Fractional elliptic reaction-diffusion systems with coupled gradient terms and different diffusion

TL;DR

This paper analyzes the existence and nonexistence of nonnegative weak solutions to a two-equation nonlocal elliptic system driven by region-specific fractional diffusion $(-igtriangleup)^{s_i}$ with different exponents $s_1,s_2$ and gradient nonlinearities. It develops new sharp weighted Lebesgue-space estimates for fractional Poisson problems, leveraging detailed Green’s function bounds and weighted regularity to control the gradient terms. Existence is established under subcritical gradient growth and small data via a Schauder fixed-point approach in weighted Sobolev spaces, while three nonexistence results demonstrate the necessity and sharpness of the data and exponent conditions. The work advances the theory of nonlocal reaction-diffusion systems with coupled gradient terms and distinct diffusions, offering a robust framework for future nonlocal models with gradient sources and potentially broad applications in physics and applied sciences.

Abstract

In this work, we study the existence and nonexistence of nonnegative solutions to a class of nonlocal elliptic systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of type $u_i\mapsto d_i(-Δ)^{s_i}u_i$ where $0<s_1\neq s_2<1$, and the gradients of the unknowns act as source terms. Existence results are obtained by proving some fine estimates when data belong to weighted Lebesgue spaces. Those estimates are new and interesting in themselves.

Theorems & Definitions (19)

• Lemma 2.1
• Definition 2.1
• Theorem 2.1: LPPS
• Proposition 2.1: CV2
• Theorem 2.2: AtmaBirDaouLaam
• Lemma 2.2: Stein
• Proposition 2.2: BL
• Theorem 2.3: Necas or EdmundsHurri
• Theorem 3.1
• proof