臺灣博碩士論文加值系統

本論文就研究課題而論，大致分為兩個部分。第一部分主要在闡釋當一個定義在具有奇異點區域上的高階非線性偏微分方程，其非線性項滿足特定前題時，那麼該方程式的解其實就會具有類似於「散度定理」的性質。而這樣的性質有助於了解這類方程式的解在奇異點附近的行為。其次，在第二部分之中，主要是從變分學的觀點來看特定方程式的多重解的存在性問題。其中同時探討解的正負變號數與其所對應的變分泛函值的關係。

以章節來分，前述第一部分的內容歸納在本文的第一章。其實在處理具奇異點的高階偏微分方程時，相對於二階且無奇異點情形而言，基本上有兩個困難點。第一、在二階方程式常用的極大值原(maximum principle)在高階方程式上往往失去預期的效果。第二、「散度定理」在具有奇異點的區域上，其使用受到很大的限制，而這對解的行為的估計，常造成瓶頸。無論如何，作者在這裡試著對該問題的困難點提供一個解決的途徑。除了證明解的「散度定理」的性質外，作者對此亦給了兩個應用。其一是探討Dirichlet 邊界問題解Laplacian 階數與其零根的關係；其次則為對特定方程式解的不存在性的驗證。詳述於該章各節。

在第二章作者用變分方法證明特定方程式多重解的存在性。事實上，該類方程式在二階的情形，其存在性問題早被廣泛探討過。在方法上其實不能算是新的。在這裡作者除了將二階的存在性結果推廣到高階的情形之外，值得一提的是，非負的解的存在性在高階的情形並不容易看得出來，然而，從上述所提的解的正負變號數與其變分泛函值的關係其實正好能夠彌補此一盲點。

另外，第二章所需要的正則性(regularity)定理，獨立論述於本文的第三章。至於比較冗長的證明以及較不重要的細節則收編在附錄之中。

This dissertation is concerned with the existence and behavior for solutions of some polyharmonic equations. It is divided into two parts according to the difference of problems to which the author has devoted. The first part includes the study of a polyharmonic problem in a punctured domain. The second contains subjects about the existence of multiple solutions of some nonlinear higher order equations whose nonlinearities are assumed to be negative near the origin.

In Chapter 1 we prove a divergence-type identity for positive solutions of a certain type of equations in punctured domains. Roughly, the usual divergence theorem is assumed to holds for functions which are defined and differentiable on a smooth domain. When the domain is punctured, the behavior of functions defined there, may be very complicated near singularities even though it is very smooth otherwhere. But if a function satisfies an equation on a domain except at some isolated singularities, its behavior near those singularities will turn to be describable. In practice, considering a positive solution in our case, its behavior near singular points is governed by divergence identities. This property is helpful to the study of some singular problems, especially when the usual maximum principle or integration by part does not work.

Applying this identity the author extends a theorem about counting zeros to its singular case. Further, a onexistence result can also be proved in this manner. The details will appear in Section 1.3.

In Chapter 2 one of the main purposes is to study the existence of multiple solutions for equations whose nonlinearities satisfy some growth conditions. The method which is applied is due to Berestycki and Lions as well as Struwe. The first result concerning existence of infinitely many radial solutions, does not seems to bring more surprise than it does in the second order cases. It is believed that one can also conclude this via the method of ordinary differential equations. The second result of this chapter is to estimate the number of nodal domains of a solution by its energy value. Even in second order problems such a result has been proved quite recently.

Finally, it is worthy to mention that in the study of higher order problems some classical tools, which is used in dealing with second order equations to construct a nonnegative solution, does not work similarly. The existence of nonnegative solution is not obvious in higher order cases. Therefore, the study of nodal structure of a solution seems to suggest a viewpoint to answer this question.

1 A divergence-type identity in punctured domains 1
§1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1.2 Behavior of solutions near singularity . . . . . . . . . . . . . . . . . . . . . 3
§1.3 Proofs of Theorem 1.1.2 and Theorem 1.1.3 . . . . . . . . . . . . . . . . . 6
§1.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Multiple solutions of some polyharmonic problems 12
§2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§2.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
§2.3 Infinitely many critical points . . . . . . . . . . . . . . . . . . . . . . . . . 22
§2.4 Nodal behavior and nonnegative solutions . . . . . . . . . . . . . . . . . . 30
3 Regularity of weak solutions 33
Appendix A 38
Appendix B 44
Bibliography 48

Berestycki, H. and Lions, P. L.,
Nolinear scalar field equations I and II.
Arch. Rat. Mech. Anal. 82 (1983), 313-345 and 347-375

Struwe, M.,
Multiple solutions of differential equations without the Palais-Smale condition.
Math. Ann. 261 (1982), 399-412.

Bartsch, T., Liu, Z. and Weth, T.,
Sign changing solutions of superlinear Schr{"o}dinger equations.
Comm. Partial Differential Equations} 29 (2004), 25-42.