Positive solution to extremal Pucci's equations with singular and gradient nonlinearity (Q1735353)

The authors establish the existence of a positive solution to \[ -\mathcal{M}^+_{\lambda,\Lambda}(D^2u) + H(x, Du) = \frac{k(x)f(u)} {u^{\alpha}} \] in a domain $\Omega$, with a solution $u>0$ in $\Omega$, and $u = 0$ on the boundary of $\Omega$. The authors assume certain conditions on $k, f$ and $H$, using the viscosity sub-and supersolution method. The main feature of this problem is that it has a singularity as well as a superlinear growth in the gradient term. The authors use the Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with a suitable stability result for viscosity solutions to outfit the singular nonlinearity. This work extends and complements recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.