Abstract
The aim of this paper is to establish a weak comparison principle for a class fractional p -Laplacian equation with weight. The nonlinear term f x , s > 0 is a Carathéodory function which is possibly unbounded both at the origin and at infinity and such that f x , s s 1 − p decreases with respect to s for a.e. x ∈ Ω .
Highlights
Introduction and Main ResultsIn this paper, we study a weak comparison principle for the following fractional p-Laplacian problem 8 >>< ð−ΔÞαp,βuðxÞ = f ðx, uÞ, x ∈ Ω, >>: uðxÞ uðxÞ > =0, 0, x ∈ Ω, ð1Þ x ∈ RN \ Ω, where Ω is a smooth bounded domain of RN containing the origin, 0 ≤ β < ððN − pαÞ/2Þ, 1 < p, pα < N, and f : Ω × ð0,+∞Þ ⟶ ð0,+∞Þ is a general Carathéodory function, which is possibly unbounded both at the origin and at infinity and such that f ðx, sÞs1−p decreases with respect to s for a.e. x ∈ Ω
We study a weak comparison principle for the following fractional p-Laplacian problem
The weighted fractional p-Laplacian ð−ΔÞαp,β is the pseudodifferential operator defined as ð−ΔÞαp,βuðxÞ
Summary
We study a weak comparison principle for the following fractional p-Laplacian problem. The fractional p-Laplacian ð−ΔÞαp,β, on one hand, is an extension of the local operator −div ðjxj−βj∇ujp−2∇uÞ. Fractional Laplace operator ð−ΔÞα can be defined using Fourier analysis, functional calculus, singular integrals, or Lévy processes. For more recent results of fractional Laplace elliptic problem, see [14,15,16] and the reference therein. In [21], Brezis and Oswald have shown the existence and uniqueness of a solution to a Laplace elliptic problem. Durastanti and Oliva [20] obtained the existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by. Assume that f is a nonnegative function such that f ðx, sÞs1−p is decreasing with respect to s for almost every x ∈ Ω.
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