  In this work, we present a systematic approach to investigate the existence,
multiplicity, and local gradient regularity of solutions for nonlocal
quasilinear equations with local gradient degeneracy. Our method involves an
interactive geometric argument that interplays with uniqueness property for the
corresponding homogeneous problem, leading with gradient H\"older regularity
estimates. This approach is intrinsically developed for nonlocal scenarios,
where uniqueness holds for the local homogeneous problem. We illustrate our
results by showing classes of exterior data that exhibit multiple solutions,
while also highlighting relevant cases where uniqueness is confirmed.
