关于一类带有非线性边界条件的拟线性椭圆方程

本文研究了一类带有非线性边界条件的拟线性椭圆方程.在比常用的经典条件弱的假设条件下,得到该方程所对应的能量泛函存在有界Palais-Smale序列.然后,利用山路引理,得到该方程非平凡解的存在性.最后,给出一个例子,说明所给的条件比经典条件弱.     

一类带一阶差分的离散问题解的存在性

In this paper, we prove the existence of a positive solution and a negative solution for a class of second order difference equations with dependence on the first order difference. Our proofs are based on the Mountain Pass Lemma and iterative methods.     

A general Fatou Lemma

A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the finite-dimensional setting even for the case of scalar measures. Counterexamples are presented to show that the results obtained here are sharp in various aspects. Applications include systematic generalizations of the distribution of correspondences from the case of scalar Loeb spaces to the case of vector Loeb spaces and a proof of the existence of a pure strategy equilibrium in games with private and public information and with compact metric action spaces.     

A (1 + ϵ)‐approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász Local Lemma*

In his seminal result, Beck gave the first algorithmic version of the Lovász Local Lemma by giving polynomial time algorithms for 2‐coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2‐coloring nonuniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a nonuniform hypergraph in which every edge e_i intersects at most |e_i|2^αk other edges of size at most k, for some small constant α, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

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.     

On Theorem of the “Mountain Impasse” Type

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EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC EQUATION WITH CONVEX AND CONCAVE NONLINEARITIES

In this paper, we consider the existence of positive solutions of the semlinear elliptic boundary value problem with convex and concave nonlinearities, the results obtained improve and generalize some known results.     

Multipartite Ramsey numbers for odd cycles

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n?1)/2, (n?1)/2, (n?1)/2, (n?1)/2, 1) there is a monochromatic C_n, a cycle of length n. This roughly says that the Ramsey number for C_n (i.e. 2n?1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K_2n?1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009