A sub-supersolution method for nonlinear elliptic singular systems with natural growth and some applications. (February 2016) - Carmona, José; Martínez-Aparicio, Pedro J.; Suárez, Antonio
Abstract: In this paper we give a sub-supersolution method for nonlinear elliptic singular systems with quadratic gradient whose model system is the following { − Δ u + v β | ∇ u | 2 u α = f 1 ( x, u, v ) in  Ω, − Δ v + u μ | ∇ v | 2 v γ = f 2 ( x, u, v ) in  Ω, u = v = 0 on  ∂ Ω, where Ω is a smooth bounded domain of R N ( N ≥ 3 ), β, μ ≥ 0, 0 < α, γ < 1 and regular f 1, f 2 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.