Singular semilinear elliptic equations in nondivergence form

We study the singular semilinear equation -Pu = f/u^γ on a bounded domain $Ω$ with Dirichlet condition u ≡ 0 on ∂ Ω , where P is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that Ω∈ C1,1 and P has C¹ coefficients, as well as the uniqueness of solutions in L¹(Ω), under the assumptions that Ω∈ C² and P has C² coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo--Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities.