Existence of positive solutions to nonlinear elliptic systems involving gradient term and reaction potential - Ahmed Attar; Rachid Bentifour (Electronic Journal of Differential Equations, 2017)
In this note we study the elliptic system $$\displaylines{ -Δ u = z^p+f(x) \quad \text{in }Ω , \cr -Δ z = |\nabla u|^{q}+g(x) \quad \text{in }Ω , \cr z,u > 0 \quad \text{in }Ω ,\cr z=u= 0 \quad \text{on }\partial Ω, }$$ where $Ω \subset \mathbb{R}^{N}$ is a bounded domain, p>0, $0<q\le 2$ with pq<1 and f,g are two nonnegative measurable functions. The main result of this work is to analyze the interaction between the potential and the gradient terms in order to get the existence of a positive solution.