INFINITELY MANY BOUND STATE SOLUTIONS OF SCHR&#214;DINGER-POISSON EQUATIONS IN R<sup>3</sup>

Xu Zhang; Shiwang May; Qilin Xie

doi:10.11948/2018.1239

2018 Volume 8 Issue 4

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Xu Zhang, Shiwang May, Qilin Xie. INFINITELY MANY BOUND STATE SOLUTIONS OF SCHRÖDINGER-POISSON EQUATIONS IN R3[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1239-1259. doi: 10.11948/2018.1239

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Xu Zhang, Shiwang May, Qilin Xie. INFINITELY MANY BOUND STATE SOLUTIONS OF SCHRÖDINGER-POISSON EQUATIONS IN R³[J]. Journal of Applied Analysis & Computation, 2018, 8(4): 1239-1259. doi: 10.11948/2018.1239

INFINITELY MANY BOUND STATE SOLUTIONS OF SCHRÖDINGER-POISSON EQUATIONS IN R³

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

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Abstract

Abstract

In this paper, we study a system of Schrödinger-Poisson equation {-∆u + a(x)u + K(x)ϕu=|u|^p-2u,x ∈ R³,-∆ϕ=K(x)u²,x ∈ R³,} where p ∈ (4,6) and K ≥ (#8802;)0. Under some suitable decay assumptions but without any symmetry property on a and K, we obtain infinitely many solutions of this system.
- Schrödinger-Poisson system /
- infinitely many solutions /
- without symmetric condition
MSC: 35J20;35J60

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References

[1]	A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 2008, 76(1), 257-274. Google Scholar
[2]	A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anual. Appl., 2008, 345(1), 90-108. Google Scholar
[3]	A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 2008, 10(3), 391-404. Google Scholar
[4]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 1998, 11(2), 283-293. Google Scholar
[5]	V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation couple with Maxwell equations, Rev. Math. Phys., 2002, 14(4), 409-420. Google Scholar
[6]	K-Ch. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkhäuser Boston, 1993. Google Scholar
[7]	G. M. Coclite, A multiplicity result for the Schrödinger-Maxwell equations, Commun. Appl. Anal., 2003, 7(3), 417-423. Google Scholar
[8]	G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 2005, 23(2), 139-168. Google Scholar
[9]	G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential. Equations., 2010, 248(3), 521-543. Google Scholar
[10]	P. d'Avenia, A. Pomponio and G. Vaira, Infinite many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal., 2010, 74(16), 5705-5721. Google Scholar
[11]	T. D'Aprile and J. C. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 2005, 37(1), 321-342. Google Scholar
[12]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Advances in Differential Equations, 2002, 7(10), 1257-1280. Google Scholar
[13]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Berlin, 1983. Google Scholar
[14]	I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part Ⅱ:Existence, Math. Models Methods Appl. Sci., 2009, 19(6), 877-910. Google Scholar
[15]	I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I:Necessary condition, Math. Models Methods Appl. Sci., 2009, 19(5), 707-720. Google Scholar
[16]	Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep well potential, J. Differential. Equations., 2011, 251(3), 582-608. Google Scholar
[17]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré,Anual. Non Linéair, 1984, 1(2), 1(4), 109-145, 223-283. Google Scholar
[18]	G. B. Li, S. J. Peng and C. H. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Maht. phys., 2011, 52(5), 053505. Google Scholar
[19]	G. B. Li, S. J. Peng and S. S. Yan, infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 2010, 12(6), 1069-1092. Google Scholar
[20]	Z. L. Liu, Z. Q. Wang and J. J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Annali di Matematica Pura ed Applicata, 2016, 195(3), 775-794. Google Scholar
[21]	D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell concentration around a sphere, Math. Models Methods Appl. Sci., 2005, 15(1), 141-164. Google Scholar
[22]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 2006, 237(2), 655-674. Google Scholar
[23]	D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential, Rev. Mat. Iberoam., 2011, 27(1), 253-271. Google Scholar
[24]	A. Salvatore, Multiple solitary waves for a non-homogeneous SchrödingerMaxwell system in R³, Adv. Nonlinear Studies, 2006, 6(2), 157-169. Google Scholar
[25]	M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential equations and Hamiltonian Systems, Springer, Berlin, 1996. Google Scholar
[26]	J. Wang, J. X. Xu, F. B. Zhang and X. M. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 2013, 26(5), 1377-1399. Google Scholar
[27]	Z. P. Wang and H. S. Zhou, Positive solutions for a nonlinear stationary Schrödinger-Poisson system in R³, Discrete Contin. Dyn. Syst., 2007, 18(4), 809-816. Google Scholar
[28]	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., 2013, 255(1), 1-23. Google Scholar
[29]	L. G. Zhao and F. K. Zhao, On the existence of solutions for the SchrödingerPoisson equations, J. Math. Anual. Appl., 2003, 346(1), 155-169. Google Scholar