\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 227, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2016/227\hfil Existence and nonexistence of solutions]
{Existence and nonexistence of solutions for semilinear equations on
exterior domains}
\author[Joseph A. Iaia \hfil EJDE-2016/227\hfilneg]
{Joseph A. Iaia}
\address{Joseph A. Iaia \newline
Department of Mathematics,
University of North Texas,
P.O. Box 311430,
Denton, TX 76203-1430, USA}
\email{iaia@unt.edu}
\thanks{Submitted July 20, 2016. Published August 22, 2016.}
\subjclass[2010]{34B40, 35B05}
\keywords{Exterior domains; semilinear; superlinear; radial}
\begin{abstract}
In this article we study radial solutions of $\Delta u + K(r)f(u)= 0$ on
the exterior of the ball of radius $R>0$ centered at the origin in
${\mathbb R}^{N}$ where $f$ is odd with $f<0$ on $(0, \beta) $, $f>0$ on
$(\beta, \delta)$, $f\equiv 0$ for $u> \delta$, and where the function
$K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty$.
The primitive $F(u) = \int_{0}^{u} f(t) \, dt$ has a ``hilltop'' at
$u=\delta$. We prove that if $K(r) \sim r^{-\alpha}$ with $\alpha> 2(N-1)$
and if $R>0$ is sufficiently small then there are a finite number of
solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of
radius $R$ such that $u \to 0$ as $r \to \infty$. We also prove the
nonexistence of solutions if $R$ is sufficiently large.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\section{Introduction}
In this article we study radial solutions of
\begin{gather}
\Delta u + K(r)f(u) = 0 \quad\text{in } \Omega, \label{1} \\
u = 0 \quad\text{on } \partial \Omega, \label{2} \\
u \to 0 \quad\text{as } |x| \to \infty \label{3}
\end{gather}
where $ x \in \Omega = {\mathbb R}^{N} \backslash B_{R}(0)$
is the complement of the ball of radius $R>0$ centered at the origin.
We assume there exist $\beta, \delta$ with $0< \beta < \delta$ such that
$f(0)=f(\beta)=f(\delta) = 0$ and $F(u) = \int_{0}^{u} f(s) \, ds$ where:
\begin{itemize} % \label{f}
\item[(H1)] $f$ is odd and locally Lipschitz,
$f<0$ on $(0, \beta)$, $f>0$ on $ (\beta , \delta)$,
$f \equiv 0$ on $(\delta, \infty)$, and
$F(\delta)>0$.
\end{itemize}
We note it follows that $F(u) = \int_{0}^{u} f(s) \, ds$ is even and has
a unique positive zero, $\gamma$, with $\beta < \gamma < \delta$ such that
\begin{itemize} % \label{F2}
\item[(H2)] $F < 0$ on $(0, \gamma)$, $F>0$ on
$(\gamma, \infty)$, and $F$ is strictly monotone on $(0, \beta)$ and on
$(\beta, \delta)$.
\end{itemize}
In earlier papers \cite{I}--\cite{I2} we studied \eqref{1}, \eqref{3}
when $\Omega={\mathbb R}^{N}$ and $K(r) \equiv 1$.
In \cite{I3} we studied \eqref{1}-\eqref{3} with $K(r) \equiv 1$ and
$\Omega = {\mathbb R}^{N} \backslash B_{R}(0)$.
In that paper we proved existence of an infinite number of solutions -
one with exactly $n$ zeros for each nonnegative integer $n$ such that
$u \to 0 \quad\text{as } |x| \to \infty$.
Interest in the topic for this paper comes from recent papers
\cite{C,C2,S} about solutions of differential equations on exterior domains.
When $f$ grows superlinearly at infinity - i.e.
$\lim_{u \to \infty} \frac{f(u)}{u} = \infty$, and
$\Omega = {\mathbb R}^{N}$ then problem \eqref{1}--\eqref{3} has been
extensively studied
\cite{BL}-\cite{B}, \cite{JK,M,ST}.
The type of nonlinearity addressed here has not been studied as
extensively \cite{I}-\cite{I3}.
Since we are interested in radial solutions of \eqref{1}-\eqref{3} we assume
that $u(x) = u(|x|) = u(r)$ where $x \in {\mathbb R}^{N} $ and
$r=|x|$=$\sqrt{x_1^2 + \cdots + x_{N}^2}$ so that $u$ solves
\begin{gather}
u''(r)+\frac{N-1}{r}u'(r)+K(r)f(u(r))=0\quad\text{on }(R, \infty)
\text{ where } R > 0, \label{DE} \\
u(R) = 0, \quad u'(R) = b>0. \label{DE2}
\end{gather}
We assume that there exist constants $c_1>0$, $c_2>0$, and $\alpha > 0$ such that
\begin{itemize} %\label{K}
\item[(H3)]
$c_1 r^{-\alpha} \leq K(r) \leq c_2 r^{-\alpha}$ for $\alpha > 2(N-1)$
on $[R, \infty)$.
\end{itemize}
In addition, we assume that
\begin{itemize}
\item[(H4)] %\label{K2}
$K, K'$ are continuous on $ [R, \infty)$,
$\lim_{r \to \infty} \frac{rK'}{K} = -\alpha$, and
$\frac{rK'}{K} + 2(N-1)<0$ on $[R, \infty)$.
\end{itemize}
Note that (H4)implies $r^{2(N-1)}K(r)$ is nonincreasing.
In papers \cite{I4}-\cite{I5} we have discussed the case when
$0< \alpha < 2(N-1)$.
\begin{theorem} \label{thm1}
Let $N \geq 2$ and $ \alpha > 2(N-1)$. Assuming {\rm (H1)--(H4)} then
if $R$ is sufficiently large then there are no solutions of
\eqref{DE}-\eqref{DE2} such that
$\lim_{r \to \infty} u(r) = 0$.
\end{theorem}
\begin{theorem} \label{thm2}
Let $N > 2$ and $ \alpha > 2(N-1)$. Assuming {\rm (H1)--(H4)}
and given a nonnegative integer $n$ then
if $R>0$ is sufficiently small then there are constants $b_{i}>0 $
and solutions $u_{i}$ with $0\leq i \leq n$ of \eqref{DE}-\eqref{DE2}
with $b=b_{i}$ such that $\lim_{r \to \infty} u_{i}(r) = 0$ and $u_{i}$
has $i$ zeros on $(R, \infty)$.
\end{theorem}
An important step in proving this result is showing that solutions can be
obtained with more and more zeros by choosing $b$ appropriately.
Intuitively it can be of help to interpret \eqref{DE} as an equation
of motion for a point $u(r)$ moving in a double-well potential $F(u)$
subject to a damping force $ -\frac{N-1}{r} u'$. This potential however
becomes flat at $u = \pm\delta$. According to \eqref{DE2} the system
has initial position zero and initial velocity $b>0$. We will see that
if $b>0$ is sufficiently small then the solution will ``fall'' into
the well at $u = \beta$ and - due to damping - it will be unable to leave
the well whereas if $b>0$ is sufficiently large the solution will reach
the top of the hill at $u=\delta$ and will continue to move to the right
indefinitely. For an appropriate value of $b$ - which we denote $b^{**}$
- the solution will reach the top of the hill at $u = \delta$ as
$r \to \infty$. For values of $b$ slightly less than $b^{**}$ the solutions
will not make it to the top of the hill at $u = \delta$ and they will nearly
stop moving. Thus the solution ``loiters'' near the hilltop at $F(\delta)$
on a sufficiently long interval and will usually ``fall'' into the positive
well at $u = \beta$ or the negative well at $u = -\beta$ after passing the
origin a finite number of times, say $n$. For the right value of $b$
- which we denote as $b_{n}$ - the solution comes to rest at the local maximum
of the function $F(u)$ at the origin as $r \to \infty$ after passing the origin
$n$ times.
In contrast to this is a double-well potential that goes off to infinity as
$|u| \to \infty$ - for example $F(u) = u^2(u^2-4)$. Here the solutions of
\eqref{DE}-\eqref{DE2} behave quite differently. As $b$ increases the
number of zeros of $u$ increases as $b \to \infty$. Thus the number
of times that $u$ reaches the local maximum of $F(u)$ at the origin
increases as the parameter $b$ increases. See for example
\cite{JK,M,ST}.
\section{Preliminaries and Proof of Theorem \ref{thm1}}
\begin{proof}[Proof of Theorem \ref{thm1}]
We observe since $\alpha> 2(N-1)$, by \eqref{DE} and (H4)
\begin{equation}
\Big( \frac{1}{2} \frac{u'^2}{K} + F(u) \Big)'
= - \frac{u'^2}{2rK}\Big(2(N-1) + \frac{rK'}{K}\Big) \geq 0. \label{RMN}
\end{equation}
Hence $\frac{1}{2} \frac{u'^2}{K} + F(u) $ is nondecreasing.
Now suppose there is a solution of \eqref{DE}-\eqref{DE2} such that
$\lim_{r \to \infty} u(r) = 0$. Then $u$ must have a first local maximum,
$M$, such that $u'>0$ on $[R, M)$.
Then since $\frac{1}{2} \frac{u'^2}{K} + F(u) $ is nondecreasing we see that
$$
\frac{1}{2} \frac{u'^2}{K} + F(u) \leq F(u(M)) \quad\text{on } (R,M).
$$
Rewriting this and using (H3) we see that
$$
\frac{|u'|}{\sqrt{2}\sqrt{F(u(M)) - F(u)}} \leq \sqrt{K}
\leq \sqrt{c_2}r^{-\alpha/2} \quad\text{on } (R,M).
$$
Integrating on $(R,M)$ and noting that $\alpha > 2$ (since $\alpha > 2(N-1)$
and $N \geq 2$) gives
\begin{equation}
\int_{0}^{u(M)} \frac{\ dt}{\sqrt{2}\sqrt{F(u(M)) - F(t)}}
\leq \frac{\sqrt{c_2}}{\frac{\alpha}{2}-1}
(R^{1- \frac{\alpha}{2}} - M^{1- \frac{\alpha}{2}})
\leq \frac{\sqrt{c_2}}{\frac{\alpha}{2}-1}
R^{1- \frac{\alpha}{2}}. \label{zombie}
\end{equation}
In addition, since $\frac{1}{2} \frac{u'^2}{K} + F(u) $ is nondecreasing
we see that
$0< \frac{1}{2}\frac{b^2}{K(R)} \leq F(u(M))$ so $u(M) > \gamma$.
Further it follows from (H1)-(H2) that $F(u(M)) \leq F(\delta)$ and
$F(t) \geq -F_{0}$ for all $t\geq 0$ where $F_{0}>0$ and therefore
$F(u(M)) - F(t) \leq F(\delta) + F_{0}$. Therefore \eqref{zombie} implies
\begin{equation}
\frac{\gamma}{\sqrt{2}\sqrt{F(\delta) + F_{0}}}
\leq \frac{\sqrt{c_2}}{\frac{\alpha}{2}-1}R^{1-\frac{\alpha}{2}}. \label{NTC}
\end{equation}
We note that the left-hand side of \eqref{NTC} is positive and independent
of $R$ but that the right-hand side goes to zero as $R \to \infty$ since
$\alpha > 2$. Thus we see that if $R$ is sufficiently large then \eqref{NTC}
is violated hence there are no solutions $u$ of \eqref{DE}-\eqref{DE2}
such that $\lim_{r \to \infty} u(r)=0$ if $R$ is sufficiently large.
This completes the proof.
\end{proof}
For the remainder of this paper we assume $\alpha > 2(N-1)$ and $N>2$.
Now we make the change of variables
$$
u(r) = w(r^{2-N}).
$$
Then \eqref{DE}-\eqref{DE2} becomes
\begin{equation}
w'' + h(t) f(w) = 0, \label{DEw}
\end{equation}
and
\begin{equation}
w(R^{2-N})=0, \quad w'(R^{2-N}) = -\frac{bR^{N-1}}{N-2}<0 \label{DE3w}
\end{equation}
where $h(t) = T(t^{\frac{1}{2-N}})$ and $T(r) = \frac{r^{2(N-1)}K(r)}{(N-2)^2}$.
Then from (H3) and (H4) we see:
\begin{equation}
h(t) = T(t^{\frac{1}{2-N}}) \sim \frac{t^{q}}{(N-2)^2} \quad\text{for }
00, \quad
\lim_{t \to 0^{+}} \frac{th'(t)}{h(t)} =q.
\]
In addition, it follows from (H3)-(H4) that
\begin{equation}
\frac{c_1}{(N-2)^2} t^{q} \leq h(t) \leq \frac{c_2}{(N-2)^2} t^{q}
\text{ and } h'>0 \quad \text{for } 00. \label{DE2w}
\end{equation}
(We note that we will occasionally write $w(t) = w(t,a)$ to emphasize
the dependence of $w$ on $a$).
We attempt now to show that if $R>0$ is sufficiently small
and $n$ is a nonnegative integer then there are $a_i>0$ with
$a_0< a_1< \cdots < a_n$ such that $ w(R^{2-N}, a_i)=0$ and $w(t,a_{i})$
has $i$ zeros on $(0, R^{2-N})$.
To proceed we temporarily extend the definition of the function $h$ so that
$$
h(t) = h(R^{2-N}) + \frac{h'(R^{2-N})}{q R^{(2-N)(q-1)}}
[t^q- R^{(2-N)q} ] \text{ for } t > R^{2-N}.
$$
Note then that \eqref{hequation2} holds on $(0, \infty)$.
A useful function in the analysis of \eqref{DEw}-\eqref{DE3w} is
\begin{equation}
E(t) = \frac{1}{2} \frac{w'^2(t)}{h(t)} + F(w(t)) \quad\text{for } t>0.
\label{energy}
\end{equation}
Using \eqref{DEw}, we obtain
\begin{equation}
E'(t) = -\frac{w'^2 h'}{2h^2} \leq 0 \quad \text{since } h'>0 \text{ for } t>0.
\label{energy'}
\end{equation}
Thus $E$ is nonincreasing. Also note that $\lim_{t \to 0^{+}} E(t) = +\infty$.
We also observe using \eqref{DEw},
\begin{equation}
\frac{1}{2} w'^2 + h(t) F(w) = \frac{1}{2} a^{2}
+ \int_{0}^{t} h'(s) F(w) \, ds. \label{energy2}
\end{equation}
Another useful equation is obtained by integrating \eqref{DEw} on $(0,t)$
and using \eqref{DE2w} which gives
\begin{equation}
w'(t) = a - \int_{0}^{t} h(x) f(w(x)) \, dx. \label{integralequation2}
\end{equation}
Integrating again on $(0,t)$ gives
\begin{equation}
w(t) = at - \int_{0}^{t} \int_{0}^{s} h(x) f(w(x)) \, dx \, ds.
\label{integralequation}
\end{equation}
\section{Proof of Theorem \ref{thm2}}
From the standard theory of ordinary differential equations there exists
a unique solution of \eqref{DEw}, \eqref{DE2w} on $[0, 2\epsilon)$ for some
$\epsilon>0$. Since $E$ is nonincreasing then
$\frac{1}{2} \frac{w'^2(t)}{h(t)} + F(w(t))= E(t) \leq E(\epsilon)$ for $t> \epsilon$
from which it follows that
$w$ and $w'$ are uniformly bounded on compact subsets of $[0, \infty)$
and thus the solution $w(t)$ of \eqref{DEw},
\eqref{DE2w} exists on all of $[0, \infty)$ and varies continuously with
respect to $a$ on compact subsets of $[0, \infty)$.
\begin{lemma} \label{lem1}
Let $\alpha> 2(N-1)$, $N>2$, and let $w$ satisfy \eqref{DEw}, \eqref{DE2w}.
Suppose {\rm (H1)--(H4)} hold. Then there exists an $r_{a}>0$ such that
$w(r_{a}) = \beta$ and $00$
is sufficiently small.
\end{lemma}
\begin{proof}
By \eqref{DE2w} we have $w'(0)=a>0$ so it follows that $w$ is initially
increasing. If $0 0$ then $f(w) < 0$ by (H1) and we see
from \eqref{integralequation} that $w(t) > a t$. Thus $w(t)$ exceeds $\beta$
for large enough $t$ contradicting that $0 < w < \beta$. Thus there is an
$r_{a}>0$ such that $w(r_{a}) = \beta$ and $00$ such that $|w(s_{a})| = \gamma$
and $|w|< \gamma$ on $(0, s_{a})$.
Evaluating \eqref{energy2} at $t=s_{a}$ gives
\begin{equation}
\frac{1}{2} w'^{2}(s_{a}) \leq \frac{1}{2} a^{2} \label{paul}
\end{equation}
since $F(w(s_{a}))= F(\gamma)=0$ and $F(w)\leq 0$ on $(0, s_{a})$.
Using \eqref{paul} and the fact that $E$ is nonincreasing gives
\begin{equation}
F(w) \leq \frac{1}{2} \frac{w'^2}{h(t)} + F(w)
=E(t) \leq E(s_{a}) = \frac{1}{2} w'^{2}(s_{a})
\leq \frac{1}{2} a^{2} \text{ for } t \geq s_{a}. \label{george}
\end{equation}
Thus if $\epsilon >0$ and $a>0$ is sufficiently small then we see from (H2)
and \eqref{george} that $|w|< \gamma + \epsilon< \delta$ for $t \geq 0$.
This proves the last statement in Lemma \ref{lem1}.
Next observe from (H1) that $|f(w)| \leq C_1|w|$ for all $w$ for some $C_1>0$.
Using this along with \eqref{hequation2} in \eqref{integralequation} and
estimating gives
$$
|w(t)| \leq at + \frac{C_1c_2}{(N-2)^2} t^{q+1} \int_{0}^{t} |w(s)| \, ds.
$$
Applying the Gronwall inequality \cite{BR} we then obtain
\begin{equation}
|w| \leq a \Big( t + p(t)\int_{0}^{t} s e^{P(t)-P(s)} \, ds \Big)
\label{upperbound}
\end{equation}
where:
$$
P(t) = \int_{0}^{t} p(s) \, ds
= \int_{0}^{t} \frac{C_1c_2s^{q+1}}{(N-2)^2} \, ds
= \frac{C_1c_2t^{q+2}}{(q+2)(N-2)^2}.
$$
Evaluating \eqref{upperbound} at $t=r_{a}$ gives
\begin{equation}
\beta \leq a\Big( r_{a}+ p(r_{a})\int_{0}^{r_{a}} s e^{P(r_{a})-P(s)} \, ds \Big).
\label{GRF}
\end{equation}
It follows from \eqref{GRF} and since $p(t)$, $P(t)$ are continuous that
$r_{a} \to \infty$ as $a \to 0^{+}$. This completes the proof.
\end{proof}
\begin{lemma} \label{lem2}
Let $\alpha> 2(N-1)$, $N>2$, and let $w$ satisfy \eqref{DEw}, \eqref{DE2w}.
Suppose {\rm (H1)--(H4)} hold. If $a>0$ is sufficiently large then there
exists a $t_{a}>0$ such that $w(t_{a})=\delta$
and $w(t)<\delta$ on $[0, t_{a})$.
\end{lemma}
\begin{proof}
It follows from (H1) that $|f(w)|\leq C_2$ for some $C_2>0$ so by
\eqref{hequation2} and \eqref{integralequation2}:
$$
w' \geq a- \frac{C_2c_2t^{q+1}}{(q+1)(N-2)^2} \quad \text{for } t\geq 0.
$$
Integrating on $(0,t)$ gives
$$
w(t) \geq at - \frac{C_2c_2t^{q+2}}{(q+2)(q+1)(N-2)^2} \quad\text{for } t \geq 0.
$$
Thus for large enough $a$ we have
$$
w(1) \geq a - \frac{C_2c_2}{(q+2)(q+1)(N-2)^2} \geq \delta.
$$
Therefore $w(t)$ exceeds $\delta$ if $a>0$ is sufficiently large.
This completes the proof.
\end{proof}
Let
$$
S = \{ a>0 :\text{ there is a $t_{a}>0$ such that $w(t_{a},a)=\delta$
and $0< w<\delta$ on $(0, t_{a})$} \}.
$$
By Lemma \ref{lem2} the set $S$ is nonempty and from Lemma \ref{lem1}
the set $S$ is bounded from below by a positive constant.
Now we let:
$$
0< a^* = \inf S.
$$
\begin{lemma} \label{lem3}
Let $\alpha> 2(N-1)$, $N>2$, and let $w$ satisfy \eqref{DEw}, \eqref{DE2w}.
Suppose {\rm (H1)--(H4)} hold. Then $w(t,a^{*}) \to \delta$ as $t \to \infty$
and $w'(t, a^{*})> 0$ on $[0, \infty)$.
\end{lemma}
\begin{proof}
We first show $w(t, a^{*}) < \delta$ on $[0, \infty)$.
If not then there is a $t_{a^*}>0$ such that $w(t_{a^*},a^*)=\delta$ and
$w(t, a^*)<\delta $ on $[0, t_{a^*})$.
Thus $w'(t_{a^*},a^*)\geq 0$. In fact $w'(t_{a^*},a^*)> 0$ for if
$w'(t_{a^*},a^*)= 0$ then by uniqueness of solutions of initial value problems
$w(t,a^*) \equiv \delta$ contradicting that $w(0,a^*)=0$.
So since $w'(t_{a^*},a^*)> 0$ and $w(t_{a^*},a^*)=\delta$
then there is an $x_{a^*}>t_{a^*}$ such that $w(x_{a^*}, a^*)> \delta+ \epsilon$
for some $\epsilon>0$.
Now for $a< a^{*}$ but $a$ close to $a^*$ then by continuity with respect
to initial conditions we have $w(x_{a^*},a)> \delta$ contradicting the
definition of $a^*$. Thus $w(t,a^*) < \delta$ on $[0, \infty)$.
Next we show
\begin{equation}
E(t, a^*) \geq F(\delta) \quad \text{for all } t> 0. \label{this}
\end{equation}
So suppose not. Then there is a $t_{0}>0$ such that
$E(t_{0}, a^*) < F(\delta)$. By continuity with respect to initial
conditions $E(t_{0}, a) < F(\delta)$ for $a>a^*$ and $a$ close to $a^*$.
However, for $a>a^*$ there is a $t_{a}>0$ such that $w(t_{a},a)= \delta$
and $w'(t_{a},a)>0$ so therefore
since $f(w)\equiv 0$ for $w> \delta$ (by (H1)) then by \eqref{DEw}
it follows that $w(t,a) = w'(t_{a},a)(t-t_{a}) + \delta \geq \delta$
for $t\geq t_{a}$ and thus $E(t,a) \geq F(\delta)$ for all $t>t_{a}$.
Then since $E$ is nonincreasing (by \eqref{energy'}) it follows that
$E(t,a) \geq F(\delta)$ for all $t > 0$ contradicting that
$E(t_{0}, a) < F(\delta)$. Thus $E(t, a^*) \geq F(\delta)$ for $t> 0$.
Next we show $w'(t, a^*)> 0$ for $t\geq 0$. First, since $w'(0,a)=a>0$
we see that $w'(t,a)>0$ for small $t>0$.
Suppose then there is an $M>0$ such that $w'(M, a^*)=0$ and $w'(t,a^*)>0$
on $[0,M)$. Then from \eqref{DEw} we have $w''(M,a^*)\leq 0$ and so
$f(w(M,a^*))\geq 0$. Thus $w(M,a^*)\geq \beta$.
Also since we showed at the beginning of the proof that $w(t,a^*)<\delta$
for $t\geq 0$ it follows that $\beta \leq w(M,a^*)<\delta$ and since $F$
is increasing on $(\beta, \delta)$ (by (H2)) then
$E(M,a^*) = F(w(M,a^*)) < F(\delta)$. On the other hand it follows from
\eqref{this} that $E(M,a^*)\geq F(\delta)$ and so we obtain a contradiction.
Thus, $w'(t, a^*)> 0$ on $[0,\infty)$.
It now follows from Lemmas \ref{lem1} and \ref{lem2} that there is an $L$ with
$\beta < L\leq \delta$ such that $\lim_{t \to \infty} w(t, a^*)=L$.
From \eqref{DEw} we see that $\frac{w''(t,a^*)}{h(t)} \to -f(L)$ as $t \to \infty$.
If $f(L)\neq 0$ then $ |w''| \geq \epsilon_{0}h(t)>0$ for large $t>0$ and for some
$\epsilon_{0}>0$. Since $h(t)\sim t^q$ with $q>0$ then
integrating the inequality $ |w''| \geq \epsilon_{0}h(t)>0$ twice on
$(t_{0},t)$ where $t_{0}$ is large we see that $|w|\to \infty $
contradicting that $w(t,a^*)\to L$. Thus $f(L)=0$ and since
$\beta 0 \text{ for } t \geq 0 \text{ and }
\lim_{t\to \infty} w(t, a) = \delta \}.
\label{lagos}
\end{equation}
By Lemma \ref{lem3} we see that
$$
a^* \in \{ a : w'(t,a) > 0 \text{ for } t \geq 0 \text{ and }
\lim_{t\to \infty} w(t, a) = \delta \}.
$$
Thus the set on the right-hand side of \eqref{lagos} is nonempty
and by Lemma \ref{lem1} it is bounded from below by a positive constant.
Thus $0< a^{**} \leq a^*$ and a similar argument as in Lemma \ref{lem3}
shows that $w(t,a^{**}) \to \delta$ as $t \to \infty$ and $w'(t, a^{**})> 0$
for $t\geq 0$.
\begin{lemma} \label{lem4}
Let $\alpha> 2(N-1)$, $N>2$, and let $w$ satisfy \eqref{DEw}, \eqref{DE2w}.
Suppose {\rm (H1)--(H4)} hold. If $00$, and $M_{a} \to \infty$ as $a\to (a^{**})^{-}$.
In addition, $w(M_{a},a) < \delta $ and $w(M_{a},a)\to \delta$
as $a \to (a^{**})^{-}$.
\end{lemma}
\begin{proof}
If $a< a^{**}$ and $w'(t,a)> 0$ for $t \geq 0$ then we see as in Lemma \ref{lem3}
that $w(t,a) \to \delta$ contradicting the definition of $a^{**}$.
Thus there exists $M_{a}>0$ such that $w'(t,a)>0$ on $[0, M_{a})$ and
$w'(M_{a},a)=0$.
Then $w''(M_{a},a)\leq 0$ and so $f(w(M_{a},a)) \geq 0$.
Thus $w(M_{a},a) \geq \beta$.
Since we know $w(t,a)$ does not attain the value $\delta$ because
$a< a^{**} \leq a^*$ we therefore have $\beta\leq w(M_{a},a)< \delta$.
Now if the $\{ M_{a} \}$ were bounded then
a subsequence would converge to some $M_{a^{**}}$ and so by the
Arzela-Ascoli theorem a subsequence of $w(t,a)$ and $w'(t,a)$ would
converge uniformly to $w(t,a^{**})$ and $w'(t,a^{**})$ on $[0, M_{a^{**}} + 1]$
as $a \to (a^{**})^-$ and $w'(M_{a^{**}},a^{**})=0$ contradicting
$w'(t,a^{**})>0$ from the remarks after Lemma \ref{lem3}.
Thus $M_{a} \to \infty$ as $a \to (a^{**})^{-}$.
Also, as $a \to (a^{**})^{-}$ with $a < a^{**}$ we know $w(t,a)$
must get arbitrarily close to $\delta$ by continuity with respect to
initial conditions and so $w(M_{a},a) \to \delta$ as $a \to (a^{**})^{-}$.
This completes the proof.
\end{proof}
\begin{lemma} \label{lem5}
Let $\alpha> 2(N-1)$, $N>2$, and let $w$ satisfy \eqref{DEw}, \eqref{DE2w}.
Suppose {\rm (H1)--(H4)} hold. Given a positive integer $n$ if
$0 < a < a^{**}$ and $a$ is sufficiently close to $a^{**}$ then $w(t,a)$
has at least $n$ zeros on $(0, \infty)$. In addition denoting the $n$th
zero as $z_{n}(a)$ then $z_{n}(a) < R^{2-N}$ if $R$ is sufficiently small
and $a$ is sufficiently close to $a^{**}$ with $a< a^{**}$.
\end{lemma}
\begin{proof}
From Lemma \ref{lem4} we know that for $a$ sufficiently close to $a^{**}$ with
$a< a^{**}$ then $w$ has a local maximum $M_{a}$ and $w(M_{a})> \gamma > \beta$.
From \eqref{DEw} it follows that $w''<0$ while $w> \beta$ and since
$w'(M_{a}) =0$ it follows that there exists $y_{a}> M_{a}$ such that
$w(y_{a}) = \beta$. Thus there is an $x_{a}$ with $M_{a} < x_{a} < y_{a}$
such that $w(x_{a}) = \gamma$.
From \eqref{energy'} we have
$$
\frac{1}{2} \frac{w'^2}{h(t)} + F(w) = E(t) \leq E(M_{a})=F(w(M_{a},a))
\quad \text{for } t \geq M_{a}.
$$
Rewriting this gives
\begin{equation}
\frac{|w'|}{\sqrt{h}} \leq \sqrt{2}\sqrt{F(w(M_{a},a)) - F(w)}. \label{ringo}
\end{equation}
Now it follows from \eqref{hequation} that $0< \frac{th'}{h} \leq c_3$
for some $c_3>0$ and $t>0$. Then from this and \eqref{hequation2}
we see that
\begin{equation}
0 < \frac{h'}{h^{3/2}} = \frac{th'}{h} \frac{1}{th^{1/2}}
\leq \frac{c_3(N-2)}{\sqrt{c_1}} \frac{1}{t^{\frac{q}{2}+1}}.
\label{hinequality}
\end{equation}
Thus from \eqref{energy'}, \eqref{ringo}-\eqref{hinequality}, and (H3)
\begin{equation}
\begin{aligned}
-E' &= \frac{w'^2 h'}{2h^2} = \frac{|w'|}{2\sqrt{h}} \ \frac{h'}{h^{3/2}} |w'| \\
&\leq \frac{c_3(N-2)}{\sqrt{2c_1}}\sqrt{F(w(M_{a},a))
- F(w)} \frac{1}{t^{\frac{q}{2}+1}} |w'|.
\end{aligned} \label{here}
\end{equation}
Suppose now that $M_{a} < s < t$ and that $w'<0$ on $(M_{a},t)$.
Then integrating \eqref{here} on $(M_{a},t)$ and estimating we obtain
\begin{equation}
E(M_{a},a) - E(t,a) \leq \frac{c_3}{\sqrt{2c_1}}
\frac{(N-2)}{M_{a}^{\frac{q}{2}+1}}
\int_{w(t,a)}^{w(M_{a},a)} \sqrt{F(w(M_{a},a)) - F(y)} \, dy. \label{mick}
\end{equation}
Let us assume $w(t,a)>0$ and $w'(t,a)<0$ for $t>M_{a}$.
Then $[w(t,a), w(M_{a},a)] \subset [0, \delta]$ and the integrand
in \eqref{mick} is bounded hence the integral in \eqref{mick} is bounded
independent of $a$.
Thus the right-hand side of \eqref{mick} goes to $0$ as $a \to (a^{**})^{-}$
because $M_{a} \to \infty$ from Lemma \ref{lem4} and the integral is uniformly bounded.
Thus since $E(M_{a},a) = F(u(M_{a},a)) \to F(\delta)$ as
$a \to (a^{**})^{-}$ by Lemma \ref{lem4} it follows from \eqref{mick} that
$ E(t,a) \to F(\delta)$ as $a \to (a^{**})^{-}$.
Thus $E(t,a) \geq \frac{1}{2} F(\delta)$ for $a$ close to $a^{**}$
and $a < a^{**}$. In particular on $(x_{a},t)$ where $ 00$.
Thus there exists $z_{a}>x_{a}$ such that $w(z_{a},a)=0$ and $w(t,a)>0$ on
$(0, z_{a})$. By uniqueness of solutions of initial value problems we have
$w'(z_{a},a)<0$ and so while $-\beta < w(t,a) < 0$ then $w''<0$
by \eqref{DEw} and so we see that there is a
$Y_{a}> z_{a}$ such that $w(Y_{a},a) = -\beta$. Now if $w(t,a)$
does not have a local minimum for $t> Y_{a}$ then we can show in a similar
way as we did in Lemma \ref{lem3} that $w\to L$ but now where $ L< -\beta$ and
$f(L) =0$ implying $L=-\delta$.
But since $E$ is nonincreasing and $F$ is even this would imply
$F(\delta)=F(-\delta) \leq \lim_{t \to \infty} E(t,a)
\leq E(M_{a},a)=F(w(M_{a},a))$ and hence by (H2) we have $w(M_{a},a) \geq \delta$.
But recall from Lemma \ref{lem4} that since $a< a^{**}$ then $w(M_{a},a)< \delta$
thus we obtain a contradiction. Therefore it must be the case that $w(t,a)$ has
a local minimum, $m_{a}>z_{a}$, and in a similar way as in Lemma \ref{lem4} it is
possible to show $m_{a} \to \infty$ and $w(m_{a},a) \to -\delta$
as $a \to (a^{**})^{-}$. Also as we did at the beginning of this lemma
we can show that $w(t,a)$ has a second zero $z_{2,a} > z_{a}$ if $a$ is
sufficiently close to $a^{**}$ and $a < a^{**}$. Similarly we can show that
$w(t,a)$ has any desired (finite) number of zeros by choosing $a$ sufficiently
close to $a^{**}$ with $a < a^{**}$. This completes the proof.
\end{proof}
Thus we see that $z_{k}(a)$ the $k$th zero of $w(t,a)$ on $(0, \infty)$
is defined as long as $a$ is sufficiently close to $a^{**}$ with
$a < a^{**}$. It follows from continuous dependence of solutions on initial
conditions that $z_{k}(a)$ is a continuous function of $a$.
In addition $\lim_{a \to (a^{**})^{-}} z_{k}(a) = \infty$. This follows for if
the $z_{k}(a)$ were bounded then for a subsequence (again labeled $a$)
we would have $z_{k}(a) \to z^{**}$ and by the Arzela-Ascoli theorem
$w(z^{**}, a^{**}) = 0$ contradicting that $w(t, a^{**})>0$ on $(0, \infty)$.
Finally suppose $R$ is sufficiently small and $a< a^{**}$ is sufficiently
close to $a^{**}$ so that $ z_{k}(a)