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Existence and Asymptotic Behavior of Singular Solutions

Jann-Long Chern

In this paper we consider the quasilinear elliptic equation
$$
\text {div}(|\nabla u|^{m-2}\nabla u)+f(u)=0	\eqno (1)
$$
where $n>m\ge 2$.  We obtain a necessary and sufficient condition for
the existence of positive radial solutions $u=u(r)$ on $[{r_0}, \infty)$,
where $r_0 > 0$. If $f$ satisfies the further condition then Eq. $(1)$
possesses infinitely many singular ground state solutions $u(r)$
satisfying $u(r)\sim r^{-{(n-m)}\over {m-1}}$ at $\infty $ and
$u(r)\sim r^{-{(n-m)}\over m}$ at $r=0$.

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