| dc.description.abstract | 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. | en_US |