scholarly journals Existence of Solutions for a Fractional Laplacian Equation with Critical Nonlinearity

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zifei Shen ◽  
Fashun Gao
Keyword(s):  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Critical Nonlinearity ◽  
Laplacian Equation

scholarly journals Solvability of Some Boundary Value Problems for Fractional -Laplacian Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Taiyong Chen ◽  
Wenbin Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Growth Conditions ◽  
Existence Results ◽  
Laplacian Equation ◽  
Nonlinear Growth

This paper considers the existence of solutions for two boundary value problems for fractional -Laplacian equation. Under certain nonlinear growth conditions of the nonlinearity, two new existence results are obtained by using Schaefer's fixed point theorem. As an application, an example to illustrate our results is given.

scholarly journals On the Existence of Solutions for the Critical Fractional Laplacian Equation inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zifei Shen ◽  
Fashun Gao
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Lower Order ◽  
Fractional Laplacian ◽  
Critical Power ◽  
Existence Of Solutions ◽  
Order Perturbation ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Laplacian Equation

We study existence of solutions for the fractional Laplacian equation-Δsu+Vxu=u2*s-2u+fx, uinℝN,u∈Hs(RN), with critical exponent2*s=2N/(N-2s),N>2s,s∈0, 1, whereVx≥0has a potential well andf:ℝN×ℝ→ℝis a lower order perturbation of the critical poweru2*s-2u. By employing the variational method, we prove the existence of nontrivial solutions for the equation.

Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth

2020 ◽  
Vol 20 (3) ◽  
pp. 579-597
Author(s):  
Guozhen Lu ◽  
Yansheng Shen
Keyword(s):  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Laplacian Equation ◽  
Concentration Compactness Principle ◽  
Sobolev Exponent ◽  
Laplacian Operators ◽  
Special Case

AbstractIn this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:\left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right.where {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, {p>1}, {s\in(0,1)}, {ps<N}, {p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in {\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with {p<q\leq p^{\ast}_{s}} and {Q_{u}} and {Q_{v}} are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., {s=1}) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian {(-\Delta)^{s}} (i.e., {p=2} and {0<s<1}) has not been studied in the literature before.

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.

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 Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

2015 ◽  
Vol 4 (1) ◽  
pp. 37-58 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Continuous Function ◽  
Weight Function ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Smooth Boundary ◽  
Existence Results ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Beta 2 ◽  
Fractional Laplace Operator

AbstractIn this article, we study the following p-fractional Laplacian equation: $ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.

Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation

2021 ◽  
pp. 2150005
Author(s):  
Wei Dai ◽  
Zhao Liu ◽  
Pengyan Wang
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equations ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Fractional Laplacian ◽  
Unbounded Domains ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Laplacian Equations

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [Formula: see text]-Laplacian: [Formula: see text] where [Formula: see text] is a bounded or an unbounded domain which is convex in [Formula: see text]-direction, and [Formula: see text] is the fractional [Formula: see text]-Laplacian operator defined by [Formula: see text] Under some mild assumptions on the nonlinearity [Formula: see text], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [Formula: see text]-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

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