| Abstract: |
| In this communication, concerning a work with S. A. Marano, we investigate the existence of positive weak solutions for a Dirichlet problem driven by the fractional $(p,q)$-Laplacian operator. The problem is characterized by a reaction term that combines a weak singularity with a non-local convection dependence, specifically involving the distributional Riesz gradient of the solutions. Due to the presence of the convective term, the problem does not possess a direct variational structure. To address this difficulty, we combine sub-supersolution methods with variational techniques and truncation arguments. Furthermore, fixed-point results are employed to handle the non-local gradient term. Our main result establishes the existence of at least one positive weak solution, extending literature on fractional $(p,q)$ equations to cases involving both singular and non-local gradient terms. |
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