Existence and regularity properties of non-isotropic singular elliptic equations

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u(-beta), 0 < beta < 1, with gradient dependence and involving a forcing term lambda f (x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter lambda > 0 is large enough, our solution is positive. For lambda small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We also establish regularity results for the free boundary and study the asymptotic behavior of the problem as beta SE arrow 0 and beta NE arrow 1. In the former, we show that our solutions u(beta) converge to a C-1,C-1 function which is a solution to an obstacle type problem. When beta NE arrow 1 we recover the Alt-Caffarelli theory.