数理解析研究所講究録 1445 巻 2005 年 12-18

12

On the uniqueness of nodal radial solutions of sublinear elliptic
equations in a ball
岡山理科大学・理学部

田中

敏 (Satoshi Tanaka)

Department of Applied Mathematics
Faculty of Science
Okayama University of Science

1. INTRODUCTION
We consider the second order ordinary differential equation$0<r$ $<1$ ,$u’+ \frac{N-1}{r}u’+K(r)f(u)=0$ ,(1.1)

with the boundary condition$u’(0)=u(1)=0$ ,(1.2), $sf(s)>0$ for
where $N\geq 2$ , $K\in C^{2}[0,1]$ , $K(r)>0$ for $0\leq r\leq 1$ ,$s\neq 0$ . Assume moreover that the following sublinear condition is satisfied:$f\in C^{1}(\mathrm{R})$(1.3)$\frac{f(s)}{s}>f’(s)$for $s\neq 0$ .

Note that a solution of problem (1.1)-(1.2) is a radial solution $u(r)(r=|x|)$ of
the Dirichlet problem of$\Delta u+K(|x|)f(u)=0$ in
,
$B$

$\{$

on

$u=0$

$\partial B$,

where $B=\{x\in \mathrm{R}^{N} : |x|<1\}$ .
We consider solutions of problem (1.1)-(1.2) satisfying $u(0)>0$ only. If is asolution of problem (1.1)-(1.2) with $u(0)<0$ , then it can be treated similarly as in
the case where $u(0)>0$ , since $v\equiv-u$ satisfies $v(0)>0$ and is a solution of
$u$

$u$

$v’+ \frac{N-1}{r}v’+K(r)f_{0}(v)=0$

,

$\{$

$0<r<1$ ,$v’(0)=v(1)=0$ ,where $f_{0}(s)=-f(-s)$ .
In this paper we study the uniqueness of solutions of the problem (1.1)-(1.2)
N.
having exactly $k-1$ zeros in $(0, 1)$ , where
Hence we consider the following problem:$k\in$$u’+ \frac{N-1}{r}u’+K(r)f(u)=0$ ,$(\mathrm{P}_{k})$$\{$$u’(0)=u(1)=0$ ,$u$$0<r<1$ ,$u(0)>0$ ,has exactly k–l zeros in $(0, 1)$ .

13under a certain condition.
It is known that there exists at least one solution of$f(u)=|u|^{p-1}u$,, $p>0$ $p\neq 1$ and $N\geq 3$ , theFor example, in the case where
were obtained by Y. Naito [4]. Assume thatexistence results of solutions ofsuch that
andthere exists limits$(\mathrm{P}_{k})$$(\mathrm{P}_{k})$$f_{0}$$f_{\infty}$$f_{0}= \lim_{uarrow 0}\frac{f(u)}{u}$,$f_{\infty}= \lim_{uarrow\infty}\frac{f(u)}{u}$$(0\leq f_{0}, f_{\infty}\leq\infty)$.

, the existenceandIn the case where there is a sufficiently large gap between
was established by Dambrosio [1],of solutions of. For the superlinear caseNow we consider the uniquness of solutions of$\mathrm{f}(\mathrm{u})=|u|^{p-1}u(p>1)$ , Yanagida [6] showed that, for each,has at$N\geq3$. For the subliear
most one solution if $rK’(r)/K(r)$ is nonincreasing and$f_{\infty}=0$,$f_{0}=$
that,provedfor each[2]
Kajikiya,oo andcase where (3),exists and is unique if $K(r)\equiv 1$ . However very little is known
the solution of
for the sublinear case and $K(r)\not\equiv 1$ .
about the uniquness of solutions of$f_{0}$$f_{\infty}$$(\mathrm{P}_{k})$$(\mathrm{p}_{k})$$k\in \mathrm{N}$$(\mathrm{P}_{k})$$k\in \mathrm{N}$$(\mathrm{P}_{k})$$(\mathrm{P}_{k})$The main result of this paper is as follows.
Theorem 1.1. Suppose that (1.3) holds. If(1.4)$3r^{2}(K’)^{2}-2r^{2}KK’+2(N$then, for each$k\in \mathrm{N}$,$(\mathrm{P}_{k})$–$1)rKK’$ $+4(N-1)K^{2}\geq 0$ ,$0\leq r\leq 1$,

has at most one solution.

In view of the following equality$3r^{2}(K’)^{2}-2r^{2}KK’+2$ (A -l)rKK’ $+4(N-1)K^{2}$$=K^{2} \ovalbox{\tt\small REJECT}(\frac{rK’}{K}+2)(\frac{rK’}{K}+2(N-1))-2r(\frac{rK’}{K})’\ovalbox{\tt\small REJECT}$,

we have the following corollary of Theorem 1.1.
Corollary 1.1. Suppose that (1.3) holds, Assume moreover that one of the following (1.5)-(1.7) is satisfied:

(1.5)
(1.6)
(1.7)
Then, for each$K’\leq 0$$N=2_{2}$$N>2$ ,,

$K’\geq 0$$( \frac{rK’}{K})’\leq 0$$\frac{rK’}{K}\geq-2$$k\in \mathrm{N}_{J}(\mathrm{P}_{k})$for,$0\leq r\leq 1$

for,$0\leq r\leq 1$$( \frac{rK’}{K})’\leq 0$

for,$0\leq r\leq 1$

has at most one solution.

2. LEMMAS

In this section we give several lemmas.
First we note that (1.1) can be rewritten as follows:$(r^{N-1}u’)’+r^{N-1}K(r)f(u)=0$ ,$0<r<1$ .
(2.1)

.

14

The proof of Theorem 1.1 is based on the method of Kolodner [3]. Namely weof (1.1) satisfying the initial condition
consider the solution$u’(\mathrm{O})=0$ ,$u(0)=\alpha$ $>0$ ,(2.2), we see that
where $\alpha>0$ is a parameter. Since $K\in C^{2}[0,1]$ and$u’\inC^{1}([0,1]\mathrm{x}(0,\infty))$$[0,1]$, and thatis unique and satisfies ,exists on
is a solution of linearized problem$r\in(0, 1]$ ,$(r^{N-1}w’)’+r^{N-1}K(r)f’(u(r, \alpha))w=0$ ,(2.3)$u(r, \alpha)$$u(r, \alpha)$$f\in C^{1}(\mathrm{R})$$u_{\alpha}(r, \alpha)$

$u$

$\{$

$w’(0)=0$ .

$w(0)=1$ ,(See, for example, [5, \S 6 and 13].). Let be the i-th zero ofis a solution of
Hereafter we assume thatsuch that, there exists. Let $t_{1}=0$ . For each$r\in(z_{i},z_{i+1})$. Therefore we find$u’(t_{i}, \alpha)=0$ , since $u(r, \alpha)(r^{N-1}u’(r, \alpha))’<0$ forthat$u(r, \alpha)$$(\mathrm{P}_{k})$$z_{i}$$t_{i}\in(z_{i-1)}z_{i})$$\mathrm{i}\in\{2,3, \ldots, k\}$$u(r, \alpha)$$0=t_{1}<z_{1}<t_{2}<z_{2}<\cdots<t_{k-1}<z_{k-1}<t_{k}<z_{h}=1$ ,$u(z_{\dot{f}}, \alpha)=0$,$u’(t_{i}, \alpha)=0$$u(r, \alpha)>0$(2.4)$(-1)^{i}u(r, \alpha)>0$(2.5)$(-1)^{i}u’(r, \alpha)>0$

(2.6)

for

,

$\mathrm{i}=1,2$$r\in[t_{1}, z_{1})$for $r\in(z_{i}, z_{i+1})$ ,,for$r\in(t_{i}, t_{i+1})$$(-1)^{k}u’(r, \alpha)>0$

for

,

, ,
$k$

$\ldots$,$\mathrm{i}=1,2$,$\mathrm{i}=1,2$,$r\in(t_{k}, z_{k}]$$\ldots$$\ldots$, $k-1$ ,, $k-1$ ,.

$r$Lemma 2.1. Assume that (1.3) holds. Let
$0$

$w$

be the solution of (2.3). Tften $w(r)>$.
for
Proof. Note that $w(0)=1$ and $w’(0)=0$ . Assume to the contrary that there$x\in[0, z_{1}]$exists a number$r_{1}\in$$(0, z_{1}]$such that $w(r)>0$ for$r\in[\mathrm{O}, r_{1})$and $w(r_{1})=0$ . Then15

we see that $w’(r_{1})<0$ . Let. An easy computation shows that$[r^{N-1}(w’u-wu’)]’=r^{N-1}K(r)[ff(u)-f’(u)u]w$ .

(2.7)$u\equiv u(r, \alpha)$

Recall that $u(r)>0$ for $r\in[0, z_{1})$ . Integrating of (2.7) over
we have$[0, r_{1}]$$r_{1}^{N-1}w’(r_{1})u(r_{1})= \oint_{0}^{r_{1}}r^{N-1}K(r)[f(u)-f’(u)u]w$

and using $(1,3)$ ,dr $>0$ ,which implies $w’(r_{1})>0$ . This is a contradiction. Consequently we find that$w(r)>0$ for $r\in(0, z_{1}]$ .

Lemma 2.2. Assume that (L3) holds. For each i.
w of (2.3) has at most one zero in

$\in$

{1,2,

\ldots ,k–1}, the solution$[z_{i}, z_{i+1}]$

Proof Note that$u\equiv u(r, \alpha)$

is a solution of$(r^{N-1}u’)’+r^{N-1}K(r) \frac{f(u)}{u}u=0$,$r\in(z_{i}, z_{i+1})$

and satisfies $u(z_{i})=u(z_{i+1})=0$ and $u(r)\neq 0$ for $r\in(z_{i}, z_{i+1})$ . From (1.3) it
follows that$r^{N-1}K(r)f’(u)<r^{N-1}K(r) \frac{f(u)}{u}$ ,.
$r\in(z_{i}, z_{\iota+1})$

Assume to the contrary that there exist numbers and such thatand $w(r_{0})=w(r_{1})=0$ . Then Sturm’s comparison theorem implies that
. This is a contradiction. The proof is complete.
at least one zero in$r_{0}$$z_{i}\leq r_{0}<r_{1}\leq$$r_{1}$$z_{i+1}$

has

$u$

$(r_{0}, r_{1})$The following identity plays a crucial part in the proof of Theorem 1.1.
Lemma 2.3. Leiu

(2.8)$\equiv u(r, \alpha)$

and let w be the solution of (2.3). Then$[r^{N-1}K^{-\frac{1}{2}}[w’u’-wu’]-r^{N-1}(K^{-\frac{1}{2}})’wu’]’$$=- \frac{r^{N-2}}{4K^{\frac{5}{2}}}[3r^{2}(K’)^{2}-2r^{2}KK’+2(N-1)rKK’$ $+4(N-1)K^{2}]w \frac{u’}{r}$.

for $0<r\leq 1$ .
. A direct calculation shows that (2.8) follows immediately.

$Proo/$

Remark 2.1. We note that
(2.9)$u’(0, \alpha)=\lim_{rarrow+0}\frac{u’(r,\alpha)}{r}=-\frac{K(0)f(\alpha)}{N}$,and hence, the right side of (2.8) is continuous for $0\leq r\leq 1$ . In fact, by integrating(2.1) over $[0, r]$ , we see that$u’(r, \alpha)=-r^{-(N-1)}\int_{0}^{r}t^{N-1}K(t)f(u(t, \alpha))dt$,$r\in[0, 1]$,

so that$- \frac{r}{N}\max_{t\in[0,r]}K(t)f(u(t, \alpha))\leq u’(r, \alpha)\leq-\frac{r}{N}\min_{t\in[0,r]}K(t)f(u(t, \alpha))$Then we obtain (2.9),$r\in[0, 1]$.

16

Lemma 2.4. Assume that (1.4) holds. Then the solution w of (2.3) has at least
one zero in
k–1}.
for each i {1, 2,$(t_{i}, t_{i+1}]$$\in$\ldots ,. We may assume that $w(r)>0$ for
can be treated similarly.
, since the case where $w(r)<0$ for
Then we have $w(t_{i})\geq 0$ , $w(t_{i+1})>0$ . In view of (1.1) we have

Proof. Suppose that $w(r)\neq 0$ for$r\in(t_{i}, t_{i+1}]$$r\in(t_{i}, t_{i+1}]$$r\in(t_{i}, t_{i+1}]$$u’(t_{j})=-K(t_{i})f(u(t_{j}))_{7}$$j=2,3$ , . . . , .
$k$From (2.4) and (2.9) it follows that $(-1)^{j}u’(t_{j})>0$ for $j=1$ , 2,
we have$(-\mathrm{l})^{}$, . Consequently$k$$\ldots$$(-g(t_{i+1})w(t_{i+1})u’(t_{i+1})+g(t_{i})w(t_{i})u’(t_{i}))>0$,. On the other hand, integrating (2.8) overwhere $g(r)$using (1.4) and (2.5), we find that$=r^{N-1}[K(r)]^{-\frac{1}{2}}$$(-\mathrm{l})^{}$$(-g(t_{i+1})w(t_{i+1})u’(t_{i+1})+g(t_{i})w(t_{i})u’(t_{i}))\leq 0$$[t_{i}, t_{i+1}]$

and.

This is a contradiction. The proof is complete.
Lemma 2.5. Let w be the solution of (2.3). Assume that (1.3) and (1.4) hold.
k.
Then $(-1)^{i}w(z_{i})<0$ for i $=1,$ 2,\ldots ,Proof. Lemma 2.1 implies that $w(z_{1})>0$ . By Lemmas 2.1 and 2.4, there exists

such that $w(r)>0$ for $r\in[0, c_{1})$ and $w(c_{1})=0$ . Then
Lemma 2.2 implies that $w(r)<0$ for $r\in(c_{1}, z_{2}]$ . Hence we have $w(z_{2})<0$ . Promsuch that $w(r)<0$ for
Lemma 2.4 it follows that there exists a number$r\in(c_{1}, c_{2})$ and $w(c_{2})=0$ . By Lemma 2.2 we see that $w(r)>0$ for $r\in(c_{2}, z_{3}]$ , sothat $w(z_{3})>0$ . By continuing this process, we conclude that $(-1)^{i}w(z_{i})<0$ for, . The proof is complete.
,

a number$c_{1}\in$$(z_{1}, t_{2}]$$c_{2}\in(z_{2}, t_{3}]$$\mathrm{i}=1,2$$k$$\ldots$

17

3. Proof OF THEOREM 1.1
In this section we give the proof of Theorem 1.1. To this end we employ the Pr\"ufertransformation for the solutionof problem (1.1)-(2.2). For the solution$\alpha>0$, we define the functions
with
by
and
$u(r, \alpha)$$u(r, \alpha)$$\rho(r, \alpha)$$\theta(r, \alpha)$$u(r, \alpha)=\rho(r, \alpha)\sin\theta(r, \alpha)$,$r^{N-1}u’(r, \alpha)=\rho(r, \alpha)\cos\theta(r_{l}\alpha)$,

where $’=d/dx$ . Sinceandare written in the forms$u(r, \alpha)$$u’$( , or) cannot vanish simultaneously,$r$$\rho(r, \alpha)$

and$\theta(r, \alpha)$$\rho(r, \alpha)=([u(r, \alpha)]^{2}+r^{2(N-1)}[u’(r, \alpha)]^{2})^{\frac{1}{2}}>0$

and

,,

$\theta(r, \alpha)=\arctan\frac{u(r,\alpha)}{r^{N-1}u(r,\alpha)}$respectively. Therefore, since , $u’\in C^{1}([0, 1] \mathrm{x} (0, \infty))$ , we find that
. From the initial condition (2.2) it follows that(mod). For simplicity we take $\theta(0, \alpha)=\pi/2$ . By a simple
andcalculation we see that$u$$\rho$$C^{1}([0, 1] \mathrm{x} (0, \infty))$$\theta\in$)$\rho(0, \alpha)=\alpha$$2\pi$$\theta(0, \alpha)\equiv\pi/2$$\theta’(r, \alpha)=\frac{1}{r^{N-1}}\cos^{2}\theta(r, \alpha)+r^{N-1}K(r)\frac{\sin\theta(r,\alpha)f(\rho(r,\alpha)\sin\theta(r,\alpha))}{\rho(r,\alpha)}>0$

for $r\in(0,1]$ , which shows that
fixed $\alpha>0$ . It is easy to see that$\theta(r, \alpha)$

is strictly increasing in $r\in(0,1]$ for each
if and only if
is a solution of$u(r, \alpha)$(3.1)$(\mathrm{P}_{k})$$\theta(1, \alpha)=k\pi_{7}$Hence the number of solutions of(3.1).

$(\mathrm{P}_{k})$is equal to the number of roots $\alpha>0$ of

Proposition 3.1. Let k
and letSuppose that (1.3) and (1.4) hold. Then$u(r, \alpha_{0})$$\in \mathrm{N}$be a solution of$\theta_{\alpha}(1, \alpha_{0})<0$$(\mathrm{P}_{k})$.

for some$\alpha_{0}>0$.

Proof. Observe that$\theta_{\alpha}(r, \alpha)=\frac{u_{\alpha}(r,\alpha)r^{N-1}u’(r,\alpha)-u(r,\alpha)r^{N-1}u_{\alpha}’(r,\alpha)}{[u(r,\alpha)]^{2}+[u’(r,\alpha)]^{2}}$.

Since $u(1, \alpha_{0})=0$ and $z_{k}=1$ , we obtain$\theta_{\alpha}(1, \alpha_{0})=\frac{u_{\alpha}(z_{k},\alpha_{0})}{u’(z_{k},\alpha_{0})}$.

Note that $(-1)^{k}u’(z_{k}, \alpha_{0})>0$ , because of (2.6). From Lemma 2.5, it follows that, which implies that ( , a ) $<0$ . The proof is complete.
$(-1)^{k}u_{\alpha}(z_{k\}\alpha_{0})<0$$\theta_{\alpha}$

$1$

$0$

Proof of Theorem 1.1. Assume to the contrary that there exist numbers

and $\alpha_{2}>0$ such thatThen
$0<\alpha_{1}<\alpha_{2}$ and

$u$

.
and
that
generality
lossof. We may assume without. In view of Proposition 3.1, wefor cu( , a1) and$r$$\theta(1, \alpha_{1})=\theta(1, \alpha_{2})=k\pi$$\alpha_{1}>0$$\theta(1, \alpha)\neq k\pi$$u(r, \alpha_{2})$are solutions of$\in(\alpha_{1}, \alpha_{2})$$(\mathrm{P}_{k})$$\alpha_{1}\neq\alpha_{2}$18. The intermediate value theorem
andconclude that)such that ( , a o) $=k\pi$ . This is aComm.
implies that there is a number
has at most one solution. The proof of Theorem
contradiction. Consequently,
1.1 is complete.
$\theta_{\alpha}(1, \alpha_{2})<0$$\theta_{\alpha}(1, \alpha_{1})<0$$\theta$$\alpha_{0}\in$$1$$\alpha_{2}$$(\mathrm{P}_{k})$REFERENCES[1] W. Dambrosio, Nodal solutions to semilinear elliptic equations in a ball, Differential Integral
Equations 15 (2002), 945-972.
[2] R. Kajikiya, Necessary and sufficient condition for existence and uniqueness of nodal solutions
to sublinear elliptic equations, Adv. Differential Equations 6 (2001), 1317-1346.
[3] I. I. Kolodner, Heavy rotating string–a nonlinear eigenvalue problem, Comm. Pure Appl.
Math. 8 (1955), 395-408.
[4] Y. Naito, Bounded solutions with prescribed numbers of zeros for the Bmden-Fowler differential equation, Hiroshima Math. J. 24 (1994), 177-220.
[5] W. Walter, Ordinary Differential Equations, Springer Verlag, New York, 1998.
, SIAM J. Math.
[6] E. Yanagida, Structure of radial solutions to $\Delta u+K(|x|)|u|^{p-1}u=0$ in
Anal, 27 (1996), 997-1014$R^{n}$