The main objective of this paper is to establish the existence of weak solutions for a class of nonlocal problems driven by the integro-differential operator \({\mathcal {L}}_K^p\) , subject to nonlocal Dirichlet boundary conditions. The novelty lies in the treatment of nonlinearities involving a perturbative term, which introduces additional analytical challenges. Addressing this perturbation requires delicate functional analytic tools, including the use of a well-adapted fractional Sobolev space where the problem is properly formulated, and the application of Young measure techniques to handle weak convergence behavior and the lack of compactness. This contributes to a deeper understanding of nonlocal variational problems involving perturbations.
