Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Definition, existence, stability and uniqueness of the solution to a
  semilinear elliptic problem with a strong singularity at $ u = 0 $   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 $s$.

15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas