Taiwanese Journal of Mathematics

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS WITH THE ASYMPTOTICAL NONLINEARITY IN $\mathbb{R}^3$

Ling Ding

Full-text: Open access

Abstract

In this paper, we study nonhomogeneous Schrödinger-Poisson systems \[ \begin{cases} -\Delta u + u + K(x) \phi(x) u = a(x) f(u) + h(x), & x \in \mathbb{R}^3, \\ -\Delta \phi = K(x) u^2, & x \in \mathbb{R}^3, \end{cases}\] where $f(t)$ is either asymptotically linear or asymptotically 3-linear with respect to $t$ at infinity. Under appropriate assumptions on $K, a, f$ and $ h$, the existence of two positive solutions of the above system is obtained by using the Ekeland's variational principle and the MountainPass Theorem in critical point theory.

Article information

Source
Taiwanese J. Math., Volume 17, Number 5 (2013), 1627-1650.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706229

Digital Object Identifier
doi:10.11650/tjm.17.2013.2798

Mathematical Reviews number (MathSciNet)
MR3106034

Zentralblatt MATH identifier
1281.35026

Subjects
Primary: 45M20: Positive solutions
Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

Keywords
nonhomogeneous Schrödinger-Poisson system variational methods positive solutions asymptotically linear

Citation

Ding, Ling. MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEMS WITH THE ASYMPTOTICAL NONLINEARITY IN $\mathbb{R}^3$. Taiwanese J. Math. 17 (2013), no. 5, 1627--1650. doi:10.11650/tjm.17.2013.2798. https://projecteuclid.org/euclid.twjm/1499706229


Export citation

References

  • T. D' Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schr$\ddot{\mbox{o}}$dinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134(5) (2004), 893-906.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
  • A. Ambrosetti and D. Ruiz, Multiple bound states for the Schr$\ddot{\mbox{o}}$dinger-Poisson problem, Commun. Contemp. Math., 10(3) (2008), 391-404.
  • V. Benci and D. Fortunato, An eigenvalue problem for the Schr$\ddot{\mbox{o}}$dinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11(2) (1998), 283-293.
  • V. Benci, D. Fortunato, A. Masiello and L. Pisani, Solitons and the electromagnetic field, Math. Z., 232(1) (1999), 73-102.
  • A. M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneousSchr$\ddot{\mbox{o}}$dinger-Maxwell equations, Mediterr. J. Math., 3(3-4) (2006), 483-493.
  • D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 126(2) (1996), 443-463.
  • G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schr$\ddot{\mbox{o}}$dinger-Poisson systems, J. Differential Equations, 248(3) (2010), 521-543.
  • K. J. Chen, Exactly two entire positive solutions for a class of nonhomogeneous elliptic equations, Differential Integral Equations, 17(1-2) (2004), 1-16.
  • S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schr$\ddot{\mbox{o}}$dinger-Maxwell equations, Nonlinear Anal., 71(10) (2009), 4927-4934.
  • G. M. Coclite, A multiplicity result for the nonlinear Schr$\ddot{\mbox{o}}$dinger-Maxwell equations, Commun. Appl. Anal., 7(2-3) (2003), 417-423.
  • G. M. Coclite, A multiplicity result for the Schr$\ddot{\mbox{o}}$dinger-Maxwell equations with negative potential, Ann. Polon. Math., 79(1) (2002), 21-30.
  • D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $\mathbb{R}^N$, J. Differential Equations, 173(2) (2001), 470-494.
  • Y. B. Deng, Existence of multiple positive solutions for a semilinear equation with critical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 122(1-2) (1992), 161-175.
  • I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Springer-Verlag, Berlin, 2001.
  • L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129(4) (1999), 787-809.
  • G. B. Li and H. S. Zhou, The existence of a positive solution to asymptotically linear scalar field equations, Proc. Roy. Soc. Edinburgh Sect. A, 130(1) (2000), 81-105.
  • Z. L. Liu and Z. Q. Wang, Existence of a positive solution of an elliptic equation on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 134(1) (2004), 191-200.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.
  • D. Ruiz, The Schr$\ddot{\mbox{o}}$dinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237(2) (2006), 655-674.
  • A. Salvatore, Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3}$, Advanced Nonlinear Studies, 6(2) (2006), 157-169.
  • J. T. Sun, H. B. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schr$\ddot{\mbox{o}}$dinger-Poisson systems, J. Differential Equations, 252(5) (2012), 3365-3380.
  • C. A. Stuart and H. S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on $\mathbb{R}^N$, Comm. Partial Differential Equations, 24(9-10) (1999), 1731-1758.
  • Z. P. Wang and H. S. Zhou, Positive solution for a nonlinear stationary Schr$\ddot{\mbox{o}}$dinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18(4) (2007), 809-816.
  • Z. P. Wang and H. S. Zhou, Positive solutions for a nonhomogeneous elliptic equation on $R^N$ without $(AR)$ condition, J. Math. Anal. Appl., 353(1) (2009), 470-479.
  • X. P. Zhu and H. S. Zhou, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 115(3-4) (1990), 301-318.
  • X. P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations, 92(2) (1991), 163-178.
  • H. B. Zhu, Asymptotically linear Schr$\ddot{\mbox{o}}$dinger-Poisson systems with potential vanishing at infinity, J. Math. Appl., 380(2) (2011), 501-510.
  • H. B. Zhu, An asymptotically linear Schr$\ddot{\mbox{o}}$dinger-Poisson systems on $\mathbb{R}^3$, Nonlinear Anal., 75(13) (2012), 5261-5269.