preprint

Inserted: 19 dec 2024
Last Updated: 19 dec 2024

Year: 2024
Doi: https://doi.org/10.48550/arXiv.2412.08427

Abstract:

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 $X=\{u\in H^s_0(\Omega): u\geq 0\}$ 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.

Keywords: Variational methods, fractional Sobolev spaces, Fractional Gradient, existence of solutions, fractional divergence, Schechter-Palais-Smale sequence