The Existence Of Positive Ground State Solutions For Nonlinear Kirchhoff Type Systems In R~3

In this paper, we study the existence of positive solutions of the following non-linear problem of Kirchhoff type with pure power nonlinearities: where a, b> 0 are constants, Qu=(?)Q(u,v)/(?)u,Qv=(?)Q(u,v)/(?)v and Q ∈ C1(R1 x R1,R1). Suppose 3< p< 6, Q satisfies(Q1) there exists C> 0 such that(Q2) Wu(0,1)＝Qv(1,0)＝0;(Q3) Qu(0,1)= Qv(l,0)= 0;(Q4)(u,v)>0 (?)u,v>0;(Q5) Qu(u,v)=Qv(u,v)≥0 (?)u,v≥0;(Q6) Q(tu,tv)=tpQ(u,v) (?t>0,u,v∈R1.(Q7) Q(-u,v),Q(u,v),Q(u,-v)=Q(u,v) (?)u,v∈R1.Qu(0, v)= 0, (>)v ∈ R1; Qv(u,0)= 0 (?)u∈e R1. and W(x),V(x) verifies the following hypotheses:(W1)(wx), V(x) ∈ C(R3, E) is weakly differentiable and satisfies (DW(x), x),(DV(x), x) L∞(R3)UL3/2(R3) and where (·,·) is the usual inner product in R3;(W2) for almost every x ∈ R3, W(x) +oo and the inequality is strict in a subset of positive Lebesgue measure;(W3) there exists a C> 0 such that(0.1)is the nonlocal nonlinear problem, we use the method of global compactness lemma and concentration compactness to obtain the existence of positive ground state solutions to (0.1).Our main results extend predecessors’ arguments from two sides:Fistly, we extend the existence of positive ground state solutions to the semilinear system (C.O. Alves, Local mountain pass for a class of ellipitc system, J. Math. Anal. Appl.335 (2007) 135-150.) to the nonlocal nonlinear Kirchhoff system;Next, we extend the single Kirchhoff equation (G. B. Li, H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3, JDE.257 (2014) 566-600.) to system.