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 Three Weak Solutions for Nonlocal Fractional Laplacian Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Weak Solutions ◽  
Fractional Laplacian ◽  
Growth Conditions ◽  
Natural Growth ◽  
Bounded Subset ◽  
Critical Point Theorem ◽  
Laplacian Equations ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator ◽  
Fractional Equation

The existence of three weak solutions for the following nonlocal fractional equation(-Δ)su-λu=μf(x,u)inΩ,u=0inℝn∖Ω,is investigated, wheres∈(0,1)is fixed,(-Δ)sis the fractional Laplace operator,λandμare real parameters,Ωis an open bounded subset ofℝn,n>2s, and the functionfsatisfies some regularity and natural growth conditions. The approach is based on a three-critical-point theorem for differential functionals.

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.

scholarly journals A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

2017 ◽  
Vol 8 (1) ◽  
pp. 645-660 ◽  
Author(s):  
Alessio Fiscella
Keyword(s):  
Singular Term ◽  
Nonlocal Problem ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Subset ◽  
Sobolev Exponent ◽  
Continuous Boundary ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator ◽  
Kirchhoff Problem

Abstract In this paper, we consider the following critical nonlocal problem: \left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{% \lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)% ^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text% {in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is an open bounded subset of {\mathbb{R}^{N}} with continuous boundary, dimension {N>2s} with parameter {s\in(0,1)} , {2^{*}_{s}=2N/(N-2s)} is the fractional critical Sobolev exponent, {\lambda>0} is a real parameter, {\gamma\in(0,1)} and M models a Kirchhoff-type coefficient, while {(-\Delta)^{s}} is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

scholarly journals On the spectrum of two different fractional operators

2014 ◽  
Vol 144 (4) ◽  
pp. 831-855 ◽  
Author(s):  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Operators ◽  
Local Operators ◽  
Non Local ◽  
Definition Of ◽  
Image Position ◽  
The Laplace Operator ◽  
Fractional Type

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

scholarly journals Continuity and differentiability properties of the Nemitskii operator in Hölder spaces

1988 ◽  
Vol 30 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Bounded Subset ◽  
Holder Space ◽  
Usual Norm ◽  
Image Position ◽  
Holder Spaces ◽  
Open Bounded Subset ◽  
Differentiability Properties

Let ℝn be the n-dimensional Euclidean space with the usual norm denoted by |·| In what follows 蒆 will denote an open bounded subset of ℝn, and its closure.For α ∊(0,1], is the space of all functions such that: is called the Holder space with exponent a and is a Banach space when endowed with the norm:where ‖u‖∞ is, as usual, defined by:

ON NONLOCAL NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN

2019 ◽  
Vol 62 (1) ◽  
pp. 75-84
Author(s):  
LI MA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Fractional Laplacian ◽  
Radial Function ◽  
Nonlocal Problem ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Nonlocal Nonlinear

AbstractIn this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem $$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$ where K(x) = K(|x|) is a non-negative non-increasing continuous radial function in Rn and p > 1.

Existence results for Kirchhoff–type superlinear problems involving the fractional Laplacian

2018 ◽  
Vol 149 (04) ◽  
pp. 1061-1081 ◽  
Author(s):  
Zhang Binlin ◽  
Vicenţiu D. Rădulescu ◽  
Li Wang
Keyword(s):  
Fractional Laplacian ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Critical Groups ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Non Local ◽  
Image Position ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator

AbstractIn this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill &amp; {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill &amp; {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$where ( − Δ)sis the fractional Laplace operator,s∈ (0, 1),N&gt; 2s, Ω is an open bounded subset of ℝNwith smooth boundary ∂Ω,$M:{\open R}_0^ + \to {\open R}^ + $is a continuous function satisfying certain assumptions, andf(x,u) is superlinear at infinity. By computing the critical groups at zero and at infinity, we obtain the existence of non-trivial solutions for the above problem via Morse theory. To the best of our knowledge, our results are new in the study of Kirchhoff–type Laplacian problems.

Relaxation in an L∞-optimization problem

2003 ◽  
Vol 133 (3) ◽  
pp. 599-615 ◽  
Author(s):  
Hitoshi Ishii ◽  
Paola Loreti
Keyword(s):  
Optimization Problem ◽  
Dirichlet Boundary ◽  
Continuous Functions ◽  
Maximization Problem ◽  
Bounded Subset ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Image Position ◽  
Open Bounded Subset ◽  
Asymptotic Problem

Let Ω be an open bounded subset of Rn and f a continuous function on Ω̄ satisfying f(x) > 0 for all x ∈ Ω̄. We consider the maximization problem for the integral ∫Ωf(x)u(x)dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on ∂Ω and to the gradient constraint of the form H(Du(x)) ≤ 1, and prove that the supremum is ‘achieved’ by the viscosity solution of Ĥ(Du(x)) = 1 in Ω and u = 0 on ∂Ω, where Ĥ denotes the convex envelope of H. This result is applied to an asymptotic problem, as p → ∞, for quasi-minimizers of the integral An asymptotic problem as k → ∞ for inf is also considered, where the infimum is taken all over and the set K is given by {ξ | H(ξ) ≤ 1}.

An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

1999 ◽  
Vol 129 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Vesa Mustonen ◽  
Matti Tienari
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Minimization Problem ◽  
Lagrange Equation ◽  
Image Position ◽  
Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

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