Positive solution to extremal Pucci’s equations with singular and gradient nonlinearity

In this paper, we establish the existence of a positive solution to − M ⁺ <inf>λ</inf> <inf>Λ</inf> (D ² u) + H(x, Du) = ^k (x)f(u ⁾ in Ω, u ^α u > 0 in Ω, u = 0 on ∂Ω, under certain conditions on k, f and H, using viscosity sub-and supersolution method. The main feature of this problem is that it has singularity as well as a superlinear growth in the gradient term. We use Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with suitable stability result for viscosity solution to outfit the singular nonlinearity. This work extends and complements the recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.