Abstract
Our purpose of this paper is to study positive solutions of Lane-Emden equation
−
Δ
u
=
V
u
p
i
n
R
N
∖
{
0
}
$$\begin{array}{}
-{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\}
\end{array}$$
(0.1)
perturbed by a non-homogeneous potential V when
p
∈
[
p
c
,
N
+
2
N
−
2
)
,
$\begin{array}{}
p\in [p_c, \frac{N+2}{N-2}),
\end{array}$
where pc
is the Joseph-Ludgren exponent. When
p
∈
(
N
N
−
2
,
p
c
)
,
$\begin{array}{}
p\in (\frac{N}{N-2}, p_c),
\end{array}$
the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk
of −Δ
u = up
in ℝ
N
∖ {0} by authors in [9]. While the fast decaying solution wk
is unstable for
p
∈
(
p
c
,
N
+
2
N
−
2
)
,
$\begin{array}{}
p\in (p_c, \frac{N+2}{N-2}),
\end{array}$
so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of
−
Δ
u
=
u
N
+
2
N
−
2
$\begin{array}{}
-{\it\Delta} u = u^{\frac{N+2}{N-2}}
\end{array}$
in ℝ
N
and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.
A Steady Flow of The Viscous Compressible Liquid in Vertical Pipe
