Integral representation of solutions using Green function for fractional Hardy equations

Bhakta, Mousomi; Biswas, Anup; Ganguly, Debdip; Montoro, Luigi

doi:10.1016/j.jde.2020.04.022

Cited by 12 publications

(13 citation statements)

References 23 publications

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“…The fact in Lemma 3.6 above is proved in [3] for Dirichlet forms generated by rotationally symmetric α-stable processes.…”

Section: Analytic Criterion For Subcriticalitymentioning

confidence: 89%

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Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Takeda¹,

Uemura²

2021

Preprint

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Introducing the notion of extended Schrödinger spaces, we define the criticality and subcriticality of Schrödinger forms in the same manner as the recurrence and transience of Dirichlet forms, and give a sufficient condition for the subcriticality of Schrödinger forms in terms the bottom of spectrum. We define a subclass of Hardy potentials and prove that Schrödinger forms with potentials in this subclass are always critical, which leads us to optimal Hardy type inequality. We show that this definition of criticality and subcriticality is equivalent to that there exists an excessive function with respect to Schrödinger semigroup and its generating Dirichlet form through h-transform is recurrent and transient respectively. As an application, we can show the recurrence and transience of a family of Dirichlet forms by showing the criticality and subcriticaly of Schrödinger forms and show the other way around through h-transform, We give a such example with fractional Schrödinger operators with Hardy potential.

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“…The fact in Lemma 3.6 above is proved in [3] for Dirichlet forms generated by rotationally symmetric α-stable processes.…”

Section: Analytic Criterion For Subcriticalitymentioning

confidence: 89%

“…Since (E µ , D(E)∩C 0 (E)) is positive semi-definite, in particular, lower semibounded, the semigroup p µ t is strongly continuous on L 2 (E; m) ([1, Theorem 4.1]). 3 For a non-negative Borel function f…”

Section: We Remark That (Sf) Impliesmentioning

confidence: 99%

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Takeda¹,

Uemura²

2021

Preprint

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“…There is a wide literature regarding problems involving the fractional Hardy potential. Avoiding to disclose the discussion we refer to the following (far from being complete) list of works and references therein [1,2,5,9,12,14,17]. In [12] Dipierro, et al study the equation (E γ 1,0,0 ) (i.e., (E γ 1,t,0 ) with t = 0) and prove existence of a ground state solution, qualitative properties of positive solutions and asymptotic behavior of solutions at both 0 and infinity.…”

Section: Introductionmentioning

confidence: 99%

Fractional Hardy-Sobolev equations with nonhomogeneous terms

Bhakta¹,

Chakraborty²,

Pucci³

2020

Preprint

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This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:whereN−2s . Here 0 < γ < γ N,s and γ N,s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on R N , with K(0) = 1 = lim |x|→∞ K(x). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣs (R N ) of Ḣs (R N ) i.e., ( Ḣs ) ′ f, u Ḣs ≥ 0, whenever u is a nonnegative function in Ḣs (R N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and f ( Ḣs ) ′ is small enough (but f ≡ 0), we establish existence of at least two positive solutions to the above equation.

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“…and its nonlinear generalizations, see e.g. [1][2][3]24,26] and the references therein. Different definitions of weak solutions of (1.25) are used in the literature, and they usually depend on µ and the integrability properties of f , see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, no distributional identities related to point measures supported at the origin have been considered in previous papers. Related to this aspect, we point out that, in the case µ ∈ (µ 0 , 0), source terms with measures supported away from the origin on the RHS of (1.25) can be treated with the help of the Green function constructed in [3], but the construction in [3] does not yield a solution of (1.22). Moreover, as we have shown in Theorems 1.3 and 1.4, the associated distributional identities are different in the more subtle case of a point measure supported at the origin.…”

Section: Introductionmentioning

confidence: 99%

The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions

Chen

Weth

2020

Preprint

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The purpose of this paper is to study and classify singular solutions of the Poisson problem2) containing the origin. Here (−∆) s , s ∈ (0, 1), is the fractional Laplacian of order 2s, and 4) < 0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case µ = µ 0 which requires more subtle estimates than the case µ > µ 0 .

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Integral representation of solutions using Green function for fractional Hardy equations

Cited by 12 publications

References 23 publications

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Fractional Hardy-Sobolev equations with nonhomogeneous terms

The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions

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