一类带奇异非线性项四阶椭圆型方程极限解的正则性及其相关问题的研究

We, in this grant, consider some properties of the solution and the solution set for a class of fourth-order elliptic equation with singular nonlinearity. This problems arise in the theory of Membrane bucking and Micro-Electromechanical System (MEMS), and have attracted the interest of many researchs. A most striking feature of such problems is the singularity and a possible failure of the basic method including Maximum Principle and Truncation Method, which cause several technical difficulties. .Since the important applications of the mathematics and physics, we first consider the regularity of the extremal solution which determine the set of solutions,i.e.,it decides whether the set solutions stops here or whether a new branch of solutions emanates from a bifurcation state. Secondly, we will investigate the classification of the entire solutions in entire space for this problems, which includes the existence, classification, asymptotic and radial symmetry et. al. Finally we will study the constructure of solutions set including exact multiplicity problem and estimates of Morse index..In order to obtain the desired results, we first use blow-up argument and shooting method to study the existence and related properties of the entire solutions. And then we use some arguments which include Hardy-Rellich inequality, integral modulus estimates, classification of entire solution to investigate the regularity of the extremal solution. For solution set, we will borrow the Crandall-Rabinowitz Bifurcation Theory and Implicit Function Theorem..Due to the singularity of the nonlinearity and the failure of the basic argument associated with corresponding second elliptic problems, we need some new ideas to overcome several technical difficulties. And so our reseacher will give us a complete.understanding for this problem. Besides,it will perfect corresponding theories and methods, and also is helpful for other problems.

本项目对一类带奇异非线性项的四阶椭圆型方程的解与解集的一些性质进行研究,其中极限解的正则性是这类方程的一个重要研究对象,因为它在物理和数学上都有重要的意义.从数学的角度来看极限解的正则性决定了这类方程的解集的构成．此外,整体解的分类以及解集的构成也是我们研究的重要课题．我们首先通过爆破的方法把方程爆破到全空间,运用打靶法对整体解的存在性及其相关性质进行研究.然后我们利用Hardy-Rellich不等式、积分模估计等方法来研究极限解的正则性,以及扰动项对极限解正则性的影响．对于解集结构,我们将通过Crandall-Rabinowitz分支定理论以及隐函数定理进行探讨．由于非线性部分带有奇性以及二阶方程的基本技巧不能运用到相关四阶方程，这给我们的研究在技术上造成了极大的困难．因而这些研究将将完善相关理论及其方法,同时这些方法将有助于解决数学,物理以及其它领域的相关问题.

项目执行期间，项目组成员对下面二类问题进行了研究：1. 带奇异非线性项的半线性椭圆型方程；2. 流体方程. 这二类方程在物理上都有着广泛的应用, 是近几年受到大家关注的热点问题. 对于第一个方程，我们主要考虑了 解的正则性, 渐近性态, 解集结构等相关问题，取得了丰富的研究成果. 特别值得一提的是，课题组基本解决了低维情况下弱解的正则性问题. 对于这方面的研究，课题组成员共发表学术论文 10 篇 (包括接受待发的), 此外还有2篇在审. 这部分研究成果使课题的研究计划得到较好完成。 对于第2个问题，我们主要考虑了自相似解的存在性, 解的正则性判别条件, 上半空间上 Stoke 算子的 Green 张量估计, 公发表学术论文2篇.