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\markboth{ A one-dimensional nonlinear degenerate elliptic equation }
{ Florin Catrina \& Zhi-Qiang Wang }
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 89--99. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
A one-dimensional nonlinear degenerate elliptic equation
%
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J70.
\hfil\break\indent
{\em Key words:} best constant, ground state solutions, wighted Sobolev
inequalities.
\hfil\break\indent
\copyright 2001 Southwest Texas State University.
\hfil\break\indent Published January 8, 2001. } }
\date{}
\author{ Florin Catrina \& Zhi-Qiang Wang }
\maketitle
\begin{abstract}
We study the one-dimensional version of the
Euler-Lagrange equation associated to finding the best constant
in the Caffarelli-Kohn-Nirenberg inequalities. We give a
complete description of all non-negative solutions which exist in a
suitable weighted Sobolev space ${\cal D}_a^{1,2}(\Omega)$.
Using these results we are able to extend the parameter range for
the inequalities in higher dimensions when we consider radial
functions only, and gain some useful information about
the radial solutions in the $N$-dimensional case.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{rem}[theorem]{Remark}
\begin{section}{Introduction}
The motivation of this paper are the following
inequalities due to Caffarelli, Kohn and
Nirenberg (see \cite{cakn}): for some positive constants $C_{a,b}$
\begin{equation}
\label{ine1}
\left( \int_{\mathbb{R}^N} |x|^{-bp}|u|^{p} \,dx \right)^{2/p}
\leq C_{a,b} \int_{\mathbb{R}^N} |x|^{-2a}|\nabla u|^2 \,dx
\end{equation}
holds for all $u \in C_0^{\infty}(\mathbb{R}^N)$, if and only if
\begin{equation}
\label{con}
\mbox{ for } N \geq 2 : \ \ -\infty < a < \frac{N-2}{2}, \ \ a
\leq b \leq a + 1, \ \ \mbox{ and } p= \frac{ 2N}{ N-2(1+a -b)},
\end{equation}
(with the case $N=2$ excluding $a=b$), and
\begin{equation}
\label{con1}
\mbox{ for } N =1 : \ \ -\infty < a < -\frac{1}{2}, \ \ a +
\frac{1}{2}
< b \leq a + 1, \ \ \mbox{ and } p= \frac{ 2}{ -1 + 2(b-a)}.
\end{equation}
Let ${\cal D}_a^{1,2}(\mathbb{R}^N)$ be the completion of
$C_0^{\infty}(\mathbb{R}^N)$, with respect to the norm $|| \cdot ||_a$
induced by the inner product
\begin{equation}
\label{inner}
(u,v) = \int_{\mathbb{R}^N} |x|^{-2a} \nabla u \cdot \nabla v \,dx.
\end{equation}
Then we see that (\ref{ine1}) holds for $u \in {\cal D}_a^{1,2} (\mathbb{R}^N)$.
Define the best embedding constants
\begin{equation}
\label{bco}
S(a,b) = \inf_{u \in {\cal D}_a^{1,2} (\mathbb{R}^N)\setminus \{0\}} E_{a,b}(u),
\end{equation}
where
\begin{equation}
E_{a,b}(u) = \frac{\displaystyle \int_{\mathbb{R}^N} |x|^{-2a}|\nabla u|^2 \,dx
}{\displaystyle \left(
\int_{\mathbb{R}^N} |x|^{-bp}|u|^{p} \,dx \right)^{2/p}}.
\end{equation}
The extremal functions for $S(a,b)$ are ground state solutions of the
Euler equation
\begin{equation}
\label{pr}
-div(|x|^{-2a}\nabla u) = |x|^{-bp}u^{p-1}, \ \ u \geq 0, \ \ \mbox{ in
} \mathbb{R}^N.
\end{equation}
This equation is a prototype of more general
nonlinear
degenerate elliptic equations from some physical phenomena related to
equilibrium of
anisotropic continuous media which
possibly are somewhere perfect insulators and
somewhere are perfect conductors (e.g.,
\cite{dautray:man85}).
Note that the classical Sobolev inequality
$(a = b = 0)$ and the Hardy inequality $(a = 0,\ b = 1)$ are
special cases of (\ref{ine1}), (see
also generalizations in \cite{lin} by Lin).
These inequalities have been studied by many authors.
In \cite{aubi}, \cite{tale}, the best constant and the
minimizers for the Sobolev inequality ($a=b=0$) were given by Aubin, and
Talenti. In \cite{lieb}, Lieb considered the case $a = 0$, $0 <
b< 1$ and gave the best
constants and explicit minimizers. In \cite{chch}, Chou and Chu
considered the case $a\geq 0$ and gave the best
constants and explicit minimizers. Also,
Lions in \cite{lion1} (for $a=0$) , and Wang and Willem in
\cite{wawi} (for $a>0$), have established the compactness of all minimizing
sequences up to dilations.
The symmetry of the minimizers has
been studied in \cite{lieb} and \cite{chch}. In fact, {\it all}
nonnegative solutions in ${\cal D}_a^{1,2}(\mathbb{R}^N)$ for the corresponding Euler equation
(\ref{pr}) are {\it radial solutions}
%(in the case $a = b = 0$, they are radial
%with respect to some point)
and explicitly given (\cite{aubi}, \cite{tale},
\cite{lieb}, \cite{chch}).
This was established in \cite{chch},
using a generalization of the moving plane method (e.g.,
\cite{ginn}, \cite{cags}, \cite{chli}).
The case $a<0$ has been studied
recently in \cite{hori}, \cite{camu}, \cite{will2};
in \cite{cawa} we have studied the case $a<0$
and
discovered some new phenomena including symmetry breaking
of the ground state solutions for certain values of the
parameters $a$ and $b$.
In this paper we concentrate on the case $N=1$.
Under conditions (\ref{con1}), equation (\ref{pr}) becomes
\begin{equation}
\label{ne1}
-(|x|^{-2a} u')' = |x|^{-bp}u^{p-1}, \;u\geq 0, \\\;\;
\mbox{in}\\\;\;\Omega \subset \mathbb{R},
\end{equation}
with $\Omega=\mathbb{R}$, and more generally, we consider this equation in an open
interval $\Omega\subset \mathbb{R}$ (possibly unbounded).
We are interested in solutions in
$${\cal D}_a^{1,2}(\Omega):=
\overline{C_0^\infty (\Omega)}^{||\cdot||_a}.
$$
This problem is of {\it critical case}
in the sense that there is a family
of dilations with {\it two} parameters
that leave the problems invariant (see (\ref{dila1}) in Section 2).
This feature
distinguishes the case $N=1$ from the case $N\geq 2$, for the
latter case a one-parameter family of dilations exists.
Nevertheless, we are able to give
a complete and detailed solution for the structure
of the solutions of (\ref{ne1}) for both
the ground states and the bound states, as well as
some interesting
qualitative properties of solutions.
First, we give the following
which is a corollary of our main results.
\begin{theorem}
\label{iff}
Let $a, b, p$ satisfy (\ref{con1})
with $b< a+1$. Then (\ref{ne1}) has a ground
state
solution in ${\cal D}_a^{1,2}(\Omega)$ if and only if $0\in \overline{\Omega}$ and
$\Omega$ is unbounded, or $0\notin \overline{\Omega}$ and $\Omega$ is
bounded.
\end{theorem}
\begin{rem}
Due to the degeneracy, the ground state solutions given here
may not be continuous at $0$ and can be identically zero
in a subinterval of $\Omega$. In fact, when
$\Omega=(-c, \infty)$ with $0 0, \ \ \mbox{ in } \mathbb{R}
\end{equation}
with $p > 2$.
The only positive solutions which are in $H^1(\mathbb{R})$, are translates of
\begin{equation}
\label{solo}
v(t) = \left( \frac{ \lambda^2 p}{2} \right)^{1/(p-2)} \left
( \cosh \left( \frac{p-2}{2} \lambda t \right)
\right)^{-2/(p-2)}.
\end{equation}
%A direct calculation gives that for the $v$ above,
%\begin{equation}
%\label{ene}
% \frac{\displaystyle \int_\mathbb{R} v_t^2 + \lambda^2 v^2 \ dt }
%{ \displaystyle \left( \int_\mathbb{R} v^p \ dt \right)^{2/p} }=
%2p \frac{\lambda^{(p+2)/p}}{(p-2)^{(p-2)/p}}\left( \frac{\Gamma^2
%(p/(p-2))}{ \Gamma (2p/(p-2))}\right)^{(p-2)/p}.
%\end{equation}
We describe briefly the results we obtained in \cite{cawa}, for
the case $N=1$.
Any nonzero solution of (\ref{pr}), $u\in {\cal D}_a^{1,2} (\mathbb{R})$ is a critical point
for the energy
\[ E_{a,b}(u) = \frac{\displaystyle \int_\mathbb{R} |x|^{-2a}| u'|^2 \,dx
}{\displaystyle \left(
\int_\mathbb{R} |x|^{-bp}|u|^{p} \,dx \right)^{2/p}}. \]
There is a
two-sided dilation invariance of (\ref{ne1}):
for $(\tau_+, \tau_-) \in (0,\infty)^2 $
\begin{equation}
\label{dila1}
u(x) \to u_{\tau_+, \tau_-} (x)=\left\{
\begin{array}{ll} \tau_+^{-\frac{1+2a}{2}} u(\tau_+ x) & x>0 \\
\tau_-^{-\frac{1+2a}{2}} u(\tau_- x) & x<0. \end{array}\right.
\end{equation}
That is, if $u$ is a solution of (\ref{ne1}) then so is $u_{\tau_+,
\tau_-}$. More generally, the energy functional
$E_{a,b}(u)$ is invariant under these two-sided dilations.
To reveal the relation with equation (\ref{ode}),
to a function $u\in {\cal D}_a^{1,2} (\mathbb{R})$, we associate a
$\mathbb{R}^2$-valued function ${\bf w}(t) = (w_1(t_1), w_2(t_2))$ for $(t_1, t_2)
\in {\cal C}$ with ${\cal C}$ being the union of two real lines $\mathbb{R}\cup \mathbb{R}$,
where
\begin{equation}
\label{tran1}
\begin{array}{ll}
u(x) = (-x)^{(1+2a)/2}w_1(-\ln (-x)), \ &\mbox{ for } \ x < 0, \\
u(x) = x^{(1+2a)/2}w_2(-\ln x), \ &\mbox{ for } \ x > 0,
\end{array}
\end{equation}
and $t_1 = -\ln (-x)$ for $x < 0$, and $t_2=-\ln x$ for $x > 0$.
Under the transformation (\ref{tran1}) we have
a Hilbert space isomorphism between ${\cal D}_a^{1,2}(\mathbb{R})$ and
$H^1({\cal C}, \mathbb{R}^2)$ (see \cite{cawa} for details) and
equation (\ref{ne1}) is equivalent to the system of
autonomous equations
\begin{equation}
\label{cy1}
-{\bf w}_{tt} + \left( \frac{1 + 2a}{2}\right)^2{\bf w} = \nabla W({\bf
w}),
\end{equation}
where $W({\bf w}) = (|w_1|^p + |w_2|^p)/p$.
Note that each of the two equations is the same as (\ref{ode}),
where $\lambda=\frac{1+2a}{2}$.
Critical points of $E_{a,b}(u)$ on
${\cal D}_a^{1,2}(\mathbb{R})$ now correspond to critical points of
a new energy functional
on $H^1({\cal C}, \mathbb{R}^2)$
\[ F_{a,b}({\bf w}) = \frac{\displaystyle \int_\mathbb{R} | {\bf w}_t|^2 +
\left( \frac{1 + 2a}{2}\right)^2|{\bf w}|^2 \ dt
}{\displaystyle \left(
\int_\mathbb{R} pW({\bf w}) \ dt \right)^{2/p}}, \;\; {\bf w} \in H^1({\cal C},
\mathbb{R}^2). \]
Each of the two ODE's of (\ref{cy1}) has the zero solution, and the only (positive) homoclinic
solutions are translates of
\begin{equation}
\label{ne1s}
v(t) = \left(\frac{ (1+2a)^2}{4(1 - 2(1 + a - b))}\right)^\frac{1 -
2(1+a - b)}{4(1 + a - b)}\left( \cosh \frac{(1 + 2a)(1 + a -
b)}{1 - 2(1+a - b)}t \right)^{-\frac{1 -2(1+a - b)}{2(1 + a -
b)}}.
\end{equation}
The minimizers of $F_{a,b}({\bf w}) $ are achieved by ${\bf w}$, for
which one of the
two components $w_1$ or $w_2$ is identically zero and the other
is a translate of $v(t)$ given above. Using $v$ and $F_{a,b}$ we have
\begin{eqnarray}
\label{noin}
S(a,b) &=&
\frac{(-1-2a)^{2(b-a)}}{2^{2(1+a-b)}(-1+2(b-a))^{-1+2(b-a)}(1+a-b)^{2(1+a-b)}}
\nonumber \\
&&\times \left( \frac{\Gamma^2
\left(\frac{1}{2(1+a-b)}\right)}{ \Gamma \left(\frac{1}{1 +
a-b}\right)}\right)^{2(1+ a -b)}.
\end{eqnarray}
We observe that as $b \searrow a+\frac{1}{2}$, we obtain $S(a,b)
\to -1 -2a$.
Note that when both $w_1$ and $w_2$ are nonzero and are (possibly different)
translates
of $v(t)$ in
(\ref{ne1s}) we get the energy $F_{a,b}({\bf w})$ to be higher
\[ R(a,b) = 2^{2(1+a-b)}S(a,b),\]
which is the least energy in the radial class.
On this energy level, there is a two parameter family of positive
solutions, according to the two parameters that control by how
much $w_1$ and $w_2$ are translated from (\ref{ne1s}).
Note that
the
two-sided dilations
(\ref{dila1}) correspond
to the translations invariance of ${\cal C}$ for
(\ref{cy1}).
Correspondingly,
$u(x)$ defined in (\ref{tran1}) is a two parameter family of solutions for
(\ref{ne1}),
which possibly after a dilation given
in (\ref{dila1}) is radial in $\mathbb{R}$.
Explicitly,
$u_{\tau_+, \tau_-} (x)$ is equal to
\begin{equation}
\label{dilre}
\left\{
\begin{array}{ll} \tau_+^{\frac{1+2a}{2}}\left(
\frac{(1+2a)^2}{1-2(1+a-b)}
\right)^{\frac{1-2(1+a-b)}{4(1+a-b)}}
\frac{x^{1+2a}}{ [
1+(\tau_+x)^{\frac{2(1+2a)(1+a-b)}{1-2(1+a-b)}}
]^{\frac{1-2(1+a-b)}{2(1+a-b)}} }
& x>0 \\
\tau_-^{\frac{1+2a}{2}}\left(
\frac{(1+2a)^2}{1-2(1+a-b)} \right)^{\frac{1-2(1+a-b)}{4(1+a-b)}}
\frac{(-x)^{1+2a}}{ [
1+(-\tau_-x)^{\frac{2(1+2a)(1+a-b)}{1-2(1+a-b)}}
]^{\frac{1-2(1+a-b)}{2(1+a-b)}}}
& x<0.
\end{array}
\right.
\end{equation}
Summarizing these, we have the following theorems from \cite{cawa}.
\begin{theorem}
\label{abc1}
(Best constants and existence of ground states)
Let $a, b, p$ satisfy (\ref{con1})
with $b< a+1$.Then $S(a,b)$ is explicitly given
in (\ref{noin}), and up to a dilation of the form (\ref{dila1})
it is achieved at a function of the form (\ref{tran1}) with either
$w_1=0$ and $w_2$ given by (\ref{ne1s}) or vice versa.
Consequently, the ground states for $S(a,b)$ are always nonradial.
\end{theorem}
\begin{theorem}
\label{nrm1}
(Bound state solutions)
Let $a, b, p$ satisfy (\ref{con1})
with $b< a+1$. Then the only bound state solutions of (\ref{ne1}) besides
the ground state solutions are given by (\ref{dilre}).
\end{theorem}
\begin{rem}
In \cite{cawa} we also proved the nonexistence of extremal functions
when $b=a+1$ for which $ S(a,a+1) =\left(\frac{1+ 2a}{2}\right)^2$,
as well as the asymptotic property of $S(a,b)$ as
$b \to \left(a+\frac{1}{2}\right)^+$.
Note that all solutions of (\ref{ne1}), possibly
after a dilation given in (\ref{dila1}), satisfy the modified
inversion symmetry $u(x) = |x|^{1+2a}u\left(\frac{x}{|x|^2}\right)$.
This was also discovered in \cite{cawa}
for bound state solutions when $N\geq 2$.
\end{rem}
\begin{rem}
\label{remre}
Standard elliptic regularity arguments break down for problem
(\ref{ne1}) at $x=0$ due to the existence of the weights.
In fact, solutions of (\ref{ne1}) need not even be continuous at
$x = 0$. We observe
that the ground states are never
continuous and that
the only bound states other than
the ground states
are continuous if and only if $\tau_+ = \tau_-$ in
(\ref{dilre}).
By a direct
computation we can see easily that
the solutions that are continuous belong
to $C^1(\mathbb{R})$ if and only if
$b < \frac{(1 + 2a)^2}{4a}$.
\end{rem}
\end{section}
\begin{section}{Half line domain}
Recall from Section 1, for
an open interval $\Omega \subset \mathbb{R}$ (possibly unbounded),
we let
\[ {\cal D}_a^{1,2}(\Omega)= \overline{C_0^\infty (\Omega)}^{||\cdot||_a}.
\]
We follow the same idea from \cite{cawa}, as used
in Section 2.
Under the transformation
\begin{equation}
\label{tran2}
u(x)=x^{(1+2a)/2} v(-\ln x),
\end{equation}
we still have
a Hilbert space isomorphism between
${\cal D}_a^{1,2}(\Omega)$ and $H^1_0(\tilde{\Omega})$ where $\tilde{\Omega}
\subset {\cal C}$ is the
image of $\Omega$.
Especially when $\Omega =(0,\infty)$, $\tilde{\Omega}$ is
$\mathbb{R}$, one of the two components of ${\cal C}$.
In this section we look at the problem
\begin{equation}
\label{neo1}
-(|x|^{-2a} u')' = |x|^{-bp}u^{p-1}, \;u\geq 0, \\\;\;
u\in {\cal D}_a^{1,2}(\Omega)
\end{equation}
with $\Omega =(0,\infty)$.
Under the above transformation,
equation
(\ref{neo1}) becomes (\ref{ode}) with $\lambda=\frac{1+2a}{2}$.
According to
(\ref{solo}), after transformed
back to $\Omega$, the
only solutions of (\ref{neo1}) are
\begin{equation}
\label{hali}
u_\tau (x)= \tau^{\frac{1+2a}{2}}\left(
\frac{(1+2a)^2}{1-2(1+a-b)}
\right)^{\frac{1-2(1+a-b)}{4(1+a-b)}}
\frac{x^{1+2a}}{ \left( 1+(\tau x)^{\frac{2(1+2a)(1+a-b)}{1-2(1+a-b)}}
\right)^{\frac{1-2(1+a-b)}{2(1+a-b)}}}.
\end{equation}
Let $E_{a,b}(u,\Omega)$ be the restriction
of $E_{a,b}(u)$ on ${\cal D}_a^{1,2}(\Omega)$.
We have the following result.
\begin{theorem}
\label{half}
Let $a, b, p$ satisfy (\ref{con1})
with $b< a+1$. Then the best constant
$\inf_{{\cal D}_a^{1,2}(\Omega)\setminus \{0\}} E_{a,b}(u,\Omega)$ is achieved
by a family of functions given by (\ref{hali}). Moreover,
all nontrivial bound state solutions of (\ref{neo1}) are given by
(\ref{hali}).
\end{theorem}
As a consequence of the above result we can prove
the inequality in Theorem \ref{radi} for radial functions in any space
dimension.
\vspace{2ex}
\noindent
{\bf Proof of Theorem \ref{radi}:}
We work in the class of radial functions $u(|x|)=u(r)$ defined on
$\mathbb{R}^N$.
Denote by $\omega_{N-1}$ the area of the unit $N-1$ dimensional
sphere in $\mathbb{R}^N$. Then
\begin{equation}
\label{norm1}
\int_{\mathbb{R}^N} |x|^{-2a}|\nabla u(x)|^2 \,dx = \omega_{N-1}
\int_0^{\infty} r^{N-1-2a} u'(r)^2 \ dr,
\end{equation}
and
\begin{equation}
\label{norm2}
\int_{\mathbb{R}^N} |x|^{-bp}|u(x)|^p \,dx = \omega_{N-1} \int_0^{\infty}
r^{N-1-bp}|u(r)|^p \ dr.
\end{equation}
If we denote
\[ \bar{a}= a - \frac{N-1}{2}, \mbox{ and } \bar{b}=b- \frac{N-1}{2} +
\frac{(1+a-b)(N-1)}{N},\]
then
\[ p= \frac{2}{-1+2(\bar{b}-\bar{a})}.\]
With a proof similar to Lemma 2.1 in \cite{cawa}, we have
\[ {\cal D}_{a,R}^{1,2}(\mathbb{R}^N) = \overline{C_{0,R}^\infty(\mathbb{R}^N \setminus \{0\})}^{|| \cdot
||_a}, \]
where $C_{0,R}^\infty(\mathbb{R}^N \setminus \{0\})$ is the space of
radial, smooth functions with compact support in $\mathbb{R}^N \setminus
\{ 0\}$.
Since inequalities (\ref{ine1}) hold for $\bar{a}+\frac{1}{2} <
\bar{b} \leq \bar{a}+1$ for functions with compact support in
$\mathbb{R}$, they also hold for functions with compact support in $(0,
\infty)$.
Hence, for $b$ in the interval $\left( a-\frac{N-2}{2}, a+1\right]$,
inequalities (\ref{ine1}) still hold in the class of radial
functions.
In fact, with $a$, $b$, and $p$ satisfying (\ref{con}),
if we denote by $R(a,b)$ the best constant for the
embedding ${\cal D}_{a,R}^{1,2}(\mathbb{R}^N)$ into the weighted $L^p_b(\mathbb{R}^N)$, we have
\begin{equation}
\label{cons}
R(a,b) = \omega_{N-1}^{\frac{p-2}{p}}S(\bar{a}, \bar{b}),
\end{equation}
where $S(\bar{a}, \bar{b})$ is given in (\ref{noin}).
When looking for solutions of $(\ref{pr})$ in ${\cal D}_{a,R}^{1,2}(\mathbb{R}^N)$, the results in
$(0, \infty)$ solve the problem by symmetry reduction.
Equation (\ref{pr}) becomes
\begin{equation}
\label{rare}
-r^{-2a}u'' -(N-1-2a)r^{-2a-1}u'=r^{-bp}u^{p-1}.
\end{equation}
Multiplying the reduced equation by $r^{N-1}$, we obtain
\begin{equation}
\label{rar}
-(r^{N-1-2a}u')' =r^{N-1-bp}u^{p-1}.
\end{equation}
Therefore, if $-\frac{N-2}{2} 0
\mbox{ on } (-\infty, m), \mbox{ and } v(m)=0,\]
where $m$ is some real number. Assuming there is a solution
$v(t)$, we multiply by $v_t(t)$ and integrate from $s\in (-\infty,
m)$ to $m$. We get,
\[ v^2_t(m)= v^2_t(s) - \left( \frac{1+2a}{2}\right)^2 v^2(s) +
\frac{2}{p}v^p(s).\]
Integrate the equality above again, from $t$ to $m$, to obtain
\begin{equation}
\label{po}
(m-t)v^2_t(m)= \int_t^m v_t^2(s) - \left( \frac{1+2a}{2}\right)^2 v^2(s) +
\frac{2}{p}v^p(s) \ ds.
\end{equation}
Since $u\in {\cal D}_a^{1,2}(\Omega)$ we have that $v\in H^1_0 (-\infty, m)$
and also in $L^p(-\infty,m)$. In equality (\ref{po}), let $t\to
-\infty$. The right hand side is bounded and this implies
$v_t(m)=0$, hence $v$ is identically zero, which provides the
necessary contradiction.
B. The second case is when zero is one of the endpoints of $\Omega$.
\hspace{2ex} $(i)$ If $\Omega$ is bounded, then after the transformation
(\ref{tros}) we have the same situation as in A$(ii)$, hence
there is no solution.
\hspace{2ex} $(ii)$ If $\Omega $ is unbounded, we are in the case of the half
line treated in Section~3.
C. Finally, the case when $0 \in \Omega$.
\hspace{2ex} $(i)$ If $\Omega$ is bounded, we have no solution by the argument
in A$(ii)$, carried out on each of the two components of
$\Omega \setminus \{0\}$.
\hspace{2ex} $(ii)$ If $\Omega$ is unbounded, and not equal to $\mathbb{R}$, then
$\Omega \setminus \{0\}$ has one component equal to the half line
and the other one being a bounded segment.
This translates into $\tilde{\Omega}$ being the union
of a copy of $\mathbb{R}$ and an unbounded interval of $\mathbb{R}$. By using
a combination of arguments above we conclude that
the only nontrivial solution is (\ref{hali}) on the half line, and
zero on the bounded segment.
Summarizing the above
we have the following
complete solution for (\ref{ne1}) as we have treated
the cases of the whole real line and the half real line
in Sections 2 and 3 respectively.
\begin{theorem}
\label{doma}
Let $a, b, p$ satisfy (\ref{con1})
with $b< a+1$.
(i) Assume $\Omega$ is not equal to $\mathbb{R}$ or $\mathbb{R}_+$ and
$0\in \overline{\Omega}$.
If $\Omega$ is bounded, (\ref{neo1}) has no solutions at all.
If $\Omega$ is unbounded (\ref{neo1})
has a unique solution (up to a dilation) which is the ground
state, ann it is zero on the bounded
component of $\Omega \setminus \{0\}$.
(ii). Assume $0\notin \overline{\Omega}$. If $\Omega$ is bounded,
(\ref{neo1}) has a unique solution which is the
ground
state.
If $\Omega$ is unbounded (\ref{neo1}) has no solution at all.
\end{theorem}
Now,
Theorem \ref{iff}
follows from Theorems \ref{abc1}, \ref{half},
and \ref{doma}.
\begin{rem}
\label{remcm}
We comment here that a result in the spirit
of Theorem \ref{iff}
was given
for the case $b=0, -1< a<-\frac{1}{2}$
in \cite{camu} (p.386, Theorem 4.1)
in which it was claimed that
(\ref{neo1}) has a ground
state
solution in ${\cal D}_a^{1,2}(\Omega)$ if and only if $\Omega=\mathbb{R}$,
$\mathbb{R}_{\pm}$, or
$\Omega$ is bounded with $0\notin \overline{\Omega}$.
Here
%Their result is incorrect as
we have shown that
(\ref{neo1}) {\it also} has ground
state
solutions in ${\cal D}_a^{1,2}(\Omega)$ when $\Omega=(-c, \infty)$ and $(-\infty, c)$
for any $c\geq 0$.
The reason is that due to the degeneracy
the maximum principle does not hold in intervals that contain
$0$.
Therefore there are nonzero
nonnegative solutions which are identically
zero in a subinterval.
\end{rem}
\begin{rem}
For completeness, we remark
that for $b=a+1$ we have $p=2$, and (\ref{neo1})
becomes a linear problem:
\begin{equation}
\label{eige}
-(|x|^{-2a} u')' = |x|^{-2(a+1)}u, \;u\geq 0, \\\;\;
\mbox{in}\\\;\;\Omega =(\alpha, \beta) \subset \mathbb{R},
\end{equation}
Using transformation (\ref{tran1}) and analyzing
the resulting equation we can easily conclude
that
(\ref{eige}) has a nonzero solution if and only if
$-\frac{3}{2} < a< -\frac{1}{2} $,
$0 \notin \Omega$
and $\frac{\beta}{\alpha}=
exp( \pm \frac{2\pi}{\sqrt{4-(1+2a)^2}})$ where
$\pm$ depends on the sign of $\beta$.
We leave the details to reader.
\end{rem}
\end{section}
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\noindent{\sc Florin Catrina \& Zhi-Qiang Wang} \\
Department of Mathematics and Statistics, \\
Utah State University, \\
Logan, UT 84322, USA. \\
e-mail: wang@sunfs.math.usu.edu
\end{document}