Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive solutions to a fractional equation with singular nonlinearity   In this paper, we study the positive solutions to the following singular and
non local elliptic problem posed in a bounded and smooth domain $\Omega\subset
\R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll}
&(-\Delta)^s u=\lambda(K(x)u^{-\delta}+f(u))\mbox{ in }\Omega &u>0 \mbox{ in
}\Omega & u\equiv\, 0\mbox{ in }\R^N\backslash\Omega. \end{array}\right.
\end{eqnarray*} % Here $0<s<1$, $\delta>0$, $\lambda>0$ and $f\,:\,
\R^+\to\R^+$ is a positive $C^2$ function. $K\,:\, \Omega\to \R^+$ is a
H\"older continuous function in $\Omega$ which behave as ${\rm
dist}(x,\partial\Omega)^{-\beta}$ near the boundary with $0\leq \beta<2s$.
First, for any $\delta>0$ and for $\lambda>$ small enough, we prove the
existence of solutions to $(P_\lambda)$. Next, for a suitable range of values
of $\delta$, we show the existence of an unbounded connected branch of
solutions to $(P_\lambda)$ emanating from the trivial solution at $\lambda=0$.
For a certain class of nonlinearities $f$, we derive a global multiplicity
result that extends results proved in \cite{peral-al}. To establish the
results, we prove new properties which are of independent interest and deal
with the behavior and H\"older regularity of solutions to $(P_\lambda)$.

15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph