In this work, we study the existence and the multiplicity of non-negative solutions for the following problem (Pλ)Lu=a(x)uq+λb(x)upinΩ,u=0,inRn\Ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} ({\mathrm{P}}_\uplambda ) \left\{ \begin{array}{ll} \mathcal {L} u = a(x) u^{q}+ \lambda b(x) u^p\quad \text {in }\Omega , \\ \\ u= 0 ,\;\; \text{ in } \,\mathbb {R}^n\setminus \Omega , \end{array} \right. \end{aligned}$$\end{document}where Ω⊂Rn(n≥2)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset \mathbb {R}^n \;(n\ge 2)$$\end{document} , is a bounded smooth domain, λ,p,q\documentclass[12pt]{minimal}
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\begin{document}$$\lambda , p, q$$\end{document} are positive real numbers, s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s\in (0,1) $$\end{document}, a,b\documentclass[12pt]{minimal}
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\begin{document}$$a,\, b$$\end{document} are continuous functions, and L\documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal {L}$$\end{document} is a nonlocal operator defined later by (1.1). We establish the existence and we give a multiplicity of solutions by constrained minimization of the Euler-Lagrange functional corresponding to the problem (Pλ)\documentclass[12pt]{minimal}
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\begin{document}$$(P_\lambda )$$\end{document}, on suitable subsets of Nehari manifold and using the fibering maps. Precisely, we show the existence of λ0>0,\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _0>0,$$\end{document} such that for all λ∈(0,λ0)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in (0,\lambda _0)$$\end{document}, problem (Pλ)\documentclass[12pt]{minimal}
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\begin{document}$$(P_\lambda )$$\end{document} has at least two non-negative solutions.
