We study the existence of least energy sign-changing solution for the fractional equation (where s ∈ (0, 1) and 2 * s is the fractional critical Sobolev exponent. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible sign-changing solutions of the equation. Keywords Fractional Laplacian, sign-changing solutions, critical exponents. 2010 Mathematical Subject Classifications: 49J35, 35S15, and 35B33. = 0, uniformly for a.e. x ∈ Ω. (H3) There exist µ ∈ (2, 2 * s ) such that 0 < µF (x, t) ≤ tf (x, t), for a.e. x ∈ Ω and all t = 0, where F (x, t) := t 0 f (x, τ )dτ .(H4) f (x,t) t is increasing in t > 0 and decreasing in t < 0, for a.e. x ∈ Ω.Remark 1.1 Note that from assumptions (H3) and (H4), we can conclude that the function H(x, t) := tf (x, t) − 2F (x, t) is non-negative, increasing in t > 0 and decreasing in t < 0. Thus, t 2 ∂ t f (x, t) − tf (x, t) = ∂ t H(x, t)t > 0, for a.e. x ∈ Ω and all t = 0.

In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p‐Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on mountain pass theorem and constrained minimization in Nehari sets.