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.

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.

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.

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 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 nonlocal problem for a differential operator of even order with involution

2020 ◽  
Vol 26 (2) ◽  
pp. 297-307
Author(s):  
Petro I. Kalenyuk ◽  
Yaroslav O. Baranetskij ◽  
Lubov I. Kolyasa
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

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 Existence of W01,1Ω Solutions to Non-Coercivity Quasilinear Elliptic Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Jimao Xiawu ◽  
Shuibo Huang ◽  
Yingyuan Mi ◽  
Maoji Ri
Keyword(s):  
Elliptic Problem ◽  
Bounded Subset ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Open Bounded Subset

In this paper we consider the existence of W01,1Ω solutions to following kind of problems −div∇up−2∇u/1+uθp−1=fx,x∈Ω;ux=0,x∈∂Ω where Ω is an open bounded subset of RNN>2, maxp−2N+1/p−1N−1,0<θ<1 and 1<p⩽1+N−1/N1−θ+θ, f is a function which belongs to a suitable integrable space.

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