Variational methods for a \(p(x,\cdot )\)-fractional bi-nonlocal problem of elliptic type

In this paper, we establish the existence of weak solutions for a class of bi-nonlocal Kirchhoff-type problems incorporating the fractional \(p(x,\cdot )\) $p(x,·)$ -Laplacian operator through variational methods. We investigate three distinct classes of Kirchhoff functions, leveraging the mountain pass theorem and the Ekeland’s variational principle to achieve our results.