EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS AND HARDY TERMS

This paper deals with the existence of solutions to the elliptic equation -△uμu/|x|2=λu |u|2*-2u f(x, u) in Ω, u = 0 on ( a)Ω, where Ω is a bounded domain in RN(N≥3),0∈Ω,2*=2N/N-2,λ＞0,λ(a)σμ, σμ is the spectrum of the operator -△- μI/|x|2with zero Dirichlet boundary condition, 0 ＜μ＜-μ,-μ=(N-2)2/4,f(x,u) is an asymmetric lower order perturbation of |u|2*-1 at infinity. Using the dual variational methods, the existence of nontrivial solutions is proved.     

THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS

This article consider, for the following heat equation ut/|x|s-△pu=uq,(x,t)∈Ω×(0,T), u(x,t)=0,(x,t)∈(?)Ω×(0,T), u(x,0)=u0(x),u0(x)≥0,u0(x)(?)0 the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, whereΩis a smooth bounded domain in RN(N>p),0∈Ω,△pu=div(|▽u|p-2▽u),0≤s≤2,p≥2,p-1相似文献   

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

具Hardy-Sobolev临界指数椭圆方程的非平凡解

运用精确估计和变分法得到具奇异位势的椭圆方程-△u-μu/|x|2=|u|2*(s)-2u/|x|s+λu,u∈H0(1,2)(Ω)的非平凡解的存在性,其中Ω是有光滑边界的有界开区域,μ,λ是两个正参数.     

一类带扰动项的奇异椭圆型方程无穷多解的存在性

文章研究了一类带扰动项的奇异椭圆型方程{-div(|x|~(-2a)▽u-uu/(|x|~(2(1+n)))=|u|~(q-1)u/|x|~(bp)+h(x),x∈Ω,u=0,x∈Ω,其中ΩR~N为一光滑有界区域,0∈Ω,N≥3,p=p(a,b)=(2N)/(N-2(1+a-b),1qp-1,h(x)∈L~2(Ω).应用扰动方法,文章证明了存在q_N1,使得对任意的q∈(1,qN),上述方程存在无穷多个不同解.     

带有位势的拟线性椭圆型方程正解的存在性

本文利用Ekeland的变分原理及山路引理,研究了以下问题在一定条件下的正解的存在性：{-△pu=λuq/|x|s+ur,u＞0,x∈Ω(∩)RN,{u(x)=0,x∈(a)Ω,其中△pu=div(| ▽ u |p-2 ▽u),u∈W1,p0(Ω),Ω是RN中的有界区域,且0∈Ω,0＜q＜p-1,N≥3,0＜s＜N(p-q-1)p-1 +q+1,p-1＜r≤p*-1,p*=Np(N-p)-1,λ＞0.此时,s可以大于p,从而推广了p=2时的某些结果.     

带Hardy项的半线性椭圆方程非球对称解的研究

本文考虑如下带Hardy项的半线性椭圆问题{-Δu-μu/|x|2=f(u), x∈Ω,/ u=0,x∈(6)Ω}非球对称解的存在性.这里Ω={x|x∈Rn,n≥3,a＜|x|＜1}是Rn≥3)中的环,其中0≤μ＜μ=(n-2/2)2,f(u)为已知函数.本文在讨论球对称解的性质的基础上,利用变分方法得到了方程的极小能量解的存在性,并且利用分支理论得到了方程的非球对称解.     

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.     

一类分数阶椭圆方程解的渐近行为

应用Stampacchia方法,研究低阶项的正则化效应和Hardy位势对如下分数阶拉普拉斯方程解的渐近行为的影响{(-△)~su-λu/|x|~~(2s)+u~p=f(x),x∈Ω,u0,x∈Ω,u=0,x∈R~N\Ω,其中(-△)~s是分数阶Laplacian算子,s∈(0,1)且N 2s,ΩR~N是具有Lipschitz边界的有界光滑区域且0∈Ω.     

含Hardy位势的拟线性椭圆型方程非平凡解的存在性

设0∈Ω∈R^N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|N,(N≥2)为有界光滑区域,利用山路定理,考虑如下一类含Hardy位势的拟线性椭圆型方程非平凡解的存在性:-△u-u△(|u|²)=μu/|x|2)=μu/|x|²+λg(x,u),x∈Ω,其中μ>0,λ>0为常数,g(x,u)为Caratheodory函数.     

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.     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

Determination of asymptotic solutions of problems of nonlinear theories of thermoviscoelasticity

It is shown that the asymptotic solution of a problem of the nonlinear theory of thermoviscoelasticity, if it exists, can be found directly from the solution of the asymptotic boundary-value problem without completely solving the starting problem.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 3, pp. 395–400, May–June, 1976.     

Emptiness of set of points of lower semicontinuity of Lyapunov exponents

We consider parametric families of differential systems with coefficients that are bounded and continuous on the half-line and uniformly in time continuously depend on a real parameter. For each Lyapunov exponent, we construct a family such that the Lyapunov exponent of its systems treated as a function of the parameter is not a lower semicontinuous function for any value of the parameter.