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

doi:10.1016/S0252-9602(10)60183-X

Abstract

Abstract:

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

{-Δu+u=f(x, v), x∈R^N,
-Δv+v=g(x, u), x∈R^N, (0.1)

by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈ C⁰(R^N×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（R^N 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

CLC Number:

35J50

LI Gong-Bao, WANG Chun-Hua. THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ[J].Acta mathematica scientia,Series B, 2010, 30(6): 1917-1936.

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THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ

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