Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

Existence and asymptotic behaviour of ground state solutions for quasilinear Schrodinger-Poisson systems in $R^3$
  • Home
  • /
  • Existence and asymptotic behaviour of ground state solutions for quasilinear Schrodinger-Poisson systems in $R^3$
  1. Home /
  2. Archives /
  3. Vol 47, No 1 (March 2016) /
  4. Articles

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

Authors

  • Ling Ding
  • Lin Li
  • Jixiang Meng
  • Chun Zhu

DOI:

https://doi.org/10.12775/TMNA.2016.004

Keywords

Schrodinger-Poisson system, variational methods, ground state solution, asymptotically linear, asymptotic behavior

Abstract

In this paper, we are concerned with existence and asymptotic behavior of ground state in the whole space $\mathbb{R}^3$ for quasilinear Schr\"odinger--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 $\varepsilon\ge 0$ 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 $\varepsilon=0$.

References

N. Akhmediev, A. Ankiewicz and, J.M. Soto-Crespo, Does the nonlinear Schrodinger equation correctly describe beam propagation? Optics Lett. 18 (1993), no. 8, 411-413.

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrodinger-Poisson problem, Commun. Contemp. Math. 10 (2008), no. 3, 391-404.

V. Benci and D. Fortunato, An eigenvalue problem for the Schrodinger-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 Schrodinger-Poisson systems in R3, Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008), no. 3, 449-470.

A.M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrodinger-Maxwell equations, Mediterr. J. Math. 3 (2006), no. 3-4, 483-493.

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrodinger-Poisson, J. Differential Equations 248 (2010), no. 3, 521-543.

G.M. Coclite, A multiplicity result for the nonlinear Schrodinger-Maxwell equations, Commun. Appl. Anal. 7 (2003), no. 2-3, 417-423.

G.M. Coclite, A multiplicity result for the Schrodinger-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 RN, J. Differential Equations 173 (2001), no. 2, 470-494.

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-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. Poincare Anal. Non Lineaire 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, C1+ff 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 Schrodinger-Poisson systems, J. Differential Equations 145 (1998), no. 1, 1-16.

R. Illner, H. Lange, B. Toomire and P.F. Zweifel, On quasi-linear Schrodinger-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 RN, 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 RN, 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 Schrodinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655-674.

D. Ruiz, Semiclassical states for coupled Schrodinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci. 15 (2005), no. 1, 141-164.

G. Siciliano, Multiple positive solutions for a Schrodinger-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 RN, 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 Schrodinger-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 Schrodinger-Poisson system in R3, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 809-816.

K. Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin, 1980.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2016-03-01

How to Cite

1.
DING, Ling, LI, Lin, MENG, Jixiang and ZHU, Chun. Existence and asymptotic behaviour of ground state solutions for quasilinear Schrodinger-Poisson systems in $R^3$. Topological Methods in Nonlinear Analysis. Online. 1 March 2016. Vol. 47, no. 1, pp. 241 - 264. [Accessed 6 July 2025]. DOI 10.12775/TMNA.2016.004.
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 47, No 1 (March 2016)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 6

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop