On the existence of positive weak solutions for a class of chemically reacting systems with sign-changing weights

Abstract

We study the existence of positive weak solutions for a class of nonlinear systems 

-∆_pu = λ a(x) (f(v)-1/u^α), x ∈ Ω

-∆_qv=λ b(x) (g(u)-1/v^β), x ∈ Ω

u = v = 0,                          x ∈ ∂Ω

where ∆_sz=div(|z|^s-2 ∇z), s > 1, λ is a positive parameter and Ω is a bounded domain with smooth boundary, α, β ∈ (0, 1). Here a(x) and b(x) are C¹ sign-changing functions that maybe negative near the boundary and f; g are C¹ nondecreasing functions such that f; g : (0,∞) → (0,∞); f(s) > 0; g(s) > 0 for s > 0 and lim_s→∞ (fMg(s)^1/q-1)=0. We discuss the existence of positive weak solutions when f,  g, a(x) and b(x) satisfy certain additional conditions. We use the method of sub-supersolution to establish our results.