for the stability of general molecular systems, Comm. Partial diferential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.

[13]

G. CeramiD. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with non symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.

[14]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.

[15]

G. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl., 7 (2003), 417-423. 

[16]

E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.  doi: 10.1216/rmjm/1181072198.

[17]

T. $ \acute{{\rm{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.

[18]

T. $ \acute{{\rm{D}}}$Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edindurgh Sect., 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.

[19]

T. $ \acute{{\rm{D}}}$Aprile and J. Wei, On bound states concentration on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.  doi: 10.1137/S0036141004442793.

[20]

M. DelpinoJ. Wei and W. Yao, Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 53 (2015), 473-523.  doi: 10.1007/s00526-014-0756-3.

[21]

B. Gidas, W. M. Ni, L. Nirernberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbb{R}^N$. In: Mathematical Analysis and Applications, part A, 369–402. Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.

[22]

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

[23]

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

[24]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson system with potential, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.

[25]

G. LiS. Peng and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.  doi: 10.1142/S0219199710004068.

[26]

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

[27]

F. LinW. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.

[28]

E. Lions and B. Simon, The Thomas-Fermi theory of atoms, moleules and solids, Adv. Math., 23 (997), 22-116.  doi: 10.1016/0001-8708(77)90108-6.

[29]

P. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1007/BF01205672.

[30]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[31]

C. Mercuri, Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 211-227.  doi: 10.4171/RLM/520.

[32]

M. MonicaF. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 14 (2012), 1923-1953.  doi: 10.4171/JEMS/351.

[33]

W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[34]

P. Poláčik, Morse indices and bifurations of positive solutions of $ \Delta u +f(u)=0$ on $ \mathbb{R}^{N}$, Indiana Univ. Math. J., 50 (2001), 1407-1432.  doi: 10.1512/iumj.2001.50.1909.

[35]

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.

[36]

D. Ruiz, The nonlinear Schrödinger-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.

[37]

Z. Wang and H. Zhou, Positive solution 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.

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]

W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with nonsymmetric potential, Calc. Var. Partial Differential Equ., 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.

[4]

W. AoJ. Wei and J. Zeng, An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.  doi: 10.1016/j.jfa.2013.06.016.

[5]

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

[6]

A. Azzollini and A. Pomponio, Ground state solutions for the non-linear Schrödinger-Maxwell equations with a singular potential, arXiv: 0706.1679[math.AP].

[7]

A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $ \mathbb{R}^{N}$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.  doi: 10.4171/RMI/92.

[8]

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.

[9]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.

[10]

R. BenguriaH. Brézis and E. Lieb, The Thomas-Fermi-Von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.  doi: 10.1007/BF01942059.

[11]

D. Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal., 58 (2004), 733-747.  doi: 10.1016/j.na.2003.05.001.

[12]

I. Catto and P. Loins, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Ⅰ. A necessary and sufficient condition for the stability of general molecular systems, Comm. Partial diferential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.

[13]

G. CeramiD. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with non symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.

[14]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.

[15]

G. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl., 7 (2003), 417-423. 

[16]

E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.  doi: 10.1216/rmjm/1181072198.

[17]

T. $ \acute{{\rm{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.

[18]

T. $ \acute{{\rm{D}}}$Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edindurgh Sect., 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.

[19]

T. $ \acute{{\rm{D}}}$Aprile and J. Wei, On bound states concentration on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.  doi: 10.1137/S0036141004442793.

[20]

M. DelpinoJ. Wei and W. Yao, Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 53 (2015), 473-523.  doi: 10.1007/s00526-014-0756-3.

[21]

B. Gidas, W. M. Ni, L. Nirernberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbb{R}^N$. In: Mathematical Analysis and Applications, part A, 369–402. Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.

[22]

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

[23]

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

[24]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson system with potential, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.

[25]

G. LiS. Peng and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.  doi: 10.1142/S0219199710004068.

[26]

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

[27]

F. LinW. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.

[28]

E. Lions and B. Simon, The Thomas-Fermi theory of atoms, moleules and solids, Adv. Math., 23 (997), 22-116.  doi: 10.1016/0001-8708(77)90108-6.

[29]

P. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1007/BF01205672.

[30]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.

[31]

C. Mercuri, Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 211-227.  doi: 10.4171/RLM/520.

[32]

M. MonicaF. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 14 (2012), 1923-1953.  doi: 10.4171/JEMS/351.

[33]

W. M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[34]

P. Poláčik, Morse indices and bifurations of positive solutions of $ \Delta u +f(u)=0$ on $ \mathbb{R}^{N}$, Indiana Univ. Math. J., 50 (2001), 1407-1432.  doi: 10.1512/iumj.2001.50.1909.

[35]

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.

[36]

D. Ruiz, The nonlinear Schrödinger-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.

[37]

Z. Wang and H. Zhou, Positive solution 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.

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