Existence of solutions to a quasilinear nonlocal PDE

TL;DR

This paper investigates the existence of solutions for a new class of nonlocal quasilinear PDEs inspired by nonlinear optics, using variational methods to analyze different growth scenarios and establishing nonexistence in some cases.

Contribution

It introduces a novel nonlocal quasilinear operator and studies solution existence under various growth conditions, extending previous models in nonlinear optics.

Findings

Existence of solutions in sublinear and linear growth cases.

Nonexistence results in the sublinear case.

Application of variational methods to nonlocal PDEs.

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.