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URN etd-0616103-180709
Author Jiunn-Yean Shiao
Author's Email Address shiaujy@math.nsysu.edu.tw
Statistics This thesis had been viewed 5059 times. Download 2055 times.
Department Applied Mathematics
Year 2002
Semester 2
Degree Master
Type of Document
Language English
Title The Structure of Radial Solutions to a Semilinear Elliptic Equation and A Pohozaev Identity
Date of Defense 2003-06-06
Page Count 34
Keyword
  • semilinear elliptic equation
  • Pohozaev identity
  • Abstract 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.
    Advisory Committee
  • J.-L. Chern - chair
  • none - co-chair
  • none - co-chair
  • C.-K. Law - advisor
  • Files
  • etd-0616103-180709.pdf
  • indicate access worldwide
    Date of Submission 2003-06-16

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