Singular semilinear elliptic equations in nondivergence form

TL;DR

This paper investigates the existence and uniqueness of solutions to singular semilinear elliptic equations in nondivergence form, employing novel inequalities and Green function estimates.

Contribution

It introduces new methods combining nonlinear Gagliardo--Nirenberg inequalities, Green function estimates, and Kato-type inequalities for analyzing such equations.

Findings

Existence of solutions under $C^{1,1}$ domain and $C^1$ coefficients.

Uniqueness of solutions in $L^1( ext{Omega})$ under $C^2$ regularity.

Development of new analytical tools for singular elliptic equations.

Abstract

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