這篇論文的目的是討論下述半線性橢圓方程的穩定解。 / -Δu = f (u) in Ω, / 這裡Ω是R{U+207F}中的光滑區域。對於非線性項f (u) = / [附圖]. / 我們得到了關於穩定解的Liouville-type結果。對於帶有更一般非線性項的半線性橢圓方程，在二到四維的情況下，我們得到一個關於穩定解的先驗估計。 / 最後，在維數很大的情形下，我們具體的給出一個關於以下雙調和方程的指數P的上界P，使得當P滿足 <1 P < Pc ‘下述雙調和方程不存在穩定解。 / Δ²u=u{U+1D56}, u>0 in R{U+207F} / The main aim of this thesis is to review recent results on stable solutions to the following semilinear elliptic equation / -Δu = f (u) in Ω, / where Ω ⊆ R{U+207F}.For the special case f (u) = / [With mathematic formula]. / For the general nonlinearity f in a bounded domain, we obtain a priori estimate for the stable solution when 2 ≤ n ≤ 4 / Finally, we give a explicit bound on the exponent for the nonexistence of stable solutions to the following biharmonic problem in large dimensions. / Δ²u=u{U+1D56}, u>0 in R{U+207F} / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yang, Wen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 66-69). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- On the Lane-Emden equation --- p.12 / Chapter 2.1 --- On stable solutions to the Lane-Emden equation --- p.12 / Chapter 2.2 --- On nite Morse index solutions to the Lane-Emden equation --- p.19 / Chapter 3 --- On general semilinear elliptic equation --- p.28 / Chapter 4 --- On the biharmonic equation --- p.44 / Chapter 4.1 --- The rst part of the proof of Theorem 1.6 --- p.44 / Chapter 4.2 --- The second part of the proof of Theorem 1.6 --- p.54 / Bibliography --- p.66
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328583 |
| Date | January 2012 |
| Contributors | Yang, Wen., 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, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (69 leaves) |
| 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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