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.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0616103-180709 |
| Date | 16 June 2003 |
| Creators | Shiao, Jiunn-Yean |
| Contributors | none, none, C.-K. Law, J.-L. Chern |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0616103-180709 |
| Rights | unrestricted, Copyright information available at source archive |
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