Nonlocal Fractional P $(Z, Y)$ -Laplacian Problems With Variable Exponents: Existence Results and a Numerical Illustration

In this work, we study the existence of weak solutions for a class of nonlocal problems driven by the fractional p(z, y)-Laplacian operator with spatially variable exponents. The considered model includes a nonlinear source term depending simultaneously on local interactions and a global integral over the domain, which introduces significant analytical difficulties. Specifically, we examine the problem: $ {equation*} (P)\{ {array}{ll} ( ∫Ω h(z,w)\,dz)β(z) (-Δp(z,y))ˢw = (h(z,w))α(z), & in Ω, \\[1.0ex] w=0, & on RN Ω,\\[0.5ex] w>0, & in Ω, {array} . {equation*} $ where $ Ω ⊂ RN $ is a bounded domain, $ (-Δp(z, y))ˢ $ denotes the fractional $ p(z, y) $-Laplacian, and $ α(z) $, $ β(z) $ are spatially varying exponents. By employing fixed-point arguments, specifically Tychonoff's and Schauder's fixed-point theorems, we establish two existence results for weak solutions. To complement the theoretical analysis, we also present a two-dimensional numerical example of a simplified fractional model (p=2) featuring a nonlinear and nonlocal source inspired by the analytical structure of the studied problem.