INFINITELY MANY SOLUTIONS FOR A ZERO MASS SCHRÖDINGER-POISSON-SLATER PROBLEM WITH CRITICAL GROWTH

Liu Yang; Zhisu Liu

doi:10.11948/20180273

2019 Volume 9 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2019 > 9(5): 1706-1718

Next Article Previous Article

Liu Yang, Zhisu Liu. INFINITELY MANY SOLUTIONS FOR A ZERO MASS SCHRÖDINGER-POISSON-SLATER PROBLEM WITH CRITICAL GROWTH[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1706-1718. doi: 10.11948/20180273

Citation:

Liu Yang, Zhisu Liu. INFINITELY MANY SOLUTIONS FOR A ZERO MASS SCHRÖDINGER-POISSON-SLATER PROBLEM WITH CRITICAL GROWTH[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1706-1718. doi: 10.11948/20180273

INFINITELY MANY SOLUTIONS FOR A ZERO MASS SCHRÖDINGER-POISSON-SLATER PROBLEM WITH CRITICAL GROWTH

Liu Yang¹,
Zhisu Liu^2,3, ,

1.
Department of Mathematics and Computing Sciences, Hengyang Normal University, Hengyang, Hunan 421001, China
2.
Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang, 421002, China
3.
School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

Corresponding author: Email address:liuzhisu183@sina.com(Z. Liu)
Fund Project: L. Yang was supported by the Science and Technology Plan Project of Hunan Province(2016TP1020) and The Science and Technology Plan Project of Hengyang City(2017KJ183), and Research Foundation of Education Bureau of Hunan Province(16A031); Z. Liu was supported by the Project of Hunan Provincial Key Laboratory (2016TP1020), Open fund project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang normal university (IIPA18K04)

Abstract

Abstract

In this paper, we are concerned with the following Schrödinger-Poisson-Slater problem with critical growth: $ -\Delta u+(u^{2}\star \frac{1}{|4\pi x|})u=\mu k(x)|u|^{p-2}u+|u|^{4}u\, \, \mbox{in}\, \, \mathbb R^{3}. $ We use a measure representation concentration-compactness principle of Lions to prove that the $(PS)_{c}$ condition holds locally. Via a truncation technique and Krasnoselskii genus theory, we further obtain infinitely many solutions for $\mu\in(0, \mu^{\ast})$ with some $\mu^{\ast}>0$.
- Schrödinger-Poisson-Slater problem /
- Zero mass /
- critical growth /
- concentration-compactness principle
MSC: 35B38, 35A15, 35B33

FullText(HTML)

References

[1]	C. Alves, M. Souto and S. Soares, Schördinger-Piosson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 2011, 377, 584-592. doi: 10.1016/j.jmaa.2010.11.031 CrossRef Google Scholar
[2]	A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 1994, 122, 519-543. doi: 10.1006/jfan.1994.1078 CrossRef Google Scholar
[3]	A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Piosson problem, Commun. Contemp. Math., 2008, 10, 391-404. doi: 10.1142/S021919970800282X CrossRef Google Scholar
[4]	T. D' Aprile and J. Wei, On bound states concentrating on spheres for Maxwell-Schrödinger equation, SIAM J. Math. Anal., 2005, 37, 321-342. doi: 10.1137/S0036141004442793 CrossRef Google Scholar
[5]	J. Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problem, J. Differential Equations, 1998, 144, 441-476. doi: 10.1006/jdeq.1997.3375 CrossRef Google Scholar
[6]	A. Azzollini and A. Pomponio, Ground state solutions for nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 2008, 11, 90-108. Google Scholar
[7]	V. Benci and D. Fortunato, An eigenvalue problem for the nonlinear Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 1998, 11, 283-293. doi: 10.12775/TMNA.1998.019 CrossRef Google Scholar
[8]	H. Bersstycki and P. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal., 1983, 82, 313-345. doi: 10.1007/BF00250555 CrossRef Google Scholar
[9]	C. Le Bris and P. Lions, From atoms to crystals: a mathematical journey, Bull. Amer. Math. Soc.(N.S.), 2005, 42(3), 291-363. doi: 10.1090/S0273-0979-05-01059-1 CrossRef Google Scholar
[10]	G. Cerami and G. Vaira, Postive solutions for some non-autonomos Schrödinger-Piosson systems, J. Differential Equations, 2010, 248, 521-543. doi: 10.1016/j.jde.2009.06.017 CrossRef Google Scholar
[11]	X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Piosson equations with ritical growth, J. Math. Phys., 2012, 53, 023702. doi: 10.1063/1.3683156 CrossRef Google Scholar
[12]	I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger-Piosson-Slater problem, Commun. Contemp. Math., 2012, 14, 1250003. doi: 10.1142/S0219199712500034 CrossRef Google Scholar
[13]	S. Kim and J. Seok, On nodal solutions of he nonlinear Schrödinger-Piosson equations, Commun. Contemp. Math., 2012, 14, 1250041. doi: 10.1142/S0219199712500411 CrossRef Google Scholar
[14]	R. Kajikiya, A critical point theorem related to the symmetric mountain-pass lemms and its applicaitions to elliptic equations, J. Funct. Analysis, 2005, 225, 352-370. doi: 10.1016/j.jfa.2005.04.005 CrossRef Google Scholar
[15]	E. Lieb and B. Simon, The thomas-Fermi theory of atoms, molecules and solids, Adv. Math., 1977, 23, 22-116. doi: 10.1016/0001-8708(77)90108-6 CrossRef Google Scholar
[16]	Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl., 2014, 412, 435-448. doi: 10.1016/j.jmaa.2013.10.066 CrossRef Google Scholar
[17]	Z. Liu, S. Guo and Y. Fang, Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth, J. Math. Phys., 2015, 56, 041505. doi: 10.1063/1.4919543 CrossRef Google Scholar
[18]	Z. Liu and J. Zhang, Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth, ESAIM: Control Optim. Calc. Var., 2017, 23, 1515-1542. doi: 10.1051/cocv/2016063 CrossRef Google Scholar
[19]	Z. Liu, Z. Zhang and S. Huang, Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations, 2019, 266, 5912-5941. doi: 10.1016/j.jde.2018.10.048 CrossRef Google Scholar
[20]	P. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Matemática Iberoamericana, 1985, 1, 145-201. Google Scholar
[21]	J. Mauser, The Schördinger-Piosson-X^α equation, Appl. Math. Lett., 2001, 14, 759-763. doi: 10.1016/S0893-9659(01)80038-0 CrossRef Google Scholar
[22]	C. Mercuri, V. Moroz and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Piosson-Slater equations at the critical frequency, Calc. Var. Partial Differential Equations, 2016, 55:146, DOI10.1007/s00526-016-1079-3. doi: 10.1007/s00526-016-1079-3 CrossRef Google Scholar
[23]	D. Ruiz, The Schrödinger-Piosson equation under the effect of a nonlinear local term, J. Funct. Anal., 2006, 237, 655-674. doi: 10.1016/j.jfa.2006.04.005 CrossRef Google Scholar
[24]	D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Piosson-Slater problem around a local minimun of potential, Rev. Mat. Iberoam, 2011, 27, 253-271. Google Scholar
[25]	D. Ruiz, On the Schrödinger-Piosson-Slater System: Behavior of Minimizers, Radial and Non-radial Cases, Arch. Rational Mech. Anal., 2010, 198, 349-368. doi: 10.1007/s00205-010-0299-5 CrossRef Google Scholar
[26]	P. Rabinowitz, Minimax Methods in Ctitical-Point Theory with Applications to Differential Equations, CBME Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Procidence, 1986. Google Scholar
[27]	G. Siciliano, Multiple positive solutions for a Schrödinger-Piosson-Slater system, J. Math. Anal. Appl., 2010, 365, 288-299. doi: 10.1016/j.jmaa.2009.10.061 CrossRef Google Scholar
[28]	Y. Song and S. Shi, Existence of infinitely many solutions for degenerate p-fractional Kirchhoff equations with critical Sobolev-Hardy nonlinearities, Z. Angew. Math. Phys., 2017, 128, 1-13. Google Scholar
[29]	J. Sun, T. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Piosson system, J. Differential Equations, 2016, 260, 586-627. doi: 10.1016/j.jde.2015.09.002 CrossRef Google Scholar
[30]	X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Piosson problems with general potentials, Discrete Contin. Dyn. Syst., 2017, 37, 4973-5002. doi: 10.3934/dcds.2017214 CrossRef Google Scholar
[31]	Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Piosson system in $\mathbb R^{3}$, Calc. Var. Partial Differential Equations, 2015, 52, 927-943. doi: 10.1007/s00526-014-0738-5 CrossRef $\mathbb R^{3}$" target="_blank">Google Scholar
[32]	J. Zhang, On the Schrödinger-Piosson equatons with a general nonlinearity in the critical growth, Nonlinear Anal., 2012, 75, 6391-6401. doi: 10.1016/j.na.2012.07.008 CrossRef Google Scholar
[33]	L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Piosson equations, J. Math. Anal. Appl., 2008, 346, 155-169. doi: 10.1016/j.jmaa.2008.04.053 CrossRef Google Scholar