\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)