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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

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

Chen, Den-bon 09 August 2004 (has links)
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.
2

The Structure of Radial Solutions to a Semilinear Elliptic Equation and A Pohozaev Identity

Shiao, Jiunn-Yean 16 June 2003 (has links)
The elliptic equation $Delta u+K(|x|)|u|^{p-1}u=0,xin mathbf{R}^{n}$ is studied, where $p>1$, $n>2$, $K(r)$ is smooth and positive on $(0,infty)$, and $rK(r)in L^{1}(0,1)$. It is known that the radial solution either oscillates infinitely, or $lim_{r ightarrow infty}r^{n-2}u(r;al) in Rsetminus {0}$ (rapidly decaying), or $lim_{r ightarrow infty}r^{n-2}u(r;al) = infty (or -infty)$ (slowly decaying). Let $u=u(r;al)$ is a solution satisfying $u(0)=al$. In this thesis, we classify all the radial solutions into three types: Type R($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is rapidly decaying at $r=infty$. Type S($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is slowly decaying at $r=infty$. Type O: $u$ has infinitely many zeros on $(0,infty)$. If $rK_{r}(r)/K(r)$ satisfies some conditions, then the structure of radial solutions is determined completely. In particular, there exists $0<al_{0}<al_{1}<al_{2}<cdots<infty$ such that $u(r;al_{i})$ is of Type R($i$), and $u(r;al)$ is of Type S($i$) for all $al in (al_{i-1},al_{i})$, where $al_{-1}:=0$. These works are due to Yanagida and Yotsutani. Their main tools are Kelvin transformation, Pr"{u}fer transformation, and a Pohozaev identity. Here we give a concise account. Also, I impose a concept so called $r-mu graph$, and give two proofs of the Pohozaev identity.

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