共查询到20条相似文献,搜索用时 31 毫秒
1.
In this article, we consider the geodesic flows induced by the natural Hamiltonian systems $H(x,p)=\frac{1}{2}g^{ij}(x) p_{i}p_{j} + V(x) $ defined on a smooth Riemannian manifold$(M = \mathbb{S}^{1} \times N, g)$, where $\mathbb {S}^{1}$ is the one dimensional torus, N is a compact manifold, g is the Riemannian metric on M and V is a potential function satisfying $V \leq 0$. We prove that under suitable conditions, if the fundamental group $\pi_{1}(N)$ has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric $(h-V)g$, i.e., $(M, (h-V)g)$, is a manifold with conjugate points for all h with $0 < h <\delta$, where $\delta$ is a small number. 相似文献
2.
Acta Mathematica Sinica, English Series - Let Gλ, λ > 0, be a Grushin type operator on $$\mathbb{R}^{n}\times\mathbb{R}^{m}$$ . In this paper, we will first establish existence and... 相似文献
3.
《复变函数与椭圆型方程》2012,57(5):409-415
Let $ \cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\lim_{r \to \infty} \prod_{|a_j|\leq r}{(1- \fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\shadC} $ , let $ \cal V $ be the set of $ f\in {\cal W} $ such that $ {\shadC} [\,f]\subset {\cal W} $ , and let $ {\cal H} $ be the $ {\shadC} $ -algebra of entire functions satisfying $ { {\lim_{r\to \infty } } ({\ln M(r,f) / r})=0} $ . Then $ \cal H $ is included in $ {\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\in {\shadN} ^* $ satisfy n > m S 3. Let $ a\in {\shadC}^* $ satisfies $ {a^n\not = \fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\neq (1+\xi )^{n-m} (\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\cal V} $ . In particular this applies to functions such as sin x and cos x . 相似文献
4.
Norihisa Ikoma 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(5):555-567
In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system:
$\left\{\begin{aligned} -\Delta u_1+V_1(|x|) u_1 &= \mu_1 u_1^3+\beta u_1u_2^2 & &\quad{\rm in} \ {\mathbb R}^{N},\\ -\Delta u_2+V_2(|x|)u_2&=\beta u_1^2u_2+\mu_2 u_2^3&&\quad{\rm in} \ {\mathbb R}^{N}.\end{aligned}\right.$\left\{\begin{aligned} -\Delta u_1+V_1(|x|) u_1 &= \mu_1 u_1^3+\beta u_1u_2^2 & &\quad{\rm in} \ {\mathbb R}^{N},\\ -\Delta u_2+V_2(|x|)u_2&=\beta u_1^2u_2+\mu_2 u_2^3&&\quad{\rm in} \ {\mathbb R}^{N}.\end{aligned}\right. 相似文献
5.
Fernando Bernal-Vílchis Nakao Hayashi Pavel I. Naumkin 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):329-355
We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }
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