ON THE SINGULAR VARIATIONAL PROBLEMS

The authors deal with the singular variational problem S(a,b,λ0):=infu∈E,u(≡／)0 ∫RN(||X|-a(△)u|m ∫|x|-(a 1)m|u|m)dx/(∫RN||X|-bU|P dx)m/p as well as (S)=(S)(a,b,λ1,λ2):=u,ν,E∈,u(u,ν)(≡／)(1,1) ∫RN J(u,ν)dx/(∫RN|x|-bp|u|α|ν|βdx)m/p, whereJ(u, v) = ||x|- au|m λ1|x|- (a 1)m|u|m ||x|- av|m λ2|x|- (a 1)m|v|m,N ≥ m 1 ＞ 2, 0 ≤ a ＜ N-m/m, a ≤ b ＜ a 1 and p = p(a,b) = α β =Nm/N-m m(b-a), α, β≥ 1, E = D1,mα(RN). The aim of this paper is to show the existence of minimizer for S(a, b, A0) and S(a, b, λ1, λ2).     

非线性临界Kirchhoff型问题的正基态解

该文研究如下Kirchhoff型方程-(a+b∫R3 |▽u|2dx)Δu+V(x)u=|u|p-2U+ε|u|4u,x∈ R3,u∈ H1(R3),其中a＞0,b＞0,4＜p＜6,V(x)∈L3/2c(R3)是一个给定的非负函数且满足 lim V(x):=V∞.对V(x)给定适当的假设条件,当ε充分小时,证明了基态解...     

带梯度项的半线性椭圆方程整体解的存在性

By a sub-supersolution method and a perturbed argument,we show the existence of entire solutions for the semilinear elliptic problem-△u + a(x)|▽u|~q = λb(x)g(u),u 0,x ∈ R~N,lim(|x|→∞) u(x) = 0,where q ∈(1,2],λ 0,a and b are locally Holder continuous,a ≥0,b 0,(?)x∈ R~N,arid g ∈ C~1((0,∞),(0,∞)) which may be both possibly singular at zero and strongly unbounded at infinity.     

含有陡峭势阱和凹凸非线性项的Kirchhoff型问题的多重正解*

在这篇文章中, 作者研究涉及凹凸非线性项的Kirchhoff型问题-(a + b ∫R3|▽u|²dx) Δu + λV (x)u = μf(x)|u|^q?2u + |u|^p?2u, x ∈ R³,u ∈ H¹(R³),其中a,b > 0 是常数, λ, μ > 0 是参数, 1 < q < 2, 4 < p < 6 且 V 是一个非负连续位势. 在f(x) 和 V 的合适条件下,此问题正解的存在性和集中性能够通过Nehari 流形和Ekeland 变分原理得到.     

带对流项的一类奇异Dirichlet问题唯一古典解的渐近行为

设Ω是RN中的C2有界区域,应用问题-p"(s)=g(p(s)),p(s)＞0,s∈(0,∞),p(0)=0,lims→∞ p'(s)=β≥0解的性质,构造比较函数,得到了奇异非线性Dirichlet问题-△u=g(u)+λ|▽u|q+σ,u＞0,x∈Ω,u|(e)Ω=0的唯一解u∈C2(Ω)∩ C(Ω)满足lim d(x)→O u(x)/p(d(x))=ξo,这里q∈[0,2],λ,σ是非负参数,T(ξ0)=lim t→O+ g(ξot)/ξog(t)=1,9(s)在(0,∞)是正的单调非增函数且lim s→O+g(s)=+∞,∫∞ 1 9(s)ds＜∞.     

R~N上临界增长p-Kirchhoff型方程的非平凡解的存在性

本文主要研究以下具临界增长的非线性p-Kirchhoff型方程的非平凡解的存在性:{-(a+b∫_(R~N)|▽u|~p)?_pu=|u|~(p*-2)u+μf (x)|u|~(q-2)u, x∈R~N,(0.1) u∈D~(1,p)(R~N),其中a≥0,b0,1pN,1qp,p*=N_p/(N-p),μ≥0,?_pu=div(|▽u|~(p-2)▽u)表示p-Laplace算子对函数u的作用, f∈L(p*/(p*-q))(R~N)\{0}且f是非负的.本文利用Ekeland变分原理和山路定理证明方程(0.1)在适当条件下至少存在两个非平凡解.     

双重退化抛物型方程整体解的存在性和L~∞估计

文考虑双重退化抛物型方程ut=div(|u|r|u|m--2u)+A(u)带有零边界条件的初边值问题的整体解存在性,唯一性和解在t=0,∞处的L∞模估计.证明了当u0∈Lq(Ω)时,整体解u(t)满足估计‖u(t)‖∞≤C(1+t-λβ)(1+t)-β/M,‖(u(t)|r/(m-1)u(t))‖m≤C(1+t-μ)(1+t)-σ,t＞0,这里λ,μ,σ,M,β为依赖于m,q,N和r的适当正常数.     

GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH NONLINEAR DAMPING AND SOURCE TERMS

The initial boundary value problem for a viscoelastic equation | u t | ρ u tt △u-△u tt + t 0 g(ts)△u(s)ds + | u t | m u t = | u | p u in a bounded domain is considered, where ρ, m, p > 0 and g is a nonnegative and decaying function. The general uniform decay of solution energy is discussed under some conditions on the relaxation function g and the initial data by adopting the method of [14, 15, 19]. This work generalizes and improves earlier results in the literature.     

一类Kirchhoff方程Neumann边值问题解的存在性

考虑如下一类Kirchhoff方程Neumann边值问题:{-(a+b∫Ω(|↓△u|²+|u|²dx)(△u-u)+=c(x)|u|^q-2u+f(x,u)■u/■v=0,其中Ω■R^N是光滑有界域,c(x)可能是变号函数,a≥0,b>0且a+b>0,1相似文献   

关于|a+b|=|a|+|b|及|a-b|=|a|-|b|的应用

1.在重要不等式|a+b|≤|a|+|b|中,当且仅当a≥0,b≥0或a≤0,b≤0时等号成立,即|a+b|=|a|+|b|的充要条件是ab≥0。因此|a+b|<|a|+|b|的充要条件是ab<0。同样,等式|a_1+a_2+…+a_n|=|a_1|+|a_2|+…+|a_n|成立的充要条件是a_1,a_2,…,a_n有相同符号。这一简单事实,在数学中有着重要的应用。 1)在解方程中的应用解方程|lg(2x-3)+lg(4-x~2)|==|lg(2x-3)|+|lg(4-x~2)|。解:根据|a+b|=|a|+|b|的充要条件是ab≥0,所以原方程等价于不等式 lg(2x-3)lg(4-x~2)≥0。解这个不等式: lg(2x-3)lg(4-x~2)≥0 lg(2x-3)lg(4-x~2)≥0 2x-3>0 4-x~2>0     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Axiomatizability of reducts of algebras of relations

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrifications. A similar result is proved for subreducts of the class of representable sequential algebras. Received October 7, 1998; accepted in final form September 10, 1999.     

Lengths of periods of continued fractions of square roots of integers

Empirical study of the period’s length T of the continued fractions of $\sqrt{Q}$ (for growing integers Q) shows several strange asymptotical results, for instance, $T\leq C\sqrt{Q}\ln{Q}$ . These results show important differences between the statistics of the elements of the continued fractions of random real numbers and of square roots of random integers.