Nonexistence results for classes of 3 × 3 elliptic systems

We consider the system −Δu=λf(v,w);x∈Ω−Δv=μg(u,w);x∈Ω−Δw=σh(u,v);x∈Ωu=v=w=0;x∈∂Ω, where ΩΩ is a ball in RN,N>1RN,N>1 and ∂Ω∂Ω is its boundary, λ,μ,σλ,μ,σ are positive parameters bounded away from zero, and f,g,hf,g,h are smooth functions that are negative at the origin (semipositone system) and satisfy certain linear growth conditions at infinity. We establish nonexistence of positive solutions when two of the parameters λ,μ,σλ,μ,σ are large. Our proofs depend on energy analysis and comparison methods.