The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means o...The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions.展开更多
In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ...In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ≥ 3） is a smooth bounded domain such that the different points ai∈Ω,i= 1,2,...,k,0≤μi〈μ^-=（N-p/p）^p,λ〉0,1≤q〈p,and p^＊=p^N/N-p.The results depend crucially cn the parameters λ,q and μi for i=1,2,...,k.展开更多
This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅)...This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.展开更多
In this article,an efficient federated learning(FL)Framework in the Internet of Vehicles(IoV)is studied.In the considered model,vehicle users implement an FL algorithm by training their local FL models and sending the...In this article,an efficient federated learning(FL)Framework in the Internet of Vehicles(IoV)is studied.In the considered model,vehicle users implement an FL algorithm by training their local FL models and sending their models to a base station(BS)that generates a global FL model through the model aggregation.Since each user owns data samples with diverse sizes and different quality,it is necessary for the BS to select the proper participating users to acquire a better global model.Meanwhile,considering the high computational overhead of existing selection methods based on the gradient,the lightweight user selection scheme based on the loss decay is proposed.Due to the limited wireless bandwidth,the BS needs to select an suitable subset of users to implement the FL algorithm.Moreover,the vehicle users’computing resource that can be used for FL training is usually limited in the IoV when other multiple tasks are required to be executed.The local model training and model parameter transmission of FL will have significant effects on the latency of FL.To address this issue,the joint communication and computing optimization problem is formulated whose objective is to minimize the FL delay in the resource-constrained system.To solve the complex nonconvex problem,an algorithm based on the concave-convex procedure(CCCP)is proposed,which can achieve superior performance in the small-scale and delay-insensitive FL system.Due to the fact that the convergence rate of CCCP method is too slow in a large-scale FL system,this method is not suitable for delay-sensitive applications.To solve this issue,a block coordinate descent algorithm based on the one-step projected gradient method is proposed to decrease the complexity of the solution at the cost of light performance degrading.Simulations are conducted and numerical results show the good performance of the proposed methods.展开更多
基金supported by the National Key Research and Development Program of China(No.2021YFA1500703 to Ganglong Cui)Distinguished Professorship of Chang Jiang Scholars of Ministry of Education of China(Ganglong Cui)+1 种基金the Fundamental Research Funds for the Central Universitiesthe National Natural Science Foundation of China(No.22233001 to Ganglong Cui,No.22003043 to Xiang-Yang Liu)。
基金supported by NSFC(10771085)Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Educationthe 985 Program of Jilin University
文摘The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions.
文摘In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ≥ 3） is a smooth bounded domain such that the different points ai∈Ω,i= 1,2,...,k,0≤μi〈μ^-=（N-p/p）^p,λ〉0,1≤q〈p,and p^＊=p^N/N-p.The results depend crucially cn the parameters λ,q and μi for i=1,2,...,k.
文摘This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in R<sup>N</sup>, s(⋅) ∈ C (R<sup>N</sup> × R<sup>N</sup>, (0,1)), (-Δ)<sup>s(⋅)</sup> is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.
基金supported by the Fundamental Research Funds for the Central Universities(No.2022YJS127)the National Key Research and Development Program under Grant 2022YFB3303702the Key Program of National Natural Science Foundation of China under Grant 61931001。
文摘In this article,an efficient federated learning(FL)Framework in the Internet of Vehicles(IoV)is studied.In the considered model,vehicle users implement an FL algorithm by training their local FL models and sending their models to a base station(BS)that generates a global FL model through the model aggregation.Since each user owns data samples with diverse sizes and different quality,it is necessary for the BS to select the proper participating users to acquire a better global model.Meanwhile,considering the high computational overhead of existing selection methods based on the gradient,the lightweight user selection scheme based on the loss decay is proposed.Due to the limited wireless bandwidth,the BS needs to select an suitable subset of users to implement the FL algorithm.Moreover,the vehicle users’computing resource that can be used for FL training is usually limited in the IoV when other multiple tasks are required to be executed.The local model training and model parameter transmission of FL will have significant effects on the latency of FL.To address this issue,the joint communication and computing optimization problem is formulated whose objective is to minimize the FL delay in the resource-constrained system.To solve the complex nonconvex problem,an algorithm based on the concave-convex procedure(CCCP)is proposed,which can achieve superior performance in the small-scale and delay-insensitive FL system.Due to the fact that the convergence rate of CCCP method is too slow in a large-scale FL system,this method is not suitable for delay-sensitive applications.To solve this issue,a block coordinate descent algorithm based on the one-step projected gradient method is proposed to decrease the complexity of the solution at the cost of light performance degrading.Simulations are conducted and numerical results show the good performance of the proposed methods.
