Uniqueness of positive solutions for a class of semipositone elliptic systems

We consider the system −Δu=λf(v);x∈Ω−Δv=μg(u);x∈Ωu=0=v;x∈∂Ω where ΩΩ is a ball in RNRN, N≥1N≥1, and ∂Ω∂Ω is its boundary, λ,μλ,μ are positive parameters, and f,gf,g are smooth functions that are negative at the origin and f(x)∼xpf(x)∼xp and g(x)∼xqg(x)∼xq for large xx for some p,q≥0p,q≥0 with pq<1pq<1. We establish the uniqueness of positive solutions when the parameters λλ and μμ are large.