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Classification of the Structure of Positive Radial Solutions to some Semilinear Elliptic EquationChen, 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.
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The Structure of Radial Solutions to a Semilinear Elliptic Equation and A Pohozaev IdentityShiao, 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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