A remark on the existence of positive weak solution for a class of (p,q)(p,q)-Laplacian nonlinear system with sign-changing weight

In this article, we study the existence of positive weak solution for a class of (p,q)(p,q)-Laplacian system {−Δpu=λa(x)f(v),x∈Ω,−Δqv=λb(x)g(u),x∈Ω,u=v=0,x∈∂Ω, where ΔpΔp denotes the p-Laplacian operator defined by Δpz=div(∣∇z∣p−2∇z), p>1,λ>0p>1,λ>0 is a parameter and ΩΩ is a bounded domain in RN(N>1)RN(N>1) with smooth boundary ∂Ω∂Ω. Here a(x)a(x) and b(x)b(x) are C1C1 sign-changing functions that maybe negative near the boundary and ff, gg are C1C1 nondecreasing functions such that f,g:[0,∞)→[0,∞)f,g:[0,∞)→[0,∞); f(s)f(s), g(s)>0g(s)>0; s>0s>0 and for every M>0M>0,limx→∞f(Mg(x)1q−1)xp−1=0.We discuss the existence of positive weak solution when ff, gg, a(x)a(x) and b(x)b(x) satisfy certain additional conditions. We use the method of sub–supersolutions to establish our results.