Singular semilinear elliptic equations in nondivergence form

We study the singular semilinear equation $-Pu=\frac{f}{u^{\gamma }}$ on a bounded domain $\mathrm{\Omega }$ with Dirichlet condition $u\equiv 0$ on $\mathrm{\partial }\mathrm{\Omega }$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $\mathrm{\Omega }\in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(\mathrm{\Omega })$, under the assumptions that $\mathrm{\Omega }\in C^2$ and $P$ has $C^2$ 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.