A priori bounds and existence result of positive solutions for fractional Laplacian systems

Abstract

In this paper, we consider the fractional Laplacian system          (−) α 2 u + N i=1 b i (x) ∂u ∂x i + C(x)u = f (x, v), x ∈ Ω, (−) β 2 v + N i=1 c i (x) ∂v ∂x i + D(x)v = g(x, u), x ∈ Ω, u > 0, v > 0, x ∈ Ω, u = 0, v = 0, x ∈ R N \ Ω, where Ω is a smooth bounded domain in R N , α ∈ (1, 2), β ∈ (1, 2), N > max{α, β}. Under some suitable conditions on potential functions and nonlinear terms, we use scaling method to obtain a priori bounds of positive solutions for the fractional Laplacian system with distinct fractional Laplacians.

Regularity and existence of positive solutions for a fractional system

“…Based on the estimate and the Leray-Schauder degree theory, they obtained the existence of positive solutions. For more research work on the topic of the a priori estimate, please see [18,21,20,23] and the references therein.…”