Positive solutions to a fractional equation with singular nonlinearity

Abstract

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 $(P_\lambda)$.