Singular semilinear elliptic equations in nondivergence form

Abstract

We study the singular semilinear equation $-Pu=\frac{f}{u^3}$ on a bounded domain $\hat{I}\text{c◯}$ with Dirichlet condition $u\equiv 0$ on $\partial \hat{I}\text{c◯}$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $\hat{I}\text{c◯}\in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(\hat{I}\text{c◯})$, under the assumptions that $\hat{I}\text{c◯}\in C^2$ 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.

33 pages