Existence and regularity properties of non-isotropic singular elliptic equations

Abstract

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u −β , 0 < β < 1, with gradient dependence and involving a forcing term λ f (x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough, our solution is positive. For λ 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 β 0 and β 1. In the former, we show that our solutions u β converge to a C 1,1 function which is a solution to an obstacle type problem. When β 1 we recover the Alt-Caffarelli theory.

Almost minimizers for semilinear free boundary problems with variable coefficients

“…Let us mention that some results for the two phase version of were obtained in and using a monotonicity formula due to Weiss . A related nonvariational problem was also considered in .…”