$W^{1,∞}$-Interior Estimates for Finite Element Method on Regular Mesh

For a large class of piecewise polynomial subspaces $S^h$ defined on the regular mesh, $W^{1,∞}$-interior estimate $\|u_h\|_{1,∞,Ω_0}$ ≤ $c\|u_h\|_{-s,Ω_1}$, $u_h\in S^h{Ω_1}$ satisfying the interior Ritz equation is proved. For the finite element approximation $u_h$ (of degree $r-1$) to $u$, we have $W^{1,∞}$-interior error estimate $\|u-u_h\|_{1,∞,Ω_0}$)≤$ch^{r-1} (\|u\|_{r,∞,Ω_1}+\|u\|_{1,Ω}$). If the triangulation is strongly regular in $Ω_1$ and $r=2$ we obtain $W^{1,∞}$-interior superconvergence.     

Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent

In this paper,we consider the following Kirchhoff type problemwith critical exponent $-(a+b∫_Ω|∇u|^2dx)Δu=λu^q+u^5, in\ Ω, u=0, on\ ∂Ω$, where $Ω⊂R^3$ is a bounded smooth domain, $0< q < 1$ and the parameters $a,b,λ > 0$. We show that there exists a positive constant $T_4(a)$ depending only on a, such that for each $a > 0$ and $0 < λ < T_4(a)$, the above problem has at least one positive solution. The method we used here is based on the Nehari manifold, Ekeland's variational principle and the concentration compactness principle.     

A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight

The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{ \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω,\end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$. We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.     

一类非线性椭圆边值问题解的存在性

目前 ,对 s——拉普拉斯算子△s的研究是较为活跃的数学课题 .原因在于算子 -△s与许多物理现象有关 .比如 :反射扩散问题 ,石油提取问题等等 .基于此因 ,在文 [3]的基础上 ,我们将继续研究以下非线性边值问题在 Ls(Ω) ,( 1 2 nn+1 )中解的存在条件 .-△su +g( x,u) =f几乎处处在Ω中-〈 ,| u|s- 2 u〉 =0几乎处处在Γ上其中 f∈Ls( Ω)给定 ,Ω Rn( n 1 ) ,△su=div( | u|s- 2 u) ,g∶Ω× R→ R满足 Caratheodory条件 .本文把文 [3]关于非线性边值问题 @在 Lp( Ω) ( 2 p<+∞ )空间中解的存在性的研究推广到 Ls( Ω) ( 1 2 nn+1 )空间中 .     

一类带扰动的非线性椭圆型方程组无穷多弱解的存在性

本文在一定条件讨论了如下一类带扰动项,且被两个Laplacian算子控制的非线性椭圆方程Dirichlet问题无穷多弱解的存在性.（-△u=∣u∣α-1∣υ∣β＋1u＋f,x∈Ω,-△υ=∣u∣α＋1∣υ∣β-1υ＋g,x∈Ω,u（x）＋ υ（x）=0,x∈（e）Ω,）其中-△u：=div（▽u）,（u,υ）∈E：=H10（Ω）× H10（Ω）,（f,g）属于E的对偶空间.     

A maximum modulus theorem for the Oseen problem

Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R ³. It is proved a maximum modulus estimate of the following form: If ${{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}$ and ${p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}$ in Ω, and if ${|{\bf u}| \le M}$ on ${\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}$ as ${|{\bf x}|\to \infty }$ , then ${|{\bf u}({\bf x})|\le c M}$ in Ω. Here the constant c depends only on Ω and λ.     

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.     

The Existence Theorem for Solutions of General Boundary Value Problem of Quasi-Linear Second Order Elliptic Differential Equations

In this paper, we provide the existence theorem for solutions of general boundary value problem of quasi-linear second order elliptic differential equations in the following form: $\[\sum\limits_{i,j = 1}^n {({a_{ij}}(x,u)\frac{{\partial u}}{{\partial {x_j}}}) + a(x,u,{u_{{x_k}}}),{\rm{ }}in} {\rm{ }}\Omega \]$, $\[\alpha (x,u)\frac{{\partial u}}{{\partial \gamma }} + \beta (x,u) = 0,{\rm{ on }}\partial \Omega \]$, where \alpha(x, u) \geq 0,\alpha_u(x, u) \leq 0 and \gamma is some direction, defining on $\[\partial \Omega \]$.     

L~∞-Bounds of Solutions for Strongly Nonlinear Elliptic Problems with Two Lower Order Terms

In this work,we will prove the existence of bounded solutions in W_0~(1,p)(Ω)∩L~∞(Ω) for nonlinear elliptic equations-div(a(x,u,▽u))+g(x,u,▽u)+H(x,▽u) =f,where a,g and H are Caratheodory functions which satisfy some conditions,and the right hand side/belongs to W~(-1,q)(Ω).     

关于非线性椭圆边值问题解的存在性的注

利用非线性增生映射值域的扰动理论,本文研究了与P拉普拉斯算子△p相关的非线性椭圆边值问题@在Ls(Ω)空间中解的存在性,其中2>sp>2nn+1且n1.@-Δpu+|u(x)|p-2u(x)+g(x,u(x))=fa.e.x∈Ω-〈υ,|u|p-2u〉=0a.e.x∈Γ其中f∈Ls(Ω)给定,ΩRn,n1,Δpu=div(|u|p-2u)为P拉普拉斯算子,υ为Γ的外法向导数,g∶Ω×R→R满足Caratheodory条件.本文所讨论的方程及所用的方法是对以往一些工作的补充和延续.     

Renormalized solutions of nonlinear parabolic equations with diffuse measure data

Given a parabolic cylinder Ω × (0, T), where Ω is a bounded domain of ${\mathbb{R}^N}$ , we consider IBV problems involving equations of the type $$b(u)_{t} - \Delta_{p} u = \mu$$ where b is a increasing C ¹-function and μ is a diffuse measure. We prove the existence and uniqueness of a renormalized solution for this class of nonlinear parabolic equations.     

一类色散方程的L~p-Besov估计

设u(x,t)=(SΩf)(x,t)是一般色散初值问题(?)tu-iΩ(D)u=0,u(x,0)=f(x),(x,t)∈Rn×R的解,SΩ*f,SΩ**f是它的局部和整体极大算子.本文给出它们范数的若干估计.     

渐近线性椭圆方程组的非负解

本文讨论了如下一类渐近线性椭圆方程组{-Δu-μΔv=g(x,v),-Δv-λΔu=f(x,u),x∈Ω,u=v=0,x∈(e)Ω在H10(Ω)×H10(Ω)中至少存在一个非负非平凡的解对(u,v),其中Ω是RN中的一个光滑有界区域,f(x,t)和g(x,t)是Ω×R上的连续函数并且在无穷远处渐近线性.     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.     

具$p$-Laplace 算子的三点边值问题正解的存在性

本文用锥上的Krasnoselskii’s不动点定理研究了具有p-Laplace算子的三点边值问题:其中0<α,β<1,0<η<1且(?)p(z)=|x|p-2z,p>1．在f满足一定的增长条件下,得到方程正解的存在性．作为应用,给出两个例子．     

On a Class of Quasilinear Elliptic Equations

We consider a class of quasilinear elliptic boundary problems, including the following Modified Nonlinear Schrödinger Equation as a special case: $$\begin{cases} ∆u+ \frac{1}{2} u∆(u^2)−V(x)u+|u|^{q−2}u=0 \ \ \ in \ Ω, \\u=0 \ \ \ \ \ \ \ ~ ~ ~ on \ ∂Ω, \end{cases}$$ where $Ω$ is the entire space $\mathbb{R}^N$ or $Ω ⊂ \mathbb{R}^N$ is a bounded domain with smooth boundary, $q∈(2,22^∗]$ with $2^∗=2N/(N−2)$ being the critical Sobolev exponent and $22^∗= 4N/(N−2).$ We review the general methods developed in the last twenty years or so for the studies of existence, multiplicity, nodal property of the solutions within this range of nonlinearity up to the new critical exponent $4N/(N−2),$ which is a unique feature for this class of problems. We also discuss some related and more general problems.     

对非线性椭圆边值问题解的存在性的研究

利用非线性增生映射值域的扰动定理 ,研究了非线性椭圆边值问题 ( @)在 L2 (Ω )中解的存在性 .( @) -△pu +g( x,u) =f a.e.在Ω中-〈v,| u|p- 2 u〉∈βx( u( x) ) a.e.在Γ上其中 f∈ L2 (Ω )给定 ,Ω RN,N 1 ,△ pu=div( | u|p- 2 u)为 P拉普拉斯算子 ,1 2 NN +1 ,v为 Γ的外法向导数 ,g:Ω× R→ R满足 Caratheodory条件 ,对 x∈ Γ,βx是正常、凸、下半连续函数 φx=φ( x,· )的次微分 ,其中 φ:Γ×R→ R.     

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.