Research Article Existence of Two Positive Solutions for Two Kinds of Fractional p -Laplacian Equations Yong Wu 1 and Said Taarabti 2 1 School of Tourism Date, Guilin Tourism University, Guilin 541006, China 2 Laboratory of Systems Engineering and Information Technologies (LISTI), National School of Applied Sciences of Agadir, Ibn Zohr University, Morocco Correspondence should be addressed to Yong Wu; wuyong@gltu.edu.cn Received 28 January 2021; Revised 5 February 2021; Accepted 6 February 2021; Published 26 February 2021 Academic Editor: Jiabin Zuo Copyright © 2021 Yong Wu and Said Taarabti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator L p K . Specifically, we get multiple solutions to the following fractional p-Laplacian equations with the help of fibering maps and Nehari manifold. ð−ΔÞ s p uðxÞ = λu q + u r , u > 0 in Ω, u = 0, in ℝ N \ Ω: ( . Our results extend the previous results in some respects. 1. Introduction In this work, we are concerned with the existence of solutions for a nonlocal integro-differential equation −L p K ux ðÞ = λu q + u r , u > 0 in Ω, u = 0, in ℝ N \ Ω, ( ð1Þ where Ω is a bounded smooth domain in ℝ n , n > ps with s ∈ ð0, 1Þ, λ >0, the exponents r and q fulfill 0< q <1< r ≤ p ∗ s − 1 with the critical fractional Sobolev exponent p ∗ s = ðnp /ðn − psÞÞðn > psÞ, and L p K is a kind of nonlocal integro- differential operator defined by: L p K ux ðÞ = 2 lim ε⟶0 + ð ℝ N \B ε x ðÞ ux ðÞ − uy ðÞ j j p−2 ux ðÞ ð − uy ð ÞÞKx − y ð Þdy, ð2Þ x ∈ ℝ N , and K : ℝ N \ f0g ⟶ ð0, +∞Þ is a measurable function with the following property: γK ∈ L 1 ℝ N  where γ x ðÞ = min x jj p ,1   , there exists a k 0 > 0 such that, Kx ðÞ ≥ k 0 x jj − N+ps ð Þ for any x ∈ ℝ N \ 0 fg, Kx ðÞ = K −x ð Þ for anyx ∈ ℝ N \ 0 fg: 8 > > > > > < > > > > > : ð3Þ In recent years, the existence and multiplicity of solutions of elliptic equations in nonlinear analysis have attracted the attention of many scholars. In particular, problems with reg- ular nolinearities like u q + λu p , p, q >0 and singular nonline- arities u −q + λu p , p, q >0. At the same time, elliptic problems can be divided into two categories according to their order: integer order and fractional order. On the one hand, when s = 1, in [1], the authors consid- ered a class of semilinear problems with singular nonlinear- ities. Many results on the existence and multiplicity of solutions for singular problems have appeared in the literature Hindawi Journal of Function Spaces Volume 2021, Article ID 5572645, 9 pages https://doi.org/10.1155/2021/5572645