【作者】 付玉霞；

【导师】 戴求亿；

【作者基本信息】 湖南大学 ， 应用数学， 2008， 博士

【摘要】 本学位论文主要研究二阶半线性椭圆方程及方程组的边值问题,运用blow-up技巧、变分理论、不动点定理、上下解、先验估计、渐近分析等相结合的方法得到了Dirichlet边值和Robin边值条件下半线性椭圆问题正解的存在性、非退化性、唯一性以及多重性等结果.第一章中,简单介绍了所研究问题的背景及主要结果,并给出了基本术语和文中需多次用到的引理,以及全文的结构安排.第二章中,首先研究了一般形式的二阶椭圆方程Dirichlet问题正解的存在性与唯一性.由于许多实际问题并不一定具有变分结构,常用于证明解的存在性的变分方法不一定适用.为了避开这点,文中采用对解做先验估计结合不动点理论的方法,得到了在Dirichlet边界条件下正解存在的充分必要条件.第二章中更主要的结果是关于解的唯一性的.对半线性椭圆问题而言,要证明解的唯一性存在相当大的困难,原因是解的个数不仅依赖于算子系数,而且也依赖于区域的几何特性.文中采用算子扰动与blow-up技巧相结合的方法,证明了如果某类半线性椭圆问题的解是唯一的和非退化的,则对问题中的微分算子做小扰动后其正解仍是唯一的和非退化的,作为推论,还证明了正解的唯一性在区域的小扰动下是不变的.对于Robin边值问题,很多学者都认为与Dirichlet边值问题差不多,研究方法也类似,但近期的研究文献表明,这两类边值问题的差别还是很大的.因此,越来越多的人开始关注Robin边值问题了.当然,解的存在性和先验估计对Robin边值问题而言是相对容易的,但有关Robin边值问题解的唯一性和对称性至今没有任何结果.第三、四、五章分别就不同模型的Robin边值问题正解的唯一性和多重性作了研究.第三章中,研究了一维情形下带Robin边值条件的二阶半线性椭圆问题正解的唯一性.在Dirichlet边值问题解的唯一性研究中,经常用到的最为有力的工具是打靶方法,但是对于一般的Robin边值问题,由于解不一定具有对称性,打靶方法不再适用.文中先证明齐次Robin问题的解是非退化的,再由隐函数定理得到了正解的唯一性,随后再利用齐次问题解的唯一性和非退化性结合blow-up技巧,给出了带变号非齐次项的模型一和带凹凸非线性项的模型二存在一个或多个正解的充分必要条件.第四章中,研究了高维一般区域上的半线性椭圆方程及方程组在Robin边值条件下正解的唯一性问题.由于边界的复杂性,常用于证明某种对称区域上的Dirichlet问题解的对称性和唯一性的移动平面法不再适用,文中利用先验估计结合解的渐近分析的方法,给出了带Robin边值条件的方程及方程组正解具有唯一性的充分条件,相比较于Dirichlet边值问题的相关结论,这一条件对区域的形状没有特殊要求.最后,第五章中研究了高维环域上??u = f(u)的Robin边值问题.在一定的假设条件下,通过先验估计结合变分方法中的Nehari技巧证明了:当边界条件中的参数β充分小时,Robin边值问题只有一个正解,又因为算子和区域是旋转不变的,此解一定是径向解,当β充分大时,Robin边值问题存在无穷多个非径向解.这一结论再次说明Robin边值问题和Dirichlet边值问题有着很大的不同.更多还原

【Abstract】 In this thesis, we deal with Dirichlet and Robin boundary value problems ofsecond order semilinear elliptic equations and systems. Some results on existence,uniqueness, multiplicity and non-degeneracy for positive solutions are demonstratedby employing blow-up technique, variational theory,fixed point theorem, sup-subsolution theory, a priori estimate and asymptotic analysis.As an introduction, in chapter one, the background and main results are brie?yaddressed. The general notations, lemmas and outline of this work are also givenin this chapter.In chapter two, we study the existence and uniqueness of positive solutionsof the general second order elliptic equations with Dirichlet boundary condition.The generally used variational theory can not be applied to prove the existenceof solutions when the problems have no variational structure. In addition to asu?cient and necessary condition for the existence of positive solutions is givenby using blow up technique and fixed point theorem, we also prove that if theuniqueness and non-degeneracy results are valid for positive solutions of a classof semi-linear elliptic equations, then they are still valid when one perturbs theoperator a little bit. As a consequence, some uniqueness results of positive solutionsunder the domain perturbation are also obtained.As Robin boundary value problems are concerned, most of mathematiciansthink that they are similar to Dirichlet boundary value problems. In fact, somerecent research works imply that these two kinds of problems are di?erent in manyaspects. Though the existence and a priori estimate of solutions for Robin bound-ary value problem are easy to obtain, the results on uniqueness and symmetryof positive solutions are very di?cult to demonstrate. We study three models ofRobin problems in chapter three, chapter four and chapter five respectively for theuniqueness and multiplicity of positive solutions.In chapter three, the uniqueness of positive solution to a second order ordinarydi?erential equations with Robin boundary condition is studied. The most powerfultool used to prove uniqueness results for Dirichlet problem of second order ordinarydi?erential equations is the so-called time mapping method. But the time mappingis not valid for general Robin problem because of the loss of symmetry. At first we proved the non-degeneracy and uniqueness of positive solution for the homogeneousRobin boundary value problem. Then, as an application of the non-degeneracy anduniqueness results, we gave a necessary and su?cient conditions for the existenceof one or two positive solutions for model one with the non-homogeneousity changesign, and for model two with concave-convex nonlinearities.In chapter four, we devote to prove uniqueness results of positive solutionsto Robin boundary value problem of semilinear elliptic equations and systems ongeneral region in n-dimension space. Since the moving plane method, which isusually applied to prove the symmetry and uniqueness of solutions for Dirichletboundary value problems on some symmetry domain, can not be used, the methodwe use is a priori estimate combining with the asymptotic analysis of solutions.Compared with the uniqueness results for Dirichlet boundary value problems, themain feature of our results is that there is no special request on the domain.In the last chapter, we consider the Robin boundary value problems ??u =f(u) on annulus in n-dimension space. Under some assumptions, we prove thatthe problem has at most one solution whenβ,the parameter in the boundarycondition, is small enough, whereas, it has at least k non-radial solutions whenβis large enough. This result gives another example which implies that Robinboundary value problems may be much di?erent from Dirichlet boundary valueproblems.更多还原