JOURNAL
OF DIFFERENTIAL
EQUATIONS
83, 368-378 (1990)
Radial Solutions of Au + f( u) = 0 with
Prescribed Numbers of Zeros*
W. C. TROY,+ AND F. B. WEISSLER+
KEVIN MCLEOD,
Department of Mathematics, University of Wisconsin, Milwaukee, Wisconsin 53201;
tDepartment of Mathematics and Statistics, University of Pittsburgh,
Pittsburgh, Pennsylvania 15260; and Departement de MathPmariques,
UniuersilP de Nantes, 44072 Names Cedex 03, France
Received January 10, 1989
1. INTRODUCTION
We consider the initial-boundary value problem
U-1
u” + u’ +f( 24)= 0
for r>O
(1.1)
as r-+03.
(1.2)
r
u’(0) = 0,
u(r) --f 0
Here n > 1 is a real parameter, and the function f satisfies the following
hypotheses:
(f 1) f: R + R is locally Lipschitz continuous,
(f2) uf(u) < 0 for (~1 small, 24# 0,
(f3) There are values /I > 0 and j?’ < 0 such that
F(u) < 0
on
(0, PI,
f(u)
’ 0
on
F(u) <0
on
(B’. 01,
f(u)
< 0
on !-a,
UC ~0)
PI,
where F(u) = f;r f(s) ds,
(f4) f(u)=k(u)
jujP-’ u+g(u), where
and
as 1~1+ co.
Ig(u)=414p)
* F. B. Weissler was partially supported by NSF Grants DMS-8201639 and DMS-8703096;
W. C. Troy was partially supported by NSF Grant DMS-8701405.
368
0022-0396190 $3.00
Copyright
0 1990 by Academic Press, Inc
All rights of reproduction
in any form reserved
RADIAL SOLUTIONS OF du+f(Zd)=O
369
It should be noted that (fl) and (f2) force f(0) = 0, while (f3) and (f4)
imply that F(U) has exactly one positive and one negative zero. Clearly, we
may take p to be the positive zero of F(U), and we shall assume this in
future. Similarly, we will take p’ to be the unique negative zero of F(U).
Our main result is the following theorem.
1. Let m 2 0 be an integer. If f satisfies (fl )-(f4) there is a
solution of ( 1.1)-( 1.2) which has exactly m zeros in (0, CC).
THEOREM
Problem (1.l)-( 1.2) has been studied extensively since it arises in the
search for radially symmetric solutions of
du+f(u)=O
U-+0
in R”
as IxJ-+oo
(1.3)
or of standing wave solutions for non-linear Schriidinger equations. (Of
course, in these applications, n must be an integer.) Strauss [S] and
Berestycki and Lions [l] have proved the existence of infinitely many
radially symmetric solutions of (1.3) by variational methods when j(u) is
an odd function. These arguments apply to very general non-linearities, but
they do not give detailed information on the shape of the solution and in
particular it was an open question for some years as to whether solutions
exist with prescribed numbers of zeros. This question was answered in the
affirmative by Jones and Kiipper [3] using a dynamical systems approach
and an application of the Conley index. Their assumptions on f are in some
respects less general than those of [I], but do cover the important model
casessuch asf(u)= --u+ (uIp-’ U. They also do not require f to be an odd
function.
The present paper was motivated partly by the wish to find a “simple”
ODE proof of the Jones-Kiipper result. It turns out that our conditions on
f are not identical with those of Jones and Kiipper, although there is
naturally some overlap. We have been able to relax slightly the regularity
required, as well as the asymptotic behaviour as 1~1-+ co. We also do not
require f ‘(0) < 0, replacing it with hypothesis (f2); we can therefore treat
examples such as f(u)= -~l~~~~‘u+Iu~~-‘u
(l<q<p)
for which
S’(O) = 0. Like Jones and Kiipper, we do not require f to be odd. On the
other hand hypothesis (f3), which seems to be necessary to the shooting
argument we employ, means that our result is not strictly more general
than that of [3].
The plan of the paper is as follows. In Section 2 we prove some elementary facts about Eq. (1.1). They are mostly standard (see for example
[4, 6, 71) but are included for completeness. We also show that there are
solutions of ( 1.1) with arbitrarily large numbers of zeros. We prove this by
370
MCLEOD, TROY, AND WEISSLER
a scaling argument: we denote by U(T,a) the solution of (1.1) together with
the initial conditions
u(0) = a > 0,
u’(0) = 0.
(1.4)
(It is shown in [6] that this initial value problem has a unique solution.)
The functions u(r, a), appropriately scaled, converge as a -+ cc to the
solution of
n-1
w” + -w’+k(w)
T
w(O)= 1,
IwIP-1 w=o
w’(0) = 0
(1.5)
(1.6)
and it is known that, for 1 <p< (n + 2)/(n -2) w has infinitely many
zeros. (This is the only place where the condition on p is used.) In case
k(u) = 1, this result may be found in [2]. The proof may be easily adapted
to the present case, or we may apply a more general result, proved in [S].
In Section 3 we prove our main result. Finally, in Section 4 we state some
results for the Dirichlet problem in finite balls and mention some other
generalizations of our results. The authors thank J. Serrin for his advice
and encouragement with this paper; in particular for showing us a
simplification of the proof of Lemma 4.
2. ELEMENTARY RESULTS AND SCALING
We begin by noting that if u is a solution of (1.1) then u attains a relative
minimum at values of r > 0 for which U’ = 0 and f(u) < 0; similarly, if
f(u) > 0 at a critical point, that critical point must be a maximum. (By the
uniqueness theorem for solutions of initial value problems a solution of
(1.1) cannot have a critical point wheref(u) =0 unless it is a constant.)
Next, we multiply (1.1) by U’ and obtain
(;d2+Fq= -+2<0.
(2.1)
The quantity :u’~ +F(u) will be called the energy of the solution and
denoted Q(r) or Q(r, a) in case u is the solution of (1.1)-( 1.4). Since the
critical points of a non-constant solution are isolated, we see that Q(l) is
strictly decreasing.
Suppose now that a non-constant solution of (1.1) has a critical point,
say at r = r0 > 0. Suppose f(u(r,,)) > 0. Then u(rO) is a local maximum for
371
RADIAL SOLUTIONS OF du+f(u)=O
u and so U(Y)< u(rO) for r slightly larger than rO. Suppose that at some
subsequent value of r, say rl , we have u(rl ) = u(rO). Then we would have
Q(rl) = h’(r, 1’ + f’(u(r,)) 2 J’lu(rd) = Q(rd,
contradicting the fact that Q strictly decreases.It follows that u(r) < u(rO)
for all r > rO. Similarly, iff(u,) < 0 we would have deduced that u(r) < u(rO)
for all r > rO. The same type of argument also shows that any solution of
(1.1) is bounded and must therefore be defined for all r > 0.
We observe finally that if u(r) = 0 for some finite value r > 0 then
Q(r) = $u’(r)’ > 0, while if u(r) + 0 as r + 0 then Q(r) +O. Thus, if the
energy of a solution ever goes negative, that solution cannot subsequently
change sign or decay to zero at co. (It is easy to see that it must approach
a zero of f as r + co.) In particular, since Q(0) = F(u(0, a)) ~0 for
O<a</l,
we have
LEMMA 1. If 0 <ad
u(r, a) -+ 0 as r + co.
fi then u(r, a) > 0 for all r > 0 and cannot satisfy
Lemma 1 tells us that there are solutions of (1.1) with no zeros at all. We
now show that if a is large u(r, a) has many zeros; indeed the number of
zeros becomes arbitrarily large as a -+ co. This clearly follows from the
following two Lemmas.
LEMMA 2.
For A > 0, define
u(y, 2) = l-21(p-
‘)u (;,
;121Cp-19.
(2.2)
Then as A -+ co, u(r, 1) + w(r) uniformly on compact subsets of [0, co),
where w is the solution of (lS)-(1.6).
For 1 <p<(n+2)/(n-2)
(1.5)-( 1.6) has infinitely many zeros.
LEMMA 3.
of
(1 <p<co
ifnG2)
the solution
The proof of Lemma 3 may be found in [2] or [S]; we prove Lemma 2
here.
Proof
of
Lemma 2. A calculation shows that u(r, 1) satisfies
VN+
n-l
__U’+~-2P/(P~l)f(~2/(P--I)U)=0
Y
u(0) = 1,
(2.3)
u’(0) = 0
(2.4)
372
MC LEOD, TROY, AND WEISSLER
We can define the energy of u(r, ,I) as we did for u(r, a):
E(r, A) =-
E(r,
1
2
o’(r, A)‘+
ip2pl’pp
‘)
u(r,i)
LZl’P- 1Is) &
J‘0
*(
(2.5)
I*) is strictly decreasing, so for J.> 0,
E(r, A) < E(0, A) = l-2p”p-
‘)
i 0'*(A
2/‘P-
I$)
ds
As 1”+ co, hypothesis (f4) shows that this last quantity remains bounded.
We deduce that E(r, 1) is bounded above, independently of r and 1. Since
also
2/‘P-
1)
s)
d
s
=
~~‘2P+z)/‘P--l)F(~2/‘P-
1)“)
--)
+
m
as [VI -+ co, uniformly in A, at least for A>, 1, it follows from (2.5) that
u(r, J.) is also bounded:
forallr>O,
Iu(r, A)I 6 M
d>l.
(2.6)
Now observe that (2.3)-(2.4) are equivalent to
r”-‘v’(r,1)=
-,lp2P/(Pp1)
's"-If(~2i'"-')v(s,i))ds.
s0
(2.7)
Using (2.6) and (f4), we see that for 2 >, 1
1*(A"'"- ')u(.s,A))(d const.( 1 + IJ.2’(p-‘)ulp)
6 const.( 1 + a2p’(p--‘)Mp)
< const. A2*lCp
- ’ ).
Hence, by (2.7),
d const. r
so that Iu’(r, ,I)\ is bounded, independently of I, on compact subsets of
[0, co). By the Arzela-Ascoli theorem and a standard diagonal argument
373
RADIAL SOLUTIONS OF dld+f(Z.l)=O
there is a sequence 1, + cc and a continuous function W(T):[0, co) + R
such that
uniformly on compacta.
u(r, &) + w(r)
(2.8)
Now we see that the right hand side of (2.7) converges to
r
ds.
- s s“-‘k(w)Iw(s)l”-lw(s)
0
In particular, the derivatives u’(Y,A,) converge (point-wise) to a function
d(y). Since w is continuous, so is 4. Since u’(r, 1,) is bounded on compacta,
we can apply the dominated convergence theorem to
u(r, A,) = 1 + 1; u’(s, &) ds
and deduce that 4(r) = W’(T).Consequently
WEC(CO,~))nC’((o,
rnp’w’(r)=
Co)),
-jrs”p’k(w)
w(0) = 1,
w’(0) = 0
and
~w(~)~~~‘w(s)ds.
0
It now follows easily that w satisfies (1.5).
3. PROOF OF THEOREM 1
One of the main tools we need for the proof of our theorem is the
following Lemma.
LEMMA 4. Let ii > /I be a value for which u(r, 2) has exactly k zeros
(k>O) and also u(r, ii) -+O as r -+ 00. If la--H/ is sufficiently small then
u(r, a) has at most k + 1 zeros on [0, 00).
Before proving Lemma 4, we will show how it is used to obtain
Theorem 1. First, define the set A, = {a > 0 1u(r, a) has no zeros}. We have
shown that (0, p] s A0 so that A, is non-empty. Also, Lemmas 2 and 3
imply that A, is bounded above. We define a, = sup A, and claim that the
solution u(r, aO) satisfies
4r, a01> 0
for all r > 0
(3.1)
ub-, ao) + 0
as r-+c0.
(3.2)
374
MC LEOD, TROY, AND WEISSLER
If u(r, a,) has a zero at some finite r then continuity of u(r, a) on a
implies that u(r, a) has a finite zero for a, - a> 0 and sufficiently small,
contradicting the definition of uO.Thus (3.1) must hold. Next, suppose that
u’(ro, a,) < 0 at some first r0 > 0. Then u”( rO, a,) > 0 so u’(r, a,) > 0 for r
slightly larger than rO. Continuity implies that if a - a, is sufficiently small,
u(r, a) has a strict local minimum at some first r,,(u). By the energy
arguments of Section 2, u(r, a) cannot have a zero, again contradicting the
definition of a,. Thus it must be the case that u’(r, a,) < 0 for all r > 0, so
that lim, _ a3u(r, uO)= ii 2 0 exists. Assume for contradiction that is> 0. We
recall that the energy Q(r, uo) is decreasing, so that lim,, o. Q(r, a,) = Q
also exists. But then
u’(r, uo)* = 2(Q -F(u)) -+ 2(&-F(u))
so lim,, m u’(r, uo) exists. Clearly, this last limit must be 0. From (1.1) we
have
U”= - n-1
24’-f(u)
r
+ -f(U)
so we need f(G) =O. Hypothesis (f3) implies Q=F(ti)<O and it follows
that for a0 - a sufficiently small, Q(r, a) becomes negative before the first
zero of u(r, a). By the arguments of Section 2, u(r, a) can have no zero at
all, which again contradicts the definition of a,. It therefore follows that
17= 0, so (3.2) holds. We now know that u(r, a,) is a positive solution of
(l.lk(1.2).
Next, we define the set A, = {u > a, 1u(r, a) has at most one zero}. It
follows from the definition of a0 and Lemmas 2 and 4 that A, is bounded
above and non-empty. We set a, = sup A, and observe that u(r, a) has
exactly one zero for a E (a,, al). We can now show, by arguments similar
to those used to prove (3.1) and (3.2) that u(r, a,) is a solution of
( 1.1) - ( 1.2) with exactly one zero. Proceeding inductively, we can produce
solutions with any given number of zeros, proving Theorem 1.
Proof of Lemma 4. We will give the proof in the case k = 0; it will be
clear how the details should be modified fcr k > 0. The key element in the
proof is a differential equation satisfied by Q(r, a). Recall that Q was
defined by
Q = id2 + F(u)
so that
RADIAL
SOLUTIONS OF dU +,f(Zd)
375
= 0
Eliminating u” between these two equations gives
Q'+ ~Q=&&).
We can multiply by the integrating factor rZn- ’ and write (3.2) as
(r 2n-2Q)‘=(2n-2)r2”--3F(‘(u),
(3.4)
an identity used by Peletier and Serrin in [6].
We now assume that a is a close to iT and that U(Y,a) has a zero; say
4R,, u)=O,
u>O
in [0, R,).
(3.5)
Lemma 4 will follow (in case k = 0) if we can show that U(Y,a) does not
have a second zero. This is turn will follow if we can show that Q(r, a) goes
negative before any possible second zero of U. In fact, we will show:
for any y <: 0, Q(r, a) has gone negative before U(Y,a) = y,
provided a is sufficiently close to ii.
(3.6)
Thus, we fix y E (B’, 0) and let 6 > 0 be such that
If’(u)l~ 6
when u E (y, iy).
(3.7)
Let R, and R, be the first values of r > R, at which u(r) = $JJand u(r) = y,
respectively, and assume for contradiction that Q(R3) 3 0. Integrating (3.4)
from R, to R, gives
R:“-‘Q(R3)
- R:,-’
Q(R2)= (2n-2)
s”’ S2Hp3F(u(s)) ds
R2
6 -(2r~-2)6(R,-R,)rnin(R:“~-~,
RFp3)
and, if Q(R3) Z 0, we obtain
R:“-2Q(R2)>(2n-2)6(R,-R2)min(R:“-3,
Rpp”).
Since U’ is bounded (at least for a close to a), R, -R, is bounded away
from 0. Also, the assumption Q(R3) k0 implies Q(r) > 0 for r d R,; by
(3.7), U’ is bounded away from 0 in (R2, R3), so that the distance R, - R2
is bounded. As a -+ ii, continuity with respect to a shows that R,, R,, and
R, become arbitrarily large so that R2/R3 + 1 and min(R:“- 2, R$-‘) 2
$R:“-3.
Thus, for a sufficiently close to ci, we see that Ryp2Q(R2) 2
CRTe3, or
QUWf-
(3.8)
2
376
MC LEOD, TROY, AND WEISSLER
We now let w be the first value of r at which u(r, ii) = /I, and we choose
a value QE (R, R, ), so that ~(0, Z) < p. (i? is a fixed value, so for a close to
2 we certainly have R, > w.) By the Mean-Value Theorem,
(3.9)
Since Q > CIR, in (0, R2) and F(u) <O in (CJ,R,), we have iu’* > C/R,, so
Iu’( > C/A
in (a, R,). Using this in (3.9) and observing that the
numerator on the left hand side of (3.9) is fixed, we find R, - o < CA.
For a close to d, R, is large, and so (r > $R,, or R, < 2a. Since a was fixed,
this gives a contradiction as a + d. The contradiction shows that (3.6) is
true, and so completes the proof of Lemma 4.
4. ADDITIONAL
RESULTS
The scaling argument of Section 2 gives rather more information than
was used there. Indeed, let z,Ju) denote the kth zero of u(r, a). For a
sufficiently large, zk(u) will exist and be a continuous function of a.
Furthermore, as a -+ cc we see from (2.2) that
u(~~‘)/~z~(u) = kth zero of u(r, &‘-‘)/*)
+ kth zero of w(r),
where w satisfies (lS)-( 1.6). In particular, we have
z/r(u)+ 0
as u+co.
(4.1)
This observation leads immediately to our second theorem, which may
be interpreted as given radial solutions for the Dirichlet problem in a ball.
Let m z 0 be an integer, and fix R E (0, 00). If f satisfies
there is a solution of the problem
THEOREM 2.
(flk(f4)
u”+-
n-l
r u'+f(u)=O
u’(0) = 0,
u(R)=0
(4.3)
which has exactly m zeros in (0, R).
Our methods may also be applied to more general equations. For
example, it causes no great difficulty to replace the non-linear term f (u) in
RADIAL SOLUTIONS OF dU +f( 24)= 0
377
(1.1) by a term r”f(u) where o > max( -2,2 - 2n). If we make the substitution s = P+ ‘)‘* the equation will become
d*u
ds2+
ldu
---~+s(u)=o,
2n+a-2
a+2
(4.4)
and our analysis can proceed as before with n replaced throughout by n’,
where
2n+a-2
a+2
n’-l=
so that
n’
2n+20>1
0+2
’
by our assumptions on 0. More general termsf(r, U) do not seemamenable
to our treatment, due to the difficulity of defining a suitable energy
function.
Finally, we remark that the coefficient of U’ in (1.5) may obviously be
replaced by a more general function g(r). The energy may be defined as
before, and will satisfy
Q’ + &dr)Q = &(r)Ft’(u).
We will not write down a complete list of hypotheses on g which will
enable the proof to go through, but it can be checked that
g(r)=;+-
n-l
r
will work. The corresponding equation
occurs in the study of self-similar, radially symmetric solutions of
semilinear heat equations [9].
J. Serrin (personal communication) has pointed out that our methods
will also apply to the p-Laplacian.
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