一个带限制的椭圆特征问题的多解和变号解

利用变分方法证明了一个带限制的半线性椭圆特征问题变号解的存在性.所获得的3个解,1个是正解、1个是负解、1个是变号解.在某些附加条件下证明了多重变号解的存在性.     

H1（RN）上带限制的椭圆特征问题的三个解

用变分方法证明H~1（R~N）上一个带限制的半线性椭圆特征问题解的存在性.所获得的三个解：一个是正解,一个是负解.对于第三个解,本文只证明了它的存在性,而没有确定它是正解,负解,还是变号解.     

次椭圆算子的第一特征值问题

本文基于三维球面的Hopf纤维定义球面上的次椭圆算子 ,研究其第一非零问题 ,得到次椭圆算子的第一非零特征值λ1 =2 ,因此有最佳Poincar啨不等式 .∫S3 ｜u- u｜2 dσ≤ 12 ∫S3 ｜ Hu｜2 dσ.     

椭圆外区域上的自然边界元法

１．引言 二十年来,自然边界元法已在椭圆问题求解方面取得了许多研究成果。它可以直接用来解决圆内（外）区域、扇形区域、球内（外）区域及半平面区域等特殊区域上的椭圆边值问题［１,２,５］,也可以结合有限元法求解一般区域上的椭圆边值问题,例如基于自然边界归化的耦合算法及区域分解算法就是处理断裂区域问题及外问题的一种有效手段［２－４,６］。 人们在设计求解外问题的耦合算法或者区域分解算法时,通常选取圆周或球面作人工边界。但对具有长条型内边界的外问题,以圆周或球面作人工边界显然并非最佳选择,它将会导致大量的…     

二维RLW方程的Cauchy问题

通过椭圆积分求出了二维RLW方程椭圆余弦波解,并用先验估计方法证明了该方程Cauchy问题关于小x、y周期解的若干性质和解的唯一性、稳定性。     

巧用转化法解决椭圆问题

椭圆是一种规则图形，但关于椭圆的问题解决起来却并不方便：通常的解析法中繁长的推导过程不仅易使人陷入困境，而且也极容易使人出错，现在我向大家介绍一种巧妙解决椭圆问题的方法——转化法。     

单位球上Yamabe方程的全局分歧

该文研究了N维单位球面S^N上的Yamabe方程■通过分歧的方法,对于任意k≥1,证明了该方程对于任意的λ>λ_k:=(k+N-1)(N-2)/4都至少有一个非常数解v_k,使得v_k-λ⁽1/(N*-1))正好有k个零点,并且它们在(-1,1)中都是单根,其中N^*是Sobolev临界指数.在应用部分,得到了当n≥4时,R^N上非线性椭圆方程非径向解的存在性.此外,还得到了乘积流形中一个流形是单位球时的Yamabe问题的全局分歧结果.     

小异大不同

本人在解题教学中 ,发现有些试题貌似相同 ,而考查的知识、思想方法却大相径庭 ,常有同学因审题不清或思维定势 ,而“以貌取题” ,导致“张冠李戴” .为了大家能更好地区分、理解、掌握此类问题 ,现列举出常见的容易混淆的几组典型问题 ,略作解析 ,供大家在教与学时作参考 .题组 1　 1)已知椭圆 x24 +y23=1内有一点P(1,- 1) ,F为椭圆右焦点 ,M是椭圆上的动点 ,求|MPⅫ +2 |MF|的最小值 .2 )已知椭圆 x24 +y23=1内有一点P(1,- 1) ,F为椭圆右焦点 ,M是椭圆上的动点 ,求 |MP |+|MF|的最小值 .解　 1)依题设得 :椭圆的右准线l的方程为x =4…     

法向荷载作用下椭圆孔不等边裂纹的应力强度因子

采用复变函数方法,研究了在法向均布荷载作用下,含两个不等边裂纹椭圆孔的无限大板平面问题,得到了裂纹尖端的应力强度因子的解析解.并通过有限元软件计算了应力强度因子的数值解,与解析解进行对比,吻合较好.另外,研究了随着裂纹和椭圆孔尺寸变化时应力强度因子的变化规律.可以看出应力强度因子随椭圆孔的长短半轴之比和裂纹长度的增大而增大.     

站得高才能看的远——谈椭圆一条性质及其应用

<正>椭圆有许多性质,这些性质揭示了椭圆的一些几何特征.利用椭圆的这些几何特征往往可以使我们在解题过程中迅速抓住问题的实质,降低运算成本,提高解题效率.本文给大家介绍一条椭圆的新性质:     

Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint

We prove the existence of sign changing solutions of a semilinear elliptic eigenvalue problem with constraint by using variational methods. Among those three solutions we obtained, one is positive, one negative and one sign changing. We also prove the existence of multiple sign changing solutions under some additional condition.     

Mathematics

We prove the existence of sign changing solutions of a semilinear elliptic eigenvalue problem with constraint by using variational methods. Among those three solutions we obtained, one is positive, one negative and one sign changing. We also prove the existence of multiple sign changing solutions under some additional condition.     

SOLUTIONS OF AN ELLIPTIC EIGENVALUE PROBLEM INVOLVING SUBCRITICAL OR CRITICAL EXPONENTS

In this paper, we prove the existence of a positive solution, a negative solution and a sign-changing solution of a semilinear elliptic eigenvalue problem with constraint involving subcritical and critical Sobolev exponents. The solutions are obtained in the ω-limit sets of some descending flow curves whose starting points are specifically chosen.     

RN上一类半线性椭圆方程组的多重解

利用对称山路引理,L2(RN)上加权Sobolev空间紧嵌入定理和一个线性耦合特征值问题,本文证明了RN上一类半线性椭圆方程组的多重解的存在性.     

On a quasilinear elliptic eigenvalue problem with constraint

Via construction of pseudo gradient vector field and descending flow argument, we prove the existence of one positive, one negative and one sign-changing solutions for a quasilinear elliptic eigenvalue problem with constraint.     

On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

无界域上半线性奇异椭圆方程的共振与非共振问题

本文考虑了无界域上一类半线性奇异椭圆方程的加权特征值问题,然后讨论了在非共振与共振情形下这类方程非平凡解的存在性．