一类带Hardy-Sobolev临界指数的奇异Kirchhoff型方程正解的存在性

本文研究了一类带Hardy-Sobolev临界指数的奇异Kirchhoff型{-(a+b∫_Ω︱▽u︱~2dx)△u=u~(5-2s)/︱x︱~s+λu~(-γ),x∈Ω,u0,x∈Ω,u=0,x∈δΩ方程其中ΩR~3是一个有界开区域且具有光滑边界δΩ,0∈Ω,a,b≥0且a+b0,λ0,0γ1,0≤s1.利用变分方法,获得了该问题的一个正局部极小解,补充了文献[1]的结果.     

p-Laplacian方程边界问题的多解性(英文)

在这一篇文章中我们讨论下面这个方程:-Δpu=λf(x,u)inΩ u=0 on Ω,其中Ω是具有光滑边界的有界开集,Ω,p>n,λ>0,且f:Ω×R→R是一个Caratheodory泛函,满足下列条件,存在t>0,使得supt∈[0,t]︱f(.,t)︱∈L∞(Ω),我们可以得出上面方程存在至少三个解。     

MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved.Furthermore,by the improved inequalities and the linking theorem,in a new k-order Sobolev-Hardy space,we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△(k)u:=-△u-(((N-2)2)/4)U/︱X︱2-1/4 sum from i=1 to(k-1) u/(︱x︱2(In(i)R/︱x︱2))=f(x,u),x ∈Ω,u=0,x ∈Ω,where 0 ∈ΩBa(0)RN,N≥3,ln(i)=i éj=1 ln(j),and R=ae(k-1),where e(0)=1,e(j) = ee(j-1) for j≥1,ln(1)=ln,ln(j)=ln ln(j-1) for j≥2.Besides,positive andnegative solutions are obtained by a variant mountain pass theorem.     

EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRDINGER-POISSON-SLATER SYSTEM

In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously.     

拟线性椭圆型问题有限元近似解的迭代加速分析

一、问题的提出 我们考察二阶拟线性椭圆型第一边值问题: -?(α(x,u)?u)=f(x,u),在Ω内, u(x)=0,在?Ω上,其中Ω是R~n(n=2,3)中有界开区域,?Ω是Ω的光滑边界。若u(x),α(x,u(x))和f(x,u(x))有足够正规性,则问题(1)的等价弱形式方程是:对于u∈H_0~1(Ω), (α(x,u)?u,?v)=(f(x,u),v),?v∈H_0~1(Ω)。 (2)这里假设α(x,u)在Ω×R中为正的且有界,内积     

A concentration behavior for semilinear elliptic systems with indefinite weight

Consider the Schrdinger system{-Δu+V1,nu=αQn(x)︱u︱α-2u︱v︱β,-Δv+V2,nv=βQn(x)︱u︱α︱v︱β-2v,u,v∈H10(Ω) where ΩR~N,α,β 1,α + β 2* and the spectrum σ(-△ + V_(i,n))(0,+∞),i = 1,2;Q_n is a bounded function and is positive in a region contained in Ω and negative outside.Moreover,the sets{Q_n 0} shrink to a point x_0∈Ω as n→+∞.We obtain the concentration phenomenon.Precisely,we first show that the system has a nontrivial solution(u_n,v_n) corresponding to Q_n,then we prove that the sequences(u_n) and(v_n) concentrate at x_0 with respect to the H~1-norm.Moreover,if the sets {Q_n 0} shrink to finite points and(u_n,v_n) is a ground state solution,then we must have that both u_n and v_n concentrate at exactly one of these points.Surprisingly,the concentration of u_n and v_n occurs at the same point.Hence,we generalize the results due to Ackermann and Szulkin.     

临界Hartree方程组基态解的存在性

本文考虑临界耦合的Hartree方程组{-△+λu=∫Ω|u(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,-△+νu=∫Ω|ν(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,其中Ω是RN中带有光滑边界的有界区域,N≥3,λ,v是常数,且满足λ,v>-λ1(Ω),λ1(Ω)是(-△,H01(Ω))的第一特征值,β> 0是耦合参数,临界指标2μ*=(2N-μ)/(N-2)来源于Hardy-LittlewoodSobolev不等式,利用变分的方法证明了临界Hartree方程组基态正解的存在性.     

网格剖分和Stokes方程的混合有限元方法

设Ω?R~2是有界区域,边界为?Ω。考虑定常Stokes方程: -γ△u+?p=f,在Ω内, divu=0, 在Ω内,(1.1) u=0, 在?Ω上,其中γ>0是常数,u代表流体速度,p为压力,f为已知的外力。这是流体力学中常见的方程,它的混合变分形式为:求u∈[H_0~1(Ω)]~2,p∈L_0~2(Ω)满足     

定常Navier-Stokes方程最小二乘Petrov-Galerkin有限元逼近的L~∞-估计

描述定常粘性不可压缩流动原始变量表述的N-S方程,为求(u,p)满足 -v△u+(u·?)u+?p=f,在Ω中, div u=0, 在Ω中, (1.1) u=0, 在?Ω上,其中u表示速度,p表示压力,f表示所给外力,v为粘性系数,Ω?R~2为有界区域。引进Sobolev空间X=(H_0~1(Ω))~2,M=L_0~2(Ω),则适合于通常混合有限元逼近的弱形式如     

哈密尔顿性和部分平方图的独立集

设G是一个图,G的部分平方图G*满足V(G*)=V(G),E(G*)=E(G)∪{uv:uv■E(G),且J(u,v)≠■},这里J(u,v)={w∈N(u)∩N(v):N(w)■N[u]∪N[v]}.利用插点方法,证明了如下结果:设G是k-连通图(k2),b是整数,0min {k,(2b-1+k)/2}(n(Y)-1),则G是哈密尔顿图.同时给出图是1-哈密尔顿的和哈密尔顿连通的相关结果.     

含有共振项的p-Laplace方程解的存在性

主要讨论一类非线性项在无穷远处渐近|u|~(p-2)u增长的p-Laplace方程的Dirichlet边值问题,利用环绕定理证明了当λ_1≤λ(λ_1为算子(-△_p,W_1,p~0(Ω))第一特征值)时,方程存在非平凡解.     

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations

ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      13. On a nonlocal problem modelling Ohmic heating in planar domains      Fei Liang  Qi Lin Liu  Yu Xiang Li 《数学学报(英文版)》2013,29(3):523-534 In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.      14. Forbidden paths and cycles in ordered graphs and matrices      János Pach  Gábor Tardos 《Israel Journal of Mathematics》2006,156(1):359-380 Assume % MathType!End!2!1! and let Ω⊂R N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.      15. Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators      Yongqiang FU  Houwang LI  Patrizia PUCCI 《数学年刊B辑(英文版)》2018,39(2):357-372 The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators: $$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$ where λ > 0 is a real parameter, Ω is a bounded domain in R N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) p s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) q s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ1 for a related system, they prove that there exists a positive solution for the problem when λ < λ1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ1-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ1.       16. Solutions to nonlinear Neumann problems with an inverse square potential      Pigong Han  Zhaoxia Liu 《Calculus of Variations and Partial Differential Equations》2007,30(3):315-352 Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem where and Q(x) is a positive continuous function on . Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions on Q,  μ, we, by means of a variational method, prove that there exists such that for every , problem (*) has a positive solution and a pair of sign-changing solutions.      17. Localization type Berezin-Toeplitz      Filippo De Mari  Krzysztof Nowak 《Journal of Geometric Analysis》2002,12(1):9-27 Let II be a bounded symmetric domain, ω ⇉ I a bounded subdomain, and let denote the weighted Bergman space of holomorphic square integrable functions on I. Let Tλ, ω be the Berezin-Toeplitz operator on with symbol χΩ and kth eigenvalue λ k (T λ,Ω). We prove that for δ1 sufficiently close to 0 and δ2 sufficiently close to 1 the estimate holds for all domains ω satisfying the condition |{z ∈ I |d(z, Ω) < ε}| ≤c|Ω|, where d is the invariant distance on I and |ω| is the invariant volume of ω. The proof is based on the fact that the operator norm of the Berezin transform is smaller than 1. Our main technical tool are some of the formulae for the Berezin transform obtained by Unterberger and Upmeier in [11].      18. Γ-Convergence of some super quadratic functionals with singular weights      Giampiero Palatucci  Yannick Sire 《Mathematische Zeitschrift》2010,266(3):533-560 We study the Γ-convergence of the following functional (p > 2) $F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega} |Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}} \int\limits_{\Omega} W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}} \int\limits_{\partial\Omega} V(Tu)d\mathcal{H}^2,$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega} |Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}} \int\limits_{\Omega} W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}} \int\limits_{\partial\Omega} V(Tu)d\mathcal{H}^2,      19. Weighted Hardy inequalities and the size of the boundary      Juha Lehrbäck 《manuscripta mathematica》2008,127(2):249-273 We establish necessary and sufficient conditions for a domain to admit the (p, β)-Hardy inequality , where d(x) = dist(x, ∂Ω) and . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ω c when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ω c was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that we try to point out by giving various examples. The author was supported in part by the Academy of Finland.      20. On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials      G. A. Afrouzi  Nguyen Thanh Chung  Z. Naghizadeh 《Mediterranean Journal of Mathematics》2014,11(3):891-903 Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C 1-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.

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On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:

$$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where λ > 0 is a real parameter, Ω is a bounded domain in R ^N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.

     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Localization type Berezin-Toeplitz

Let II be a bounded symmetric domain, ω ⇉ I a bounded subdomain, and let denote the weighted Bergman space of holomorphic square integrable functions on I. Let T_{λ, ω} be the Berezin-Toeplitz operator on with symbol χ_Ω and k^th eigenvalue λ _k (T _λ,Ω). We prove that for δ₁ sufficiently close to 0 and δ₂ sufficiently close to 1 the estimate

Γ-Convergence of some super quadratic functionals with singular weights

We study the Γ-convergence of the following functional (p > 2)

Weighted Hardy inequalities and the size of the boundary

We establish necessary and sufficient conditions for a domain to admit the (p, β)-Hardy inequality , where d(x) = dist(x, ∂Ω) and . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ω ^c when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ω ^c was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that we try to point out by giving various examples. The author was supported in part by the Academy of Finland.     

On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C ¹-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.