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⊂RN, N>2s:(Pλ){(−Δ)su=λ(K(x)u−δ+f(u)) in Ωu>0 in Ωu≡0 in RN\\Ω. Here 0<s<1, δ>0, λ>0 and f:R+→R+ is a positive C2 function. K:Ω→R+ is a Hölder continuous function in Ω which behave as dist(x,∂Ω)−β near the boundary with 0≤β<2s.First, for any δ>0 and for λ> small enough, we prove the existence of solutions to (Pλ). Next, for a suitable range of values of δ, we show the existence of an unbounded connected branch of solutions to (Pλ) 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 [2]. To establish the results, we prove new properties which are of independent interest and deal with the behavior and Hölder regularity of solutions to (Pλ).