Existence Results for a Class of Semilinear Elliptic Systems

In this paper, we study the existence of nontrivial solutions for the problem
{-△u=f（x,u,v）＋h1（x）in Ω
-△v=g（x,u,v）＋h2（x）inΩ
u=v=0 onδΩ
where Ω is bounded domain in R^N and h1,h2 ∈ L^2 （Ω）. The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g：
{lim s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,t→＋∞g（x,s,t）/t=λ＋1 uniformly on Ω,
lim -s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,-t→＋∞g（x,s,t）/t=λ-,uniformly on Ω,
where λ＋,λ-∈（0）∪σ（-△）,σ（-△）denote the spectrum of -△. The cases （i） where λ＋ = λ_ and （ii） where λ＋≠λ_ such that the closed interval with endpoints λ＋,λ_ contains at most one simple eigenvatue of -△ are considered.     

一类带Hardy-Sobolev临界指数的奇异Kirchhoff型方程正解的存在性

本文研究了一类带Hardy-Sobolev临界指数的奇异Kirchhoff型{-(a+b∫_Ω︱▽u︱~2dx)△u=u~(5-2s)/︱x︱~s+λu~(-γ),x∈Ω,u0,x∈Ω,u=0,x∈δΩ方程其中ΩR~3是一个有界开区域且具有光滑边界δΩ,0∈Ω,a,b≥0且a+b0,λ0,0γ1,0≤s1.利用变分方法,获得了该问题的一个正局部极小解,补充了文献[1]的结果.     

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＜∞.     

Sobolev-Hardy不等式与临界双重调和问题

该文讨论一类带有奇异系数的双重调和方程{△^2u-μu/|x|^s=f(x,u),x∈Ω,u=δu/δv=0,x∈δΩ，这里Ω包含R^N是包含0的有界光滑区域，u∈H0^2(Ω），μ∈R是参数，0≤s≤2,△^2=△△表示双重拉普拉斯算子，当f(x,u)=u^p,p=2N/N-4时，上述问题就是一个临界双重调和问题，该文运用Sobolev-Hardy不等式和变分方法，得到它的解的存在性的一些结果。     

EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/|x|s|u|q-2u,(x,t)∈Ω×(0,∞),u(x,0)=f(x),x∈Ω,u(x,t)=0,(x,t)∈Ω×(0,∞),(P)}where-△pu ≡-div(|▽u|p-2▽u),1相似文献   

The existence of nontrivial solutions to a semilinear elliptic system on without the ambrosetti-rabinowitz condition

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.     

REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.     

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.     

The Jackson Inequality for the Best L~2-Approximation of Functions on [0,1] with the Weight x

Let L^2（[0, 1], x） be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[0, 1], x）, i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T（t）f（x）=1/π∫0^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：0≤ t ≤τ] of order r are defined in the standard way, where T^0（t） = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E（f, n） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.     

BOUND STATES FOR A STATIONARY NONLINEAR SCHRODINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R^3△

We study the following Schrodinger-Poisson system where （Pλ）{-△u＋ V（x）u＋λФ（x）u^p=x∈R^3,-△Ф=u^2,lim│x│→∞Ф（x） =0,u〉0,where λ≥0 is a parameter,1 〈 p 〈 ＋∞, V（x） and Q（x）=1 ,D.Ruiz[19] proved that（Pλ）with p∈ （2, 5） has always a positive radial solution, but （Pλ） with p E （1, 2] has solution only if λ 〉 0 small enough and no any nontrivial solution if λ≥1/4.By using sub-supersolution method,we prove that there exists λ0〉0 such that（Pλ）with p ∈（1＋∞）has alaways a bound state（H^1（R^3）solution for λ∈[0,λ0）and certain functions V（x）and Q（x）in L^∞（R^3）.Moreover,for every λ∈[0,λ0）,the solutions uλ of （Pλ）converges,along a subsequence,to a solution of （P0）in H^1 as λ→0     

RESEARCH ANNOUNCEMENTS——Dynamical Behavior for the Three-dimensional Generalized Hasegawa-Mima Equations

We consider the following generalized three-dimensional （3-D） dissipative Hasegawa-Mima equations： △ut - ut ＋ {u, △u} ＋ knuy - vz ＋ α△（u - △u） ＋ f（x, y, z） = 0, （1） vt ＋ {u, v} ＋ uz ＋ γv - β△v = g（x, y, z） （2） with initial datum v｜t=0=u0（x,y,z）,v｜t=0=v0（x,y,z）,（x,y,z）∈Ω∈R^3 （3）.     

带有Sobolev-Hardy临界指标项的非齐次椭圆方程的解

考虑带有Hardy和Sobolev-Hardy临界指标项的非齐次椭圆方程{-Δu-u(u/(|x|~2))=λu+(((|u|~(2~*(s)-2))/(|x|~s))u+f,在Ω中,u=0,在Ω上,这里2~*(s)=(2(N-s))/(N-2)是临界Sobolev-Hardy指标,N≥3,0≤s2,0≤μ=((N-2)~2)/4,ΩR~N是一个开区域.假设0≤λ≤λ_1时,λ_1是正算子-△-μ/(|x|~2)的第一特征值.f∈H~1_0(Ω)~*,f(x)≠0.当f满足适当的条件时,此方程在H~1_0(Ω)中至少具有两个解u_0和u_1.而且,当f≥0时,有u_0≥0和u_1≥0.     

一类二阶中立型微分差分方程周期解的存在性

考虑如下二阶中立型微分差分方程的边值问题：{x(t-τ)-x(t-τ） f(t,x(t),x(t-τ）,x(t-2τ）=0 x(0)=x(2kτ），x（0）=x（2kτ）其中k是任意给定的正整数,τ 为正实数，利用含有偏差变元的变分结构及临界点理论，作给出了判定上述方程存在非平凡周期解的判定准则。     

具边界反馈时滞的粘弹方程的能量衰减(英文)

考虑如下具边界反馈时滞的粘弹方程ut(x,t)-Δu(x,t)+∫0tg(t-s)Δu(x,s)ds=0,x∈Ω,t0,u(x,t)=0,x∈Γ0,t0,?u /?v=∫0tg(t-s)/vu(s)ds-μ1ut(x,t)-μ2ut(x,t-τ),x∈Γ1,t0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,ut(x,t-τ)=f0(x,t-τ),x∈Ω,0tτ,其中Ω∈Rn(n≥1)是具C2类边界Ω的有界域.此外,g是所谓的"记忆核",μ1,μ2是两个实数,τ为时滞.在假设|μ2|μ1下,通过构造合适的Lyapunov函数,证明上述问题能量的一般衰减性,使得指数型衰减和多项式衰减仅仅是其特殊情况.     

积分微分方程有限元逼近的强超收敛性

考虑下面的抛物型积分微分方程初边值问题: 　（a） ut+A（t）u+∫0tB（t,s）u（s）ds=f, （x,t）∈Q=Ω×J,J=（0,T] （b） u=0,（x,t）∈　Ω×J,（1） （c）　u（x,0）=u0,x∈Ω,其中Ω为Rd（d≤4）中具有分片光滑边界　Ω的有界域,A（t）是一致正定的二阶椭圆微分算子     

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

第二类Feigenbaum函数方程凸解的构造

考虑第二类Feigenbaum函数方程{f（x）=1/λf（f（λx））,0〈λ〈1,f（0）=1,0≤f（x）≤1,x∈[0,1]对于给定的初始函数,利用构造性方法讨论上述方程的连续凸解、C^1-凸解和C^2-凸解的存在性及唯一性.     

认真当好一个医生

考虑如下具边界反馈时滞的粘弹方程ut(x,t)-Δu(x,t)+∫0tg(t-s)Δu(x,s)ds=0,x∈Ω,t>0,u(x,t)=0,x∈Γ0,t>0,?u /?v=∫0tg(t-s)/vu(s)ds-μ1ut(x,t)-μ2ut(x,t-τ),x∈Γ1,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,ut(x,t-τ)=f0(x,t-τ),x∈Ω,0相似文献   

W0^1p（x） Versus C^1 Local Minimizers for a Functional with Critical Growth

Let Ω IR^N, （N ≥ 2） be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- ：= inf pΩ（x） ≤ p＋ = supp（x） Ω〈∞,
and f：Ω^-× IR be a C^1 function with f（x,s） ≥ 0, V （x,s） ∈Ω × R^＋ and sup ∈Ωf（x,s） ≤ C（1＋s）^q（x）, Vs∈IR^＋,Vx∈Ω for some 0〈q（x） ∈C（Ω^-） satisfying 1 〈p（x） 〈q（x） ≤p^＊ （x） -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^＋ ≤ q- ≤ q＋. As usual, p＊ （x） = Np（x）/N-p（x） if p（x） 〈 N and p^＊ （x） = ∞- if p（x） if p（x） 〉 N. Consider the functional I： W0^1,p（x） （Ω） →IR defined as
I（u） def= ∫Ω1/p（x）｜△｜^p（x）dx-∫ΩF（x,u^＋）dx,Vu∈W0^1,p（x）（Ω）,
where F （x, u） = ∫0^s f （x,s） ds. Theorem 1.1 proves that if u0 ∈ C^1 （Ω^-） is a local minimum of I in the C1 （Ω^-） ∩C0 （Ω^-）） topology, then it is also a local minimum in W0^1,p（x） （Ω）） topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation （P） defined below.     

连续函数图象的分解

本文证明:给定1≤s≤t≤2,对于区间[0,1]上的任意连续函数f,如果f的图象的Hausdorff维数不小于t,那么存在连续函数g,h使得f=g+h,并且dimHGg([0,1])=s,dimHGh([0,1])=t,其中dimH表示Hausdorff维数,Gg([0,1])={(x,g(x)|x∈[0,1])}表示函数g的图象.