Browse > Article
http://dx.doi.org/10.4134/JKMS.j190833

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR KIRCHHOFF-SCHRÖDINGER-POISSON SYSTEM WITH CONCAVE AND CONVEX NONLINEARITIES  

Che, Guofeng (School of Applied Mathematics Guangdong University of Technology)
Chen, Haibo (School of Mathematics and Statistics Central South University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.6, 2020 , pp. 1551-1571 More about this Journal
Abstract
This paper is concerned with the following Kirchhoff-Schrödinger-Poisson system $$\begin{cases} -(a+b{\displaystyle\smashmargin{2}\int\nolimits_{\mathbb{R}^3}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+{\mu}{\phi}u={\lambda}f(x){\mid}u{\mid}^{p-2}u+g(x){\mid}u{\mid}^{p-2}u,&{\text{ in }}{\mathbb{R}}^3,\\-{\Delta}{\phi}={\mu}{\mid}u{\mid}^2,&{\text{ in }}{\mathbb{R}}^3, \end{cases}$$ where a > 0, b, µ ≥ 0, p ∈ (1, 2), q ∈ [4, 6) and λ > 0 is a parameter. Under some suitable assumptions on V (x), f(x) and g(x), we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.
Keywords
Kirchhoff-$Schr{\ddot{o}}dinger$-Poisson system; concave and convex nonlinearities; Mountain Pass Theorem; Ekeland's variational princie;
Citations & Related Records
연도 인용수 순위
  • Reference
1 C. O. Alves, F. J. S. A. Correa, and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), no. 1, 85-93. https://doi.org/10.1016/j.camwa.2005.01.008   DOI
2 A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7   DOI
3 A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), no. 1, 305-330. https://doi.org/10.1090/S0002-9947-96-01532-2   DOI
4 A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrodinger-Maxwell equations, J. Math. Anal. Appl. 345 (2008), no. 1, 90-108. https://doi.org/10.1016/j.jmaa.2008.03.057   DOI
5 T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725-1741. https://doi.org/10.1080/03605309508821149   DOI
6 V. Benci and D. Fortunato, An eigenvalue problem for the Schrodinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 283-293. https://projecteuclid.org/euclid.tmna/1476842831   DOI
7 M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001), no. 6, 701-730.
8 G. Che and H. Chen, Existence and multiplicity of systems of Kirchhoff-type equations with general potentials, Math. Methods Appl. Sci. 40 (2017), no. 3, 775-785. https://doi.org/10.1002/mma.4007   DOI
9 G. Che and H. Chen, Infinitely many solutions for the Klein-Gordon equation with sublinear nonlinearity coupled with Born-Infeld theory, Bull. Iran. Math. Soc. (2019). https://doi.org/10.1007/s41980-019-00314-3
10 G. Che and H. Chen, Existence and multiplicity of positive solutions for Kirchhoff-Schrodinger-Poisson system with critical growth, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 78.   DOI
11 G. Che, H. Chen, and T. Wu, Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling, J. Math. Phys. 60 (2019), no. 8, 081511, 28 pp. https://doi.org/10.1063/1.5087755   DOI
12 I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0   DOI
13 G. Che, H. Chen, and T. Wu, Bound state positive solutions for a class of elliptic system with Hartree nonlinearity, Commun. Pure Appl. Anal. 19 (2020), no. 7, 3697-3722. https://doi.org/10.3934/cpaa.2020163   DOI
14 G. Che, H. Shi, and Z. Wang, Existence and concentration of positive ground states for a 1-Laplacian problem in $R^N$, Appl. Math. Lett. 100 (2020), 106045, 7 pp. https://doi.org/10.1016/j.aml.2019.106045   DOI
15 P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), no. 2, 247-262. https://doi.org/10.1007/BF02100605   DOI
16 J.-L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), 284-346, North-Holland Math. Stud., 30, North-Holland, Amsterdam, 1978.
17 L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrodinger equation on $R^N$, Indiana Univ. Math. J. 54 (2005), no. 2, 443-464.   DOI
18 R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), no. 2, 352-370. https://doi.org/10.1016/j.jfa.2005.04.005   DOI
19 G. Kirchhoff, Mechanik, Teubner, 1883.
20 Z. Liu and S. Guo, On ground state solutions for the Schrodinger-Poisson equations with critical growth, J. Math. Anal. Appl. 412 (2014), no. 1, 435-448.   DOI
21 Z. Liu and S. Guo, Existence of positive ground state solutions for Kirchhoff type problems, Nonlinear Anal. 120 (2015), 1-13. https://doi.org/10.1016/j.na.2014.12.008   DOI
22 Z. Liu, S. Guo, and Z. Zhang, Existence of ground state solutions for the Schrodinger- Poisson systems, Appl. Math. Comput. 244 (2014), 312-323.   DOI
23 A. Mao and S. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl. 383 (2011), no. 1, 239-243. https://doi.org/10.1016/j.jmaa.2011.05.021   DOI
24 A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 (2009), no. 3, 1275-1287. https://doi.org/10.1016/j.na.2008.02.011   DOI
25 J. Sun, H. Chen, and L. Yang, Positive solutions of asymptotically linear Schrodinger- Poisson systems with a radial potential vanishing at infinity, Nonlinear Anal. 74 (2011), no. 2, 413-423.   DOI
26 D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3, 379-391, and a comment on the generalized Ambrosetti- Rabinowitz condition, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 299-301. https://doi.org/10.1007/s00030-011-0129-y   DOI
27 D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 (2014), no. 4, 1168-1193. https://doi.org/10.1016/j.jde.2014.05.002   DOI
28 S. I. Pohozaev, A certain class of quasilinear hyperbolic equation, Mat. Sb. (NS). 96 (1975), 152-168,
29 D. Ruiz, The Schrodinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655-674.   DOI
30 J. Sun, H. Chen, and J. J. Nieto, On ground state solutions for some non-autonomous Schrodinger-Poisson systems, J. Differential Equations 252 (2012), no. 5, 3365-3380.   DOI
31 J. Sun and T. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations 256 (2014), no. 4, 1771-1792. https://doi.org/10.1016/j.jde.2013.12.006   DOI
32 J. Sun and T. Wu, Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 2, 435-448. https://doi.org/10.1017/S0308210515000475   DOI
33 X. H. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations 261 (2016), no. 4, 2384-2402. https://doi.org/10.1016/j.jde.2016.04.032   DOI
34 W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.
35 M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhauser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
36 L. Xu and H. Chen, Multiplicity of small negative-energy solutions for a class of nonlinear Schrodinger-Poisson systems, Appl. Math. Comput. 243 (2014), 817-824.   DOI
37 G. Zhao, X. Zhu, and Y. Li, Existence of infinitely many solutions to a class of Kirchhoff- Schrodinger-Poisson system, Appl. Math. Comput. 256 (2015), 572-581. https://doi.org/10.1016/j.amc.2015.01.038   DOI
38 W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001), no. 3, 343-358. https://doi.org/10.1007/s002290170032   DOI