INFINITELY MANY SOLUTIONS FOR AN ELLIPTIC PROBLEM INVOLVING CRITICAL NONLINEARITY

We study the following elliptic problem:{-div(a(x)Du)=Q(x)|u|2-2u+λu x ∈Ω ,u=0 on ΩUnder certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.     

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT

This paper is concerned with the following nonlinear Dirichlet problem:{-Δpu=|u|^p*-2 u λf(x,u) x∈Ω；u=0 x∈эΩ} whereΔp^u = div(|∧u|^p-2∧u) is the p-Laplacian of u,Ω is a bounded in R^n(n≥3），1＜p＜n, p=pn/n-p is the critical exponent for the Sobolev imbedding,λ＞0 and f(x,u)satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p=2 or f(x,u) = |u|^q-2 u, where 1＜q＜p, are generalized.     

THE SINGULAR SECOND ORDER NONLINEAR EIGENVALUE PROBLEM WITH INFINITELY MANY POSITIVE SOLUTIONS

1 IntroductionIn this paper we consider positive solutions for second order non1ineareigenVaue problem of the fOrmwhere A > 0, a,P,7,b 2 0 and p = 7P a7 6cr > 0.Positive solutions of nonlinear eigenvalue problem such as (P) have beenstudied extensiveIy by many authors (see tll-[61 and the references therein).However, to my best knowledge, less attention has been given to infinitelymany positive solutions. It must be pointed out that in all existing literaturesconcerning multiple positiv…     

关于无穷小量乘积的讨论

本文由有限个无穷小量的乘积仍是无穷小量的证明入手 ,给出无穷多个无穷小量的乘积不一定是无穷小量的例子 ,并根据这种方法得到无穷多个无穷大量的和也不一定是无穷大量的结论     

类p-Laplace方程的无穷多解的存在性

本文研究了类p-Laplace方程,利用证明其对应的变分泛函满足Cerami条件,得到了无穷多个大能量解的存在性,推广并改进了已有结果.     

Existence of equilibrium for economies with infinitely many commodities and infinitely countable consumers

This paper examines the existence of equilibria for double infinite eonomies. S.F. Richard and S. Srivastava have established the existence of equilibria for economies with infinitely countable consumers when commodity space isL ^∞ (M, M, μ). However, most Banach Lattices as commodity spaces haven’t interior points in their positive cones, so their result can’t be applied to many cases. We here consider a general Banach Lattice as commodity space and introduce a concept of equiproperness on preferences. Under the assumption the existence of equilibrium for economy is established. Finally, we examine the existence of equilibria for production economies.     

Equivalence of measures induced by infinitely divisible processes

We find sufficient conditions for the equivalence of two measures on function space induced by infinitely divisible processes. The processes are not assumed to be stochastically continuous or to have independent increments. The theorem proved here is equivalent to known results in the special case of stochastically continuous processes with independent increments.     

The limiting distribution for the infinitely deep dam with a Markovian input

The paper outlines a case for taking greater interest in the bottomless, or infinitely deep, dam model in Hydrology. It then shows that for such a model with unit withdrawals and an ergodic Markov chain input process the limiting distribution of depletion, when this exists, is a zero modified geometric distribution. This result generalises the well known result for independent inputs. The technical conditions required for the proof are satisfied for finite state space input processes and are shown to be satisfied by certain infinite state space input processes. These include as special cases examples which have a negative binomial limiting input distribution.     

p阶平均曲率算子Dirichlet问题的无穷多个解

§ 1　 IntroductionSince the Mountain Pass Theorem came out,the existence of nontrivial solutions,pos-sibly multiple,ofnonlinear elliptic equations has been extensively studied.In this paper,weconsider the following Dirichlet problem for p-(generalized) mean curvature operator:-div((1 +| u|2 ) p- 22 u) =f(x,u) ,　 x∈Ω,u∈ W1 ,p0 (Ω ) , (1 .1 )whereΩ is a bounded domain in Rn(n>p>1 ) with smooth boundary Ω.First let us recall the following Dirichletproblem for p-Laplacian:-Δpu≡ -div…     

On Powers of Stieltjes Moment Sequences,I

For a Bernstein function f the sequence s_n=f(1)·...· f(n) is a Stieltjes moment sequence with the property that all powers s_n^c,c>0 are again Stieltjes moment sequences. We prove that is Stieltjes determinate for c≤ 2, but it can be indeterminate for c>2 as is shown by the moment sequence , corresponding to the Bernstein function f(s)=s. Nevertheless there always exists a unique product convolution semigroup such that ρ_c has moments . We apply the indeterminacy of for c>2 to prove that the distribution of the product of p independent identically distributed normal random variables is indeterminate if and only if p≥ 3     

一类拟线性奇异椭圆方程无穷多解的存在性

本文研究有界区域Ω()RN上拟线性奇异椭圆方程.利用变分法,在f满足非二次条件的情况下,运用对偶喷泉定理证明了存在λ*0使得当λ∈(0,λ*)时,该方程存在无穷多个具有负能量的弱解{uk}.推广了s=0时的相应结果.     

无穷大量的性质与应用

对无穷大量进行比较,给出无穷大量等价的性质、相应的结论,并将其用于求极限,可解决形如∞∞、0?∞、1∞、00、∞0等极限未定型中的无穷小或无穷大的等价替换问题。     

On the marginal distribution of a first order autoregressive process

The question of what marginal distributions are possible for a first order autoregressive process is addressed. Results concerning the possible multimodality of the marginal distribution are obtained.     

一种无穷和的定义、性质及其在马尔科夫链中的应用

对形式级数 ∑ui无穷和的定义进行了推广 ,给出蔡查罗无穷和的概念 ,讨论了它们之间的关系 .对正项级数、交错级数的蔡氏收敛性做了完整的讨论 .并将其应用于马尔科夫链的渐近性态的研究 ,得到了更为简洁、完整的结果     

INFINITELY MANY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH STRONGLY INDEFINITE VARIATIONAL STRUCTURE

Based on the multiplicity results of Benci and Fortunato [4], we consider some elliptic systems with strongly indefinite quadratic part, and establish the existence of infinitely many nontrivial solutions in a suitable family of products of fractional Sobolev spaces.     

INFINITELY MANY SOLUTIONS WITH PEAKS FOR A FRACTIONAL SYSTEM IN R~N

In this article, we consider the following coupled fractional nonlinear Schrodinger system in R~N■,where N ≥ 2, 0 s 1, 1 p N/(N-2 s), μ1 0, μ2 0 and β∈ R is a coupling constant.We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x), Q(x), p and β. More precisely, we will show that for the attractive case,it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that P(x) and Q(x) satisfy some algebraic decay at infinity.     

Explicit estimates for the asymptotics of subexponential infinitely divisible distribution functions

In this paper, order-sharp explicit estimates are first obtained for the asymptotics at infinity of a wide class of subexponential infinitely divisible distributions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 295–301, February, 2000.     

退缩拟线性椭圆型方程某类边值问题的无穷多解

文[1]证明了如下D氏问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, u=0,x∈Ω存在非平凡解,本文讨论上述方程的另一类边界问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, g(|Du|~2)D_iu(0)(n,x_i)+h(x,u)=0,x∈Ω, (1)其中Ω∈R~n是具有光滑边界的有界区域,n(x)是Ω在x点的外法向,D_iu=u/x_i,Du=gradu=u,重复指标表示求和,与问题(1)相应的泛函为:     

INFINITELY MANY HOMOCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS WITH SYMMETRY

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓＴｈｉｓｐａｐｅｒｉｓａｎｅｘｔｅｎｓｉｏｎｏｆｔｈｅｗｏｒｋ［８］．Ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｘｉｓｔｅｎｃｅｏｆｉｎｆｉｎｉｔｅｌｙｍａｎｙｈｏｍｏｃｌｉｎｉｃｏｒｂｉｔｓｆｏｒｔｈｅｆ...     

Equivalence of the Nash conjecture for primitive and sandwiched singularities

We show that in order to prove the Nash Conjecture for sandwiched singularities it is enough to prove it for primitive singularities.