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Positive solutions of second order quasilinear equations corresponding to p-harmonic maps
NonlinearAnalysis, Theory,Methods&Applications,Vol. 31, No. 5/6, pp. 717-733, 1998 1998 El~vier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/98 $19.00+ 0.00
Pergamon PII: S0362-546X(97)00435-5
POSITIVE SOLUTIONS OF SECOND ORDER QUASILINEAR EQUATIONS CORRESPONDING TO p-HARMONIC MAPS M A N - C H U N LEUNG Department of Mathematics, National University of Singapore, Singapore 119260
(Received 1 May 1996; received for publication 18 February 1997) Key words and phrases: p-harmonic map, positive solution, rotationally symmetric. 1. I N T R O D U C T I O N
In this paper we study positive solutions u(r) of the following differential equation
1
OP/2- l(r) c~" (r) + [ ( n -- l)OP/2-1(r) . . ' ~f ' ( r ) + (0 p/2- I),(r)jot,(r )
- (n - l)Op/2-t(r) g(ot(r))g'(ct(r)) f2(r ) = 0,
(1.1)
where
O(r) = (n - 1) f2(r----g2(a(r)) ~ + (a'(r)) 2,
r > 0,
(1.2)
with p > 0 and n _> 2. In the above equation, we assume that f , g • C2([0, oo)) and f(0) = g(0) = 0,
f ' ( 0 ) = g'(0) = 1
and
f(r), g(r) > 0
(1.3)
for all r > 0. Equation (1.1) arises as the Euler-Lagrange equation of the p-energy functional for rotationally symmetric maps between certain complete noncompact Riemannian manifolds. Let (S n-I, d0 2) be the unit sphere in IPn with the induced Riemannian metric d02. Consider the following model Riemannian manifolds [1]
M ( f ) = ([0, oo) x S n-I, dr 2 + f2(r) d02), N(g) = ([0, oo) × S n-I, dr 2 + g2(r ) d02), where f , g • C2([0, oo)) satisfy the conditions in (1.3). M ( f ) and N(g) are complete noncompact Riemannian manifold. The Euclidean space and the hyperbolic space are corresponding to f ( r ) = r and g(r) = sinh r, respectively. A map F: M n ( f ) --" Mn(g) is called a rotationally symmetric map if
F(r, O) = (or(r), 0)
for all r > 0 and 0 • S"- 1,
where c~: [0, oo) --* [0, oo) with c~(0) = 0. The energy density [2] of a rotationally symmetric map F: M " ( f ) --" M"(g) is then given by (a'(r)) 2
(n +
-
1" g2(°t(r)) O(r). ) f2(r ) 717
718
MAN-CHUN LEUNG
Given p > 0, the p-energy functional [3] for C2-rotationally symmetric maps from M(f) to M(g) is given by
i'ooI
ot'2(r) + (n -
°2[alr~P/2J wx
~. l J I
l ) ~ J
~rl--I4
[r)dr.
(1.4)
t, 0
It is known that equation (1.1) is the Euler-Lagrange equation of the p-energy functional with respect to compactly supported variations. Critical points of the p-energy functional (1.4) are known as rotationally symmetric p-harmonic map from M(f) to N(g). For p = 2, equation (1.1) becomes a"(r) + (n - l ) ~ a ' ( r )
jtr)
- (n - 1) g(~(r))g'(a(r)) f2(r)
=
O.
(1.5)
Equation (1.5) corresponds to rotationally symmetric harmonic maps, which have been studied by Ratto and Rigoli [4] and Cheung and Law [5]. For p > 2, the existence of entire positive solutions to equation (1.1) together with a Liouville's type theorem have been studied in [6]. In [7], we discuss asymptotic properties of entire positive solutions to equation ( l . l ) when f(r) = g(r) = sinh r, that is, M(f) and M(g) are hyperbolic spaces. In this paper we study the local and asymptotic properties of positive solutions to equation (1.1). For p > 2, we show that if c~(r) ~ C2(0, e) is a nonnegative solution to (1.1) with limr~o÷ or(r) = 0, and a(r) goes to zero at arbitrary high order, then a -= 0 on (0, e) (Lemma 2.5). This unique continuation property is well-known for harmonic maps [8] but is unknown to us for p-harmonic maps in general with p ~ 2 (cf. [9]). By refining the argument, we can estimate the order that ct can approach zero. We show that, under the assumption that g"(y) >_ 0 for all y > 0, two positive solutions of equation (1.1) have to coincide if the solutions have the same value at one point (Theorem 2.14). The condition that g"(y) >_ 0 for all y > 0 is equivalent to the condition that the radial Ricci curvature of N(g) is nonpositive. We discuss asymptotic properties of positive entire solutions to equation (l.1). We obtain a Louville's type theorem for certain rotationally symmetric p-harmonic maps into the Euclidean space or the hyperbolic space (Theorem 2.24). When M(f) is similar to the hyperbolic space and N(g) is similar to the Euclidean space, we show that a(r) either goes to infinity exponentially ot it is bounded (Theorem 3.12). If g grows at most exponentially, then we show that either c~ grows at least linearly or ot is bounded (Theorem 3.19). We show that, under certain conditions on f and g, all rotationally symmetric p-harmonic maps with p > 2 are bounded (Corollary 3.22). Equation (l. 1) is a quasilinear second order differential equation, where the nonlinear term 0 p/E-l(r) makes the equation complicated. We introduce a first order equation on O(r) and ct(r). This first order equation is useful to obtain information about the solutions. 2. U N I Q U E C O N T I N U A T I O N
AND LIOUVILLE'S
TYPE THEOREM
For p > 0, let
6)(r) = OP/2-1(r),
r > O.
(2.1)
We have
®'(r)O(r)=(2-1)®(r)[(n-l'[g2(a(r))~'
2e~'(r)c,"(r)]
Positive solutions of second order quasilinear equations
719
Using equation (1.1) to replace the term O(r)a"(r) we obtain g2(~(r))] O'(r) (p - 1)(ot'(r)) 2 + (n - 1 ) ~ j
= (p - 2)(n - l ) O t r ) (
f2g(cl(r))g'(et(r))c '(r) f2(r )
f'(r) r g=t tr)) ~-~
[ f2tr )
]1
+ (a'(r)) 2
.
(2.2)
E q u a t i o n (2.2) involves only first derivatives o f ® and ~. For some e > 0, assume that a nonnegative CZ-solution ~ o f (1.1) exists on (0, e) with limr,0+ t~(r) = 0. Given x > 0, we can find positive constants ao, b® and Co depending on x and e, such that [f'(r)[ <_ ao,
f(r) >_ Cor
and
[g'(y(r))[ < b® for a l l r ¢
0 , ~ e
.
Then (2.2) gives
O'(r)[(p
- l)((x'(r)) 2 + (n - 1") gZ(c~(r))] ~ ]
g2(a(r)) + (a'(r))2] . < (p - 2)(n - l)®(r) (a° + bo) [ Cor I_ f 2 ( r )
(2.3)
T h e r e f o r e there is a positive constant C = C(ao, b0, c o , P , n, x) such that for p > 2 we have C ®'(r) _< - ®(r) r
on
(
X +X, 0 , ~ e
)
.
(2.4)
We have the following unique continuation p r o p e r t y for solutions o f (1.1). LEMMA 2.5. F o r p > 2 and n >_ 1, let c¢ e CZ(0, e) be a nonnegative solution to (1.1) with limr.o* a(r) = 0. If a(r) = O(r k) near 0 for all k > 0, then a - 0 on (0, e).
Proof. For n _> 2, assume that a ~ 0 on (0, e/2). We first show that a cannot be zero on (0, t~) for any di e (0, e/2) (cf. [6]). Suppose that c~ - 0 on (0, di) for some ~ e (0, e/2). Since a ~ 0 on (0, e/2), we m a y assume that a ( r ) > 0 on (di, e/2). Integrating (2.4) we have In O(r)lg -< C I n rig, where t~ < b < a < e/2. T h a t is
(°)c
®(a) _ ®(b) ~
.
(2.6)
Let b --* ~ > 0, we have ®(b) ---, 0, but ®(a) > 0, contradicting (2.6). Thus et cannot be zero on (0,,~) for a n y ,~ ~ (0, e/2). We can find a point r® e (0, e/2) such that a(ro) > 0 and a ' ( r o) > 0. Suppose that there exists a point r~ e (0, e/2) such that r~ > r o and ct(q) = 0, then there exists a point r ' e (ro, rl) such that a ( r ' ) > 0, c ~ ' ( r ' ) = 0 and
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MAN-CHUN LEUNG
~"(r) <_ O. At r ' , equation (1.1) gives O P / 2 - 1 ( r ' ) o t " ( r ') = (rt -
1)OP/2-1(r ')
g(a(r'))g'(et(r')) fZ(r )
and
O(r') = (n - 1) fZ(r,------g2(°e(r')) ~ + (od(r')) z > 0. Therefore we have a " ( r ' ) > 0, contradiction. Therefore cKr) > 0 for all r e (ro, e/2). As l i m r . 0 , o4r) = 0 and o~ ~ 0 on (0, 0) for all 0 > 0, we can let ro ~ 0. Thus o~(r) > 0 on (0, e/2). Similarly we can show that o~'(r) > 0 on (0, e/2). Integrating (2.4) we obtain O(a)_< O(r)
r
'
where in this case e / 2 > a > r > 0 and a is a constant. We have
®(r) >_ C'r c
(2.7)
for some positive constant C'. That is,
O(r) = (n - I) g2(c~(r)) fZ(r----~ + (ot'(r)) z ~ C"r 2c/0'-2),
(2.8)
where C" is a positive constant. Take a n u m b e r k such that C
k>
p-2
+1.
Since o~(r) _< Ck rk for r small, where Ck is a positive constant, a n d f ( r ) , g(r) - r when r is small, we have (n - 1)g2(°t(r)~) < C ' r 2k-2, f2(r ) where C " is a positive constant. Hence by (2.8) we have (2.9)
(o~'(r)) 2 ~ Co r 2 c / h ° - 2 )
for some positive constant Co. Therefore we have
a(r) - a(b) =
a '(s) ds ~ Cn ,,b
S 2C/(p-2)
(IS
Jb
= Co(r(2C/(p_2))+l)
_ b(2C/~_2))+1) '
~8 > r > b > O .
As limb~ o, or(b) = O, we have a positive constant such that
a(r)
>
C'r (2C/(p-2))+
1,
(2.10)
c o n t r a d i c t i n g that ct(r) - O(r k) f o r all k > O. T h e r e f o r e we have or(r) -- 0 on (0, e/2). Similar argument shows that ~x(r) ~, 0 on (0, (t¢/0¢ + l))e) f o r all K > O. Let x ~ oo, we have ix(r) - 0 on (0, e).
Positive solutions of second order quasilinear equations
721
For n = l, equation (l.1) shows that
[(ot'(r))2]p-2/Eot'(r) = C 2 for all r • (0, e), where C2 is a constant. If or(r) _> 0 and o~ , 0, then the constant C2 has to be positive. Thus a(r) = cr for some positive n u m b e r c. If el(r) _< C3r k for some k > l and for all r in a n e i g h b o r h o o d o f zero, then a - 0 on (0, R). • Let a • C2((0, e)) be a nonnegative solution to (1.1) with limr.o+ a ( r ) = 0. If we assume that a , 0, then the p r o o f o f L e m m a 2.5 shows that a(r) > 0, hence et'(r) > 0 on (0, e) [6]. Given any 6 > 0, as lim~_~0+ o~(r) = 0, we can find eo • (0, e) such that 0 < c~(r) < 6
on (0, to).
As f , g • Cz(O, ~ ) with f ' ( 0 ) = g'(0) = 1, given el • (0, I), we can assume that ~ and to are small e n o u g h such that
O < f ' ( r ) < _ 1 + el,
f(r)>(l
- el)r
and
Ig'(y(r))[ < 1 + el
for a l l r • ( 0 ,
eo).
If we assume that p > 2 and n > 2, then (2.2) gives @'(r) _< (p - 2)(n - 1)(.1+ ~ e ~ ' ~ ® ( r ) \l el/
on (0, eo).
(2.11)
-
That is, C = ( p - 2 ) ( n - l ) ( l + e l ) / ( l L e m m a 2.5 and inequality (2.9) that
-el)
in (2.4). It follows as in the p r o o f o f
or(r) >_ C i r(2c/(p-2))+ l = C l rE(n-l)((l +~O/(I-tt))+ l
(2.12)
As el can be made arbitrarily small, therefore we have the following refinement o f L e m m a 2.5. THEOREM 2.13. F o r p > 2 and n _> 1, let a • C2(0, e) be a nonnegative solution to (1.1) with lim,~o- a(r) = 0. If there exist constants k > 2n - l, C' > 0 and eo • (0, e) such that, c~(r) _< C'r k for all r • (0, eo), then ct == 0 on (0, e). We have the following uniqueness theorem for rotationally symmetric p - h a r m o n i c maps. THEOREM 2.14. F o r p _> 2 and n _> 1, let c~, fl • C2(0, R) be positive solutions to (1.1) for some constant R > 0, with limr~o+ a(r) = limr-o+ fl(r) = 0. Suppose that g"(y) >_ 0 for all y > 0. If there is a n u m b e r R o • (0, R) such that ot(Ro) = fl(R0), then ~ - ,8 on (0, R).
Proof. We first assume that p > 2 and n > 1. Suppose that ct ~ ,8 on (0, Ro). We m a y assume that there is a point rl < Ro such that c~(r~) > fl(r0. Let r o be the biggest n u m b e r in 10, rl) such that lim et(r) = lim ,8(r). r~r 0
r~r~
If a ' ( r ) _< fl'(r) for all r • (ro, r 0, then an integration shows that a ( r 0 -
c~(r') > fl(r')
and
ta'(r') > f l ' ( r ' ) .
722
MAN-CHUN LEUNG
As g"(y) >_ 0 for all y > 0 and g'(0) = 1, g and g' are positive and nondecreasing functions. As p > 2, we have O,~(r') > Oa(r'), where O,~(r)
=
®o(r)=
[(n
_
1" gZ(c~(r)) ]p/2-1 , ) f 2 ( r ) + (ot'(r))2]
[ ( n - - l ) g2(B(r)) (fl'(r))2] p/z-I ~ +
for r ~ (0, R). We have ln[®,~(r')c~'(r')] > ln[®~(r')fl'(r')].
(2.15)
Let R' 6 (r', Ro] be the smallest number such that c~(R') = fl(R'). A similar argument as above shows that a ' ( R ' ) _< p ' ( R ' ) . If a'(R') = fl'(R'), we already have c~(R') = fl(R'), then the uniqueness result for equation (1.1) (cf. [10], p. 259) implies that a---/~ on a neighborhood of R'. We note that in this case R' > 0 a n d f is positive in a neighborhood of R'. A continuation argument shows that c~ - fl on (0, R). We may assume that a'(R') < fl'(R'). Then In[O,(R')cC(R')] < In[Oo(R')B'(R')]. (2.16) By (2.15) and (2.16), there exists a point P 6 (r', R') such that In[O,(~)~'(e)l < In[O~(f)/~'(~)]
(2.17)
(ln[O~(e)cx'(e)])' _< (lnIOe(f)Jl'(t:)])'.
(2.18)
and
From equation (1.1) we have (lnlO,,(e)c~'(F)])' =
O~(e)a"(e) + O~(e)a'(e) o~(e)~'(e)
1" (g(c~(f))g'(ot(f)) (.
=
-
f'(?)'~
(2.19)
f(,:))
Similarly (ln[Oe(~)fl'(t~)])'
=
(n
-
1"j[(g(J](f))g'(fl(f)) ~
f'(f!'~ f(p) ).
(2.20)
From (2.17) we have ®~(F)~'(~) < O~(f)fl'(~).
(2.21)
As at ~, we have ~(f) > fl(~), therefore (2.21) implies that a'(~) < fl'(~). As g is an increasing function and g' is a nondecreasing function, (2.19) and (2.20) show that (ln[O,~(~)~'(f)])' > (ln[Oa(f)ff(~)])',
(2.22)
which contradicts (2.18). Hence a -= fl on (0, Ro). A continuation argument shows that - • on (0, R). In case n = 1, then a(r) = cr and fl(r) = c'r for some constants c, c' _> 0. Thus oL(r) = B(r) on (0, R).
Positive solutions of second order quasilinear equations
723
In the case of harmonic maps (p = 2) and n > 1, we assume that ot(R') = fl(R'), c~'(r') > / ~ ' ( r ' ) and u ( r ) > B(r) on ( r ' , R ' ) . There is a point ? ~ ( r ' , R ' ) such that a ( f ) > f l ( f ) , a ' ( f ) = / ~ ' ( ~ ) and a"(f)<_fl"(f). Equation (1.5) together with the assumption that g"(y) >_ 0 for all y > 0 give
.. (g(ot(f))g'(ot(r))
f ,(f ) .... )
"" (g(fl(f))g'(fl(f))~
f'(f)f(f)
~l
= p"(~),
which is a contradiction. Therefore a - B.
•
Remark. For n > l, the radial Ricci curvature of the Riemannian manifold N(g) is given by - ( n - l)g"(r)/g(r). The assumption in Theorem 2.14 is equivalent to the assumption that the radial Ricci curvature of N(g) is nonpositive. We note that for harmonic maps, m a n y uniqueness results require the curvature to be nonpositive (cf. [l l, 12]). Let ot G C2((0, oo)) be a nonnegative solution to equation (1.1). It follows from (2.2) that if f '(r) >_ 0, then we have
g2(°t(r)) ]
®'(r) (p - 1)(ta'(r)) z + (n - 1) f2(r ) j <_ (p - 2)(n - l)®(r)2g(ot(r))g'(u(r))ot'(r)f2(r) In addition, if p > 2 and n > 2 and g'(r) <_ ag(r) for all r > 0, where a is a positive constant, then there exists a constant Co such that
®'(r) <_ CoO(r)a'(r)
for all r > 0.
(2.23)
We use this to prove the following Liouville's type theorem. THEOREM 2.24. Assume that there exists positive constants c o and c~ such that 0 _< f ' r ) _< c 2 and f(r) <_ cl for all r _> r 0 . For p > 2 and n >_ 2, let F(r, 0) = (ct(r) 0) be a rotationally symmetric p-harmonic maps from M ( f ) to the Euclidean space or the hyperbolic space. If there exists a positive constant C' such that ct'(r) _< C ' for all r > 0, then o~ - 0 on ~+.
Remarks. In fact we prove a more general statement. The conditions below on g are satisfied for the Euclidean and hyperbolic metrics, where g(r) = r and g(r) = sinh r, respectively. Conic type metrics, where f(r) = k for all r large, satisfy the conditions on f in the above theorem.
Proof. If c ~ , 0, it follows from the p r o o f of L e m m a 2.5 (cf. [6]) that c~ > 0 and hence a ' ( r ) > 0 for all r > 0. Assume that (I)
g'(r) <_ ag(r)
for all r > 0.
Then (2.23) implies that
®'(r) < CO(r)a'(r),
724
MAN-CHUN LEUNG
w h e r e we m a y t a k e C = Co + 1. S u b s t i t u t e the a b o v e i n e q u a l i t y i n t o (1.1) a n d we h a v e C®[cx'(r)] 2 + ® ~ " ( r ) + (n - 1)
O ~ ' - (n -
1)j
g ( c 0 g ' ( ~ ) > 0.
(2.25)
A s 0 _< f ' ( r ) <_ c z, we h a v e f'(r) f ( r ) <- Cl
for all r > r o,
w h e r e cn is a p o s i t i v e c o n s t a n t . Since ®(r) > 0 a n d et'(r) < C ' for all r > ro, we h a v e
c~" tx' - g ( a ) g ' ( a ) + C " g(~)g'(~) -
-
1 (n - l ) f - ~ > 0
(2.26)
for all r _> r o, w h e r e C" is a p o s i t i v e c o n s t a n t . F o r r _> r o, let
H(r) =
ot'(r) g(y)g'(y)"
(2.27)
We have tt
g(~)g'(~)
- H ' + H 2 [ g ' Z ( ~ ) + g(ot)g"(ot)].
S u b s t i t u t e i n t o (2.26) we o b t a i n [g,2(a) + g ( a ) g , , ( a ) l H 2 + H ' + C ' H
(n - 1) -
f2 -
> 0.
(2.28)
A s a ' > 0, we h a v e e i t h e r limr~,~ a(r) = oo or ~ is b o u n d e d - - i n the l a t t e r case a - 0 b y a result in [6]. T h e r e f o r e we m a y a s s u m e t h a t limr.oo c~(r) = oo. A s s u m e t h a t
g"(r) >_ 0
(II)
for all r___r 0. A s g ' ( 0 ) = 1, we h a v e g'(r)_> 1 for all r > 0 . F u r t h e r m o r e , a s s u m e t h a t f o r s o m e p o s i t i v e c o n s t a n t c,
g'Z(r) - g(r)g"(r) >__c 2
(III)
T h u s l i m r ~ ® g ( r ) = o0.
for all r > 0.
T h e n we have 0 -< [g'Z(ot) + g(et)g"(cO]H 2 =
Ot'2[g'2(a) + g ( a ) g " ( a ) l g2(oOg,E(ot) ct '2
<_
ct'Zg(a)g"(a) + g3(oOg,,(c0
as b y a s s u m p t i o n ( I I I ) g ' 2 ( a ) > g(a)g"(ot). S i m i l a r l y we h a v e lim H ( r ) = O. T h e n (2.28) gives lim H ' ( r ) > k > O.
--,0
as r - ~ oo,
Positive solutions of second order quasilinear equations
725
As H(r) > 0, we have
i,r H(s) ds = O(r).
(2.29)
v ro
On the other h a n d by (III) we have
_ (\l n g 'g( (YY) )~/ ' = H[g'2(y) - g(y)g"(y)] >_ c2H for all r _> r o. Thus -In
g'(y(r))
= - ( ' (In g'(Y)']' dr >_ c 2
g(y(r))
,]ro \
g(Y) /
H(s) ds.
(2.30)
.... o
Assume that (IV)
g'(s) _> a ' g(s)
--
for all s _> 1,
where a ' is a positive c o n s t a n t , or, (IV')
g'(s) C' _> - g(s) s
--
for alls > 1.
In the first case we have a' <_ g'(y)/g(y) <_ a for all r large enough. Therefore (2.30) implies that
i'*~H(s) ds < oo. ,. r 0
Since c~'(r) < C, which implies that ¢x(r) _< C~r for all r >_ r o, where CI is a positive constant. As limr,=Cx(r) = oo, we m a y assume that o~(r) _> 1 for all r large enough. Therefore in the second case we have
a>_ g'(y) > - - c>'- g(y) c~(r)
c"
r
for all r large enough, where C" is a positive constant. We have
,t r H(s) ds <_ C" In r rO
for some constant C". In both cases, we have a contradiction with (2.29). For p = 2, Liouville's type theorems for h a r m o n i c maps have been discussed in [4]. In [13], Liouville's type theorems for p - h a r m o n i c maps are obtained under assumptions on the curvature o f the manifolds. The conditions in T h e o r e m 2.24 involve the first derivative o f the solution and g r o w t h o f f ( r ) , which relax assumptions on the curvature. On the other hand, under certain conditions o n f a n d g, there exist b o u n d e d positive entire solutions to equation (1.1) [6].
726
MAN-CHUN LEUNG 3.
ASYMPTOTIC PROPERTIES
Let f , g be functions in C2([0, oo)) which satisfy the conditions in (1.3). We study asymptotic properties o f solutions to (1.1). W h e n f ( r ) = g(r) = sinh r for all r >__0. In [71, conditions are given for positive entire solutions to equation (1.1) to be b o u n d e d , to grow to infinity fast or to be asymptotic to the solution c~(r) = r. In this section we assume that f is similar to the hyperbolic space and g is similar to the Euclidean space. More precisely, we assume that there are constants a, C > 0 such that 1
-- e ar <_f(r) <_ C e at,
(A)
C
1
-- e"" <_f ' ( r ) <_ C e ~r C
for all r > 1. A n d for all y > 1, we assume that
g(y) <_ C ' y m,
(B)
0 _< g ' ( y ) _< C ' y ' -
for some constants m _> l and C' > O. LEMMA 3.1. F o r p > 2 and n _> 2, let a(r) ~ C2(0, co) be a positive solution to (1.1) with lim~.o, a(r) = O. Suppose that f and g satisfy the conditions (A) and (B), respectively. For m > 1 let c < a/(m - 1) be a positive constant. Then there exists a positive constant ro such that either o~(r) _> e "~ or a(r) _< e "r for all r > r 0 .
Proof. If c~ is bounded, then we have c~(r) _< e ~ for r large enough. Therefore we m a y assume that lim~.= a(r) = oo. Hence there exists a constant r ' > 1 such that c~(r) > 1 for all r > r ' . For m > I, suppose that we cannot find such a r o as in the statement o f the theorem, then for any r o > r ' , there exists two points r 2 > r, > r o such that a ( r 0 = e or', or(r2) = e c~2 and a(r')<_ C r' for r ' e [rl, r2]. There is a point r E [rl, r2] such that c4r) -< e c', ot'(r) = c e cr and c~"(r) _> c2e or. Since c < a/(m - 1), we have 2g(a(r))g'(et(r))ct'(r)
f ' ( r ) [ g2(a(r)) ] f(r) f 2 ( r ~ + (~'(r))2
fE(r) <--
2C,2C2c e 2mcr e 2at
c 2 e 2cr - C2 <
(3.2)
--C'
if ro is large enough. Here c ' is a positive constant. By (2.2) and (3.2) we have
g2(a(r))] ®'(r) ( p -
l)(t~'(r))2 + ( n -
1) f 2 ( r ) j _ < - c ' ( p - 2 ) ( n -
l)O(r).
(3.3)
And
F
®'(r)](p - l)(ot'(r)) 2 + (n - l ) =
g2(ot(r))]
f Z(r) ]
- 10p/2-Z(r) (p - l)(t~'(r)) 2 + (n -
l~j
x[(n-l){g2ta(r))-'~'k, / f2tr)
->(P-l)O(r)[(n-I
+ 2ct'(r)et"tr)]
"l/gz(et(r))'~')~ f----~ )
+ 2a'(r,t~"(r)] ,
(3.4)
Positive solutions of second order quasilinear equations as O(r)= ( n - l)(g2(ct(r))/f2(r))+ (a'(r)) z a n d ® ( r ) = c o n s t a n t c" > 0 such that at r we have
727
oplZ-I(r). By (3.3), there is a
/ g 2 (t~(r))'~ # (n - 1 ) t f ~ ) + 2a'(r)~"(r)_<-c".
(3.5)
O n the o t h e r h a n d
( g2(et)'~ '
~.,]
2 f(r)g(oOIf(r)g'(oOot'(r) - f'(r)g(oO]
=
fa(r) 2 f '(r)gZ(t~)
>
f 3(r)
> -2C'2C
2 e 2met e 2at
-> - e e 2~',
(3.6)
as c < a / ( m - 1). H e r e e is a positive c o n s t a n t . A n d by c h o o s i n g r o large, we m a y a s s u m e that e < 2c3/(n - l). T h e r e f o r e if r 0 is large a n d c < a / ( m - 1), then (3.5) a n d (3.6) i m p l y that
2a'(r)a"(r) <_ (n -
l)~,c 2cr.
T h a t is
a "(r) <_ -(n -- l)e cCr < 2c
C2eCr,
which is a c o n t r a d i c t i o n . H e n c e there exists a positive c o n s t a n t r o > r ' such that either u(r) _> e cr o r c~(r) _< e ¢~ for all r > r o. •
LEMMA 3.7. F o r p > 2 a n d limr-o÷ o~(r) = 0. S u p p o s e respectively. Let c < a / ( m such that offr) _< e c" for all
n _> 2, let o~(r) ~ C2(0, oo) be a positive s o l u t i o n to (1.1) with that f a n d g satisfy the c o n d i t i o n s (A) a n d (B) with m > 1, - 1) be a positive c o n s t a n t . S u p p o s e that there exists a ro > 0 r > ro, then o~(r) _< C for s o m e positive c o n s t a n t C.
Proof. A s s u m e that or(r) < e c' for all r > r 0, where c < a / ( m - 1). W e m a y a s s u m e that a ~ 0. W e can find positive c o n s t a n t s r ' a n d d~such that o~(r) _> 6 for all r _> r ' . T h e r e is a positive c o n s t a n t C" such that g(y) <_ C " y m,
0 < g ' ( y ) < C " y m-I
for a l l y _ > f i . W e m a y t a k e r o > r ' > l. T h e n f o r r > r
oand 1 > r>O,
wehave
728
MAN-CHUN LEUNG
2g(a(r))g'(eO(r))eg(r) fZ(r)
f'(r) [ gZ(c~(r)) ] f(r) fZ(r~ + (c~'(r))2
2g(et(r))g'(e~(r))I f'(r)(e~'(r))Z]i/2 f,(r)g2(ot(r)) f'(r) = f3/2(r)f't/Z(r)
f(r)
1 gZ(o~(r))g'Z(oL(r)) < 1~ f 3(r)f'(r) + (1
]
f3(r)
f(r)
(~'(r)) z
f'(r)gZ(ot(r)) f 3(r)
f'(r) - r) ~
gZ(~(r)) = (1 -
r)f3(r)f'(r)[g'Z(a(r)) -
- (1 -
r)f3(r)f'(r)
r)2f'(r)21 - r
(1 -
~
(~'(r))2
f'(r) f(r-----)(~'(r))2
[ f'(r)gE(~(r)) f'(r) 1 f 3(r) + ~ (o~'(r))z
e ]- r
f3(r )
f(r) (a'(r))2 ' (3.8)
where we have used the inequality 2 A B _< (1 - r)A z + 1/(1 - r)B 2 for 0 < r < 1. H e n c e there exists a c o n s t a n t r" > r o such that for r > r" we have
C,,2 e2ctm- l)r
(1 -- r) 2 e2ar
U
o.
F o r r > r" we have
,o,r,,o,r, S,r,I2,o,r,,
fZ(r)
f(r)
f~
]
+ (c~'(r))2 = - - r - ~
]
f2(r~-)- + (c~'(r)) 2
- r [ gZ(c~(r)) ] _<~-g[ ~ + (~'(r)) z .
(3.9)
T o g e t h e r with (2.2) we have ®'(r) [ ( p - l)(~'(r)) 2 + (n - l ) f~ ]( )
_< - ~ - r~ ( p
- fz( - r) - 2)(n - 1) [gZ(c~(r))
+ (o~'(r))Z]O(r). (3.10)
T h u s we have
®'(r) <_ -c'®(r)
for all r > r " ,
(3.11)
where c' = (p - 2)(n - 1)r/(bC z) a n d b = m a x l p - 2, n - 11 are positive constants. I n t e g r a t i n g b o t h sides o f the a b o v e inequality we have
®(r) <_e-CV-~)O(r"),
r > r".
So we have od(r) _< C~ e -c'(t-r'')/(p-2) where C~ = (O(r")) ~/tp-2) is a positive c o n s t a n t . c o n c l u d e that a is b o u n d e d . •
for all r > r " , As c ~ ' ( r ) > 0, u p o n integration we
Positive solutions of second order quasilinear equations
729
C o m b i n i n g the previous two l e m m a s we have the following. THEOREM 3.12. For p > 2 and n _> 2, let a(r) e C2(0, oo) be a positive solution to (1.1) with limr,o* a(r) = 0. Suppose that f and g satisfy the conditions (A) and (B) with m > l, respectively. Let c < a/(m - l) be a positive constant. Then either there exists a positive constant ro such that a(r) >_eCrfor all r > r o or there is a finite posiive n u m b e r C such that o~(r) _< C for all r > 0. In the following we assume that
g'(Y) <- CEg(Y)
(C)
for all y _> 1,
where Cz is a positive constant. Condition (C) implies that g grows at most exponentially. Condition (B) is stronger than condition (C). The following can be considered as a generalization o f L e m m a 3.7. LEMMA 3.13. F o r p > 2 and n _> 2, let a(r) e C2(0, oo) be a positive solution to (1.1) with limr.o* o~(r) = 0. Suppose that f and g satisfy the conditions (A) and (C), respectively, and g'(y) > 0 for all y > 0. There is a positive constant c > 0 such that if there exists a positive constant r 0 such that c~(r) _< g-l(¢f(r)) for all r > ro, then a is a b o u n d e d function on ~+.
Proof.
We can find positive constants r ' and ~ such that t x ( r ) ~ 6 for all r _> r ' . There is a positive constant C3 such that
g'(Y) <- Cag(Y)
for all y >_ di.
For 0 < r < 1 and r > r ' , f r o m (3.8) we have
2g(a(r))g'(°l(r))°L'(r) f'(r) [ g2(a(r)) ] f2(r) f(r) fZ(r~ + (cz'(r))2 < g2(Or(r)) -- (1 - r)f3(r)f'(r)[g'2(c~(r)) - (1 -
f'(r)g2(ot(r)) f3(r )
- r
< (I - g2(c~(r))r)f 3(r)f'(r)
[ f'(r)g2(ot(r))
-r L
~
f'(r)
]
+ f - - ~ (a'(r)) 2
[ C2g2(ct(r)) f'(r)
(1 ~-z--r)2 Jtr)]~2" ,]
]
+ f - ~ -(c~'(r))2 "
If we choose a positive constant c such that c _< (1 then we have
2g(ot(r))g'(et(r))ot'(r)f'(r)[g2(ot(r)) f2(r) f(r) [ f 2 ( r )
r)Ef'(r) 2]
+ (c~'(r)) 2
We can process as in the p r o o f o f L e m m a 3.7.
r)/(CC3),
and if ~(r) _< g-~(cf(r)),
][f'(r)g2(ot(r))f'(r)] <_-z f3(r ) + ~ •
(ot'(r)) 2 •
730
MAN-CHUN LEUNG
The previous lemma provides asymptotic information a b o u t a(r). We can also obtain medium range information about ~(r). LEMMA 3.14. F o r p > 2 and n _> 2, let et(r) ~ C2(0, ~ ) be a positive solution to (1.1) with limr,0÷ a(r) = 0. Suppose that f and g satisfy the conditions (A) and (C), respectively, and g'(y) >_ 0 for all y > 0. Assume that ~ is not a b o u n d e d function on R ÷ and let a(t~) = 1 for some ~ > 0. Then there exists a positive constant 6 such that the energy density O(r) >_ 6 for all r >__L
Proof. F r o m (2.2) we have ®'(r) (p - l)(a'(r)) 2 + (n - 1)
= (p - 2 ) O ( r ) l 2(n
g2(a(r))] f2(r ) ]
l )g(et(r))g t(o~(r))o[f(r) f2(r)
2 f(r) [ ~
+ (a'(r))2 -
n-
- f 2- ( r ) + (°~'(r))2
-~
. (3.15)
At a point r _> ?, we have o~(r) _> 1. So at the point r > f, we have either (i)
l)g(~(r))g'(~(r))~'(r) >
2(n -
f2(r)
-- 2
_ _ f2(r)
f(r)
+ (c~,(r)) 2 '
or
(ii)
O'(r)
I
1) g2(°~(r))]
(p - l)(a'(r)) 2 + (n -
_< - ( p - 2) ( ~n ) -
f--~-~-]
O(r) ff'(r) ~-
Ig2(a(r))f2~(r---
+ ( c ~ ' ( r ) ) 2] .
In case o f (i), from condition (A) and (C) we have 2C2(n - l)a'(r) >
f'(r)f(r)(~'(r)) 2 g2(o~(r))
+
f'(r) 1 > -2f(r) 2C 2"
By choosing a bigger constant if necessary, we m a y assume that (A) holds for r > ~. Thus if (i) holds at r > f, we have
1 a ' ( r ) _> 4(n
-
1)C2 C2"
(3.16)
Take
6 = (8(/7
1
Suppose that there is a point r" _> ro such that 0 = {t e [r",
~2 .
_ T)C2 C2/
O(r") <_ 6. Let
oo) lO(r) <_ 6 for all r e [r", t]].
(3.17)
Positive solutions of second order quasilinear equations
731
Therefore r' ~ 0 and a'(r') <
8(n - 1)C2 C 2"
We can find t such that t > r ' and a'(r) <
4(n-
1)C"C 2
for all r e [r", t]. Hence we have (ii) for all r e [r", t]. Thus ®'(r) _< 0 and hence O'(r) <_ 0 for all r ~ [r", t]. We have O(r) <_ 6 for all r ~ [r", t]. Suppose that s u p O = a < ~ . Then at the point a we have O(a) <_ ~. The same argument shows that there exists a positive n u m b e r e such that ®'(r) < 0 for all r e (a, a + e). Then a + e e O, which is a contradiction. Therefore we have sup O = co, or O(r) < (5
for all r e [r', oo).
(3.18)
In particular, c~'(r) <
4(n-
l)C"C 2
for all r ~ It', 00). By (ii) we have O'(r)
[
g2(ct(r))] < ( p - 2 ) ( n -2{ ) C (p - l)(c~'(r)) 2 + (n - l ) ~ j _ -
, ,[-g2(a(r))~ ] OtO[ + (~'(r)) 2 •
A similar a r g u m e n t as in the p r o o f o f L e m m a 3.7 shows that there is a positive constant c > 0 such that or(r) _< c for all r ~ (0, oo). This contradicts the assumption that c~ is not b o u n d e d . Therefore O(r) >_ 3 for all r _> f. •
THEOREM 3.19. For p > 2 and n _> 2, let o~(r) ~ C2(0, oo) be a positive solution t o (1.1) with iimr,o÷ a ( r ) = 0. Suppose that f and g satisfy the conditions (A) and (C), respectively, and g ' ( y ) > 0 for all y > 0. Then either there exist positive constants r o and c o such that either or(r) _> co(r - to) for all r > r o, or c~ is a b o u n d e d function on R ÷. P r o o f . If c~ is not b o u n d e d , then there is a positive constant f such that c~(?) = 1. By L e m m a 3.14, we have O(r) >_ ~ for all r _> f. With the conditions (A) and (C), we can find positive constants c' and r6 such that
1(
~
g- \x/2(n-
1)
f(r)) > c ' r
for all r > r~. We can choose c o < m i n t c ' , ~ J and r o > max[?, r 6 I. If there is a point r ' > r o such that a ( r ' ) < co(r' - ro). Then we can find a point r > r o such that ol(r) < cor and od(r) < Co. Thus 0 < c~'(r') < / ~ .
732
MAN-CHUN LEUNG
And g2(°~(r')) (n 1" g2(cr) ~ _< -
.. g2(g-l((O/2x~ -
)f---~<_(n-|~
(n-1)
-S~)
l))f(r)))
as g ' _> 0 implies that g is n o n d e c r e a s i n g . T h e r e f o r e 0 < ~, c o n t r a d i c t i o n .
6
<-2' •
In T h e o r e m 3.12, we a s s u m e that m > 1 in c o n d i t i o n (B). W e discuss the case when m=l. THEOREM 3.20. A s s u m e that f ' ( r ) > 0 for all r > 0 a n d l i m , ~ o o f ' ( r ) = ~ . S u p p o s e that g ' ( y ) _< k for all y _> 0, where k is a positive c o n s t a n t . Let a ( r ) e C2(0, oo) be a positive solution to ( l . 1 ) with f a n d g as a b o v e a n d limr-0÷ or(r) = 0. If 2 < p _< n, then for a n y e e (0, 1), there exist positive c o n s t a n t s r o a n d C such that a ' ( r ) _< C/fl-e(r) for all r _> r o . I f p > n _> 2, then for any e > 0, there exist positive c o n s t a n t s r o a n d C s u c h that C
a'(r) <_ (f(r))O -o((.for all
I)/(p-1))
r >_ to.
Proof.
Given 1 > e > 0, as in (3.8) (with e = 1 - r) we can find a positive n u m b e r r0
such that
2g(et(r))g'(~(r))ot'(r)
f'(r) [ g2(c~(r)) f(r) L f - ~
fZ(r)
] + (°t'(r))2
gE(~(r)) [k z - ezf'(r) 2] - (1 - e) + -~7~-( -< ef3(r)f,(r) [f'(r)gZ(a(r)) )
f'(r) [ c~'(r))z~
<_
-(1 - c)~Tf~
for all r > r o, as
g'(s) <_ k
( )
]
+ (~'(r)) ~
a n d limr~®
f'(r)
= oo By (2.2) we have
gZ(ot(r)) ]
®'(r)
( p - l ) ( a ' ( r ) ) z + (n - 1 ) ~ |
<
-
-
-
)f(r)
+ (~'(r))2
®(r)
(3.21)
for all r _> f o . T h u s if 2 < p _<_ n, then (3.21) implies that ®'(r) < - ( 1 -
e)(p -
f'(r) 2 ) - ~ - ~ - ®(r)
for all r >__ r 0. A n i n t e g r a t i o n gives c~'(r)_< c/fl-~(r) a n d for all r _> r 0. I f p > n > 2, then (3.19) gives O ' ( r ) _< - ( 1 - e)
( p - 2)(n - 1)
p-
1
for s o m e positive c o n s t a n t C
f'(r) f(r)
--O(r)
Positive solutions of second order quasilinear equations
733
for all r _> r 0. A n i n t e g r a t i o n gives of(r) _<
C
(f(r))tt-e)<~,-l)/w-I))
for s o m e positive c o n s t a n t C a n d for all r > r o.
•
As a c o r o l l a r y , we have the following result (cf. [6]). COROLLARY 3.22. A s s u m e that f ' ( r ) > 0 for all r > 0 a n d limr~® f ' ( r ) = oo. S u p p o s e that g ' ( y ) _< k for all y > 0, where k is a positive c o n s t a n t . A s s u m e that f ( r ) >_ Cr s for s o m e positive c o n s t a n t C a n d for all r > f > 0, where s > 1 if 2 < p _< n, a n d (p-
i)
S > - (n -- l)
i f p > n _> 2. Let ~(r) e C2(0, oo) be a positive s o l u t i o n to (1.1) w i t h f a n d g as a b o v e a n d limr~0÷ ~(r) = 0. T h e n o~ is a b o u n d e d f u n c t i o n on ~ + .
Proof. F r o m T h e o r m 3.20, we can c h o o s e r 0 > 0 a n d e > 0 so that C
~'(r) <_ rl+6(r ) for all r > r o. H e r e d~ is a positive c o n s t a n t . A n i n t e g r a t i o n shows that c~ is a b o u n d e d f u n c t i o n on ~ + . • In p a r t i c u l a r , for p > 2 a n d n _> 2, all r o t a t i o n a l l y s y m m e t r i c p - h a r m o n i c m a p s f r o m the h y p e r b o l i c space to the E u c l i d e a n space a r e b o u n d e d . Acknowledgement--We would like to thank Professor Ravi Agarwal for valuable suggestions. REFERENCES I. Greene, R. and Wu, H., Function Theory on Manifolds which Possess a Pole. Lecture Notes in Math, Vol. 699. Springer-Verlag, Berlin, 1979. 2. Eells, J. and Lemaire, L., Another report of harmonic maps. Bulletin of the London Math. Soc., 1988, 20, 385-524. 3. Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems. Acta Math., 1970, 138, 219-240. 4. Ratto, A. and Rigoli, M., On the asymptotic behaviour of rotationally symmetric harmonic maps. J. Diff. Eqns., 1993, 101, 15-27. 5. Cheung, L.-F. and Law, C.-K., An initial value approach to rotationally symmetric harmonic maps. Preprint, 1995. 6. Cheung, L.-F., Law, C.-K., Leung, M.-C. and McLeod, J. Entire solutions of quasilinear differential equations corresponding to p-harmonic maps. Nonlinear Analysis, Theory, Methods and Applications (to appear). 7. Leung, M.-C., Asymptotic behavior of rotationally symmetric p-harmonic maps. Applicable Analysis 1996, 61, 1-15. 8. Leung, M.-C., On the infinitesimal rigidity of harmonic maps. Preprint, 1995. 9. Heinonen, J., Kilpel~.inen, T. and Martino, O., Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, Oxford University Press, Oxford, 1993. 10. Coddington, E"AnIntr°ducti°nt°OrdinaryDifferemialEquati°ns" Prentice-Hall, Englewood Cliffs, 1962. 11. Hamilton, R., Harmonic Maps of Manifolds with Boundary. Lecture Notes in Math., Vol. 471. SpringerVerlag, Berlin, 1975. 12. J~iger, W. and Kaul, H., Uniqueness of harmonic mappings and of solutions of elliptic equations on Riemannian manifolds. Math. Ann., 1979, 240, 231-250. 13. Takakuwa, S., Stability and Liouville theorems of p-harmonic maps. Japan J. Math., 1991, 17, 317-332.
Pergamon PII: S0362-546X(97)00435-5
POSITIVE SOLUTIONS OF SECOND ORDER QUASILINEAR EQUATIONS CORRESPONDING TO p-HARMONIC MAPS M A N - C H U N LEUNG Department of Mathematics, National University of Singapore, Singapore 119260
(Received 1 May 1996; received for publication 18 February 1997) Key words and phrases: p-harmonic map, positive solution, rotationally symmetric. 1. I N T R O D U C T I O N
In this paper we study positive solutions u(r) of the following differential equation
1
OP/2- l(r) c~" (r) + [ ( n -- l)OP/2-1(r) . . ' ~f ' ( r ) + (0 p/2- I),(r)jot,(r )
- (n - l)Op/2-t(r) g(ot(r))g'(ct(r)) f2(r ) = 0,
(1.1)
where
O(r) = (n - 1) f2(r----g2(a(r)) ~ + (a'(r)) 2,
r > 0,
(1.2)
with p > 0 and n _> 2. In the above equation, we assume that f , g • C2([0, oo)) and f(0) = g(0) = 0,
f ' ( 0 ) = g'(0) = 1
and
f(r), g(r) > 0
(1.3)
for all r > 0. Equation (1.1) arises as the Euler-Lagrange equation of the p-energy functional for rotationally symmetric maps between certain complete noncompact Riemannian manifolds. Let (S n-I, d0 2) be the unit sphere in IPn with the induced Riemannian metric d02. Consider the following model Riemannian manifolds [1]
M ( f ) = ([0, oo) x S n-I, dr 2 + f2(r) d02), N(g) = ([0, oo) × S n-I, dr 2 + g2(r ) d02), where f , g • C2([0, oo)) satisfy the conditions in (1.3). M ( f ) and N(g) are complete noncompact Riemannian manifold. The Euclidean space and the hyperbolic space are corresponding to f ( r ) = r and g(r) = sinh r, respectively. A map F: M n ( f ) --" Mn(g) is called a rotationally symmetric map if
F(r, O) = (or(r), 0)
for all r > 0 and 0 • S"- 1,
where c~: [0, oo) --* [0, oo) with c~(0) = 0. The energy density [2] of a rotationally symmetric map F: M " ( f ) --" M"(g) is then given by (a'(r)) 2
(n +
-
1" g2(°t(r)) O(r). ) f2(r ) 717
718
MAN-CHUN LEUNG
Given p > 0, the p-energy functional [3] for C2-rotationally symmetric maps from M(f) to M(g) is given by
i'ooI
ot'2(r) + (n -
°2[alr~P/2J wx
~. l J I
l ) ~ J
~rl--I4
[r)dr.
(1.4)
t, 0
It is known that equation (1.1) is the Euler-Lagrange equation of the p-energy functional with respect to compactly supported variations. Critical points of the p-energy functional (1.4) are known as rotationally symmetric p-harmonic map from M(f) to N(g). For p = 2, equation (1.1) becomes a"(r) + (n - l ) ~ a ' ( r )
jtr)
- (n - 1) g(~(r))g'(a(r)) f2(r)
=
O.
(1.5)
Equation (1.5) corresponds to rotationally symmetric harmonic maps, which have been studied by Ratto and Rigoli [4] and Cheung and Law [5]. For p > 2, the existence of entire positive solutions to equation (1.1) together with a Liouville's type theorem have been studied in [6]. In [7], we discuss asymptotic properties of entire positive solutions to equation ( l . l ) when f(r) = g(r) = sinh r, that is, M(f) and M(g) are hyperbolic spaces. In this paper we study the local and asymptotic properties of positive solutions to equation (1.1). For p > 2, we show that if c~(r) ~ C2(0, e) is a nonnegative solution to (1.1) with limr~o÷ or(r) = 0, and a(r) goes to zero at arbitrary high order, then a -= 0 on (0, e) (Lemma 2.5). This unique continuation property is well-known for harmonic maps [8] but is unknown to us for p-harmonic maps in general with p ~ 2 (cf. [9]). By refining the argument, we can estimate the order that ct can approach zero. We show that, under the assumption that g"(y) >_ 0 for all y > 0, two positive solutions of equation (1.1) have to coincide if the solutions have the same value at one point (Theorem 2.14). The condition that g"(y) >_ 0 for all y > 0 is equivalent to the condition that the radial Ricci curvature of N(g) is nonpositive. We discuss asymptotic properties of positive entire solutions to equation (l.1). We obtain a Louville's type theorem for certain rotationally symmetric p-harmonic maps into the Euclidean space or the hyperbolic space (Theorem 2.24). When M(f) is similar to the hyperbolic space and N(g) is similar to the Euclidean space, we show that a(r) either goes to infinity exponentially ot it is bounded (Theorem 3.12). If g grows at most exponentially, then we show that either c~ grows at least linearly or ot is bounded (Theorem 3.19). We show that, under certain conditions on f and g, all rotationally symmetric p-harmonic maps with p > 2 are bounded (Corollary 3.22). Equation (l. 1) is a quasilinear second order differential equation, where the nonlinear term 0 p/E-l(r) makes the equation complicated. We introduce a first order equation on O(r) and ct(r). This first order equation is useful to obtain information about the solutions. 2. U N I Q U E C O N T I N U A T I O N
AND LIOUVILLE'S
TYPE THEOREM
For p > 0, let
6)(r) = OP/2-1(r),
r > O.
(2.1)
We have
®'(r)O(r)=(2-1)®(r)[(n-l'[g2(a(r))~'
2e~'(r)c,"(r)]
Positive solutions of second order quasilinear equations
719
Using equation (1.1) to replace the term O(r)a"(r) we obtain g2(~(r))] O'(r) (p - 1)(ot'(r)) 2 + (n - 1 ) ~ j
= (p - 2)(n - l ) O t r ) (
f2g(cl(r))g'(et(r))c '(r) f2(r )
f'(r) r g=t tr)) ~-~
[ f2tr )
]1
+ (a'(r)) 2
.
(2.2)
E q u a t i o n (2.2) involves only first derivatives o f ® and ~. For some e > 0, assume that a nonnegative CZ-solution ~ o f (1.1) exists on (0, e) with limr,0+ t~(r) = 0. Given x > 0, we can find positive constants ao, b® and Co depending on x and e, such that [f'(r)[ <_ ao,
f(r) >_ Cor
and
[g'(y(r))[ < b® for a l l r ¢
0 , ~ e
.
Then (2.2) gives
O'(r)[(p
- l)((x'(r)) 2 + (n - 1") gZ(c~(r))] ~ ]
g2(a(r)) + (a'(r))2] . < (p - 2)(n - l)®(r) (a° + bo) [ Cor I_ f 2 ( r )
(2.3)
T h e r e f o r e there is a positive constant C = C(ao, b0, c o , P , n, x) such that for p > 2 we have C ®'(r) _< - ®(r) r
on
(
X +X, 0 , ~ e
)
.
(2.4)
We have the following unique continuation p r o p e r t y for solutions o f (1.1). LEMMA 2.5. F o r p > 2 and n >_ 1, let c¢ e CZ(0, e) be a nonnegative solution to (1.1) with limr.o* a(r) = 0. If a(r) = O(r k) near 0 for all k > 0, then a - 0 on (0, e).
Proof. For n _> 2, assume that a ~ 0 on (0, e/2). We first show that a cannot be zero on (0, t~) for any di e (0, e/2) (cf. [6]). Suppose that c~ - 0 on (0, di) for some ~ e (0, e/2). Since a ~ 0 on (0, e/2), we m a y assume that a ( r ) > 0 on (di, e/2). Integrating (2.4) we have In O(r)lg -< C I n rig, where t~ < b < a < e/2. T h a t is
(°)c
®(a) _ ®(b) ~
.
(2.6)
Let b --* ~ > 0, we have ®(b) ---, 0, but ®(a) > 0, contradicting (2.6). Thus et cannot be zero on (0,,~) for a n y ,~ ~ (0, e/2). We can find a point r® e (0, e/2) such that a(ro) > 0 and a ' ( r o) > 0. Suppose that there exists a point r~ e (0, e/2) such that r~ > r o and ct(q) = 0, then there exists a point r ' e (ro, rl) such that a ( r ' ) > 0, c ~ ' ( r ' ) = 0 and
720
MAN-CHUN LEUNG
~"(r) <_ O. At r ' , equation (1.1) gives O P / 2 - 1 ( r ' ) o t " ( r ') = (rt -
1)OP/2-1(r ')
g(a(r'))g'(et(r')) fZ(r )
and
O(r') = (n - 1) fZ(r,------g2(°e(r')) ~ + (od(r')) z > 0. Therefore we have a " ( r ' ) > 0, contradiction. Therefore cKr) > 0 for all r e (ro, e/2). As l i m r . 0 , o4r) = 0 and o~ ~ 0 on (0, 0) for all 0 > 0, we can let ro ~ 0. Thus o~(r) > 0 on (0, e/2). Similarly we can show that o~'(r) > 0 on (0, e/2). Integrating (2.4) we obtain O(a)_< O(r)
r
'
where in this case e / 2 > a > r > 0 and a is a constant. We have
®(r) >_ C'r c
(2.7)
for some positive constant C'. That is,
O(r) = (n - I) g2(c~(r)) fZ(r----~ + (ot'(r)) z ~ C"r 2c/0'-2),
(2.8)
where C" is a positive constant. Take a n u m b e r k such that C
k>
p-2
+1.
Since o~(r) _< Ck rk for r small, where Ck is a positive constant, a n d f ( r ) , g(r) - r when r is small, we have (n - 1)g2(°t(r)~) < C ' r 2k-2, f2(r ) where C " is a positive constant. Hence by (2.8) we have (2.9)
(o~'(r)) 2 ~ Co r 2 c / h ° - 2 )
for some positive constant Co. Therefore we have
a(r) - a(b) =
a '(s) ds ~ Cn ,,b
S 2C/(p-2)
(IS
Jb
= Co(r(2C/(p_2))+l)
_ b(2C/~_2))+1) '
~8 > r > b > O .
As limb~ o, or(b) = O, we have a positive constant such that
a(r)
>
C'r (2C/(p-2))+
1,
(2.10)
c o n t r a d i c t i n g that ct(r) - O(r k) f o r all k > O. T h e r e f o r e we have or(r) -- 0 on (0, e/2). Similar argument shows that ~x(r) ~, 0 on (0, (t¢/0¢ + l))e) f o r all K > O. Let x ~ oo, we have ix(r) - 0 on (0, e).
Positive solutions of second order quasilinear equations
721
For n = l, equation (l.1) shows that
[(ot'(r))2]p-2/Eot'(r) = C 2 for all r • (0, e), where C2 is a constant. If or(r) _> 0 and o~ , 0, then the constant C2 has to be positive. Thus a(r) = cr for some positive n u m b e r c. If el(r) _< C3r k for some k > l and for all r in a n e i g h b o r h o o d o f zero, then a - 0 on (0, R). • Let a • C2((0, e)) be a nonnegative solution to (1.1) with limr.o+ a ( r ) = 0. If we assume that a , 0, then the p r o o f o f L e m m a 2.5 shows that a(r) > 0, hence et'(r) > 0 on (0, e) [6]. Given any 6 > 0, as lim~_~0+ o~(r) = 0, we can find eo • (0, e) such that 0 < c~(r) < 6
on (0, to).
As f , g • Cz(O, ~ ) with f ' ( 0 ) = g'(0) = 1, given el • (0, I), we can assume that ~ and to are small e n o u g h such that
O < f ' ( r ) < _ 1 + el,
f(r)>(l
- el)r
and
Ig'(y(r))[ < 1 + el
for a l l r • ( 0 ,
eo).
If we assume that p > 2 and n > 2, then (2.2) gives @'(r) _< (p - 2)(n - 1)(.1+ ~ e ~ ' ~ ® ( r ) \l el/
on (0, eo).
(2.11)
-
That is, C = ( p - 2 ) ( n - l ) ( l + e l ) / ( l L e m m a 2.5 and inequality (2.9) that
-el)
in (2.4). It follows as in the p r o o f o f
or(r) >_ C i r(2c/(p-2))+ l = C l rE(n-l)((l +~O/(I-tt))+ l
(2.12)
As el can be made arbitrarily small, therefore we have the following refinement o f L e m m a 2.5. THEOREM 2.13. F o r p > 2 and n _> 1, let a • C2(0, e) be a nonnegative solution to (1.1) with lim,~o- a(r) = 0. If there exist constants k > 2n - l, C' > 0 and eo • (0, e) such that, c~(r) _< C'r k for all r • (0, eo), then ct == 0 on (0, e). We have the following uniqueness theorem for rotationally symmetric p - h a r m o n i c maps. THEOREM 2.14. F o r p _> 2 and n _> 1, let c~, fl • C2(0, R) be positive solutions to (1.1) for some constant R > 0, with limr~o+ a(r) = limr-o+ fl(r) = 0. Suppose that g"(y) >_ 0 for all y > 0. If there is a n u m b e r R o • (0, R) such that ot(Ro) = fl(R0), then ~ - ,8 on (0, R).
Proof. We first assume that p > 2 and n > 1. Suppose that ct ~ ,8 on (0, Ro). We m a y assume that there is a point rl < Ro such that c~(r~) > fl(r0. Let r o be the biggest n u m b e r in 10, rl) such that lim et(r) = lim ,8(r). r~r 0
r~r~
If a ' ( r ) _< fl'(r) for all r • (ro, r 0, then an integration shows that a ( r 0 -
c~(r') > fl(r')
and
ta'(r') > f l ' ( r ' ) .
722
MAN-CHUN LEUNG
As g"(y) >_ 0 for all y > 0 and g'(0) = 1, g and g' are positive and nondecreasing functions. As p > 2, we have O,~(r') > Oa(r'), where O,~(r)
=
®o(r)=
[(n
_
1" gZ(c~(r)) ]p/2-1 , ) f 2 ( r ) + (ot'(r))2]
[ ( n - - l ) g2(B(r)) (fl'(r))2] p/z-I ~ +
for r ~ (0, R). We have ln[®,~(r')c~'(r')] > ln[®~(r')fl'(r')].
(2.15)
Let R' 6 (r', Ro] be the smallest number such that c~(R') = fl(R'). A similar argument as above shows that a ' ( R ' ) _< p ' ( R ' ) . If a'(R') = fl'(R'), we already have c~(R') = fl(R'), then the uniqueness result for equation (1.1) (cf. [10], p. 259) implies that a---/~ on a neighborhood of R'. We note that in this case R' > 0 a n d f is positive in a neighborhood of R'. A continuation argument shows that c~ - fl on (0, R). We may assume that a'(R') < fl'(R'). Then In[O,(R')cC(R')] < In[Oo(R')B'(R')]. (2.16) By (2.15) and (2.16), there exists a point P 6 (r', R') such that In[O,(~)~'(e)l < In[O~(f)/~'(~)]
(2.17)
(ln[O~(e)cx'(e)])' _< (lnIOe(f)Jl'(t:)])'.
(2.18)
and
From equation (1.1) we have (lnlO,,(e)c~'(F)])' =
O~(e)a"(e) + O~(e)a'(e) o~(e)~'(e)
1" (g(c~(f))g'(ot(f)) (.
=
-
f'(?)'~
(2.19)
f(,:))
Similarly (ln[Oe(~)fl'(t~)])'
=
(n
-
1"j[(g(J](f))g'(fl(f)) ~
f'(f!'~ f(p) ).
(2.20)
From (2.17) we have ®~(F)~'(~) < O~(f)fl'(~).
(2.21)
As at ~, we have ~(f) > fl(~), therefore (2.21) implies that a'(~) < fl'(~). As g is an increasing function and g' is a nondecreasing function, (2.19) and (2.20) show that (ln[O,~(~)~'(f)])' > (ln[Oa(f)ff(~)])',
(2.22)
which contradicts (2.18). Hence a -= fl on (0, Ro). A continuation argument shows that - • on (0, R). In case n = 1, then a(r) = cr and fl(r) = c'r for some constants c, c' _> 0. Thus oL(r) = B(r) on (0, R).
Positive solutions of second order quasilinear equations
723
In the case of harmonic maps (p = 2) and n > 1, we assume that ot(R') = fl(R'), c~'(r') > / ~ ' ( r ' ) and u ( r ) > B(r) on ( r ' , R ' ) . There is a point ? ~ ( r ' , R ' ) such that a ( f ) > f l ( f ) , a ' ( f ) = / ~ ' ( ~ ) and a"(f)<_fl"(f). Equation (1.5) together with the assumption that g"(y) >_ 0 for all y > 0 give
.. (g(ot(f))g'(ot(r))
f ,(f ) .... )
"" (g(fl(f))g'(fl(f))~
f'(f)f(f)
~l
= p"(~),
which is a contradiction. Therefore a - B.
•
Remark. For n > l, the radial Ricci curvature of the Riemannian manifold N(g) is given by - ( n - l)g"(r)/g(r). The assumption in Theorem 2.14 is equivalent to the assumption that the radial Ricci curvature of N(g) is nonpositive. We note that for harmonic maps, m a n y uniqueness results require the curvature to be nonpositive (cf. [l l, 12]). Let ot G C2((0, oo)) be a nonnegative solution to equation (1.1). It follows from (2.2) that if f '(r) >_ 0, then we have
g2(°t(r)) ]
®'(r) (p - 1)(ta'(r)) z + (n - 1) f2(r ) j <_ (p - 2)(n - l)®(r)2g(ot(r))g'(u(r))ot'(r)f2(r) In addition, if p > 2 and n > 2 and g'(r) <_ ag(r) for all r > 0, where a is a positive constant, then there exists a constant Co such that
®'(r) <_ CoO(r)a'(r)
for all r > 0.
(2.23)
We use this to prove the following Liouville's type theorem. THEOREM 2.24. Assume that there exists positive constants c o and c~ such that 0 _< f ' r ) _< c 2 and f(r) <_ cl for all r _> r 0 . For p > 2 and n >_ 2, let F(r, 0) = (ct(r) 0) be a rotationally symmetric p-harmonic maps from M ( f ) to the Euclidean space or the hyperbolic space. If there exists a positive constant C' such that ct'(r) _< C ' for all r > 0, then o~ - 0 on ~+.
Remarks. In fact we prove a more general statement. The conditions below on g are satisfied for the Euclidean and hyperbolic metrics, where g(r) = r and g(r) = sinh r, respectively. Conic type metrics, where f(r) = k for all r large, satisfy the conditions on f in the above theorem.
Proof. If c ~ , 0, it follows from the p r o o f of L e m m a 2.5 (cf. [6]) that c~ > 0 and hence a ' ( r ) > 0 for all r > 0. Assume that (I)
g'(r) <_ ag(r)
for all r > 0.
Then (2.23) implies that
®'(r) < CO(r)a'(r),
724
MAN-CHUN LEUNG
w h e r e we m a y t a k e C = Co + 1. S u b s t i t u t e the a b o v e i n e q u a l i t y i n t o (1.1) a n d we h a v e C®[cx'(r)] 2 + ® ~ " ( r ) + (n - 1)
O ~ ' - (n -
1)j
g ( c 0 g ' ( ~ ) > 0.
(2.25)
A s 0 _< f ' ( r ) <_ c z, we h a v e f'(r) f ( r ) <- Cl
for all r > r o,
w h e r e cn is a p o s i t i v e c o n s t a n t . Since ®(r) > 0 a n d et'(r) < C ' for all r > ro, we h a v e
c~" tx' - g ( a ) g ' ( a ) + C " g(~)g'(~) -
-
1 (n - l ) f - ~ > 0
(2.26)
for all r _> r o, w h e r e C" is a p o s i t i v e c o n s t a n t . F o r r _> r o, let
H(r) =
ot'(r) g(y)g'(y)"
(2.27)
We have tt
g(~)g'(~)
- H ' + H 2 [ g ' Z ( ~ ) + g(ot)g"(ot)].
S u b s t i t u t e i n t o (2.26) we o b t a i n [g,2(a) + g ( a ) g , , ( a ) l H 2 + H ' + C ' H
(n - 1) -
f2 -
> 0.
(2.28)
A s a ' > 0, we h a v e e i t h e r limr~,~ a(r) = oo or ~ is b o u n d e d - - i n the l a t t e r case a - 0 b y a result in [6]. T h e r e f o r e we m a y a s s u m e t h a t limr.oo c~(r) = oo. A s s u m e t h a t
g"(r) >_ 0
(II)
for all r___r 0. A s g ' ( 0 ) = 1, we h a v e g'(r)_> 1 for all r > 0 . F u r t h e r m o r e , a s s u m e t h a t f o r s o m e p o s i t i v e c o n s t a n t c,
g'Z(r) - g(r)g"(r) >__c 2
(III)
T h u s l i m r ~ ® g ( r ) = o0.
for all r > 0.
T h e n we have 0 -< [g'Z(ot) + g(et)g"(cO]H 2 =
Ot'2[g'2(a) + g ( a ) g " ( a ) l g2(oOg,E(ot) ct '2
<_
ct'Zg(a)g"(a) + g3(oOg,,(c0
as b y a s s u m p t i o n ( I I I ) g ' 2 ( a ) > g(a)g"(ot). S i m i l a r l y we h a v e lim H ( r ) = O. T h e n (2.28) gives lim H ' ( r ) > k > O.
--,0
as r - ~ oo,
Positive solutions of second order quasilinear equations
725
As H(r) > 0, we have
i,r H(s) ds = O(r).
(2.29)
v ro
On the other h a n d by (III) we have
_ (\l n g 'g( (YY) )~/ ' = H[g'2(y) - g(y)g"(y)] >_ c2H for all r _> r o. Thus -In
g'(y(r))
= - ( ' (In g'(Y)']' dr >_ c 2
g(y(r))
,]ro \
g(Y) /
H(s) ds.
(2.30)
.... o
Assume that (IV)
g'(s) _> a ' g(s)
--
for all s _> 1,
where a ' is a positive c o n s t a n t , or, (IV')
g'(s) C' _> - g(s) s
--
for alls > 1.
In the first case we have a' <_ g'(y)/g(y) <_ a for all r large enough. Therefore (2.30) implies that
i'*~H(s) ds < oo. ,. r 0
Since c~'(r) < C, which implies that ¢x(r) _< C~r for all r >_ r o, where CI is a positive constant. As limr,=Cx(r) = oo, we m a y assume that o~(r) _> 1 for all r large enough. Therefore in the second case we have
a>_ g'(y) > - - c>'- g(y) c~(r)
c"
r
for all r large enough, where C" is a positive constant. We have
,t r H(s) ds <_ C" In r rO
for some constant C". In both cases, we have a contradiction with (2.29). For p = 2, Liouville's type theorems for h a r m o n i c maps have been discussed in [4]. In [13], Liouville's type theorems for p - h a r m o n i c maps are obtained under assumptions on the curvature o f the manifolds. The conditions in T h e o r e m 2.24 involve the first derivative o f the solution and g r o w t h o f f ( r ) , which relax assumptions on the curvature. On the other hand, under certain conditions o n f a n d g, there exist b o u n d e d positive entire solutions to equation (1.1) [6].
726
MAN-CHUN LEUNG 3.
ASYMPTOTIC PROPERTIES
Let f , g be functions in C2([0, oo)) which satisfy the conditions in (1.3). We study asymptotic properties o f solutions to (1.1). W h e n f ( r ) = g(r) = sinh r for all r >__0. In [71, conditions are given for positive entire solutions to equation (1.1) to be b o u n d e d , to grow to infinity fast or to be asymptotic to the solution c~(r) = r. In this section we assume that f is similar to the hyperbolic space and g is similar to the Euclidean space. More precisely, we assume that there are constants a, C > 0 such that 1
-- e ar <_f(r) <_ C e at,
(A)
C
1
-- e"" <_f ' ( r ) <_ C e ~r C
for all r > 1. A n d for all y > 1, we assume that
g(y) <_ C ' y m,
(B)
0 _< g ' ( y ) _< C ' y ' -
for some constants m _> l and C' > O. LEMMA 3.1. F o r p > 2 and n _> 2, let a(r) ~ C2(0, co) be a positive solution to (1.1) with lim~.o, a(r) = O. Suppose that f and g satisfy the conditions (A) and (B), respectively. For m > 1 let c < a/(m - 1) be a positive constant. Then there exists a positive constant ro such that either o~(r) _> e "~ or a(r) _< e "r for all r > r 0 .
Proof. If c~ is bounded, then we have c~(r) _< e ~ for r large enough. Therefore we m a y assume that lim~.= a(r) = oo. Hence there exists a constant r ' > 1 such that c~(r) > 1 for all r > r ' . For m > I, suppose that we cannot find such a r o as in the statement o f the theorem, then for any r o > r ' , there exists two points r 2 > r, > r o such that a ( r 0 = e or', or(r2) = e c~2 and a(r')<_ C r' for r ' e [rl, r2]. There is a point r E [rl, r2] such that c4r) -< e c', ot'(r) = c e cr and c~"(r) _> c2e or. Since c < a/(m - 1), we have 2g(a(r))g'(et(r))ct'(r)
f ' ( r ) [ g2(a(r)) ] f(r) f 2 ( r ~ + (~'(r))2
fE(r) <--
2C,2C2c e 2mcr e 2at
c 2 e 2cr - C2 <
(3.2)
--C'
if ro is large enough. Here c ' is a positive constant. By (2.2) and (3.2) we have
g2(a(r))] ®'(r) ( p -
l)(t~'(r))2 + ( n -
1) f 2 ( r ) j _ < - c ' ( p - 2 ) ( n -
l)O(r).
(3.3)
And
F
®'(r)](p - l)(ot'(r)) 2 + (n - l ) =
g2(ot(r))]
f Z(r) ]
- 10p/2-Z(r) (p - l)(t~'(r)) 2 + (n -
l~j
x[(n-l){g2ta(r))-'~'k, / f2tr)
->(P-l)O(r)[(n-I
+ 2ct'(r)et"tr)]
"l/gz(et(r))'~')~ f----~ )
+ 2a'(r,t~"(r)] ,
(3.4)
Positive solutions of second order quasilinear equations as O(r)= ( n - l)(g2(ct(r))/f2(r))+ (a'(r)) z a n d ® ( r ) = c o n s t a n t c" > 0 such that at r we have
727
oplZ-I(r). By (3.3), there is a
/ g 2 (t~(r))'~ # (n - 1 ) t f ~ ) + 2a'(r)~"(r)_<-c".
(3.5)
O n the o t h e r h a n d
( g2(et)'~ '
~.,]
2 f(r)g(oOIf(r)g'(oOot'(r) - f'(r)g(oO]
=
fa(r) 2 f '(r)gZ(t~)
>
f 3(r)
> -2C'2C
2 e 2met e 2at
-> - e e 2~',
(3.6)
as c < a / ( m - 1). H e r e e is a positive c o n s t a n t . A n d by c h o o s i n g r o large, we m a y a s s u m e that e < 2c3/(n - l). T h e r e f o r e if r 0 is large a n d c < a / ( m - 1), then (3.5) a n d (3.6) i m p l y that
2a'(r)a"(r) <_ (n -
l)~,c 2cr.
T h a t is
a "(r) <_ -(n -- l)e cCr < 2c
C2eCr,
which is a c o n t r a d i c t i o n . H e n c e there exists a positive c o n s t a n t r o > r ' such that either u(r) _> e cr o r c~(r) _< e ¢~ for all r > r o. •
LEMMA 3.7. F o r p > 2 a n d limr-o÷ o~(r) = 0. S u p p o s e respectively. Let c < a / ( m such that offr) _< e c" for all
n _> 2, let o~(r) ~ C2(0, oo) be a positive s o l u t i o n to (1.1) with that f a n d g satisfy the c o n d i t i o n s (A) a n d (B) with m > 1, - 1) be a positive c o n s t a n t . S u p p o s e that there exists a ro > 0 r > ro, then o~(r) _< C for s o m e positive c o n s t a n t C.
Proof. A s s u m e that or(r) < e c' for all r > r 0, where c < a / ( m - 1). W e m a y a s s u m e that a ~ 0. W e can find positive c o n s t a n t s r ' a n d d~such that o~(r) _> 6 for all r _> r ' . T h e r e is a positive c o n s t a n t C" such that g(y) <_ C " y m,
0 < g ' ( y ) < C " y m-I
for a l l y _ > f i . W e m a y t a k e r o > r ' > l. T h e n f o r r > r
oand 1 > r>O,
wehave
728
MAN-CHUN LEUNG
2g(a(r))g'(eO(r))eg(r) fZ(r)
f'(r) [ gZ(c~(r)) ] f(r) fZ(r~ + (c~'(r))2
2g(et(r))g'(e~(r))I f'(r)(e~'(r))Z]i/2 f,(r)g2(ot(r)) f'(r) = f3/2(r)f't/Z(r)
f(r)
1 gZ(o~(r))g'Z(oL(r)) < 1~ f 3(r)f'(r) + (1
]
f3(r)
f(r)
(~'(r)) z
f'(r)gZ(ot(r)) f 3(r)
f'(r) - r) ~
gZ(~(r)) = (1 -
r)f3(r)f'(r)[g'Z(a(r)) -
- (1 -
r)f3(r)f'(r)
r)2f'(r)21 - r
(1 -
~
(~'(r))2
f'(r) f(r-----)(~'(r))2
[ f'(r)gE(~(r)) f'(r) 1 f 3(r) + ~ (o~'(r))z
e ]- r
f3(r )
f(r) (a'(r))2 ' (3.8)
where we have used the inequality 2 A B _< (1 - r)A z + 1/(1 - r)B 2 for 0 < r < 1. H e n c e there exists a c o n s t a n t r" > r o such that for r > r" we have
C,,2 e2ctm- l)r
(1 -- r) 2 e2ar
U
o.
F o r r > r" we have
,o,r,,o,r, S,r,I2,o,r,,
fZ(r)
f(r)
f~
]
+ (c~'(r))2 = - - r - ~
]
f2(r~-)- + (c~'(r)) 2
- r [ gZ(c~(r)) ] _<~-g[ ~ + (~'(r)) z .
(3.9)
T o g e t h e r with (2.2) we have ®'(r) [ ( p - l)(~'(r)) 2 + (n - l ) f~ ]( )
_< - ~ - r~ ( p
- fz( - r) - 2)(n - 1) [gZ(c~(r))
+ (o~'(r))Z]O(r). (3.10)
T h u s we have
®'(r) <_ -c'®(r)
for all r > r " ,
(3.11)
where c' = (p - 2)(n - 1)r/(bC z) a n d b = m a x l p - 2, n - 11 are positive constants. I n t e g r a t i n g b o t h sides o f the a b o v e inequality we have
®(r) <_e-CV-~)O(r"),
r > r".
So we have od(r) _< C~ e -c'(t-r'')/(p-2) where C~ = (O(r")) ~/tp-2) is a positive c o n s t a n t . c o n c l u d e that a is b o u n d e d . •
for all r > r " , As c ~ ' ( r ) > 0, u p o n integration we
Positive solutions of second order quasilinear equations
729
C o m b i n i n g the previous two l e m m a s we have the following. THEOREM 3.12. For p > 2 and n _> 2, let a(r) e C2(0, oo) be a positive solution to (1.1) with limr,o* a(r) = 0. Suppose that f and g satisfy the conditions (A) and (B) with m > l, respectively. Let c < a/(m - l) be a positive constant. Then either there exists a positive constant ro such that a(r) >_eCrfor all r > r o or there is a finite posiive n u m b e r C such that o~(r) _< C for all r > 0. In the following we assume that
g'(Y) <- CEg(Y)
(C)
for all y _> 1,
where Cz is a positive constant. Condition (C) implies that g grows at most exponentially. Condition (B) is stronger than condition (C). The following can be considered as a generalization o f L e m m a 3.7. LEMMA 3.13. F o r p > 2 and n _> 2, let a(r) e C2(0, oo) be a positive solution to (1.1) with limr.o* o~(r) = 0. Suppose that f and g satisfy the conditions (A) and (C), respectively, and g'(y) > 0 for all y > 0. There is a positive constant c > 0 such that if there exists a positive constant r 0 such that c~(r) _< g-l(¢f(r)) for all r > ro, then a is a b o u n d e d function on ~+.
Proof.
We can find positive constants r ' and ~ such that t x ( r ) ~ 6 for all r _> r ' . There is a positive constant C3 such that
g'(Y) <- Cag(Y)
for all y >_ di.
For 0 < r < 1 and r > r ' , f r o m (3.8) we have
2g(a(r))g'(°l(r))°L'(r) f'(r) [ g2(a(r)) ] f2(r) f(r) fZ(r~ + (cz'(r))2 < g2(Or(r)) -- (1 - r)f3(r)f'(r)[g'2(c~(r)) - (1 -
f'(r)g2(ot(r)) f3(r )
- r
< (I - g2(c~(r))r)f 3(r)f'(r)
[ f'(r)g2(ot(r))
-r L
~
f'(r)
]
+ f - - ~ (a'(r)) 2
[ C2g2(ct(r)) f'(r)
(1 ~-z--r)2 Jtr)]~2" ,]
]
+ f - ~ -(c~'(r))2 "
If we choose a positive constant c such that c _< (1 then we have
2g(ot(r))g'(et(r))ot'(r)f'(r)[g2(ot(r)) f2(r) f(r) [ f 2 ( r )
r)Ef'(r) 2]
+ (c~'(r)) 2
We can process as in the p r o o f o f L e m m a 3.7.
r)/(CC3),
and if ~(r) _< g-~(cf(r)),
][f'(r)g2(ot(r))f'(r)] <_-z f3(r ) + ~ •
(ot'(r)) 2 •
730
MAN-CHUN LEUNG
The previous lemma provides asymptotic information a b o u t a(r). We can also obtain medium range information about ~(r). LEMMA 3.14. F o r p > 2 and n _> 2, let et(r) ~ C2(0, ~ ) be a positive solution to (1.1) with limr,0÷ a(r) = 0. Suppose that f and g satisfy the conditions (A) and (C), respectively, and g'(y) >_ 0 for all y > 0. Assume that ~ is not a b o u n d e d function on R ÷ and let a(t~) = 1 for some ~ > 0. Then there exists a positive constant 6 such that the energy density O(r) >_ 6 for all r >__L
Proof. F r o m (2.2) we have ®'(r) (p - l)(a'(r)) 2 + (n - 1)
= (p - 2 ) O ( r ) l 2(n
g2(a(r))] f2(r ) ]
l )g(et(r))g t(o~(r))o[f(r) f2(r)
2 f(r) [ ~
+ (a'(r))2 -
n-
- f 2- ( r ) + (°~'(r))2
-~
. (3.15)
At a point r _> ?, we have o~(r) _> 1. So at the point r > f, we have either (i)
l)g(~(r))g'(~(r))~'(r) >
2(n -
f2(r)
-- 2
_ _ f2(r)
f(r)
+ (c~,(r)) 2 '
or
(ii)
O'(r)
I
1) g2(°~(r))]
(p - l)(a'(r)) 2 + (n -
_< - ( p - 2) ( ~n ) -
f--~-~-]
O(r) ff'(r) ~-
Ig2(a(r))f2~(r---
+ ( c ~ ' ( r ) ) 2] .
In case o f (i), from condition (A) and (C) we have 2C2(n - l)a'(r) >
f'(r)f(r)(~'(r)) 2 g2(o~(r))
+
f'(r) 1 > -2f(r) 2C 2"
By choosing a bigger constant if necessary, we m a y assume that (A) holds for r > ~. Thus if (i) holds at r > f, we have
1 a ' ( r ) _> 4(n
-
1)C2 C2"
(3.16)
Take
6 = (8(/7
1
Suppose that there is a point r" _> ro such that 0 = {t e [r",
~2 .
_ T)C2 C2/
O(r") <_ 6. Let
oo) lO(r) <_ 6 for all r e [r", t]].
(3.17)
Positive solutions of second order quasilinear equations
731
Therefore r' ~ 0 and a'(r') <
8(n - 1)C2 C 2"
We can find t such that t > r ' and a'(r) <
4(n-
1)C"C 2
for all r e [r", t]. Hence we have (ii) for all r e [r", t]. Thus ®'(r) _< 0 and hence O'(r) <_ 0 for all r ~ [r", t]. We have O(r) <_ 6 for all r ~ [r", t]. Suppose that s u p O = a < ~ . Then at the point a we have O(a) <_ ~. The same argument shows that there exists a positive n u m b e r e such that ®'(r) < 0 for all r e (a, a + e). Then a + e e O, which is a contradiction. Therefore we have sup O = co, or O(r) < (5
for all r e [r', oo).
(3.18)
In particular, c~'(r) <
4(n-
l)C"C 2
for all r ~ It', 00). By (ii) we have O'(r)
[
g2(ct(r))] < ( p - 2 ) ( n -2{ ) C (p - l)(c~'(r)) 2 + (n - l ) ~ j _ -
, ,[-g2(a(r))~ ] OtO[ + (~'(r)) 2 •
A similar a r g u m e n t as in the p r o o f o f L e m m a 3.7 shows that there is a positive constant c > 0 such that or(r) _< c for all r ~ (0, oo). This contradicts the assumption that c~ is not b o u n d e d . Therefore O(r) >_ 3 for all r _> f. •
THEOREM 3.19. For p > 2 and n _> 2, let o~(r) ~ C2(0, oo) be a positive solution t o (1.1) with iimr,o÷ a ( r ) = 0. Suppose that f and g satisfy the conditions (A) and (C), respectively, and g ' ( y ) > 0 for all y > 0. Then either there exist positive constants r o and c o such that either or(r) _> co(r - to) for all r > r o, or c~ is a b o u n d e d function on R ÷. P r o o f . If c~ is not b o u n d e d , then there is a positive constant f such that c~(?) = 1. By L e m m a 3.14, we have O(r) >_ ~ for all r _> f. With the conditions (A) and (C), we can find positive constants c' and r6 such that
1(
~
g- \x/2(n-
1)
f(r)) > c ' r
for all r > r~. We can choose c o < m i n t c ' , ~ J and r o > max[?, r 6 I. If there is a point r ' > r o such that a ( r ' ) < co(r' - ro). Then we can find a point r > r o such that ol(r) < cor and od(r) < Co. Thus 0 < c~'(r') < / ~ .
732
MAN-CHUN LEUNG
And g2(°~(r')) (n 1" g2(cr) ~ _< -
.. g2(g-l((O/2x~ -
)f---~<_(n-|~
(n-1)
-S~)
l))f(r)))
as g ' _> 0 implies that g is n o n d e c r e a s i n g . T h e r e f o r e 0 < ~, c o n t r a d i c t i o n .
6
<-2' •
In T h e o r e m 3.12, we a s s u m e that m > 1 in c o n d i t i o n (B). W e discuss the case when m=l. THEOREM 3.20. A s s u m e that f ' ( r ) > 0 for all r > 0 a n d l i m , ~ o o f ' ( r ) = ~ . S u p p o s e that g ' ( y ) _< k for all y _> 0, where k is a positive c o n s t a n t . Let a ( r ) e C2(0, oo) be a positive solution to ( l . 1 ) with f a n d g as a b o v e a n d limr-0÷ or(r) = 0. If 2 < p _< n, then for a n y e e (0, 1), there exist positive c o n s t a n t s r o a n d C such that a ' ( r ) _< C/fl-e(r) for all r _> r o . I f p > n _> 2, then for any e > 0, there exist positive c o n s t a n t s r o a n d C s u c h that C
a'(r) <_ (f(r))O -o((.for all
I)/(p-1))
r >_ to.
Proof.
Given 1 > e > 0, as in (3.8) (with e = 1 - r) we can find a positive n u m b e r r0
such that
2g(et(r))g'(~(r))ot'(r)
f'(r) [ g2(c~(r)) f(r) L f - ~
fZ(r)
] + (°t'(r))2
gE(~(r)) [k z - ezf'(r) 2] - (1 - e) + -~7~-( -< ef3(r)f,(r) [f'(r)gZ(a(r)) )
f'(r) [ c~'(r))z~
<_
-(1 - c)~Tf~
for all r > r o, as
g'(s) <_ k
( )
]
+ (~'(r)) ~
a n d limr~®
f'(r)
= oo By (2.2) we have
gZ(ot(r)) ]
®'(r)
( p - l ) ( a ' ( r ) ) z + (n - 1 ) ~ |
<
-
-
-
)f(r)
+ (~'(r))2
®(r)
(3.21)
for all r _> f o . T h u s if 2 < p _<_ n, then (3.21) implies that ®'(r) < - ( 1 -
e)(p -
f'(r) 2 ) - ~ - ~ - ®(r)
for all r >__ r 0. A n i n t e g r a t i o n gives c~'(r)_< c/fl-~(r) a n d for all r _> r 0. I f p > n > 2, then (3.19) gives O ' ( r ) _< - ( 1 - e)
( p - 2)(n - 1)
p-
1
for s o m e positive c o n s t a n t C
f'(r) f(r)
--O(r)
Positive solutions of second order quasilinear equations
733
for all r _> r 0. A n i n t e g r a t i o n gives of(r) _<
C
(f(r))tt-e)<~,-l)/w-I))
for s o m e positive c o n s t a n t C a n d for all r > r o.
•
As a c o r o l l a r y , we have the following result (cf. [6]). COROLLARY 3.22. A s s u m e that f ' ( r ) > 0 for all r > 0 a n d limr~® f ' ( r ) = oo. S u p p o s e that g ' ( y ) _< k for all y > 0, where k is a positive c o n s t a n t . A s s u m e that f ( r ) >_ Cr s for s o m e positive c o n s t a n t C a n d for all r > f > 0, where s > 1 if 2 < p _< n, a n d (p-
i)
S > - (n -- l)
i f p > n _> 2. Let ~(r) e C2(0, oo) be a positive s o l u t i o n to (1.1) w i t h f a n d g as a b o v e a n d limr~0÷ ~(r) = 0. T h e n o~ is a b o u n d e d f u n c t i o n on ~ + .
Proof. F r o m T h e o r m 3.20, we can c h o o s e r 0 > 0 a n d e > 0 so that C
~'(r) <_ rl+6(r ) for all r > r o. H e r e d~ is a positive c o n s t a n t . A n i n t e g r a t i o n shows that c~ is a b o u n d e d f u n c t i o n on ~ + . • In p a r t i c u l a r , for p > 2 a n d n _> 2, all r o t a t i o n a l l y s y m m e t r i c p - h a r m o n i c m a p s f r o m the h y p e r b o l i c space to the E u c l i d e a n space a r e b o u n d e d . Acknowledgement--We would like to thank Professor Ravi Agarwal for valuable suggestions. REFERENCES I. Greene, R. and Wu, H., Function Theory on Manifolds which Possess a Pole. Lecture Notes in Math, Vol. 699. Springer-Verlag, Berlin, 1979. 2. Eells, J. and Lemaire, L., Another report of harmonic maps. Bulletin of the London Math. Soc., 1988, 20, 385-524. 3. Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems. Acta Math., 1970, 138, 219-240. 4. Ratto, A. and Rigoli, M., On the asymptotic behaviour of rotationally symmetric harmonic maps. J. Diff. Eqns., 1993, 101, 15-27. 5. Cheung, L.-F. and Law, C.-K., An initial value approach to rotationally symmetric harmonic maps. Preprint, 1995. 6. Cheung, L.-F., Law, C.-K., Leung, M.-C. and McLeod, J. Entire solutions of quasilinear differential equations corresponding to p-harmonic maps. Nonlinear Analysis, Theory, Methods and Applications (to appear). 7. Leung, M.-C., Asymptotic behavior of rotationally symmetric p-harmonic maps. Applicable Analysis 1996, 61, 1-15. 8. Leung, M.-C., On the infinitesimal rigidity of harmonic maps. Preprint, 1995. 9. Heinonen, J., Kilpel~.inen, T. and Martino, O., Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, Oxford University Press, Oxford, 1993. 10. Coddington, E"AnIntr°ducti°nt°OrdinaryDifferemialEquati°ns" Prentice-Hall, Englewood Cliffs, 1962. 11. Hamilton, R., Harmonic Maps of Manifolds with Boundary. Lecture Notes in Math., Vol. 471. SpringerVerlag, Berlin, 1975. 12. J~iger, W. and Kaul, H., Uniqueness of harmonic mappings and of solutions of elliptic equations on Riemannian manifolds. Math. Ann., 1979, 240, 231-250. 13. Takakuwa, S., Stability and Liouville theorems of p-harmonic maps. Japan J. Math., 1991, 17, 317-332.