Kirchhoff方程单峰解的局部唯一性

该文主要证明了以下非线Kirchhoff问题的单峰解的局部唯一性-(∈^2a+∈b∫R^3|▽u|^2dx)△u+u=K(x)|u|^p-1u,u> 0,x∈R^3,其中∈>0任意小,a,b> 0,1相似文献   

一类非线性Schrodinger-Maxwell方程组半经典解的存在性

本文将研究如下非线性Schrodinger—Maxwell方程组问题 {-ε^2△u＋V（x）u＋K（x）Фu=｜u｜^p-2u, x∈R^3, -△Ф=4πK（x）u^2, x∈R^3. 当势函数V（x）和电量函数K（x）满足一定假设条件时,作者利用变分法证明了ε充分小时,该方程组半经典解的存在性．     

带有非紧条件的拟线性Schrodinger-Poisson系统非平凡解的存在性

本文研究了如下带有非紧条件的拟线性Schrodinger-Poisson系统{-△u+V(x)u+Фu+k/2u△u2=λ|u|^p-2u+f(u),x ∈R^3,-ΔФ=u^2,x∈R^3, 其中κ<0,λ>0,p≥12,f∈C(R,R),V∈C(R3,R).文中首先构造截断函数,利用集中紧性原理和逼近的方法,得到了截断后系统非平凡解的存在性;然后利用Moser迭代技巧,讨论上述系统非平凡解的存在性.     

含奇异项的Schrodinger-Poisson系统正解的多重性

本文研究如下含奇异项的Schr?dinger-Poisson系统{u=φ=0,/-ΔФ=u^2,-Δu=φu=|u|^(p-2)u+λu^(=γ),x∈ЭΩ,x∈Ω,x∈Ω,正解的存在性,其中ΩСR^(3)是光滑有界域,λ是正参数,γ∈(0,1),p∈(2,6).首先将"扰动"技巧用以解决带奇异项问题所对应泛函在零点处不可微的难点,其次应用Ekeland变分原理和山路引理得到该问题对应的扰动泛函存在局部极小和山路型的临界点,最后通过估计序列有一致的下界并对扰动取极限后得到两个正解的存在性.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampere equations is studied. Let Ω R^2 be smooth and convex. Suppose that u ∈ C^2（^-Ω） is a solution to the following problem： det（uij） = K（x）f（x,u, Du） in Ω with u = 0 on аΩ. Then u ∈ C^∞（f）） provided that f（x,u,p） is smooth and positive in ^-Ω × R × R^2, K〉0 in Ω and near αΩ, K = d^m ^-K, where d is the distance to αΩ, m some integer bigger than 1 and ^-K smooth and positive on ^-Ω.     

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。     

全空间上一类奇异p-Laplace方程无穷多解的存在性北大核心CSCD

本文研究了全空间上一类带奇异系数及其扰动的椭圆型p-Laplace问题-△_pu-μ(|u|^(p-2)u)/(|x|~p)=λ(u^(p*(t)-2))/(|x|~t)u+βf(x,u),x∈R^N,u∈D_0^(1,p)(R^N),其中N≥3,D_0^(1,p)(R^N)是C_0~∞(R^N)的闭包,△_pu=-div(|▽u|^(p-2)▽u),2     

带有超线性项或次线性项的Schrdinger-Poisson系统解的存在性和多重性

该文研究如下Schrdinger-Poisson系统解的存在性和多重性-△u+V(x)u+K(x)φu=f(x,u),x∈R~3,-△φ=K(x)u~2,x∈R~3,其中V∈C(R~3,R)并且K∈L~2∪L~∞满足K0.在没有Ambrosetti-Rabinowitz型超二次条件以及映射t→(f(x,t))/t~3的单调性假设下,利用对称山路引理证明了无穷多个高能量解的存在性.此外,考虑了非线性项f次线性增长的情形并获得了解的存在性和多重性.     

EXISTENCE OF A NONTRIVIAL SOLUTION FOR CHOQUARD＇S EQUATION

In this article, the authors consider the existence of a nontrivial solution for the following equation： -△u＋u=q（x）（｜u｜^2＊1/｜x｜）u, x∈R^3, where q（x） satisfies some conditions. Using a Min-Max method, the authors prove that there is at least one nontrivial solution for the above equation.     

含有对数非线性项Kirchhoff方程多解的存在性北大核心CSCD

该文考虑如下带有对数非线性项的Kirchhoff方程-(a+b∫R^(3)|▽u|2 dx)Δu+V(x)u=|u|p-2ulogu2,x∈R^(3),其中p∈(4,6),a,b>0为常数,位势函数V(x)∈C(R^(3),R).运用约束变分法,形变引理和度理论,该文证明了上述问题在不同的位势条件下存在正解和变号解.     

二维具临界指数增长的椭圆方程基态解的存在性北大核心CSCD

本文利用临界点理论研究半线性Schrodinger方程{u=0,x∈Ωσ -△u=f(x,u),x∈Ω这里,Ω是R^(2)中的有界区域,f(x,u):Ω×R满足Trudinger-Moser不等式意义下的临界指数增长.通过对极小极大水平值进行精细估计,结合非Nehari流形方法和Trudinger-Moser不等式,获得了以上问题存在Nehari型基态解以及非平凡解的结果,改进了已有文献中的相应结果.     

非线性梁振动方程的周期解

一、引言考虑下述问题Ku″ A~2u M(‖A~1/2u‖~2)Au Au′=f(x,t),t>0,x∈Ω,(1.1)u|_t=0~=u_0(x),x∈Ω,(1.2)Ku′|_(t=0)=u_1(x),x∈Ω,(1.3)u=0,x∈(?)Ω,t≥0 (1.4)的ω-周期解的存在性.其中 Ω(?)R~n 为一有界光滑区域,u′=((?)u)/((?)t),u_″=((?)u)/((?)t)~2,K 为有界线性对称算子且满足(Ku,u)≥0,M∈C~1[0,∞),M(ξ)≥-β,ξ≥0.此模型最初由Woinowsky 和 Krieger 提出,方程形式为     

一类非线性Maxwell-Dirac系统的驻波解

对如下非线性Maxwell-Dirac系统{3Σk=1a_k(-iδ_k+K(x)Ak)u+aβu+M(x)u-K(x)A_0u=G_u(x,u),-△A_0=4πK(x)|u|~2,-△A_k=4πK(x)|a_ku),k=1,2,3进行了研究,其中x∈R~3.由于Dirac算子是上方和下方无界,相应的能量泛函是强不定的.假设非线性项满足次临界超二次的增长条件,运用强不定泛函的广义环绕定理,证明了系统驻波解的存在性.     

一类半线性波动方程的Sobolev指数

Gaustavo Ponce与Thomas C.Sideris猜测：对一些具有特殊非线性项的半线性波动方程,如utt-△u=u^k(Du)^αx∈R^n,k∈Z^ ,ρ＝│α│≥2,其中Sobloev指数会在[n/2,n/2 1]中,他们在x∈R^3时回答了这一问题,本文在R^n(n≥4）中得到了半线性波动方程utt-△u=u^k(Du)^α（x∈R^n,k∈R^n,k∈Z^ ,p=│α│≥2）的Sobolev指数为max{n/2,(n/2-1)1-3/l-1 2},此数确实在区间[n/2,n/2 1]中,特别当ρ≤n-1时,我们得到了此半线性波动方程的Sobolev指数为n/2。     

一类非线性Schrdinger-Maxwell方程组半经典解的存在性

本文将研究如下非线性Schrdinger-Maxwell方程组问题{-ε2△u+V(x)u+K(x)φu=|u|p-2u,x∈R3,-△φ=4πK(x)u2,x∈R3.当势函数V(x)和电量函数K(x)满足一定假设条件时,作者利用变分法证明了ε充分小时,该方程组半经典解的存在性.     

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRDINGER-POISSON EQUATIONS WITH SIGN-CHANGING POTENTIAL

In this article, we study the following nonhomogeneous Schr¨odinger-Poisson equations -?u + λV(x)u + K(x)φu = f(x, u) + g(x), x ∈ R~3,?-?φ = K(x)u~2, x ∈ R~3,where λ 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be signchanging.     

Asymptotically Self-Similar Global Solutions for a Higher-Order Semilinear Parabolic System

In this paper, we study the higher-order semilinear parabolic system{ut＋（-△）^mu=a｜v｜^p-1v,（t,x）∈R^1＋×R^N,vt＋（-△）^mv=b｜u｜^q-1u,（t,x）∈R^1＋×R^N,u（0,x）=φ（x）,v（0,x）=ψ（x）,x∈R^N, where m, p,q 〉 1, a,b ∈R. We prove that the global existence of mild solutions for small initial data with respect to certain norms. Some of these solutions are proved to be asymptotically self-similar.     

Nonexistence and symmetry of solutions to some fractional laplacian equations in the upper half space

In this article, we consider the fractional Laplacian equation(-△)~(α/2)u = K(x)f(u), x ∈ R_+~n,u ≡ 0, x/∈R_+~n,where 0 α 2, R_+~n:= {x =(x_1, x_2, ···, x_n)|x n 0}. When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x_1, x_2, ···, x_(n-1)) ∈R~(n-1). When K is strictly increasing with respect to x n or only depend on x n, the nonexistence of positive solutions is obtained.     

LOCAL WELL-POSEDNESS OF INTERACTION EQUATIONS FOR SHORT AND LONG DISPERSIVE WAVES

The well-posedness of the Cauchy problem for the system{iδtu+δx^2u=uv+|u|^2u,t,x∈IR，δtv+δxHδxv=δx|u|^2,u(0,x)=u0(x),v(0,x)=v0(x),is considered. It is proved that there exists a unique local solution (u(x,t), v(x,t))∈C（[0,T);H^s)&#215;C([0,T);Hs^-1/2) for any initial data (u0,v0)∈H^s(IR)&#215;H^s-1/2(IR)(s≥1/4) and the solution depends continuously on the initial data.