Return to search

p- Laplacian operators with L^1 coefficient functions

In this thesis, we consider the following one dimensional p-Laplacian eigenvalue problem:
-((y¡¦/s)^(p-1))¡¦+(p-1)(q-£fw)y^(p-1)=0 a.e. on (0,1) (0.1)
and satisfy
£\y(0)+ £\ ¡¦ (y¡¦(0)/s(0))=0
£]y(1)+£]¡¦ (y¡¦(1)/s(1))=0 (0.2)
where f^(p-1)=|f|^p-2 f=|f|^p-1 sgnf; £\, £\¡¦, £], £]¡¦ ∈R
such that £\^2+£\¡¦^2>0 and£]^2+£]¡¦^2>0;
and the functions s,q,w are required to satisfy
(1) s,q,w∈L^1(0,1);
(2) for 0≤x≤1, we have s≥0,w≥0 a.e.;
(3) for any x∈ (0,1), ¡ì_0^1 s(t)dt>0, ¡ì_0^x w(t)dt>0,and¡ì_x^1 w(t)dt>0;
(4) if for some x_1<x_2,we have¡ì_ x1^x2 w(t)dt=0,then¡ì_ x1^x2 |q(t)|dt=0;
(5) for all n∈N, there is a partition {£a_i^(n)}_i=1 ^2n of [0,1] such that for any 0<k≤n-1, ¡ì_£a_2k^(n)^ £a_2k+1^(n) w>0 and ¡ì_£a_2k+1^(n)^ £a_2k+2^(n) s>0.
We call the above conditions Atkinson conditions, first introduce in [1].There conditions include the case when s,q,w∈L^1(0,1) and s,w>0 a.e.
We use a generalized Prufer substitution and Caratheodory theorem to prove the existence and uniqueness for the solution of the initial value problem of (0.1) above. Then we generalize the Sturm oscillation theorem to one dimensional p-Laplacian and establish the Sturm-Liouville properties of the p-Laplacian operators with L^1 coefficient functions. Our results filled up some gaps in Binding-Drabek [3].

Identiferoai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0727111-224827
Date27 July 2011
CreatorsWang, Wan-Zhen
ContributorsTsung-Lin Lee, Tzon-Tzer Lu, W.C. Lian, Chun-Kong Law
PublisherNSYSU
Source SetsNSYSU Electronic Thesis and Dissertation Archive
LanguageEnglish
Detected LanguageEnglish
Typetext
Formatapplication/pdf
Sourcehttp://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0727111-224827
Rightswithheld, Copyright information available at source archive

Page generated in 0.0014 seconds