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 \({\beta\searrow 0}\) and \({\beta\nearrow 1}\). 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 \({\beta\nearrow 1}\) we recover the Alt-Caffarelli theory.
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Montenegro, M., de Queiroz, O.S. & Teixeira, E.V. Existence and regularity properties of non-isotropic singular elliptic equations. Math. Ann. 351, 215–250 (2011). https://doi.org/10.1007/s00208-010-0591-6
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DOI: https://doi.org/10.1007/s00208-010-0591-6