Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at $ u = 0 $

TL;DR

This paper investigates the existence, stability, and uniqueness of solutions to a semilinear elliptic equation with a strong singularity at zero, introducing a new solution concept and analyzing conditions for well-posedness.

Contribution

It introduces a novel solution framework for semilinear elliptic problems with strong singularities and establishes existence, stability, and uniqueness results under specific conditions.

Findings

Existence of solutions under certain conditions.

Stability of solutions with respect to data.

Uniqueness when the nonlinearity is nonincreasing in s.

Abstract

In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely $ \displaystyle u\geq 0 \mbox{ in } \Omega$, $ \displaystyle - div \,A(x) D u = F(x,u) \mbox{ in} \; \Omega$, $u = 0 \mbox{ on} \; \partial \Omega$, with $F(x,s)$ a Carath\'eodory function such that $$ 0\leq F(x,s)\leq \frac{h(x)}{\Gamma(s)}\,\,\mbox{ a.e. } x\in\Omega,\, \forall s>0, $$ with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0,+\infty[)$ function such that $\Gamma(0)=0$ and $\Gamma'(s)>0$ for every $s>0$. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allows us to prove the existence and the stability of this solution, as well as its uniqueness when $F(x,s)$ is nonincreasing in $s$.