Please use this identifier to cite or link to this item: http://hdl.handle.net/2307/5099
Title: Sign-changing solutions of the Brezis-Nirenberg problem : asymptotics and existence results
Authors: Iacopetti, Alessandro
Keywords: asympotic analysis
semilinear elliptic equations
critical exponent
tower of bubbles
blow-up
Issue Date: 9-Apr-2015
Publisher: Università degli studi Roma Tre
Abstract: In this PhD thesis we show some recent results about sign-changing solutions for the Brezis{ Nirenberg problem _􀀀_u = _u + juj2_􀀀1u in u = 0; on @; (0.1) where is a bounded smooth domain of RN, N _ 3, _ is a positive parameter, and 2_ = 2N N􀀀2 is the critical Sobolev exponent for the embedding of H1 0 () into Lp(). In the _rst part we analyze the asymptotic behavior of least-energy radial sign-changing solu-tions in the ball for N _ 7, as _ ! 0, and prove that their positive and negative part concentrate and blow up (with di_erent concentration speeds) at the same point, which is the center of the ball. This provides the _rst existence result of sign-changing bubble-tower solutions for the Brezis{Nirenberg problem. For the lower dimensions N = 4; 5; 6 we analyze the asymptotic behavior of radial sign- changing solutions (with two nodal regions) as _ goes to some strictly positive limit value ob- tained by studying the associated ordinary di_erential equation. We prove that the positive part concentrate and blows-up at the center of the ball, and its limit pro_le is that of a standard bubble in RN. On the contrary, the negative part converges to zero, when N = 4; 5, and it converges to the unique positive radial solution of (0.1) in the ball, for _ = _0, when N = 6, where _0 2 (0; _1), being _1 the _rst eigenvalue of 􀀀_. In view of the results obtained in the radial case for N _ 7, by applying a variant of the Lyapunov-Schmidt reduction method, we prove that such sign-changing bubble tower solutions exist in symmetric bounded domains, as _ ! 0. On the other hand, for the low dimensions N = 4; 5; 6, by applying the Pohozaev's identity and _ne estimates, we prove that such solutions cannot exist for _ close to zero.
URI: http://hdl.handle.net/2307/5099
Access Rights: info:eu-repo/semantics/openAccess
Appears in Collections:T - Tesi di dottorato
Dipartimento di Matematica e Fisica

Files in This Item:
File Description SizeFormat
TesiPhD_iacopetti.pdf1.09 MBAdobe PDFView/Open
SFX Query Show full item record Recommend this item

Page view(s)

11
checked on Sep 23, 2020

Download(s)

23
checked on Sep 23, 2020

Google ScholarTM

Check


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.