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 $\Omega \subset {\mathbb{R}}^N$, $N>2s$: % \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 $0<s<1$, $\delta >0$, $\lambda >0$ and $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a positive $C^2$ function. $K:\Omega \to {\mathbb{R}}^{+}$ is a Hölder continuous function in $\Omega$ which behave as $\mathrm{dist}(x,\partial \Omega {)}^{-\beta }$ near the boundary with $0\le \beta <2s$. 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 Hölder regularity of solutions to $(P_{\lambda })$.

28 pages