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.

A bifurcation result for non-local fractional equations

2015 ◽  
Vol 13 (04) ◽  
pp. 371-394 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Raffaella Servadei
Keyword(s):  
Existence Result ◽  
Growth Conditions ◽  
Bounded Subset ◽  
Bifurcation Theorem ◽  
Fractional Equations ◽  
Non Local ◽  
Critical Point Result ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator ◽  
Fractional Equation

In the present paper, we consider problems modeled by the following non-local fractional equation [Formula: see text] where s ∈ (0, 1) is fixed, (-Δ)sis the fractional Laplace operator, λ and μ are real parameters, Ω is an open bounded subset of ℝn, n > 2s, with Lipschitz boundary and f is a function satisfying suitable regularity and growth conditions. A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least one non-trivial and non-negative (non-positive) solution, provided the parameters λ and μ lie in a suitable range. The existence result obtained in the present paper may be seen as a bifurcation theorem, which extends some results, well known in the classical Laplace setting, to the non-local fractional framework.

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 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 Multiple Periodic Solutions of a Class of Fractional Laplacian Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ying-Xin Cui ◽  
Zhi-Qiang Wang
Keyword(s):  
Periodic Solutions ◽  
Morse Theory ◽  
Fractional Laplacian ◽  
Clark’S Theorem ◽  
Multiple Periodic Solutions ◽  
Large Period ◽  
Double Well Potential ◽  
Laplacian Equations ◽  
Fractional Equation

AbstractIn this paper, we study the existence of multiple periodic solutions for the following fractional equation:(-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}.For an even double-well potential, we establish more and more periodic solutions for a large period T. Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

scholarly journals Nontrivial solutions of superlinear nonlocal problems

2016 ◽  
Vol 28 (6) ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš ◽  
Raffaella Servadei
Keyword(s):  
Weak Solutions ◽  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Fountain Theorem ◽  
Nonlocal Equations ◽  
Laplacian Equations

AbstractWe study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

scholarly journals Regularity of Weak Solutions to Elliptic Problem with Irregular Data

2022 ◽  
Author(s):  
Abdelaziz Hellal
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Growth Conditions ◽  
Variable Exponents ◽  
Bounded Subset ◽  
Lebesgue Spaces ◽  
Nonlinear Elliptic ◽  
Regularity Of Weak Solutions ◽  
Irregular Data

This paper is concerned with the study of the nonlinear elliptic equations in a bounded subset Ω ⊂ RN Au = f, where A is an operator of Leray-Lions type acted from the space W1,p(·)0(Ω) into its dual. when the second term f belongs to Lm(·), with m(·) > 1 being small. we prove existence and regularity of weak solutions for this class of problems p(x)-growth conditions. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents.

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:$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$where ( − Δ)sis the fractional Laplace operator,s∈ (0, 1),N> 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.

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.

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.

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