次椭圆性

Some nonexistence results for p-degenerate sub-elliptic first order evolution inequalities associated with the generalized Baouendi-Grushin vector fields are given. The method is an improvement of the admissible function method in Euclidean space. The proof hardly depends on the properties of the generalized Baouendi-Grushin vector fields.

通过改进欧氏的容许函数法，应用广义Baouendi-Grushin向量场的一些性质，选取特殊的容许函数，利用Hlder不等式和Young不等式，证明了由广义Baouendi-Grushin向量场构成的p-退化次椭圆一阶发展不等方程，在适当条件下非平凡弱解的不存在性。

We prove the solvability of the first eigenvalue problem for a class of quasilinear elliptic equations including distance potential with homogeneous Dirichlet boundary condition.

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性。

A class of elliptic problems in Orlicz-Sobolev space with Neumann boundary conditions are studied. The sublinear case and the superlinear case are discussed, and using the classical critical point theory with the Cerami compactness condition two existence theorems are given.

研究一类在Orlicz-Sobolev空间上具有Neumann边界条件的椭圆问题，分别讨论了次线性和超线性情形，利用满足Cerami紧性条件的经典临界点理论给出两个解的存在性定理。

In part V using Lions' second Concentrate Compactness Lemma andvariational method,we prove the existence of minimal positive solutionof a class of quasilinear elliptic obstacle problems〓in space〓.

第五部分使用Lions的集中紧性原理和变分方法，我们证明了一类非齐次拟线性椭圆型方程对应的障碍问题〓在〓中极小正解的存在性。

The numercial simulation has been carried out, and the experimental results show that this method has been improved significantly comparing with the traditional one in nonlinear optics. Secondly, a second order system of semilinear elliptic equations, which comes from a mathematics model of the electric potential distribution of a media surrounded with the protein, is transformed into a variational problem, the existence of solution of it is proved by using variational principle and Trudinger—Moser inequality, and the relation between the solutions of variational problem and the solutions of its dual problem is obtained.

首先，讨论了非线性光学中的二次谐波产生的耦合方程组，利用变分法证明了耦合方程组非平凡解的存在性，然后进行了数值模拟，实验结果表明文中方法比经典的非线性光学中的方法有较大的改进，这对优化光倍频器件的设计将有所帮助；其次，将一类二阶半线性椭圆型方程组转化为一个变分问题，然后利用变分法和

In Chapter 4, we consider the qusilinear elliptic systems with the homogeneous Neumann boundary condition, and obtain the existence and multiplicity of positive solutions.

在第四章中，我们讨论带有齐次Neumann边界条件的拟线性椭圆型方程组(方程和第三章的一样，边界条件不同)，研究了正解的存在性和多解性。

Firstly, the eigenvalue problem of a class of second order elliptic equation with critical potential and indefinite weights is considered. Then, using critical point theory, Trudinger-Moser inequality and the properties of the first eigenvalue, we prove the existence of a nontrivial solution for a class of nonlinear elliptic with critical potentialand indefinite weights in R~2. Secondly, we prove the existence of nontrivial solutions for a class of subcritical and critical elliptic systems with indefinite part in R~2 byusing a generalized linking theorem, Trudinger-Moser inequality and concentration-compactness principle.5. The existence of at least three weak solutions for discrete boundary value problem is established by using a three critical point theorem introduced by Ricceri.

首先，讨论了R~2中一类带不定权且含临界位势的二阶椭圆型方程的特征值问题，并借此特征值问题的第一特征值性质，利用山路引理及Trudinger-Moser不等式，证明了R~2中一类带不定权且含临界位势的非线性椭圆型方程非平凡解的存在性；其次，利用广义环绕定理，Trudinger-Moser不等式及集中列紧原理，得到了R~2上一类具有强不定部分的半线性椭圆型方程组在非线性项分别为次临界增长和临界增长情形下非平凡解的存在性。

Secondly, by using the Sobolev inequality on the C-C spaces we prove a Fefferman-Phong type lemma. Based on which we provide nonexistence theorems to three kinds of nonlinear subelliptic parabolic equations through a relatively unified method.

其次，利用Carnot-Caratheodory空间上的Sobolev不等式证明了一个Fefferman-Phong型引理，并用比较统一的方法证明了三类非线性次椭圆抛物方程解的不存在性判定定理。

This paper discusses the existence of positive solutions for sublinear semipositone elliptic problems.

借助-ΔDirichlet边值问题的主特征值λ1相应的特征函数的性质，利用上、下解方法，讨论了一类次线性半正单调椭圆问题正解的存在性。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。