A sub-supersolution method for nonlinear elliptic singular systems with natural growth and some applications

In this paper we give a sub-supersolution method for nonlinear elliptic singular systems with quadratic gradient whose model system is the following {−Δu+vβ|∇u|2uα=f1(x,u,v)in  Ω,−Δv+uμ|∇v|2vγ=f2(x,u,v)in  Ω,u=v=0on  ∂Ω, where ΩΩ is a smooth bounded domain of RNRN (N≥3N≥3), β,μ≥0β,μ≥0, 0<α,γ<10<α,γ<1 and regular f1,f2f1,f2 functions. Moreover, we apply it to prove existence of solution for some systems, including the classical Lotka–Volterra models with gradient terms. Specifically, we study the competition and the symbiotic Lotka–Volterra systems.