Acta mathematica scientia, Series B >
THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ
Received date: 2010-07-02
Online published: 2010-11-20
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).
In this paper, we prove the existence of at least one positive solution pair (u, v)∈H1(RN)×H1(RN) to the following semilinear elliptic system
{-Δu+u=f(x, v), x∈RN,
-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,
u∈H10(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.
LI Gong-Bao , WANG Chun-Hua . THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ[J]. Acta mathematica scientia, Series B, 2010 , 30(6) : 1917 -1936 . DOI: 10.1016/S0252-9602(10)60183-X
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