scholarly journals On the Existence of a Non-trivial Non-negative Global Radial Weak Solution to a Fractional Laplacian Problem with a Singular Potential

2016 ◽  
Vol 64 (2,3) ◽  
pp. 175-183
Author(s):  
Masoud Bayrami ◽  
Mahmoud Hesaaraki
Keyword(s):  
Weak Solution ◽  
Fractional Laplacian ◽  
Singular Potential

scholarly journals Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 851
Author(s):  
Robert Stegliński
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Variational Methods ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Fractional Laplacian ◽  
Unique Weak Solution ◽  
Fractional Partial Differential Equation ◽  
Integrodifferential Operator

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods.

EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS

2016 ◽  
Vol 102 (3) ◽  
pp. 392-404
Author(s):  
V. RAGHAVENDRA ◽  
RASMITA KAR
Keyword(s):  
Weak Solution ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Bounded Subset ◽  
Lipschitz Boundary ◽  
Integrodifferential Operator ◽  
Image Position ◽  
Laplacian Equations ◽  
Open Bounded Subset ◽  
Fractional Type

We study the existence of a weak solution of a nonlocal problem$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$where${\mathcal{L}}_{k}$is a general nonlocal integrodifferential operator of fractional type,$\unicode[STIX]{x1D707}$is a real parameter and$\unicode[STIX]{x1D6FA}$is an open bounded subset of$\mathbb{R}^{n}$($n>2s$, where$s\in (0,1)$is fixed) with Lipschitz boundary$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. Here$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$and$h:\mathbb{R}\rightarrow \mathbb{R}$are functions satisfying suitable hypotheses.

Applications of monotone operators to a class of fractional Laplacian equation

2015 ◽  
Vol 26 (07) ◽  
pp. 1550043
Author(s):  
V. Raghavendra ◽  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Differential Operator ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Monotone Operators ◽  
Bounded Subset ◽  
Laplacian Equation ◽  
Continuous Boundary ◽  
Open Bounded Subset ◽  
Fractional Type

In this study we establish the existence of a weak solution for a class of nonlocal problem [Formula: see text] where [Formula: see text] is a general nonlocal integro-differential operator of fractional type, λ is a real parameter, Ω is an open bounded subset of ℝn(n > 2s, where s ∈(0, 1) is fixed) with continuous boundary ∂Ω. Here f, g1: Ω → ℝ and h : ℝ → ℝ are functions satisfying suitable hypotheses.

scholarly journals Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation

Annals of PDE ◽  
2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Maria Colombo ◽  
Silja Haffter
Keyword(s):  
Weak Solution ◽  
Hausdorff Dimension ◽  
Fractional Laplacian ◽  
Compact Set ◽  
Singular Set ◽  
Suitable Weak Solution

AbstractWe consider the SQG equation with dissipation given by a fractional Laplacian of order $$\alpha <\frac{1}{2}$$ α < 1 2 . We introduce a notion of suitable weak solution, which exists for every $$L^2$$ L 2 initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most $$\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) $$ 1 2 α 1 + α α ( 1 - 2 α ) + 2 .

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

2019 ◽  
Vol 19 (1) ◽  
pp. 197-217 ◽  
Author(s):  
Caifeng Zhang
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Existence Of Weak Solutions ◽  
Laplacian Equation ◽  
Trudinger Inequality ◽  
Dimension One

Abstract In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in {W^{s,p}(\mathbb{R}^{N})} . Define \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where {s\in(0,1)} , {sp=N} , {\alpha\in[0,\alpha_{*})} and \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where {V(x)} has a positive lower bound, {f(x,t)} behaves like {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and {\varepsilon>0} . Moreover, we also derive a weak solution with negative energy.

scholarly journals Extinction behavior for a parabolic $ p $-Laplacian equation with gradient source and singular potential

2021 ◽  
Vol 7 (1) ◽  
pp. 915-924
Author(s):  
Dengming Liu ◽  
◽  
Luo Yang
Keyword(s):  
Weak Solution ◽  
Sobolev Inequality ◽  
Singular Potential ◽  
Energy Estimate ◽  
Differential Inequalities ◽  
Laplacian Equation ◽  
Solution Methods

<abstract><p>We concern with the extinction behavior of the solution for a parabolic $ p $-Laplacian equation with gradient source and singular potential. By energy estimate approach, Hardy-Littlewood-Sobolev inequality, a series of ordinary differential inequalities, and super-solution and sub-solution methods, we obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.</p></abstract>

Boundary blow-up solutions to nonlocal elliptic equations with gradient nonlinearity

2016 ◽  
Vol 19 (05) ◽  
pp. 1650051 ◽  
Author(s):  
Huyuan Chen ◽  
Guangying Lv
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Elliptic Equations ◽  
Fractional Laplacian ◽  
Blow Up ◽  
Normal Vector ◽  
Open Domain ◽  
Extra Condition ◽  
Hölder Continuous ◽  
Gradient Nonlinearity

Let [Formula: see text], [Formula: see text] be a bounded open domain in [Formula: see text] ([Formula: see text]) with a [Formula: see text] boundary [Formula: see text] and [Formula: see text] be a Hausdorff measure on [Formula: see text]. We denote by [Formula: see text] [Formula: see text] where [Formula: see text] is the unit inward normal vector of [Formula: see text] at point [Formula: see text] and [Formula: see text] Our purpose of this paper is to study a nonlocal elliptic problem involving gradient nonlinearity [Formula: see text] where [Formula: see text], the operator [Formula: see text] is the fractional Laplacian and [Formula: see text] is a locally Hölder continuous function satisfying some extra condition. We show that the above problem admits a nonnegative weak solution [Formula: see text], which is a classical solution of [Formula: see text]

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

Sign in / Sign up

Export Citation Format

Share Document