In this paper, we study the multiplicity of solutions to equations driven by a nonlocal integro-differential operator $${{\mathcal{L}}_K}$$ L K with homogeneous Dirichlet boundary conditions. In particular, using fibering maps and Nehari manifold, we obtain multiple solutions to the following fractional elliptic problem $$\left\{\begin{array}{ll}(-\triangle)^su(x)=\lambda u^q+ u^p,\quad u > 0 \; {\rm in}\; \Omega;\\ u=0, \qquad\qquad\qquad\qquad\quad \,\,\,{\rm in}\; {\mathbb{R}}^N\backslash\Omega,\end{array}\right.$$ ( - ▵ ) s u ( x ) = λ u q + u p , u > 0 in Ω ; u = 0 , in R N \ Ω , where Ω is a smooth bounded set in $${{\mathbb{R}}^n}$$ R n , n > 2s with $${s \in (0,1)}$$ s ∈ ( 0 , 1 ) , λ is a positive parameter, the exponents p and q satisfy $${0 < q < 1 < p\; \leqslant \; 2_s^\ast-1}$$ 0 < q < 1 < p ⩽ 2 s * - 1 with $${2_s^\ast=\frac{2n}{n-2s}}$$ 2 s * = 2 n n - 2 s .