In this paper, we discuss the existence and non-existence of weak solutions to the non-linear problem with a fractional p-Laplacian introduced by the ψ-Hilfer fractional operator, by combining the technique of Nehari manifolds and fibering maps. Also, we obtain some results on the ψ-Hilfer fractional operator and the Nehari manifold via the Euler functional.

The paper deals with the following Kirchhoff‐type problem M()∬ℝ2N1pfalse(x,yfalse)false|vfalse(xfalse)−vfalse(yfalse)false|pfalse(x,yfalse)false|x−yfalse|N+pfalse(x,yfalse)sfalse(x,yfalse)dxdyfalse(−normalΔfalse)pfalse(·false)sfalse(·false)vfalse(xfalse)=μgfalse(x,vfalse)+false|vfalse|rfalse(xfalse)−2vin0.1em0.1emnormalΩ,v=0in0.1em0.1emℝN\normalΩ, where M models a Kirchhoff coefficient, false(−normalΔfalse)pfalse(·false)sfalse(·false) is a variable s(·)‐order p(·)‐fractional Laplace operator, with sfalse(·false):ℝ2N→false(0,1false) and pfalse(·false):ℝ2N→false(1,∞false). Here, normalΩ⊂ℝN is a bounded smooth domain with N > p(x, y)s(x, y) for any false(x,yfalse)∈truenormalΩ¯×truenormalΩ¯, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent ps∗false(xfalse)=Ntruep¯false(xfalse)false/false(N−trues¯false(xfalse)truep¯false(xfalse)false), given with truep¯false(xfalse)=pfalse(x,xfalse) and trues¯false(xfalse)=sfalse(x,xfalse) for x∈truenormalΩ¯. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.