一类含Caffarem-Kohn-Nirenberg不等式的奇异椭圆型方程正解的存在性

本文研究一类带奇异项及临界指数的方程 利用纤维方法证明方程在满足一定条件下正解的存在性．     

关于奇异非线性多调和方程的正整体解

该文主要研究形如Δ((Δ\+nu)\+\{p-1*\}) = f(|x|, u, |u|)u\+\{-β\},\ x∈R\+2的奇异非线性多调和方程在R\+2上的正整体解，此处p>1，β≥0是常数，n是自然数，f: [AKR-]\-+×R\-+×[AKR-]\-+→R\-+是 一个连续函数,ξ\+\{α*\}:=|ξ|\+\{α-1\}ξ,ξ∈R，α>0 . 证明了这种解 u必无界且其渐进阶（当n→∞时u作为无穷大量的阶）不低于|x|\+\{2n\}log|x| ，给 出了该方程具有无穷多个其渐进阶刚好为 |x|\+\{2n\}log|x| 的正整体解的充分与充分必要条件. 这些结论可以推广到更一般的方程中去.      

Time dependent nonlinear Schrödinger equations

We prove the existence of global classical solutions of the Cauchy-problem for nonlinear Schrödinger equations u+iAu+f(|u|²)u or u+Au+if(|u|²)u respectively. We need various growth conditions on the nonlinearity f and some restrictions on the admissible space dimension n.     

奇异非线性$p-$调和方程的一类正整体解

设p>1,β≥0是常数, n是自然数, 是一个连续函数.本文研究形如的奇异非线性p-调和方程的正整体解,给出了该类方程具有无穷多个其渐近阶刚好为|x|(2n-2)(当|x|→∞时)的径向对称的正整体解的若干充分条件.     

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   

方程△u+g(|X|)f(u)=0的环上Dirichlet边值问题的多重正对径解

考察了二阶半线性椭圆边值问题△u+g(|X|)f(u)=0,R     

二阶拟线性椭圆型方程振动性的积分平均方法

§ 1.　IntroductionThepurposeofthispaperistostudytheoscillatorybehaviorofsolutionsofcertainquasi linearellipticequationsdiv( ｜Du｜m -2 A(x)Du) + p(x) ｜u｜m -2 u=0 ,x∈Ω Rn,(E)whereΩisanexteriordomain ,m >1 ,andfunctionsA(x) ,p(x)aretobespecifiedinthefollowingtext.Recently ,USAMI [6]consideredEq .(E)whenA(x)≡I (identitymatrix) ,andob tainedoscillationcriteriaforEq .(E)with“infiniteintegral”coefficient [cf.[6],Theorem 4].However,asfarasthepresentreferencesisconcerned ,therearefewo…     

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.     

一类奇异拟线性椭圆方程正解的多重性

研究奇异拟线性椭圆型方程{-div(|x|~(-ap)|▽u|~(p-2)▽u) + f(x)|u|~(p-2) = g(x)\u|~(q-2)u + λh(x)|u|~(r-2),x R~N,u(x) 0,x∈ R~N,其中λ0是参数,1pN(N3),1rpgp*=0a(N—p)/p,p*=Np/{N~pd),aa+l,d=a+l-60,权函数f(x),g(x),h(x)满足一定的条件.利用山路引理和Ekeland变分原理证明了问题至少有两个非平凡的弱解.     

一类退缩椭园型方程弱解的正则性

本文得出一类形如：－Ｄｉｖ（ｇ（｜Ｄｕ｜）｜Ｄｕ｜＾ｐ－２Ｄｕ＋ｆ（ｘ,ｕ））＝Ｂ（ｘ,ｕ,Ｄｕ）在一定的条件下在Ｗ＾１．ｐ空间中的弱解的Ｈｏｌｄｅｒ连续性。     

广义Davey-Stewartson系统驻波的强不稳定性

本文研究方程驻波的强不稳定性iu_t+△u+a|u|~(p-1)u+E_1(|u|~2)u=0,t≥0,x∈R~n,其中a0,1p(n+2)/(n+2)~+,n∈{2,3}.当1+4/n≤pn+2/(n-2)~+)时,文[Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,Commun.Math.Phys.,2008,283:93-125]在驻波的频率满足一定假设条件下,证明了此方程驻波的强不稳定性.本文去掉这个假设,得到相同的结论.     

拟线性椭圆型方程带有混合奇异项的正整体解

在本文中,研究了方程div(｜u｜p-2u) f(x,u)=0,x∈RN,N≥3的正整体解,其中f(x,u)在u=0未假定是正则的,且f(x,u)可以同时包含超线性,亚线性项和奇异项.     

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.     

Explicit Formulas of the Bergman Kernel for Some Reinhardt Domains

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains \(\{|z_3|^{\lambda } < |z_1|^{2p} + |z_2|^2, \ |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}\) and \(\{|z_4|^{\lambda } < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \ (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}\).     

Regularity Results for Nonlinear Systems of Partial Differential Equations Under Weak Ellipticity Conditions

We prove C^{1,α} almost everywhere regularity for weak solutions in the space W^{1,k} (Ω, R^N) of the systems - D_αA^i_α(x,u,Du) = B^i(z,u,Du) under the weak ellipticity condition ∫A(x_0 ,u, p + DΦ) ⋅ DΦdy ≥ λ ∫ (|DΦ|² + |DΦ|^k)dy.     

含临界指数的类p-Laplacian方程无穷多解的存在性

考虑如下一类含临界指数的类p-Laplacian方程-div(a(|Du|~p)|Du|~(p-2)Du)=:-- |u|~(p~*-2)u+λf(x,u),u∈W_0~(1,p)(Ω),其中Ω∈R~N(N≥2)为有界光滑区域,a:R~+→R为连续函数.由于问题失去紧性,对Palais-Smale序列的分析需要一点技巧.本文利用Lions的集中紧原理,证明了相应泛函I_λ满足(PS)_c条件,再应用Clark临界点定理和亏格的性质,证明了方程无穷多解的存在性.进一步,得到当λ充分小时一个特殊的特征函数的存在性.     

Existence of C1-solutions to Certain Non-uniformly Degenerate Elliptic Equations

We are concerned with the Dirichlet problem of {div A(x, Du) + B(z) = 0 \qquad in Ω u= u_0 \qquad \qquad on ∂ Ω Here Ω ⊂ R^N is a bounded domain, A(x, p) = (A¹ (x, p), ... >A^N (x, p}) satisfies min{|p|^{1+α}, |p|^{1+β}} ≤ A(x, p) ⋅ p ≤ α_0(|p|^{1+α}+|p|^{1+β}) with 0 < α ≤ β. We show that if A is Lipschitz, B and u_0 are bounded and β < max {\frac{N+2}{N}α + \frac{2}{N},α + 2}, then there exists a C¹-weak solution of (0.1).     

对非线性椭圆边值问题解的存在性的研究

利用非线性增生映射值域的扰动定理 ,研究了非线性椭圆边值问题 ( @)在 L2 (Ω )中解的存在性 .( @) -△pu +g( x,u) =f a.e.在Ω中-〈v,| u|p- 2 u〉∈βx( u( x) ) a.e.在Γ上其中 f∈ L2 (Ω )给定 ,Ω RN,N 1 ,△ pu=div( | u|p- 2 u)为 P拉普拉斯算子 ,1 2 NN +1 ,v为 Γ的外法向导数 ,g:Ω× R→ R满足 Caratheodory条件 ,对 x∈ Γ,βx是正常、凸、下半连续函数 φx=φ( x,· )的次微分 ,其中 φ:Γ×R→ R.     

方程-▽(|▽u|p-2▽u)=ρ(x)uα在RN存在有界解的充分必要条件

考虑全空间Rn上的方程:-▽(|▽u|p-2▽u)=ρ(x)uα存在有界解的充分必要条件.