On a singular $$p(x,\mathbin {\cdot })$$ -integro-differential elliptic problem

The present paper aims to establish the existence of at least two weak solutions of a nonlocal singular problem governed by a generalized integro-differential operator with singular kernel in a bounded domain \(\Omega \) of \(\mathbb {R}^N\) with Lipschitz boundary. The main variational tool is based on the Nehari manifold approach and the fibering maps analysis. Moreover, we state and prove two embedding results of the generalized fractional Sobolev spaces into generalized weighted Lebesgue spaces, which serve as pivotal components in our principal proof.