Nonlinear fractional elliptic problem with singular term at the boundary

In this work, we consider a nonlocal semilinear parabolic problem related to a fractional Hardy inequality with singular weight at the boundary. More precisely, we consider the problem (P)      ut + (−∆) s u = λ u p d 2s in Ω T = Ω × (0, T), u(x, 0) = u 0 (x) in Ω, u = 0 in (IR N \ Ω) × (0, T), where 0 < s < 1, Ω ⊂ IR N is a bounded regular domain, d(x) = d(x, ∂Ω), p > 0 and λ > 0 is a positive constant. The initial data u 0 0 is a nonnegative function in a suitable Lebesgue space that we make precise later. The main goal of this work is to analyze the interaction between the parameters s, p and λ in order to show the existence or the nonexistence of solution to problem (P) in a suitable sense. We will show that our results have a significative difference with respect to the local case s = 1.

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“…Finally, we conclude this section with the next Picone's inequality that the proof can be found in [26] and [5].…”

Section: The Functional Setting and Toolsmentioning

confidence: 84%

Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

2020

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“…For general domains with some boundary regularity, the fractional Hardy-Sobolev inequality is proved for s ∈ [ 1 2 , 1). See [66,67,68]. But in [65], the authors proved the fractional Hardy-Sobolev inequality for any s ∈ (0, 1), by using the fact that the domain is a convex set and its distance from the boundary is a superharmonic function.…”

Section: Some Uniqueness Results and The Rate Of The Growth Of Solutionsmentioning

confidence: 99%

Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence

Bayrami-Aminlouee¹,

Hesaaraki²,

Hamdani³

et al. 2020

Preprint

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“…For general domains with some boundary regularity, the fractional Hardy-Sobolev inequality is proved for s ∈ [ 1 2 , 1). See [41,39,40]. But in [38], the authors proved the fractional Hardy-Sobolev inequality for any s ∈ (0, 1), by using the fact that the domain is a convex set and its distance from the boundary is a superharmonic function.…”

Section: Proof Of Theorem 23mentioning

confidence: 99%

Existence of a unique positive entropy solution to a singular fractional Laplacian

Bayrami-Aminlouee

Hesaaraki

2020

Complex Variables and Elliptic Equations

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In this paper, we study the existence of a positive solution to the following elliptic problem:Hereis an open bounded domain with smooth boundary, s ∈ (0, 1), and q ∈ (0, 1). For s ∈ (0, 1 2 ), we take advantage of the convexity of Ω. The operator (−∆) s indicates the restricted fractional Laplacian, and µ is a non-negative bounded Radon measure as a source term. The assumptions on f and h will be precise later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures.2010 Mathematics Subject Classification. Primary 35R11, 35J75, 35B09, 35A01.

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Nonlinear fractional elliptic problem with singular term at the boundary

Abstract: Let Ω ⊂ R N be a bounded regular domain, 0 < s < 1 and N > 2s. We considerThe main goal of this paper is to analyze existence and non existence of solution to problem (P ) according to the values of s and q.

Cited by 11 publications

References 32 publications

Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence

Existence of a unique positive entropy solution to a singular fractional Laplacian

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