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In this paper we consider a semilinear elliptic equation with a strongsingularity 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 $$0\leq F(x,s)\leq \frac{h(x)}{\Gamma(s)}\,\,\mbox{ a.e. } x\in\Omega,\, \foralls>0, $$ 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 $s>0$. Weintroduce a notion of solution to this problem in the spirit of the solutionsdefined by transposition. This definition allows us to prove the existence andthe stability of this solution, as well as its uniqueness when $F(x,s)$ isnonincreasing in $s$.

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