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

Firstly, we consider the following critical elliptic Neumann problem --Deltau + muu = uN+2N-2 , u > 0 in O; 6u6n = 0 on &part;O, where O is a smooth bounded domain in RN , N &ge; 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 &ge; 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 &isin; &part;O with the property that umQmj &sim;mN-22 ,Qmj&rarr;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&ne;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 &phiv;(Q) = Q T GQ with G = (&nabla;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 &sub; 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 &part;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 &sub; RN . We prove that, along some sequence {&egr;j} with &egr;j &rarr; 0, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary &part;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.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1928. / "Sonderdruck aus "Mathematische Annalen", Bd. 101, Heft 1"--T.p. verso. Vita. Includes bibliographical references.

Not available / text

Thesis (Ph. D.)--West Virginia University, 2009. / Title from document title page. Document formatted into pages; contains v, 58 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 56-58).

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [113]-116).

Originally presented as the author's thesis, Bonn. / Added t.p. with thesis statement inserted. Bibliography: p. 78-80.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 84-85).

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1965. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.