NONLOCAL FRACTIONAL P $(Z, Y)$ -LAPLACIAN PROBLEMS WITH VARIABLE EXPONENTS: EXISTENCE RESULTS AND A NUMERICAL ILLUSTRATION

Abstract

In this work, we study the existence of weak solutions for a class of nonlocal problems driven by the fractional $ \mathbf{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: $ \begin{equation*} (P)\left\{ \begin{array}{ll} \left( \int_{\Omega} \mathrm{h}(z,\mathrm{w})\,dz\right)^{\beta(z)} (-\Delta_{\mathbf{p}(z,y)})^{s}\mathrm{w} = \big(\mathrm{h}(z,\mathrm{w})\big)^{\alpha(z)}, & \text{in } \Omega, \\[1.0ex] \mathrm{w}=0, & \text{on } \mathbb{R}^{N} \setminus \Omega,\\[0.5ex] \mathrm{w}>0, & \text{in } \Omega, \end{array} \right. \end{equation*} $ where $ \Omega \subset \mathbb{R}^{N} $ is a bounded domain, $ (-\Delta_{\mathbf{p}(z, y)})^{s} $ denotes the fractional $ \mathbf{p}(z, y) $-Laplacian, and $ \alpha(z) $, $ \beta(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 ($ \mathbf{p}=2 $) featuring a nonlinear and nonlocal source inspired by the analytical structure of the studied problem.