scholarly journals Uniqueness of minimal energy solutions for a semilinear problem involving the fractional Laplacian

2019 ◽  
Vol 147 (7) ◽  
pp. 2925-2936 ◽  
Author(s):  
Julián Fernández Bonder ◽  
Analía Silva ◽  
Juan Spedaletti
Keyword(s):  
Fractional Laplacian ◽  
Minimal Energy ◽  
Semilinear Problem

Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian

2021 ◽  
Vol 33 (1) ◽  
pp. 141-153
Author(s):  
N. Ustinov
Keyword(s):  
Ground State ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Sufficient Conditions ◽  
Ground State Solution ◽  
Content Type ◽  
Local Case ◽  
Neumann Laplacian ◽  
Laplacian Operators ◽  
Semilinear Problem

Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω ∈ C 2 : \Omega \in C^2: ( − Δ ) S p s u ( x ) + u ( x ) = u 2 s ∗ − 1 ( x ) (-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x) . Here ( − Δ ) S p s (-\Delta )_{Sp}^s stands for the s s th power of the conventional Neumann Laplacian in Ω ⋐ R n \Omega \Subset \mathbb {R}^n , n ≥ 3 n \geq 3 , s ∈ ( 0 , 1 ) s \in (0, 1) , 2 s ∗ = 2 n / ( n − 2 s ) 2^*_s = 2n/(n-2s) . For the local case where s = 1 s = 1 , corresponding results were obtained earlier for the Neumann Laplacian and Neumann p p -Laplacian operators.

Eco-Friendly Synthesis of Pyrazoline Derivatives

2017 ◽  
Vol 9 (04) ◽  
Author(s):  
Mahesh G. Kharatmol ◽  
Deepali Jagdale
Keyword(s):  
Ionic Liquids ◽  
Microwave Irradiation ◽  
Organic Synthesis ◽  
Ultrasonic Irradiation ◽  
Minimal Energy ◽  
Anticancer Activities ◽  
Conventional Methods

Pyrazoline class of compounds serve as better moieties for an array of treatments, they have antibacterial, antifungal, antiinflammatory, antipyretic, diuretic, cardiovascular activities. Apart from these they also have anticancer activities. So, pertaining to its importance, many attempts are made to synthesize pyrazolines. Since conventional methods of organic synthesis are energy and time consuming. There are elaborate pathways for green and eco-friendly synthesis of pyrazoline derivatives including microwave irradiation, ultrasonic irradiation, grinding and use of ionic liquids which assures the synthesis of the same within much lesser time and by use of minimal energy

scholarly journals Uniform Equicontinuity for a Family of Zero Order Operators Approaching the Fractional Laplacian

2015 ◽  
Vol 40 (9) ◽  
pp. 1591-1618 ◽  
Author(s):  
Patricio Felmer ◽  
Erwin Topp
Keyword(s):  
Fractional Laplacian ◽  
Zero Order ◽  
Uniform Equicontinuity

Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian

2021 ◽  
Vol 72 (1) ◽  
Author(s):  
Shengda Zeng ◽  
Jinxia Cen ◽  
Abdon Atangana ◽  
Van Thien Nguyen
Keyword(s):  
Qualitative Analysis ◽  
Fractional Laplacian ◽  
Inclusion Problem ◽  
Elliptic Inclusion

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

On the strong maximum principle for a fractional Laplacian

2021 ◽  
Author(s):  
Nguyen Ngoc Trong ◽  
Do Duc Tan ◽  
Bui Le Trong Thanh
Keyword(s):  
Maximum Principle ◽  
Fractional Laplacian ◽  
Strong Maximum Principle

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Existence of the Gauge for Fractional Laplacian Schrödinger Operators

2021 ◽  
Author(s):  
Michael W. Frazier ◽  
Igor E. Verbitsky
Keyword(s):  
Fractional Laplacian ◽  
Schrödinger Operators ◽  
Schrodinger Operators
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