Positive solutions to a fractional equation with singular nonlinearity

TL;DR

This paper investigates positive solutions for a fractional elliptic equation with singular nonlinearity, establishing existence, multiplicity, and regularity results for solutions under various conditions.

Contribution

It introduces new existence and multiplicity results for fractional equations with singular nonlinearities, including properties of solutions and their regularity.

Findings

Existence of solutions for small mbda and any elta.

Unbounded connected branch of solutions from trivial solution at mbda=0.

Global multiplicity results for certain nonlinearities.

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 $(P_\lambda)$.