数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (6): 1917-1936.doi: 10.1016/S0252-9602(10)60183-X

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THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

李工宝|王春花   

  1. School of Mathematics and Statistics, |Huazhong Normal University, Wuhan 430079, |China
  • 收稿日期:2010-07-02 出版日期:2010-11-20 发布日期:2010-11-20
  • 通讯作者: Gongbao Li; E-mail: ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
  • 基金资助:

    Partially  supported  by  NSFC (10571069, 10631030) and  Hubei Key Laboratory of Mathematical Sciences.

    Partially  supported  by the fund of CCNU for PHD students(2009019).

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ

 LI Gong-Bao, WANG Chun-Hua   

  1. School of Mathematics and Statistics, |Huazhong Normal University, Wuhan 430079, |China
  • Received:2010-07-02 Online:2010-11-20 Published:2010-11-20
  • Contact: Gongbao Li; E-mail: ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
  • Supported by:

    Partially  supported  by  NSFC (10571069, 10631030) and  Hubei Key Laboratory of Mathematical Sciences.
    Partially  supported  by the fund of CCNU for PHD students(2009019).

摘要:

In this paper, we prove the existence of at least one positive solution pair (u, v)∈H1(RNH1(RN) to the following semilinear elliptic system

{-Δu+u=f(x, v),        xRN
  -Δv+v=g(x, u),      x∈RN,                      (0.1)

by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g C0(RN×R1) are that, f(x, t ) and g(x, t) are superlinear at t=0 as well as at t=+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.

Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem 

{-Δu+u=f(x, u),       x∈Ω, 
 uH10(Ω) 
where Ω(RN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 & 6.pp.925--954, 2004] concerning (0.1)  when f and g are asymptotically linear.

关键词: existence, nontrivial solution, semilinear elliptic system, without the (AR) condition

Abstract:

In this paper, we prove the existence of at least one positive solution pair (u, v)∈H1(RNH1(RN) to the following semilinear elliptic system

{-Δu+u=f(x, v),        xRN
  -Δv+v=g(x, u),      x∈RN,                      (0.1)

by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g C0(RN×R1) are that, f(x, t ) and g(x, t) are superlinear at t=0 as well as at t=+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.

Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem 

{-Δu+u=f(x, u),       x∈Ω, 
 uH10(Ω) 
where Ω(RN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 & 6.pp.925--954, 2004] concerning (0.1)  when f and g are asymptotically linear.

Key words: existence, nontrivial solution, semilinear elliptic system, without the (AR) condition

中图分类号: 

  • 35J50