Firstly, we consider the following critical elliptic Neumann problem --Deltau + muu = uN+2N-2 , u > 0 in O; 6u6n = 0 on ∂O, where O is a smooth bounded domain in RN , N ≥ 7, mu is a large positive number and nu denotes exterior unit normal vector. We show that at a positive nondegenerate local minimum point Q0 of the mean curvature function, for any fixed integer K ≥ 2, there exists a mu K > 0 such that for mu > muK, the above problem has K -- bubble solution umu concentrating at the same point Q 0. Precisely, we show that umu has K local maximum points Qm1,...,Qm K ∈ ∂O with the property that umQmj ∼mN-22 ,Qmj→Q0 , j = 1, ..., K, and mN-3N Q'1 m,...,Q'K m approaches an optimal configuration that minimizes the following functional RQ'1,...,Q 'K=c1 j=1K4Q' j+c2 i≠j1&vbm0;Q' i-Q'j&vbm0;N-2 where Qmi=Qm i,1,...,Qmi,N-1 ,Qmi,N:= Q'i m,Qmi,N , c1, c2 > 0 are two generic constants and ϕ(Q) = Q T GQ with G = (∇ijH(Q0)). / In my thesis, I will address different concentration phenomena for some second order elliptic problems. / Lastly, we consider the problem &egr;2Delta u -- u + uq = 0 in a smooth bounded domain O ⊂ R2 with Neumann boundary condition where &egr; > 0 is a small parameter and q > 1. We prove for some explicit &egr;'s the existence of positive solution u&egr; concentrating at any connected component of ∂O, exponentially small in &egr; at any positive distance from it. / Secondly, we study positive solutions of the equation &egr;2Delta u -- u + uN+2N-2 = 0, where N = 3, 4, 5, and &egr; > 0 is small, with Neumann boundary condition in a smooth bounded domain O ⊂ RN . We prove that, along some sequence {&egr;j} with &egr;j → 0, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary ∂O. / Wang, Liping. / "June 2008." / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1707. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 107-117). / 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, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_344169 |
| Date | January 2008 |
| Contributors | Wang, Liping, 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 (vi, 117 p. : 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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