一类带调和势的非线性Schr(O)dinger方程在RN中的整体解存在的门槛

本文考虑一类带调和势的非线性Schrodinger方程iψt=-△ψ+|x|2ψ-μ|ψ|p-1ψ-λ|ψ|q-1ψ,x∈RN,t≥0,其中μ＞0,λ＞0.当N=1,2时,1＜p＜q＜∞;当N≥3时,1＜p＜q＜N+2/N-2.运用精巧的变分方法、势井方法和凸方法,得到了方程的整体解和爆破解存在的门槛.进一步回答了:当q＞p＞1+4/N时,方程的Cauchy问题的初值小到什么程度,其整体解存在?.     

一类带调和势的非线性Schrdinger方程在R~N中的整体解存在的门槛

本文考虑一类带调和势的非线性 Schrdinger 方程 it=-△ |x|~2-μ||~(p-1)-λ||~(q-1),x∈R~N,t≥0, 其中μ＞0,λ＞0．当 N=1,2时,1＜p＜q＜∞；当 N≥3时,1＜p＜q＜(N 2)/(N-2)．运用精巧的变分方法、势井方法和凸方法,得到了方程的整体解和爆破解存在的门槛．进一步回答了：当 q＞p＞1 4/N 时,方程的 Cauchy 问题的初值小到什么程度,其整体解存在?     

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.     

Schrdinger-Poisson-Slater方程驻波的强不稳定性

本文研究Schr¨odinger-Poisson-Slater(SPS)方程i?ψ?t+△ψ-χ(|x|-1*|ψ|2)ψ+|ψ|p-1ψ=0,χ0,x∈R3.当p73时,运用变分法证明SPS方程的基态驻波强不稳定,并进一步用基态驻波解刻画SPS方程的解在有限时间爆破的门槛初始条件;在p=73时,建立了基态驻波解的L2范数的下界.     

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.     

一类奇异p-Laplace方程无穷多解的存在性

讨论了一类具有奇异系数的p-Laplace问题-Δpu-μ|u|u|x|p=u|x|tu+λuq-2u,x∈Ω,u=0,x∈Ω无穷多解的存在性,其中N≥3,Ω是RN中一有界光滑区域,0∈Ω,Δpu=-div(|▽u|p-2▽u),0≤μ<μ=(N-p)ppp,10,1相似文献   

一类两点边值问题的正解个数

本文讨论边值问题y"+λ(yp+μy+yq)=0,y(-1)=y(1)=0,其中λ＞0是正参数,μ≥0.对(1-p)(1-q)＞0的情形得出了正解的存在唯一性.对(1-p)(1-q)＜0的情形,其主要结论是若p＞1＞q＞-(25+23p)/(23+25p),μ≥0,则存在λ*＞0,使得当0＜λ＜λ*时,此边值问题恰好存在两个正解,当λ=λ*时,存在唯一正解,当λ＞λ*时,不存在正解.     

凸区域上拟线性椭圆型方程的尖峰解

本文讨论奇异扰动的拟线性椭圆型方程-ε△pu(x)=f(u(x)),u(x)≥0,x∈Ω;u=0,x∈Ω在Dirichlet边值条件下极小能量解的存在性和结构.其中ε＞0是小参数,p＞2,△pu=div(|Du|p-2Du),f(s)=sq-sp-1,p-1＜q＜Np/N-p-1.Ω RN(N≥2)是有界光滑区域.当ε→0时,方程存在一个极小能量解,应用移动平面方法可以证明此解在凸区域上会变成一个尖峰解.     

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＜∞.     

非线性薛定谔方程组的散射

本文主要考虑如下非线性薛定谔方程组的柯西问题:{-iu1t=△u1-μ|u1 |p1u1--α |u1 | q1-2 |u2 |q2u1,(x,t)∈RN×(0,T),-iu2t=△u2-ν |u2 |p2u2-β|u1|q1|u2 | q2-2u2, (x,t)∈RN×(0,T),u1 (x,0)=φ(x),u2(x,0)=φ2(x), x∈RN,其中μ,ν,α,β＞0,q1+q2=p3+2,且α/q1=β/q2=b.本文主要研究一些渐近性质,并分别在Sobolev空间、Σ空间及L2(RN)中建立散射理论,这里三={u∈H1(RN),|x|u∈L2 (RN)}.     

Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance

We study the asymptotic behavior at large time of a solution to a system of nonlinear integro-differential equations which arises in mathematical modeling of diffusion of a magnetic field into a substance. We establish the corresponding stabilization rate.     

Determining a Rotation of a Tetrahedron from a Projection

The following problem, arising from medical imaging, is addressed: Suppose that T is a known tetrahedron in ?³ with centroid at the origin. Also known is the orthogonal projection U of the vertices of the image ?T of T under an unknown rotation ? about the origin. Under what circumstances can ? be determined from T and U?     

Construction of a Formula from a Syntactic Graph of a Sentence

The paper considers the construction of formulas describing relations between entities from a Russian sentence. We describe rules which help constructing such formulas from a syntax graph.     

Symmetry-based solution of a model for a combination of a risky investment and a riskless investment

Benth and Karlsen [F.E. Benth, K.H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl. 23 (2005) 687-704] treated a problem of the optimisation of the selection of a portfolio based upon the Schwartz mean-reversion model. The resulting Hamilton-Jacobi-Bellman equation in 1+2 dimensions is quite nonlinear. The solution obtained by Benth and Karlsen was very ingenious. We provide a solution of the problem based on the application of the Lie theory of continuous groups to the partial differential equation and its associated boundary and terminal conditions.     

Motion of a container as a nonholonomous system in a curved section of a horizontal pipeline

The problem of the motion of a container in a curved section of a horizontal pipeline is solved using second-order Lagrange equations in the presence of nonholonous couplings. The special case of the motion of a container in a circular curve is examined.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 90–95, 1987.