1 - Department of Mathematics

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KDAM
PUBLISHING
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Differential and Integral Equations, Volume 8, Number 6, July 1995, pp, 1369- 1383.
EXISTENCE AND NONEXISTENCE OF POSITIVE
SINGULAR SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS
WITH APPLICATIONS IN ASTROPHYSICS
"',
YI LI1
Department of Mathematics, University of Rochester, Rochester, NY 14627-0001
JAIRO SANTANILLA2
,--, -, .. "
---"'.'",....
"<, ,",""
"cc,'.;'
,
,
.-
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Departmentof Mathematics,Universityof New Orleans, New Orleans,LA 70I48
,~,~~;",:,.::"",'.':O
,
,
'
'..
(Submitted by: James Serrin)
.. ",',
,-,
<,;' ":,::,::,'2,,"::';:c:'l
~:':':-:.;:~,;~~~~6~>;i.{-=:1
1. Introduction. Stationary radially symmetric models in stellar dynamics have
been studied extensively. Eddington [II] in 1915 introduced the equation
/::iu+
e2u
1+ Ixl2
= 0 in JR3
to study the gravitational potential u of a globular cluster of stars. Fifteen years later,
Matukuma [24, 25] proposed the equation
/::iu+
uP
1+ Ixl2
=0
in JR3
to improve Eddington's model. Here u > 0 represents the gravitational potential,
p = (4n)-I(1 + IxI2)-lup is the density and JR.3pdx represents the total mass. (See
also Ni and Yotsutani [34]). In 1972, Peebles [35, 36] gives for the first time a derivation
of the steady-state distribution of stars near a massive collapsed object, such as a black
hole, located at the center of a globular cluster. The same year, Peebles [35] motivated
the observer and theoretician with the title of his paper, "Black holes are where you
find them" and concluded, that "there can be no conclusions until we find a black hole".
Since then, a great deal has been written about black holes by astrophysicists (see the
recent review article by Shapiro [42]). However, the question of the existence of a black
hole in a globular cluster is still open. Hubble Space Telescope (HST) observations of
globular cluster cores should improve the observational basis for confirming or denying
the presence of massive black holes in globular clusters (see, e.g., Cohen [10]). Core
collapse does occur; for instance, using the HST, Bendinelli etal. [5, May 1993] have
~~;"-"-~;~i'.~;:;"'c"s.-::~,~'~.,;~,~'-i
;:-",:'"
""'0
"","
..
",
,
",
"",
'.-,
..
",'
..,
--
'
",..~..,,~.:..,-,;,
"""-'
'
Received March 1994,
This work was reported on at the International Conference on Differential Equations in August, 1993.
ISupported in part by National Science Foundation Grant DMS-9225 145,
2Supported in part by National Science Foundation Grant DMS-91O8021.
AMS Subject Classification: 35A20, 35805, 35840, 35J6O, 35Q75.
1369
.. ,
"-'-',,",
"",.
. ,
1370
,'C.
c. '0, "0'"
",'
,-,-
/0,-
":.;"'.';:';~~;\'"
- '-0
,','o-'c'o
,','
','
- 0
'-
'"'*-~~';':
,0'
C"
,C'
'
-.
,
'0""":"'"
coo"
,:,1
",{<'
"-.0',
~~A ~~ ~~5,i~~.~~\~~
YI LI AND JAIRO SANTANILLA
documented the first detection of a collapsed core globular cluster in M31. It is also
probable that M15 is in a current state of collapse [18].
From the theoretical point of view, it is of interest to prove the existence of solutions
of appropriate models for black holes. For a relativistic model this was first done (unintentionally) by Schwarzschild [38] in 1916 within a month of Einstein's publications of
the theory of general relativity. However, neither Einstein nor Schwarzschild knew at
the time that Schwarzschild's solutions contained a complete description of a black hole.
A very recent study of the existence of black hole solutions for the Einstein -YanglMills
equations is due to Smoller, Wasserman, and Yau [43].
This paper is concerned with the existence and non-existence of "black hole solutions" of two different models; i.e., Matukuma's, as well, Henon's-type equations. In
contrast to known models for black holes located at the center of globular clusters, we
do not assume the existence of a black hole, instead we impose a restriction on the model
which a black hole must satisfy; i.e., the gravitational potential of the cluster behaves
like l/r(r = Ixl) near the center. We then study the existence and nonexistence of
solutions of the model.
To study the existence of stationary radially symmetric solutions of the standard
system involving the Boltzmann's equation, it is sufficient to solve an equation of the
form
1lu + h(lxl. u) = O.
(See, e.g., Batt et al. [4, p. 170]).
Motivated by the above remarks we consider the n-dimensional version of
1lU + A(lxl)uP
=0
u>O
in JR3\ {OJ
near x
U'" 1/lxl
1 u '" 1/lx
-'.
in JR3\ {OJ
=0
(1.1)
at 00,
where p > 1 and A is nonnegative and locally HOlder continuous on (0,00). The
presence of A indicates that the model is not necessarily isotropic. It follows from our
result in JRn(Theorem 2.1 below) that (1.1) has infinitely many solutions with finite total
mass provided
-.oo
100
r2-p A(r)dr
< 00.
(1.2)
We will use the super-sub solution approach as in Ni [28], Naito [26] and Kawano [16]
to obtain the result. Consequently, our result can be extended to include nonradial
A = A(x), if we add conditions like
- '-'t'
""..- "'--",
- -0'0--' -,-'
rOOr2-p( sup A(x) )dr < 00.
10
Ixl=r
Our main result applied to Matukuma's equation implies that (1.1) with A(r) = I/O +
r2) has infinitely many radial solutions with finite total mass provided 1 < p < 3
..
I
I ._- ..- "-.'.
SEMILINEAR
ELLIPTIC
PROBLEMS
1371
(Corollary 2.2 below). This is in contrast to the uniqueness result for an entire solution in IRn obtained recently by Yanagida [44] and Kawano, Yanagida, and Yotsutani
[17]. Furthermore, we shall observe that the stellar density profile p (r) for (1.1) with
Matukuma's equation satisfies, near the center,
r -(m+3/2)
(*)
p(r) '" 4n(1 + r2)
and
Cl
C2
< p (r) < ,
r(1 + r2)
r3(1 + r2)
.
.
:.,
,.
.
:~,f~~ ~~~;£\~~~ \{$~.j~.;;:4.. ~0.
~
\
/}.u + Ixl-£uP
=0
in Q \ {OJ
u>O
{ u-+oo
"-"-".
""-'" .,'
for some Cl, C2 > 0 and -112 < m < 312 (cf. Ni and Yutsutani [34, p. 30]). We point
out that the above inequality is valid for r > 0 sufficiently small. In a similar model, for
r away from zero (0 < rD S r S ra), Bahcall and Wolf [2] obtained the density profile
r-7/4. It is interesting to note the density we obtain in (*) with m = 1/2 resembles that
of Lightman and Shapiro [20], where rD is "extremely small" and 0 < rD S r S ra.
We shall further observe that the integrability condition in (1.2) is sharp (Theorem 2.3
below). Consequently, we obtain that (1.1) with Matukuma's equation has no solution
provided p ::: 3 (Corollary 2.4 below).
Singular positive solutions for elliptic problems in IRn related to (1.1) have been
studied recently by Ni and Serrin [31], Bandle and Marcus [3], Serrin [40, 41], Guedda
and Veron [13], Schoen [37], Gidas and Sprock [12], Johnson, Pan and Yi [15], and
Caffarelli, Gidas and Sprock [7]. The main feature in our results is the prescribed
behavior of solutions near the origin.
It is also well known (see, e.g., Ni [29]) that
in Q \ {O}, u
(1.3)
as Ixl -+ 0,
.(':..:
".
= 0 on aQ
where Q is an open ball in IRncontaining the origin, has infinitely many singular radial
solutions provided 1 < p < (n + 2 - 2£)I (n - 2) and f!.< 2. If u is a C2 solution of
(1.3) with 1 < p < (n - f!.)/(n - 2) and a nonremovable singularity at x = 0, then there
exist positive constants C 1 and C2 such that
C1
~
Ixln-2 s u(x) S Ixln-2
near x = 0,
(1.4)
and singular solutions are not distribution solutions at x = 0 (Gidas and Sprock [12]).
We shall generalize the above existence result for solutions satisfying (1.4). More
specifically we consider
. '"
/}.u+ f(lxl, u)
u>O
{ u=O
'. '1
=0
in Bro \ {OJ
in
Bro
\ {OJ
on aBro'
(1.5)
.,
1372
","
--
YI LI AND JAIRO SANTANILLA
where fiscontinuouson
(0, ro] x [0,00), and Brois the ball centered atO and radius ro in
JRn(n ~ 3). When n = 3, (1.5) includes the special case f(lxl, u) = Ixl-lup, where u
represents the gravitational potential of a stationary rotating stellar system (Henon [14]).
It follows from our result (Theorem A below) that (1.5) in this situation has infinitely
many (radial) solutions behaving like 1/lxl near the origin, provided 1 < p < 3 - f.
Furthermore, there are no such solutions if p ~ 3 - f (Corollary 3.6 and Theorem 3.8
below).
The explicit dependence on IxIof the nonlinearity in (1.5) is of interest in applications
to real clusters in which the velocity distribution may not be isotropic (cf. Henon [14,
p. 233]).
Recent works for singular solutions in bounded domains include Ni [27, 29], Senba,
Ebihara and Furosho [39], Caffarelli, Gidas and Sprock [7], Aviles [1], Brezis and
Oswald [6], and Lions [21].
Our main result concerning (1.5) is
,W
-..,--,"'>""',C',.~,'",""~:""',".'.'
," o;,o;=-;'<."~:,~c:,~~" ',-:;,"",<:"::;'!
',C,' ","".":"':
b'f,o'>',
~;:o_~~:;'b~c;"::';c,,:;{
jj~'i'X:~:
~
Theorem A. Let f(r, u) ~ Of or u ~ 0 and r E (0, ro]. Assume there exists afunction
j (r, s), nondecreasing in s ~ 0, for each r E (0, ro], such that f (r, s) ~ s j (r, s) for
s ~ 0, r E (0, ro]. Moreover, suppose there exists 0 > 0 with
(0
d ==(n - 2)-1 Jo r j(r, or2-n)dr < 1.
Thenfor each a E (0,0] there exists a radial solution u of(1.5) with finite mass, such
that
a(1 - d)«J(r) ~ u(r) ~ a«J(r),
0 < r ~ ro,
where «J(r)= r2-n - rg-n. Furthermore, u'(ro) = -(arJ-n)(n - 2).
We point out that the inequality d < 1 in Theorem A is sharp in some sense (Theorem 3.5 below). Theorem A applied to Matukuma's equation implies that (1.5) with
f(lxl, u) = uP /(1 + Ix12) in Bro \ {O}has infinitely many radial solutions behaving like
r2-n near r = 0, provided 1 < p < n/(n - 2).
Let A be continuous on (0, ro] with 0 ~ A(r) = O(r-l) near r = 0, f < 2. As a
corollary of Theorem A we also have that
b.u + A(r)uP
u>O
u
I
=0
u '" r2-n
=0
in Bro \ {O}
in Bro \ {O}
(1.6)
on a Bro
near r = 0
has infinitely many radial solutions provided 1 < P < (n - f)/(n - 2). (See also
Corollary 3.2 below).
If f < -1 and p ~ (n - f)/(n - 2), it follows from a known result (see, e.g., Ni and
Weprovide(Theorem3.8below)
Sacks [30])that (1.6) has no solution in C2(Bro\ {O}).
a proof of the case -1 ~ f < 2, p ~ (n - f)/(n - 2). Other nonexistence results have
been obtained by Aviles [1] , Ni [27,29] and Gidas and Sprock [12]
.
We further prove a nonexistence result for the equation in (1.6) with no restriction at
the origin. More specifically we have
..
SEMILINEAR
Theorem B. Let A : (0, 00)
ELLIPTIC
PROBLEMS
1373
[0, 00) be continuous with
~
r-'1A( l/r)
nondecreasing where'fJ == n+2-i(n-2). Thenfor p > 1,
-.
--
{
flu + A(r)uP = 0
u>o
in Br0 \ {OJ
u=o
on aBr0
(1.7)
in Br0 \ {OJ
has no radial solution.
, '-,', , ,",'" ,'".., ,-.. - "..
'".. -- , ---- '",-,-- '-- ---' -" - -,--
#.%~~~b;~~~;2~~:i~:~:~
~~
-"'--'
,
- d"-..'
,',', ' ---, "'---',---,-,
--- "--,- --,,-- '-,'" ,- ,- --, .. -- W'----------------.-- -..
-" -', --- '"
"---
-------------------
;'0-~~~;:1;-W~-:-~ =ioyJ;~~-';6~
The interesting feature of (1.7) is that no behavior of u at the origin is required.
Another general nonexistence result of this type can be easily obtainedfrom Lemma 1.1
by Ni and Sacks [30].
It follows from Theorem B and Ni [29] that (1.7) with A(r)
=
r-l has no radial
solution provided p ~ (n + 2 - 2f.)/(n - 2), f. < 2. When f. = 0 the result is due to Ni
and Serrin [31].
2. Singular positive solutions for Matukuma's type equation in ffi.n\ {OJ.Let
p > 1 and n ~ 3. In this section we consider the problem
+ A(lxl)uP = 0
flU
!
in ffi.n \ {OJ
u>O
in ffi.n \ {OJ
u "-'
r2-n
near r
u "-' r2-n
at 00.
Theorem 2.1. Let A : (0, 00)
~
=0
(2.1)
[0, 00) be a locally Holder continuous function such
that
---,
---
- --
(2.2)
100 rn-l-p(n-2) A(r)dr < 00.
'--- ,
Then (2.1) has infinitely many solutions with finite total mass.
Proof. Similar to the work by Li and Ni [19], we will make the following Kelvintransform. Let
y = x/lxI2, V(x) = IxI2-nu(y)= IYln-2u(y);
the desiredsingularityat y
satisfies
= 0 is guaranteed if V(x)
fl V + Ixl-n-2+p(n-2) A(1/lxl)VP
C(> 0) as Ix! ~
= 0 in ffi.n\ {OJ.
00. V
(2.3)
The result is then an immediate consequence of the following claim. There exists a
(3 > 0 such that for any constant C E (0, (3), there exists a positive solution V of (2.3)
in ffi.nwith V E C(ffi.n) n C2(ffi.n\ {O}),V(x) ~
Cas Ixl ~
00. The proof of this
assertion is essentially an adoption of results by Ni [28], Naito [26], Kawano [16], and
Ni and Yotsutani [34].
'..--
.
---
~
-
-
-
- - -
. ,
YI LI AND JAIRO SANTANILLA
1374
Let
A == 100
A(r) dr
rn-l-p(n-2)
and
.,' .."
"
- '"~.
~
,.
fJ =
Consider g(t)
C = a - ~aP
n-2
.
- l/(p-l)
mIll {(n - 2)/A)
,(1-
1
n - 2 ---L
l/P)(pl/(P-l»(A)P-I}.
=t-
n~2tP. Then for each C E (0, fJ), there exists a > o such that
and C - ~CP
> 0. Let
n-2
s
T
aP
1
W(r) = a--=sn-3(
tp(n-2)-n-l A(1/t) dt) ds,
rn 2 0
0
1
~";2~i:'"~O~~;i
h(s) = is tp(n-2)-n-lA(1/t) dr.
~~_~.~i:~~~~~;~~~.~;j
Then
T
aP
W(r)
=a- _
2
1
rn0
aP
= a - -h(r)
n - 2
aP
sn-3h(s)ds
=a -
T sn-2
aP
+ _sp(n-2)-n-lA(1/s)ds
n - 2 0 rn-2
00
aP
1
-h(r)
+ n-2
n-2
aP + a - -A
=C
n-2
10
(r
(s/rt-2X[o,Tj(s)sp(n-2)-n-l A(1/s)ds
+ 00)
by the Lebesgue Dominated Convergence Theorem. Now,
aP n - 2
W'(r) = -[r ~rn-
10
T
aP n - 2
T
sn-3h(s)ds-h(r)] = -[r ~rn-
10
sn-\h(s)-h(r»ds]
::::°
with
(rn-lW'(r»)' = -aPrn-2h'(r) = -aPrP(n-2)-3 A(l/r).
Thus W satisfies
{
W" + n~l W' + aPrP(n-2)-n-2 A(~) = o in (0, 00)
W(O) =a > 0,
W -!-Catoo.
Since a ::: W ::: c, we have
n-l
W"+-W'+rP(n-2)-n-2A(1/r)WP::::
r
n-l
W"+-W'+rP(n-2)-n-2A(1/r)aP
r
= 0.
It is obvious that W ==C is a subsolution of (2.3). Thus a pair of super-subsolutions of
(2.3) is obtained. Hence there exists a solution V of (2.3) which satisfies C ::::V ::::W
--
---
~ -
-
- -
--
- -
~
. .
SEMILINEAR ELLIPTIC
1375
PROBLEMS
and Y E C(JRn) n C2(JRn\ {On. Finally, to show that the solutions obtained above have
finite total mass we use the Kelvin transfonn introduced at the beginning of the proof
and the fact that dy
{
~"\~
saP
= Ix1-2ndx.We obtain
A(y)uP(y)dy = ( A(1/lxIHlxln-2Y(x))Plxl-2ndx
~"
100 rn-l A(1/r)rP(n-2)-2ndr
= aP 100 rn-l-p(n-2)
A(r)dr
< 00.
This completes the proof of the theorem.
".'
':':~k,c,.:,.;.,,<;'e~'~)'~:~,:~
"
.'
~.;.A:;::;,..;C.:.'I
Remark 2.1. Asymptotic expansion of solutions of (2.1) near x = 0 can be obtained
for suitable A's for which one can find an expansion at 00 of solutions of (2.3) via the
fonnulas in Li and Ni [19]. For instance, this can be done for (2.1) with Matukuma's
equation.
Remark 2.2. The solutions obtained by Naito in [26] are radially symmetric about the
origin. And radial symmetry of the solutions of (2.1), including Matukuma' s equation,
can be obtained via the techniques in [8, 19,23]. For instance, if A(r) is nonincreasing
and r(n-2)(1-p) A(r) = O(r-2) at infinity, then all solutions of (2.1) must be radially
symmetric. This follows from the works [8, 19,23].
Remark 2.3. It is easy to show that the solutions obtained in Theorem 2.1 have infinite
energy:
( I V ul2dx = 00.
JJR"
= O(r-£)
Corollary 2.2. Let A be as above with A(r)
at 00, A(r)
= O(r-a)
at O,for
some 0 S (j < f and
n-f
max{l, -}
n-2
< p <-.
n-(j
n-2
Then (2.1) has infinitely many radial solutions with finite total mass.
As a simple consequence of this corollary «(j = 0, f
Matukuma's equation, i.e.,
I
= 2) we obtain
\ {OJ,
tJ.u + 1:1:12= 0
u>O
in JRn
u
"-'
near x = 0,
u
"-' r2-n
r2-n
that (2.1) with
in JRn\ {OJ,
(2.4)
at 00,
has infinitely many radial solutions with finite total mass provided 1 < P < n/(n - 2).
Next we show that the integrability condition (2.2) in Theorem 2.1 is sharp in the
following sense.
f
..
(
1376
~)
s.;.
YI LI AND JAIRO SANTANILLA
Theorem 2.3. Let A : (0, 00)
[0, 00) be continuous with
~
100
rn-l-p(n-2)
A(r)dr
= 00.
(2.5)
Then (2.1) has no radial solutions.
Proof. Suppose that (2.1) has a radial solution u. Then as in the proof of Theorem 2.1,
the Kelvin transform V of u satisfies (2.3). It suffices to show that (2.3) has no bounded
positive radial solutions which are bounded away from zero at 00. To this end, we apply
Theorem 4.1 by Kawano [16] with a minor modification. The result implies that if (2.3)
has a positive bounded solution which is bounded away from zero, then
"W,
,',
,
-, "
.. ,
"""""'"
)':~;o"",::.">::'~"-~:OY.:;;:;i,,,,:~::::,;>'~,~~
00 >
100 rrP(n-2)-n-2A(1/r)dr = 100 rn-l-p(n-2)A(r) dr,
contradicting (2.5).
if p ~ n/(n - 2).
Corollary 2.4. Problem (2.4) has no radial solutions
Corollary 2.5. The equation t::.u+ uP = 0 in JRn\ {O}(n ~ 3), (p > 1) has no positive
radial solutions behaving like r2-n near r = 0 and r = 00.
Remark 2.4. For classical solutions in an exterior domain, Noussair and Swanson [32]
have obtained the sharper result
100
rn-l-p(n-2)
A(r) dr <
00
for some c > o.
f;;..::,:,;' ,~;":c",o~",,,;,o,,_,""":-'~-~1
c- , ;,,"'-',':';."
Remark 2.5. Theorem 4.1 by Kawano [16] is for entire solutions in JRn. However, the
arguments in his proof are valid when the coefficient of VP is continuous on (0, 00).
' . ',,'," ','",",','0,',',
3. Singular solution for Henon's type equations in Bra \ {OJ. Our objective in this
section is to prove Theorems A and B in the Introduction. To prove Theorem A we start
with a lemma. Consider
v"+g(t,v)
=0,
v(to) = 0,
t > to(> 0)
(3.1)
f
f
!
;
,.',
.",
~-""'~;'
...'"
where g is continuous on (to, 00) x [0,00).
The following lemma is an improved version of Theorem 3.1 of Noussair and Swanson
[33].
Lemma3.1. Letg(t,s) ~ Ofors ~ Oandt ~ to.Assumethereexistsafunctiong(t,s),
nondecreasing in s ~ 0, for each t ~ to, such that
g(t, s) ::: sg(t, s)for t ~ to, s ~ O.
"
.
SEMILINEAR ELLIPTIC
1377
PROBLEMS
Moreover, suppose there exists 8 > 0 with
[00
tg(t, 8t)dt < 1.
lto
...
"
.: .:.;.
...
Thenfor each a E (0, 8] there exists a solution v of (3.1) such that vi(to)
. ,,;' :','
a(1 -
[00
tg(t, 8t)dt)(t
-
= a and
to) :::: vet) ::::a(t - to), t ::::to.
lto
Proof. For each a E (0, 8],
'"5.',,':: 'c"'...,.~:.'::.'."-O'-":',;", ,.
~~:'~:,:c::,~,,;::~':'
v" + g(t, v) = 0,
",
,
~-;:~>i}~'k$,<Pf/'O--:~~i,"0F,~-:;:g
v(to) = 0, vi(to) = a
has a local solution v. Let [to,ta) be the maximal interval on which v is positive. We
assert that ta = +00. Supposeta < 00. Since vet) :::: v'(to)(t - to) ::: at on [to,ta),
then for to < t < ta we have
a
= vi (to) = viet) +
~t g(s, v(s»ds
::::viet) + a [00 sg(s, 8s)ds.
~
Thus
v'(t)
Thereforelimt-w
"
a
::::a(1
-
[00
(3.2)
tg(t, 8t)dt) > 0, to < t < ta.
lto
vi (t) exists and is finite. Furthermore,
his easily seen thatlimt-w
exists and is finite, proving continuability of v through ta. We must have v(ta)
then vi (ta) ::::0, contradicting (3.2).
a
vet)
= 0, and
Remark 3.1. The existence of 8 in Lemma 3.1 is guaranteed if J;~ tg(t, O)dt < 1.
Let Brobe the ball centered at 0 with radius ro in IRn(n ::::3). We consider
in Br0 \ {a}
!::"u+ f(lxl, u) = 0
u>O
{ u=O
..
(3.3)
in Br0 \ {O}
on aBr0'
where f is continuous on (0, ro] x [0,00).
Proof of Theorem A. Let
w(y)
-
. . ---.
'-'.' .,,-..
= IYI2-nu(x),
X
= 4.
Iyl
Then (3.3) is transformed into
!::"w+ IYI-2-nf(I~I' IYln-2w)
w > 0,
{ w =0,
<- I
= 0,
Iyl >
l/ro
Iyl>
l/ro
Iyl = l/ro.
(3.4)
.
YI LI AND JAIRO SANTANILLA
1378
Thus, we consider
n-1
wiler) + -W'(T)
T
W > 0,
",',
( w(llro)
'c"',,
"
1
+ T-Z-nf( -,
T Tn-ZW(T»)= 0,
T > 11ro
(3.5)
T > 1lro,
= O.
or equivalently,
V"+g(t,V)
v(t) > 0,
{ v(to) = 0,
t > to
t > to
=0,
(3.5)'
where t = (n - 2)Tn-Z, v(t) = W(T)t, to = (n - 2)rJ-n and
,',:
"
~t.:,,\;;;o:"
"
~'~.'.:~!.<:
1
g(t, v) = t-3[a(t)r-4f(-, a(t) [a(t)]n-Zvlt),
',,;
~:;;:~X;.:;':':~b'~
with a(t)
= [tl(n
Letg(t,s)
- 2)]1/n-Z.
= t-3[a(t)r-4f(a~t)'
g(t, s) = r3[a(t)r-4f(
and
n~Z)(n~Z)' We have
1
s
t-3[a(t)]n-4s - 1
s
'
_
-at ' _
2
) ~
f( a- (t) n- 2 )
n- 2
( ) n-
00
1~
= sg(s,
t),
~
tg(t,8t)dt=(n-2)-1
10
rf(r,8rZ-n)dr
< 1.
Thus the assumptions of Lemma 3.1 are satisfied and the result follows.
Corollary 3.2. Let f be continuous on (0, TO]x [0,00), with 0 ~ f(r, u) ~ B(r)u +
A (r )uP, p > I. Then the conclusion of Theorem A holds provided
to
10 rB(r)dr
:~;~y;;~~~i~~;~~;;Xi
,","
,,'
,','1
< n - 2
and
to
10 rl+(p-l)(Z-n) A(r)dr
< 00.
(3.6)
Proof. Let f(r, u) = B(r) + A(r)up-l.
Corollary 3.3. Let A be continuous on (0, TO]with 0 ~ A(r) = O(r-f) near r = 0,
.e < 2. Then
tJ.u+ A(r)uP = 0 in Bro \ {OJ
" ,,','"
u > 0
in Br0 \ {OJ
u =0
onlJBr 0
[ u '" rz-n
'
nearr = 0
has infinitely many radial solutions provided 1 < p < (n - .e)I (n - 2).
,.
~-
-
SEMILINEAR
ELLIPTIC
1379
PROBLEMS
Corollary 3.4. For I < p < n/(n - 2), Matukuma's equation in Br0 \ {OJhas infinitely
many positive radial solutions vanishing on a Br0 and behaving like r2-n near r = O.
Proof.
.
:~/:. ~;:;:~,~;?3::'B.~~io~:~
",
, ,
,,',
, .Yf.--:i..
= 0 in Corollary
3.3.
Remark 3.2. Corollary 3.3 includes Henon's model [14] where A(r) = Ixl-l and u
represents the gravitationalpotential of a stationary rotating stellar system.
Remark 3.3. It is interesting to note that the integrand in the integrability condition
(3.6) is the same as in (2.2) for Matukuma's type equations.
.
"C :':, ,
,.:.,'::~:;
Take £
,
','
,
"",'".
Remark 3.4. It is easy to show that the solutions obtained in Theorem A have finite
total mass (in Bro\ {O}).
Remark 3.5. A more detailed asymptotic expansion near the origin can also be given
using the results by Li and Ni [19], and Li [22] applied to (3.4).
Next we show that the integrability condition in Theorem A is sharp in the following
sense.
',: :.:~~+i.-;ji;c;;. ,o:~:'o'?,,",,!;;;:,~;;j
Theorem 3.5. Let A be Cion [0, ro] and nonnegative. If
(0
10 rl+(p-l)(2-n) A(r) dr
= 00,
then
6,.u+ A(r)uP = 0
u>O
in Br 0
(3.7)
in Bro\{OJ,
near r
{ u '" r2-n
\ to},
=0
has no radial solution.
Proof. This follows easily from Lemma 1.1 in Ni and Sacks [30]. In fact,
'-,
...
,
,
"'--"""""'
'-"""'~-"--"""""""'"..
'--"
,
,'..
".0,
""
,
'~-'
,
00 >
'-.
,
,.""
,.'
"
"
1~\~
A(lxl)uP(x)
dx
=
(0 rn-l A(r)uP(r)
h
dr,
contradicting the integrability assumption.
Corollary
3.6. If A is Cion
[0, ro], nonnegative,
and A(r)
2: cr-l
near r
positive constant c, then (3.7) has no radial solution provided p 2: (n
,.."", '
,-'..
~
-'-'~"
' '...
, '
-
= Of or
-
£)/(n
some
2).
This corollary shows the sharpness of the range ofp(p < (n-f.)/(n-2),£::::-1)
in Henon's model (Corollary 3.4 above). It also applies to Matukuma's equation (£ = 0)
in the punctured ball.
If A is not necessarily C1, we may still have nonexistence results; e.g., Theorem B.
Proof of Theorem B. It sufficesto show that Problem (3.5)' with g(t, v) = a(t)vP has
no solution. Here,
a(t) ==t-3-p[a(t)t-4A(1/a(t»[a(t)]p(n-2),
,
."""
, ',...,,
1380
YI LI AND JAIRO SANTANILLA
2)rn-2, r = a(t).
Applying Corollary 10of Coffman and Wong [9], any solution of (3.5)' is oscillatory
if t(p+3}/2a(t)is nondecreasing. A simple calculation shows that
t
= (n -
t(p+3}/2a(t)
where c is a positive constant and 1] =
Corollary
n+2-~(n-2}.
3.7. Let P > 1 and £ < 2. Then
t:w + Ixl-lup
','
.. '"
,
"
0,'
"""'-"
,"
",'
,
"'-"',
,
'",
..
"
-,'
"
','..
..0,"
'
'
0..
""..
"""
'..,
,',
= cr-'1 A(l/r),
=0
u>O
{ u=O
,
"
in Bra \ {O},
in Bra
\ {O},
(3.8)
on aBra ,
~~~~'~~~~~f~~~~~E'~~
has no radial solutionprovided p ::: (n + 2 - U)/(n - 2).
;<~~~~i~~!'~i~1~
Note that in Corollary 3.7, no restriction is imposed at the origin.
The nonexistenceresult of singular solutions for (3.8) with £ ::: 0 and p ::: (n + 2 2£)/ (n - 2) is due to Ni and Sacks [30]. If 0 .:::£ < 2 and
(n - £)/(n - 2) < P < (n + 2 - U)/(n - 2),
then it follows from Theorem 3.3 by Gidas and Sprock [12] that
t..u + Ixl-lup = 0
u>O
r u
'" r2-n
In Bra \ {O}
in Bra \ {O}
near r
(3.8)'
=0
has no solution.
If p
= (n -
£)/(n - 2) and -2 < £ < 2, the nonexistence result for (3.8)' follows
from the work by Aviles [1]. Finally, if £ ::: 2 and p > 1, it is known [27] that
t..u + A(x)uP = 0 in Bra \ {O},where A '" Ixl-l, does not possess any solutions. We
do not know of any results for (3.8)' covering the case -1 .:::£ < 0 and
(n - £)/(n - 2) < P < (n + 2 - 2£)/(n - 2).
The next result closes the gap.
Theorem 3.8. Let A be continuous and nonnegative on (0, To]with A(r) '" r-l, -1 .:::
£<Oand
(n - £)/(n - 2) < P < (n + 2 - 2£)/(n - 2).
~~~=~.~-:~>t.
.'~':~ ..>"
Then
f
,..
,.
t..u + A(r)uP = 0
u>O
in Br 0 \ {O}
u '" r2-n
near r = 0
,
'j
-
-~
-
-
in Br a
\ {O}
SEMILINEAR
ELLIPTIC
PROBLEMS
1381
solution.
has no
Proof. Suppose there is a solution. Let
=
u(r)
'~~~}1
~~.~.
1Ixl=Tu(x)dsx;
u is radial and
n-l
U"+ -u'
+ A(r)uP(r) :::0
r
0 or u(r) :::Clr2-n, A(r) :::C2r-f..
with u(r) "-' r2-n near
For a fixed rl > 0, and any 0 < r < rl, we have
(rn-Iu')' + rn-I A(r)uP(r)
.. .' -~':'-'
or
,~~:~\-~
.,
.:::0
. ..
' "
y"~,~<..~,:,,.,;;,::.:j¥ ;;:~:-.-,~~~~;i~~
i
~
T
rn-Iu'(r)
I
rn-I
:::u'(r)
i
rn-Iu'(r)
-
r~-Iu'(rl)
T
tn-
i
= U(rl ) + rlu
-'
(rl )
n - 2
sn-I A(s)uP(s)ds
dt
.::: 0,
i i
i i
T!
dt + T ~tnI
n-I-,
sn-I A(s)uP(s)ds ::: u'(rl) - u(r),
t
)
TI
I I dt -
tn-
rl
-
T
T!
TI rn-Iu'(r
T
i
sn-I A(s)uP(s) ds
T
TI rn-Iu'(r
)
I
I I
u(r) ::: u(rl) -
-
i
1
rn-I
+ -
I
+
~
~
u (rl )
T
tn-
P
- C2C
(n - 2)rn-2
dt
~
T!
sn-I A(s)uP(s)ds
t
i i
~
-
dt
I T tn-I
T!
n-I-f.- p (n-2) d
S
S
t
'
where
l-
3 < n - I -l-
pen - 2) < -1,
and hence
,".--,'." "";""":':-.,'"':;C".,,).,:?2.":':'--':~
u(r) ::: u(rl) + rlu'(rl) - r~-Iu'(rl)
n -2
(n -2)rn-2
, '::0"~\-~~~~~;d
.-.
'. _. .
-'
(
u(r) :::u(rd + rlu rl
n - 2
cPc
+
.:,
"'.
.,
',"
t...,
,.'
"
P
cI C2
pen - 2) -l-
+
"
,.",
+
""I
)
n-I -'
-
(
n
)
rl
U rl
(n - 2)rn-2
1
-
T
dt
tn-I
[sn-I-f.-p(n-2)
1;1],
P
CI C2
- ~r2-f.-p(n-2)
pen - 2) + l - n
n - 2 I
I 2
- -rn-f.-p(n-2)
pen - 2) + .e - n
n - 2 I
P
2-f.-p(n-2)
CI c2rl
i
T!
1
.rn-2
-
[pen - 2) + l - n][p(n - 2) + .e- 2]
cf C2
1
[pen - 2) + l - n][p(n - 2) + l - 2] rP(n-2)+f.-2
/
/
"
1382
YI LI AND JAIRO SANTANILLA
and since n - f > p(n - 2) + f - 2> n - 2, we have u(r) < 0 for small r > 0, which
is a contradiction.
Acknowledgment. The authors thank Gregory Seab for his initial input on the problem,
and Ivan King and Stuart Shapiro for interesting references and comments on black holes
in globular clusters.
,',
"Y?'-:"7c,~~T~..;
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...
~.2k~ftk:~.
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,;."*~,.:""j'f:"""":""""":~",,,
",-,oj
:.':.",.-0,,';,.-,""""<':""",
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"
,.
SEMILINEAR
ELLIPTIC
1383
PROBLEMS
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68-89.
.'.
,..',
",;"
'.
..
,: :':""",','>""',"'"""'","'".,,
""'~~;;":;o~;;;~/',~"-:~iC7~,,
,
,,'"
"
"i.:!¥¥j--~.';i~;"~,:l
" '~.:-.',~'i
'
"',,,
~~;;.~;~;;~~
=
:'.-,':."'-,~,""'i.::"",,;:~,,~~?<:;,:,...o~
,.-.".-,-..,.., '. ""'H--..
'
, -._,
"'~.f
';""',
,""'-;
.~
..,
."...
"',""
[25] T. Matukuma, Sur la dynamique des amas globulaires stellaires, Proc. Imp. Acad., 6 (1930), 133-136.
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.
Added in Proof. On May 25, 1994 astronomers at NASA headquarters announced the Hubble Space Telescope
finding of a supermassiye black hole in the heart of the giant galaxy M87, more than 50 million light-years
away.
~. ,.
- .v. .. "" .."
--
-
-----
-~--
'. -'