Topological Methods in Nonlinear Analysis

Existence and asymptotic behaviour of ground state solutions for quasilinear Schrödinger-Poisson systems in $\mathbb R^3$

Ling Ding, Lin Li, Yi-Jie Meng, and Chang-Ling Zhuang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we are concerned with existence and asymptotic behavior of ground state in the whole space $\mathbb{R}^3$ for quasilinear Schrödinger-Poisson systems $$ \begin{cases} -\Delta u+u+K(x)\phi(x)u=a(x)f(u), & x\in \mathbb{R}^3, \\ -\mbox{div}[(1+\varepsilon^4|\nabla\phi|^2)\nabla\phi]=K(x)u^2, & x\in \mathbb{R}^3, \end{cases} $$ when the nonlinearity coefficient $\varepsilon\gt 0$ goes to zero, where $f(t)$ is asymptotically linear with respect to $t$ at infinity. Under appropriate assumptions on $K$, $a$ and $ f$, we establish existence of a ground state solution $(u_\varepsilon, \phi_{\varepsilon, K}(u_\varepsilon))$ of the above system. Furthermore, for all $\varepsilon$ sufficiently small, we show that $(u_\varepsilon, \phi_{\varepsilon, K}(u_\varepsilon))$ converges to $(u_0, \phi_{0, K}(u_0))$ which is the solution of the corresponding system for $\varepsilon=0$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 1 (2016), 241-264.

Dates
First available in Project Euclid: 23 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458740738

Digital Object Identifier
doi:10.12775/TMNA.2016.004

Mathematical Reviews number (MathSciNet)
MR3469056

Zentralblatt MATH identifier
1367.35153

Citation

Ding, Ling; Li, Lin; Meng, Yi-Jie; Zhuang, Chang-Ling. Existence and asymptotic behaviour of ground state solutions for quasilinear Schrödinger-Poisson systems in $\mathbb R^3$. Topol. Methods Nonlinear Anal. 47 (2016), no. 1, 241--264. doi:10.12775/TMNA.2016.004. https://projecteuclid.org/euclid.tmna/1458740738


Export citation

References

  • N. Akhmediev, A. Ankiewicz and, J.M. Soto-Crespo, Does the nonlinear Schrödinger equation correctly describe beam propagation? Optics Lett. 18 (1993), no. 8, 411–413.
  • A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger–Poisson problem, Commun. Contemp. Math. 10 (2008), no. 3, 391–404.
  • V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 283–293.
  • K. Benmlih and O. Kavian, Existence and asymptotic behaviour of standing waves for quasilinear Schrödinger–Poisson systems in $\mathbb{R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 3, 449–470.
  • A.M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrödinger–Maxwell equations, Mediterr. J. Math. 3 (2006), no. 3–4, 483–493.
  • G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson, J. Differential Equations 248 (2010), no. 3, 521–543.
  • G.M. Coclite, A multiplicity result for the nonlinear Schrödinger–Maxwell equations, Commun. Appl. Anal. 7 (2003), no. 2–3, 417–423.
  • ––––, A multiplicity result for the Schrödinger–Maxwell equations with negative potential, Ann. Polon. Math. 79 (2002), no. 1, 21–30.
  • D.G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $\mathbb{R}^N$, J. Differential Equations 173 (2001), no. 2, 470–494.
  • T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 5, 893–906.
  • T. D'Aprile and J. Wei, Clustered solutions around harmonic centers to a coupled elliptic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 4, 605–628.
  • ––––, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations 226 (2006), no. 1, 269–294.
  • E. DiBenedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850.
  • I. Ekeland, Convexity methods in Hamiltonian mechanics,Springer–Verlag, Berlin, 1990.
  • M. Ghimenti and A.M. Micheletti, Number and profile of low energy solutions for singularly perturbed Klein–Gordon–Maxwell systems on a Riemannian manifold, J. Differential Equations 256 (2014), 2502–2525.
  • H.A. Hauss, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 1984.
  • R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger–Poisson systems, J. Differential Equations 145 (1998), no. 1, 1–16.
  • R. Illner, H. Lange, B. Toomire and P.F. Zweifel, On quasi-linear Schrödinger–Poisson systems, Math. Methods Appl. Sci. 20 (1997), no. 14, 1223–1238.
  • 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 (1999), no. 4, 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 (2000), no. 1, 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 (2004), no. 1, 191–200.
  • P.A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.
  • D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655–674.
  • ––––, Semiclassical states for coupled Schrödinger–Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci. 15 (2005), no. 1, 141–164.
  • G. Siciliano, Multiple positive solutions for a Schrödinger–Poisson–Slater system, J. Math. Anal. Appl. 365 (2010), no. 1, 288–299.
  • 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 (1999), no. 9–10, 1731–1758.
  • J.T. Sun, H.B. Chen and J.J. Nieto, On ground state solutions for some non-autonomous Schrödinger–Poisson systems, J. Differential Equations 252 (2012), no. 5, 3365–3380.
  • P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150.
  • N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747.
  • Z.P. Wang and H.S. Zhou, Positive solution for a nonlinear stationary Schrödinger–Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 809–816.
  • K. Yosida, Functional Analysis, 6th ed., Springer–Verlag, Berlin, 1980.