NON-EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR ELLIPTIC INEQUALITIES OF P-LAPLACIAN

In this paper,the nonexistence of positive entire solutions for div(|Du|^1-2Du)≥q(x)f(u),x∈R^N,is establisbed,where p＞1,DU=(D,u……dnu),qsR^N→(o,∞)and f2(0,∞)→(o,∞)are continuous functions.     

凸区域上拟线性椭圆型方程的尖峰解

本文讨论奇异扰动的拟线性椭圆型方程-εΔ_pu(x)=f(u(x)),u(x)≥0,x∈Ω;u=0,x∈Ω在 Dirichlet边值条件下极小能量解的存在性和结构。其中ε>0是小参数,p>2,Δ_pu=div(|Du|~(p-2)Du),f(s)=s~q-s~(p-1),p-1相似文献   

The Cauchy Problem for the p-Laplacian Equation with a Nonlinear Source

In this paper we study the existence and uniqueness of positive solutions for the p-Laplaeian equation with nonlinear source eu/et=div(|Du|^p-2Du) u^-q,p＞2,0＜q＜∞ in the class of functions with some prescribed growth rate as|x|→∞. We also give a deseription of the large time behaviour and show that it is determined by thep ecompetition between the diffusion and the souree.     

BV solutions to a degenerate parabolic equation for image denoising

In this article, the authors consider equation ut = div(ψ(Гu)A(|Du|2)Du) -(u- I), where ψ is strictly positive and Г is a known vector-valued mapping, A: R → R is decreasing and A(s) ～ 1/√a as s → ∞. This kind of equation arises naturally from image denoising. For an initial datum I ∈ BVloc ∩ L∞, the existence of BV solutions to the initial value problem of the equation is obtained.     

Solution to Nonlinear Parabolic Equations Related to the P-Laplacian

Consider the following Cauchy problem:u_t = div(|▽u ~m |~ p-2▽u~m),(x,t) ∈ST=R~N ×(0,T),u(x,0) = μ,x ∈R~N,where 1相似文献   

NON-NEWTON FILTRATION EQUATION WITH SPECIAL MEDIUM VOID

This paper considers the existence and asymptotic estimates of global solutions and finite time blowup of local solution of non-Newton filtration equation with special medium void of the following form:(*)where△pu=div(|(△)u|p-2(△)u),Ωis a smooth bounded domain in RN(N≥3),0∈Ω,2〈p〈N,p-1〈q〈Np/N-p-1.The result of asymptotic estimate of global solution depends on the best constant in Hardy inequality.     

Boundedness of the Littlewood—Paley Operator,the Function g（f）,on BMO（T）

Littlewood-Paley operator,the function g(f),is considered as an operator on BMO(T).It is proved that if f∈BMO(T),then g(f)∈BMO(T) and there is a constant C that is independent of f such that ||g(f)||*≤C||f||*.Moreover,we have got the further results.     

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTIONS OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR SOME DEGENERATE HYPERBOLIC EQUATION

The author considers the global existence and global nonexistence of the initial-boundary value problem for some degenerate hyperbolic equation of the form utt- div(|▽u|p-2▽u)= |u|mu, (x,t) ∈ [0, ∞) ×Ωwith p ＞ 2 and m ＞ 0. He deals with the global solutions by D.H.Sattinger's potential well ideas. At the same time, when the initial energy is positive, but appropriately bounded,the global nonexistence of solutions is verified by using the analysis method.     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

凸区域上拟线性椭圆型方程的尖峰解

本文讨论奇异扰动的拟线性椭圆型方程-ε△pu(x)=f(u(x)),u(x)≥0,x∈Ω;u=0,x∈Ω在Dirichlet边值条件下极小能量解的存在性和结构.其中ε＞0是小参数,p＞2,△pu=div(|Du|p-2Du),f(s)=sq-sp-1,p-1＜q＜Np/N-p-1.Ω RN(N≥2)是有界光滑区域.当ε→0时,方程存在一个极小能量解,应用移动平面方法可以证明此解在凸区域上会变成一个尖峰解.     

Sharp Inequalities for BMO Functions

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 p ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ : I → R and any subset E I of positive measure,For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ~p and exponential bounds for the BMO class.     

Elliptic Systems with a Partially Sublinear Local Term

Let $1 0.$ This is in sharp contrast to D'Aprile and Mugnai's non-existence results.     

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri

In this paper we deal with the existence of weak solutions for the following Neumann problem¶¶$ \left\{{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) $ \left\{\begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) &; $ \mbox{in $ \mbox{in \Omega$}\\ {\partial u \over \partial \nu} = 0 $}\\ {\partial u \over \partial \nu} = 0 &; $ \mbox{on $ \mbox{on \partial \Omega$} \right. $}\end{array} \right. ¶¶ where $ \nu $ \nu is the outward unit normal to the boundary $ \partial\Omega $ \partial\Omega of the bounded open set _boxclose^N \Omega \subset \mathbb{R}^N . The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.     

新的齐型BMO, Lipschitz空间上的奇异积分算子

Let(X, d, μ) be a space of homogeneous type, BMO_A(X) and Lip_A(β,X) be the space of BMO type,lipschitz type associated with an approximation to the identity {A_t}_t0 and introduced by Duong,Yan and Tang, respectively. Assuming that T is a bounded linear operator on L~2(X), we find the sufficient condition on the kernel of T so that T is bounded from BMO(X) to BMO_A(X) and from Lip(β, X) to Lip_A(β, X). As an application, the boundedness of Calderón-Zygmund operators with nonsmooth kernels on BMO(R~n) and Lip(β, R~n) are also obtained.     

Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems

?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      17. 一类含测度的非强制拟线性椭圆方程解的存在性和不存在性      日毛吉  李翔睿  田巧玉  黄水波 《数学研究及应用》2022,42(2):173-188 本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1相似文献    18. Existence and Comparison Results for Non-Uniformly Parabolic Problems      Francesco Della Pietra  Giuseppina di Blasio 《Mediterranean Journal of Mathematics》2010,7(3):323-340 In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem $\left\{{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\right.$\left\{\begin{array}{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\end{array}\right.      19. 齐型空间上$L^1$ 与{\rm BMO}的内插空间      李刚  李文明 《数学研究及应用》2007,27(3):525-532 本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于摘要：本文讨论齐型空间上L^1与BMO的内插空间，得到下列结果：对于0〈θ〈1，1≤q≤∞，有（L^1，BMO）θ，q=Lpq，其中θ=1-1/p。      20. 具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果      储昌木  唐春雷 《数学年刊A辑(中文版)》2011,32(4):443-458 研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献

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一类含测度的非强制拟线性椭圆方程解的存在性和不存在性

本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1相似文献   

Existence and Comparison Results for Non-Uniformly Parabolic Problems

In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem

齐型空间上$L^1$ 与{\rm BMO}的内插空间

本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于摘要：本文讨论齐型空间上L^1与BMO的内插空间，得到下列结果：对于0〈θ〈1，1≤q≤∞，有（L^1，BMO）θ，q=Lpq，其中θ=1-1/p。     

具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果

研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献