On finite Morse index solutions of higher order fractional Lane-Emden equations

Fazly, Mostafa; Wei, Juncheng

doi:10.1353/ajm.2017.0011

Cited by 40 publications

(55 citation statements)

References 17 publications

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“…Note also that it is wellknown that there is a correspondence between the regularity of stable solutions on bounded domains and the Liouville theorems for stable solutions on the entire space, via rescaling and a blow-up procedure. For the classification of solutions of above nonlocal equations on the entire space we refer interested readers to [12,16,24,30], and for the local equations to [19][20][21] and references therein. For the case of systems, as discussed in [14,22,33], set Q := {(λ, γ) : λ, γ > 0} and define…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this section, we establish some technical integral estimates for stable solutions of systems. Most of the ideas and methods applied in this section are inspired by the ones developed in the literature, see for example [17,20,21,23,24]. We start with the Gelfand system.…”

Section: Integral Estimates For Stable Solutionsmentioning

confidence: 99%

“…As s → 1, the above inequality is consistent with the dimensions given in (1.15). For more information, we refer interested readers to Wei and the author in [24] and Davila et al in [16]. Given above, proof of the optimal dimension for regularity of extremal solutions remains an open problem.…”

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confidence: 99%

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Regularity of extremal solutions of nonlocal elliptic systems

Fazly¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem    Lu = λF (u, v) in Ω, Lv = γG (u, v) in Ω,with an integro-differential operator, including the fractional Laplacian, of the formwhen J is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of J(y) = a(y/|y|) |y| n+2s where s ∈ (0, 1) and a is any nonnegative even measurable function in L 1 (S n−1 ) that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions n < 10s and n < 2s + 4s p∓1 [p + p(p ∓ 1)] for the Gelfand and Lane-Emden systems when p > 1 (with positive and negative exponents), respectively. When s → 1, these dimensions are optimal. However, for the case of s ∈ (0, 1) getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions n < 4s. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Integral Estimates For Stable Solutionsmentioning

confidence: 99%

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Regularity of extremal solutions of nonlocal elliptic systems

Fazly¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…Here, η ǫ is a standard cut-off function η ǫ ∈ C 1 c (R + ) at the origin and at infinity that is η ǫ = 1 for ǫ < r < ǫ −1 and η ǫ = 0 for either r < ǫ/2 or r > 2/ǫ. Applying similar ideas provided in [24], we get…”

Section: Local Case: Bi-laplacian Operatormentioning

confidence: 97%

“…For the case of 1 < s < 2, there are various definitions for the fractional operator (−∆) s , see [9,12,24,47]. From the Fourier transform one can define the fractional Laplacian by…”

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Monotonicity formulas for coupled elliptic gradient systems with applications

Fazly

Shahgholian²

2019

Advances in Nonlinear Analysis

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Consider the following coupled elliptic system of equations (−∆) s u i = (u 2 1 + · · · + u 2 m )The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when m = 1 and s = 1, Gidas and Spruck in [26] and later Caffarelli, Gidas and Spruck in [6] provided the classification of solutions for Sobolev sub-critical and critical exponents. More recently, for the case of local system of equations that is when m ≥ 1 and s = 1 a similar classification result is given by Druet, Hebey and Vétois in [17] and references therein. In this paper, we derive monotonicity formulae for entire solutions of the above local, when s = 1, 2, and nonlocal, when 0 < s < 1 and 1 < s < 2, system. These monotonicity formulae are of great interests due to the fact that a counterpart of the celebrated monotonicity formula of Alt-Caffarelli-Friedman [1] seems to be challenging to derive for system of equations. Then, we apply these formulae to give a classification of finite Morse index solutions. In the end, we provide an open problem in regards to monotonicity formulae for Lane-Emden systems.

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On stable solutions of a weighted elliptic equation involving the fractional Laplacian

Quynh Nguyen,

Tuan Duong

2023

Math Methods in App Sciences

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In this paper, we study the following fractional Choquard equation with weight where and is a positive weight function satisfying at infinity, for some . We establish, in this paper, a Liouville type theorem saying that if then the above equation has no nontrivial stable solution. Our result, in particular, extends the result in [Le, Phuong. Bull. Aust. Math. Soc. 102 (2020), no. 3, 471–478.] from the Laplace operator to the fractional Laplacian.

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On finite Morse index solutions of higher order fractional Lane-Emden equations

Cited by 40 publications

References 17 publications

Regularity of extremal solutions of nonlocal elliptic systems

Regularity of extremal solutions of nonlocal elliptic systems

Monotonicity formulas for coupled elliptic gradient systems with applications

On stable solutions of a weighted elliptic equation involving the fractional Laplacian

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