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In this paper we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given aswhere Ω is an open bounded domain in R N with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, p − 1 < q ≤ p * s = N p N −ps . We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some β ∈ (0, 1), the weak solution to the problem is in C 1,β (Ω).

Abstract. In this work we study the following singular problem involving the fractional Laplace operator:where Ω ⊂ R N , N 2 be a bounded smooth domain, a ∈ C(Ω), λ is a positive parameter and 0 < γ < 1, 2 < r < 2 * s where 2 * s = N2 N−2s . Under appropriate assumptions on the function K and the function f and we employ the method of Nehari manifold in order to show the existence of T r,γ such that for all λ ∈ (0,T r,γ ) , problem (P λ ) has at least two solutions.Mathematics subject classification (2010): 34B15, 37C25, 35R20.

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