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 $Ω\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_λ)\left\{\begin{array}{lll} &(-Δ)^s u=λ(K(x)u^{-δ}+f(u))\mbox{ in }Ω&u>0 \mbox{ in }Ω& u\equiv\, 0\mbox{ in }\R^N\backslashΩ. \end{array}\right. \end{eqnarray*} % Here $0<s<1$, $δ>0$, $λ>0$ and $f\,:\, \R^+\to\R^+$ is a positive $C^2$ function. $K\,:\, Ω\to \R^+$ is a Hölder continuous function in $Ω$ which behave as ${\rm dist}(x,\partialΩ)^{-β}$ near the boundary with $0\leq β<2s$. First, for any $δ>0$ and for $λ>$ small enough, we prove the existence of solutions to $(P_λ)$. Next, for a suitable range of values of $δ$, we show the existence of an unbounded connected branch of solutions to $(P_λ)$ emanating from the trivial solution at $λ=0$. 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 Hölder regularity of solutions to $(P_λ)$.

28 pages