In Chapter 2 we study the uniqueness problem of sign-changing solutions for a nonlinear scalar equation. It is well-known that positive solution is radially symmetric and unique up to a translation. Recently, there are many works on the existence and multiplicity of sign-changing solutions. However much less is known for uniqueness, even in the radially symmetric class. In Chapter 2, we solve this problem for nearly critical nonlinearity by Lyaponov-Schmidt reduction. Moreover, we can also prove the non-degeneracy. / In Chapter 3 we are concerned with the uniqueness problem for coupled nonlinear Schrodinger equations. The problem is to classify all positive solutions. In Chapter 3, some sufficient conditions are given. In particular, we have a sufficient and necessary condition in one dimension. The proof is elementary because only the implicit function theorem, integration by parts, and the uniqueness for scalar equation are needed. / In Chapter 4 we go back to the nonlinear scalar equation and consider the traveling wave solutions. Using an infinite dimensional Lyaponov-Schmidt reduction, new examples of traveling wave solutions are constructed. Our approach explains the difference between two dimension and higher dimensions, and also explores a connection between moving fronts and the mean curvature flow. This is the first such traveling waves connecting the same states. / This thesis is devoted to the study of nonlinear elliptic equations and systems. It is divided into two parts. In the first part, we study the uniqueness problem, and in the second part, we are concerns with traveling wave solutions. / Yao, Wei. / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 132-142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_344848 |
| Date | January 2011 |
| Contributors | Yao, Wei, Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, theses |
| Format | electronic resource, microform, microfiche, 1 online resource (vii, 142 leaves : ill.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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