Existence of solutions to a quasilinear nonlocal PDE

In this paper, we introduce a new class of quasilinear operators, which represents a nonlocal version of the operator studied by Stuart and Zhou [1], inspired by models in nonlinear optics. We will study the existence of at least one or two solutions in the cone $X=\{u\in H_0^s(\mathrm{\Omega }):u\ge 0\}$ using variational methods. For this purpose, we analyze two scenarios: the asymptotic sublinear and linear growth cases for the reaction term. Additionally, in the sublinear case, we establish a nonexistence result.