Acta mathematica scientia,Series B ›› 2013, Vol. 33 ›› Issue (4): 913-928.doi: 10.1016/S0252-9602(13)60050-8

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EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

 LI Gong-Bao*, YE Hong-Yu   

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • Received:2012-07-05 Revised:2012-09-27 Online:2013-07-20 Published:2013-07-20
  • Contact: LI Gong-Bao,ligb@mail.ccnu.edu.cn E-mail:ligb@mail.ccnu.edu.cn
  • Supported by:

    Partially supported by NSFC (11071095) and Hubei Key Laboratory of Mathematical Sciences.

Abstract:

In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ D1,2(RN) × D1,2(RN) to the following semilinear elliptic system
{−△u = K(x)f(v), x ∈ RN,
−△v = K(x)g(u), x ∈ RN               (0.1)
by using a linking theorem, where K(x) is a positive function in Ls(RN) for some s > 1 and the nonnegative functions f, gC(R, R) are of quasicritical growth, superlinear at infinity. We do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.
Our main result can be viewed as a partial extension of a recent result of Alves, Souto and Montenegro in [1] concerning the existence of a positive solution to the following semilinear elliptic problem

−△u = K(x)f(u), x ∈ RN,

and a recent result of Li and Wang in [22] concerning the existence of nontrivial solutions to a semilinear elliptic system of Hamiltonian type in RN.

Key words: existence, positive solutions, elliptic system, linking geometric structure, zero mass case

CLC Number: 

  • 35J60
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