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 and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

scholarly journals Existence of Nontrivial Solutions for Perturbed Elliptic System inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Juan Jiang
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Elliptic System ◽  
Existence Result ◽  
Nonlinear Elliptic System ◽  
Nontrivial Solutions ◽  
Nonlinear Elliptic ◽  
Sobolev Critical Exponent

We consider the perturbed nonlinear elliptic system-ε2Δu+V(x)u=K(x)|u|2*-2u+Hu(u,v),  x∈ℝN,-ε2Δv+V(x)v=K(x)|v|2*-2v+Hv(u,v),  x∈ℝN, whereN≥3,2*=2N/(N-2)is the Sobolev critical exponent. Under proper conditions onV,H, andK, the existence result and multiplicity of the system are obtained by using variational method providedεis small enough.

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 Existence of Nontrivial Solutions for Perturbedp-Laplacian Equation inℝNwith Critical Nonlinearity

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huixing Zhang ◽  
Xiaoqian Liu
Keyword(s):  
Variational Method ◽  
Positive Solution ◽  
Critical Nonlinearity ◽  
Nontrivial Solutions ◽  
Laplacian Equation

We consider a perturbedp-Laplacian equation with critical nonlinearity inℝN. By using variational method, we show that it has at least one positive solution under the proper conditions.

scholarly journals Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jiafeng Zhang ◽  
Wei Guo ◽  
Changmu Chu ◽  
Hongmin Suo
Keyword(s):  
Variational Method ◽  
Existence Of Solutions ◽  
Nonlocal Term ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Analysis Technique ◽  
General Nonlinearity

We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.

scholarly journals A new variational method for thep(x)-Laplacian equation

2005 ◽  
Vol 72 (1) ◽  
pp. 53-65 ◽  
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Existence Of Solutions ◽  
Laplacian Equation ◽  
Palais Smale Condition ◽  
Dual Variational Method

Using a dual variational method we shall show the existence of solutions to the Dirichlet problemwithout assuming Palais-Smale condition.

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.

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

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