具有Dirichlet边界条件的一类拟线性椭圆方程组的多重解的存在性

运用Ricceri三临界点定理,研究了一类具有Dirichlet边界条件的拟线性椭圆方程组问题,证明了该方程组在其非线性项满足某些新的条件时至少存在三个解.     

一般散度型拟线性弱椭圆方程组解的Hlder连续性

本文考虑一般散度型拟线性弱椭圆方程组弱解的正则性。利用文[6]中所发展的 Moser 迭代技巧,在一定条件下,得到弱解的局部 Hlder 连续性;特别地,对两个自变量的情形,证明了 S.Hildebrant 猜想对一般散度型拟线性弱椭圆型方程组也是对的。     

一类拟线性椭圆方程的多重解(英文)

本文考虑一类包含拟线性椭圆算子当非线性项在无穷远处是(p-1)-次线性增长时多重解的存在性.结果,利用三临界点定理,我们证明了该类方程多重解的存在性.     

拟线性椭圆方程组特征问题

在本文中,我们考虑了一类自然增长条件的拟线性椭圆方程组的特征(值)问题.这问题是二次泛函 I(u)=integral a_(αβ)(x,u)D_αu~iD_βu~idx 在限制 E 上的欧拉方程,这里 a_(αβ)(x,u)不要求关于 u 是一致有界的.我们用变分方法证明了特征问题的弱解存在性定理.证明中的困难在于 I(u) 是不可微的,除非知道 u 的有界性.     

RN上一类半线性椭圆方程组的多重解

利用对称山路引理,L2(RN)上加权Sobolev空间紧嵌入定理和一个线性耦合特征值问题,本文证明了RN上一类半线性椭圆方程组的多重解的存在性.     

大气动力学方程的周期边界问题

在本文中,我们研究一类多自变量拟线性双曲方程组的周期边界问题。利用积分估计和Schauder不动点定理,在较弱的条件下,证明了其局部连续可微解的存在性与唯一性。     

二阶非线性偏微分方程组的“拟椭圆”条件和部分正则性

如果对某λ>0、所有有界开域 T■R~n、所有矩阵 A∈M~(n×N)所有φ∈C_0~1(T,R~N),成立,∫λ|Dφ|~2dx≤∫_TA_i~α(A+Dφ)D_αφ~idx,我们就称二阶非线性偏微分方程组(*)-D_aA_i~a(Du)=0,i=1,2,…,N 满足“拟椭圆”条件。这个条件远远弱于通常的椭圆条件。■本文只假定方程组(*)满足“拟椭圆”条件(不需要椭圆条件),我们用 blowup 技巧证明了方程组的部分正则性。     

具有球对称结构的相对论欧拉方程组稳态解的存在性

对于等温气体,作者将考虑具有球对称结构的相对论欧拉方程组经典稳态解(定理1.1)和弱稳态解(定理1.2)的存在性.通过求解两个常微分方程以及比较马赫数M和1的关系,定理1.1证明了相对论欧拉方程组经典稳态解的存在性.在一个C~(2,α),α∈(0,1)稳态背景解的扰动下,定理1.2将证明高维球对称跨音速(双曲-椭圆)解的存在性.     

具Neumann边界条件的拟线性椭圆方程组的多解存在性

通过运用Ricceri的一个三临界点定理，得到了一类具变分结构的拟线性椭圆方程组的多解的存在性．     

共振下椭圆偏微分方程边值问题的可解性:无标准Landesman-Lazer条件情形

本文研究如下共振下的椭圆偏微分方程边值问题的可解性：△u＋λku＋g（x，u）＝h（x）；u＝0，x∈Ω．提出了三类非标准的Landesman－Lazer条件，证明了上述问题弱解存在性的非常一般性的结果．最有意义的应用是关于λk＝λ1的情形，在此我们使用了几种易于验证的非标准的Landesman－Lazer条件（或拟Landesman－Lazer条件）．进一步，我们提出了新的符号条件从而全面推广了Figueiredo和倪维明（［1］）的一个主要结果．     

非齐次马尔可夫链的Gesaro—极限定理

本提出了新的非齐次马尔可夫链的遍历条件，并在此条件下，首次证明了非齐次马尔可夫链转移概率的Ｇｅｓａｒｏ－极限定理。另外，本还给出了一个例子，用它表明本的条件确实比Ｄｏｅｂｌｉｎ条件要弱。     

Infinitely many solutions for polyharmonic equations of p(x)-Laplace type

We show the existence of infinitely many weak solutions to a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations via the mountain pass principle without the (AR) condition. Furthermore, we obtain infinitely many solutions to this equation based on the genus theory, introduced by Krasnoselskii and the abstract critical point theorem (a variant of Ljusternik-Schnirelman theory) under Cerami condition.     

Some remarks on nonsmooth critical point theory

A general min–max principle established by Ghoussoub is extended to the case of functionals f which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, when f satisfies a compactness condition weaker than the Palais–Smale one, i.e., the so-called Cerami condition. Moreover, an application to a class of elliptic variational–hemivariational inequalities in the resonant case is presented.      

有序Banach空间中的拟弱收敛及其应用

在有序Ｂａｎａｃｈ 空间中引入拟弱收敛,并表明拟弱收敛弱于弱收敛·　由此,在非紧性条件下,获得了非连续增算子的不动点,并将它应用于Ｈａｍｍｅｒｓｔｅｉｎ 型非线性积分方程     

Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

In this paper, we prove the existence of nontrivial nonnegative solutions to a class of elliptic equations and systems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth at ∞. The known results without the AR condition in the literature only involve nonlinear terms of polynomial growth. We will use suitable versions of the Mountain Pass Theorem and Linking Theorem introduced by Cerami (Istit. Lombardo Accad. Sci. Lett. Rend. A, 112(2):332–336, 1978 Ann. Mat. Pura Appl., 124:161–179, 1980). The Moser–Trudinger inequality plays an important role in establishing our results. Our theorems extend the results of de Figueiredo, Miyagaki, and Ruf (Calc. Var. Partial Differ. Equ., 3(2):139–153, 1995) and of de Figueiredo, do Ó, and Ruf (Indiana Univ. Math. J., 53(4):1037–1054, 2004) to the case where the nonlinear term does not satisfy the AR condition. Examples of such nonlinear terms are given in Appendix A. Thus, we have established the existence of nontrivial nonnegative solutions for a wider class of nonlinear terms.     

Homoclinic solutions for fourth order differential equations with superlinear nonlinearities

In this paper we investigate the existence of homoclinic solutions for a class of fourth order differential equations with superlinear nonlinearities. Under some superlinear conditions weaker than the well-known (AR) condition, by using the variant fountain theorem, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions.     

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

In this article, by employing Dhage iterative method embodied in current hybrid fixed point theorem (HFPT) of Dhage, we derive an algorithm for the numerical solutions via construction of a sequence of successive approximations for a fractional order boundary value problem (FBVP) with finite delay. By using this technique, we obtain existence as well as approximation of solutions under weaker partial Lipschitz and partial compactness type conditions in a partially ordered Banach space. Additionally, we prove an existence and uniqueness theorem under a weaker partial nonlinear Lipschitz condition. The assumptions and main outcomes are also illustrated by two examples.     

Sobolev空间W^m,p（Ω）中残差泛函的极值理论

本文在Sobolev空间中讨论残差泛函J(u)的概念及性质,论证了残差泛函J(u)的弱紧性、强制性和下半连续性及凸性条件.根据临界点理论在Sobolev空间中建立起该残差泛函的极值原理,给出J(u)=0极小值存在定理.此外还证明了等价定理和J(R_n(c))=0的五种等价形式.     

On a semilinear elliptic problem without (PS) condition

An existence result for semilinear elliptic problems whose associated functionals do not satisfy a Palais-Smale condition is proved. The nonlinearity of our problem fits none of the conditions in Ambrosetti and Rabinowitz (J. Funct. Anal. 14 (1973) 349), de Figueiredo et al. (J. Math. Pures Appl. 61 (1982) 41) and Gidas and Spruck (Comm. Part. Diff. Eq. 6 (1981) 883). Some truncation happens to be essential, and in the argument some new results on Liouville-type theorems are established.     

随机积分和微分方程解的存在性准则

在[1]中我们已证明了一个一般的随机不动点定理并给出了某些应用,在本文中我们将给出该结果的进一步应用.首先证明了一随机Darbo不动点定理,然后利用此定理在紧性假设下给出了非线性随机Volterra积分方程和非线性随机微分方程Cauchy问题随机解的存在性准则.我们的定理改进和推广了Lakshmikantham[3,4],Vaugham[2],De Blasi和Myjak[5]等人的结果.