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Classification of the Structure of Positive Radial Solutions to some Semilinear Elliptic Equation

In this thesis, we shall give a concise account for the classification of the structure of positive radial solutions of the semilinear elliptic equation$$Delta u+K(|x|)u^{p}=0 .$$ It is known that a radial solution $u$ is crossing if $u$ has a zero in $(0, infty)$; $u$
is slowly decaying if $u$ is positive but $displaystylelim_{r
ightarrow{infty}}r^{n-2}u=infty$; u is rapidly decaying if $u$ is positive,
$displaystylelim_{r
ightarrow{infty}}r^{n-2}u$ exists and is positive. Using some Pohozaev identities, we show that under certain condition on $K$, by comparing some parameters $r_{G}$ and $r_{H}$, the structure of positive radial solutions for various initial conditions can be classified as Type Z ($u(r; alpha)$ is crossing for all $r>0$ ), Type S ($u(r; alpha)$ is slowly decaying for all $r>0$), and Type M (there is some $alpha_{f}$ such that
$u(r; alpha)$ is crossing for $alphain(alpha_{f},
infty)$, $u(r; alpha)$ is slowly decaying for
$alpha=alpha_{f}$, and $u(r; alpha)$ is rapidly decaying for $alphain(0, alpha_{f})$). The above work is due to Yanagida and Yotsutani.

Identiferoai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0809104-143110
Date09 August 2004
CreatorsChen, Den-bon
Contributorsnone, none, none, 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-0809104-143110
Rightsunrestricted, Copyright information available at source archive

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