Positive solutions to a fractional equation with singular nonlinearity

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 of R-N, N > 2s: (P-lambda) {(-Delta)(s)u = lambda (K (x)u(-delta) + f (u)) in Omega u > 0 in Omega u (math) 0 in R-N\Omega. Here 0 < s < 1, delta > 0, lambda > 0 and f : R+ -> R+ is a positive C-2 function. K : Omega -> R+ is a Holder continuous function in Omega which behave as dist(x, partial derivative Omega)(-beta) near the boundary with 0 <= 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 [2]. To establish the results, we prove new properties which are of independent interest and deal with the behavior and Holder regularity of solutions to (P-lambda). (C) 2018 Elsevier Inc. All rights reserved.