In this paper,we study the uniqueness of positive solutions to the following semilinear equations{-Δu=λ|x|^(α)ue^(u^(2)),in B_(1),u=0,onδB_(1)ueu2;in B_(1);u=0;on@B_(1);whereλ>0,α>-2;B_(1)denotes the unit ...In this paper,we study the uniqueness of positive solutions to the following semilinear equations{-Δu=λ|x|^(α)ue^(u^(2)),in B_(1),u=0,onδB_(1)ueu2;in B_(1);u=0;on@B_(1);whereλ>0,α>-2;B_(1)denotes the unit disk in R^(2):By delicate and relatively complicated computation of radial solutions to the above equation and the asymptotic expansion of solutions near the boundary of B_(1),the uniqueness of positive solutions is obtained.The results of this paper extend the uniqueness result for the semilinear equation with critical exponential growth in CHEN et al.(2022)to the case that includes a Henon term.展开更多
适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领...适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想(源于几何测度论,分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究,发展了诸如解析插值方法、正交性与双线性方法,Heisenberg不确定原理与局部化方法、微局部分析与驻相分析,催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是"Wolff及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具.展开更多
This paper investigates the following mixed local and nonlocal elliptic problem fea-turing concave-convex nonlinearities and a discontinuous right-hand side:{L(u)=H(u−μ)|u|^(p−2)u+λ|u|^(q−2)u,x∈Ω,u≥0,x∈Ω,u=0,x...This paper investigates the following mixed local and nonlocal elliptic problem fea-turing concave-convex nonlinearities and a discontinuous right-hand side:{L(u)=H(u−μ)|u|^(p−2)u+λ|u|^(q−2)u,x∈Ω,u≥0,x∈Ω,u=0,x∈R^(N)\Ω,where Ω R^(N)(N>2)is a bounded domain,μ≥0 and λ>0 are real parameters,H denotes the Heaviside function(H(t)=0 for t<0,H(t)=1 for t>0),and the mixed local and nolocal operator is defined as L(u)=−Δu+(−Δ)^(s)u with(−Δ)^(s) being the restricted fractional Laplace(0<s<1).The exponents satisfy 1<q<2<p.By employing a novel non-smooth variational principle,we establish the existence of an M-solution for this problem and identify a range for the exponent p.展开更多
基金Supported by the Natural Science Foundation of China(12571122,12061010)。
文摘In this paper,we study the uniqueness of positive solutions to the following semilinear equations{-Δu=λ|x|^(α)ue^(u^(2)),in B_(1),u=0,onδB_(1)ueu2;in B_(1);u=0;on@B_(1);whereλ>0,α>-2;B_(1)denotes the unit disk in R^(2):By delicate and relatively complicated computation of radial solutions to the above equation and the asymptotic expansion of solutions near the boundary of B_(1),the uniqueness of positive solutions is obtained.The results of this paper extend the uniqueness result for the semilinear equation with critical exponential growth in CHEN et al.(2022)to the case that includes a Henon term.
文摘适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想(源于几何测度论,分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究,发展了诸如解析插值方法、正交性与双线性方法,Heisenberg不确定原理与局部化方法、微局部分析与驻相分析,催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是"Wolff及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具.
基金Supported by the National Natural Science Foundation of China(Grant No.12361026)the Discipline Con-struction Fund Project of Northwest Minzu University.
文摘This paper investigates the following mixed local and nonlocal elliptic problem fea-turing concave-convex nonlinearities and a discontinuous right-hand side:{L(u)=H(u−μ)|u|^(p−2)u+λ|u|^(q−2)u,x∈Ω,u≥0,x∈Ω,u=0,x∈R^(N)\Ω,where Ω R^(N)(N>2)is a bounded domain,μ≥0 and λ>0 are real parameters,H denotes the Heaviside function(H(t)=0 for t<0,H(t)=1 for t>0),and the mixed local and nolocal operator is defined as L(u)=−Δu+(−Δ)^(s)u with(−Δ)^(s) being the restricted fractional Laplace(0<s<1).The exponents satisfy 1<q<2<p.By employing a novel non-smooth variational principle,we establish the existence of an M-solution for this problem and identify a range for the exponent p.
