A remark on the existence of positive weak solution for a class of image-Laplacian nonlinear system with sign-changing weight Original Research Article

In this article, we study the existence of positive weak solution for a class of (p,q)(p,q)-Laplacian system View the MathML source{−Δpu=λa(x)f(v),x∈Ω,−Δqv=λb(x)g(u),x∈Ω,u=v=0,x∈∂Ω, Turn MathJax on where ΔpΔp denotes the p-Laplacian operator defined by View the MathML sourceΔpz=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, View the MathML sourcelimx→∞f(Mg(x)1q−1)xp−1=0. Turn MathJax on 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.