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On Singular Boundary Value Problems for the Monge–Ampére Operator
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
197 341]362 Ž1996.
0024
On Singular Boundary Value Problems for the Monge]Ampere ´ Operator A. C. Lazer Department of Mathematics and Computer Science, Uni¨ ersity of Miami, Coral Gables, Florida 33124
and P. J. McKenna Department of Mathematics, Uni¨ ersity of Connecticut, Storrs, Connecticut 06269 Submitted by Jack K. Hale Received February 28, 1995
We consider different types of singular boundary value problems for the Monge]Ampere ´ operator. The approach is based on existing regularity theory and a subsolution]supersolution method. Nonexistence and uniqueness results are also given. Q 1996 Academic Press, Inc.
1. INTRODUCTION AND SUMMARY We consider singular boundary value problems for the P. D. E. M w u x Ž x . s f Ž x, u . ,
x g V,
Ž 1.1.
where V is a smooth, strictly convex domain in R N , N G 2, and M w u x s detŽ u x i x j ., 1 F i, j F N is the so-called Monge]Ampere ´ operator. Here x s Ž x 1 , . . . , x N . is a generic point in R N . We consider two types of singular problems for Ž1.. In the first type, which we call boundary blowup problems, we look for solutions which are smooth and satisfy Ž1.1. in V and tend to infinity as x approaches V, the boundary of V. Problems of this type in which the Monge]Ampere ´ operator is replaced by the Laplacian have been the subject of several investigations in recent years}see for example w1, 10, 12, 13, and 16x. The first paper concerning boundary 341 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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blowup problems for the Laplace operator appears to be a 1916 paper by Bieberbach w2x. Bieberbach considers the problem Du s eu,
in V ,
Ž 1.2.
where V is a bounded domain in R 2 with C 2 boundary, and shows that there is a unique solution such that the difference u Ž x . y ln Ž d Ž x .
y2
.
is bounded in V, where d Ž x . s distance Ž x, V . . Extensions of this result are given in w12x and w13x. We shall show that if V is a smooth, strictly convex, bounded domain in R N, N G 2, and f Ž x, j . s p Ž x . e j , where p is a smooth function which is positive on V, then there exists a unique solution u of Ž1.1. in C`Ž V . such that uŽ x . ª ` as x ª `. Moreover, it is shown that the difference u Ž x . y ln Ž d Ž x .
y Ž Nq1 .
.
is bounded in V. In work related to a problem of Fefferman w9x, Cheng and Yau w5x considered a similar blowup problem in which M is replaced by the complex Monge]Ampere ´ operator. Another type of boundary blowup problem for the Laplacian which seems to be well understood is the problem
½
D u s p Ž x . ug uŽ x . ª `
in V as d Ž x . ª `.
Ž 1.3.
where g ) 1. It follows from w1, 10, and 13x that if V ; R N is sufficiently regular and p is positive and smooth on V, then there exists a unique solution of Ž1.3. and for x g V, c1 Ž d Ž x . .
y2r Ž g y1 .
F u Ž x . F c2 Ž d Ž x . .
y2r Ž g y1 .
,
where c1 and c 2 are positive constants. Results of this type are extended to the case where D is replaced by the p-Laplacian in w8x. We shall show that if V is a smooth, strictly convex, bounded domain in R N , N G 2, f Ž x, j . s pŽ x . j g , where g ) N, then there exists a unique
MONGE ] AMPERE ´ OPERATOR
343
solution of Ž1.1. in C`Ž V . such that uŽ x . ª ` as distŽ x, V . ª 0. Moreover, there exist positive constants k 1 and k 2 such that for x g V, k1 d Ž x .
ya
F uŽ x . F k2 dŽ x .
ya
where
as
Nq1
gyN
.
We also show that if 0 - g F N and f Ž x, j . s pŽ x . j g , where p is as above, then there does not exist a solution of Ž1.1. in V such that uŽ x . ª ` as dŽ x . ª 0. Our tools in studying the two above-mentioned boundary blowup problems consist of a simple comparison result for the Monge]Ampere ´ operator ŽLemma 2.1 below., a subsolution]supersolution method, motivated in part by work of Lions w14x, existence theory for smooth solutions for Monge]Ampere ´ equations developed by Caffarelli et al. in w4x, and regularity theory due to Pogorelov w17x in a form used by Tso in w18x. A second type of singular problem for Ž1.1. is concerned with finding a solution u of Ž1.1., which is continuous on V and smooth on V, such that u s 0 on V and f Ž x, j . becomes infinite as j ª 0. Problems of this type have also been considered when M w u x is replaced by D u. It is known Žsee w7x and w11x. that if pŽ x . is positive and smooth on V and if V ; R N , N ) 1, is bounded and sufficiently regular, then for g ) 1 there exists a unique u g C Ž V . l C 2 Ž V . such that
½
yg
Ž D u. Ž x . q p Ž x . uŽ x . s 0 uŽ x . s 0
in V for x g V .
Moreover, there are positive constants b1 and b 2 such that for x g V, b1 d Ž x .
2r Ž1q g .
F u Ž x . F b2 d Ž x .
2r Ž1q g .
.
In the final section we indicate briefly how the same subsolution] supersolution method used to study boundary blowup problems can be used to establish the existence of a unique function u g C`Ž V . l C Ž V . such that
¡M w u x Ž x . s p Ž x . Ž yuŽ x . . ~ uŽ x . - 0 ¢ uŽ x . s 0
yg
in V in V x g V
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provided that g ) 1; V is smooth, strictly convex, and bounded; pŽ x . ) 0 on V; and p is smooth. We show that there are negative constants c1 and c 2 such that c1 dŽ x . a F uŽ x . F c 2 dŽ x . a in V where
as
Nq1 Nqg
.
We remark that a problem of this type has been considered by Cheng and Yau w6x. They show that if V in R N is convex and bounded but not necessarily strictly convex then there exists a unique u g C`Ž V . l C Ž V . which is negative on V such that
¡ 1 ~M w u x Ž x . s ž y uŽ x . / ¢ uŽ x . s 0
Nq 2
in V on V .
For clarity of presentation we first prove the existence of solutions of various singular problems under the assumption that V is a strictly convex bounded domain with V of class C`. In the final section using a result of Lions w15x and ideas of the authors w13x, we show that it is enough that V is of class C 2 .
2. BOUNDARY BLOWUP PROBLEMS Our first main tool in this section, whose proof is given for completeness, is the following elementary comparison result. LEMMA 2.1. Let V be a bounded domain in R N, N G 2, and let u k g C 2 Ž V . l C Ž V . for k s 1, 2. Let f Ž x, j . be defined for x g V and j in some inter¨ al containing the ranges of u1 and u 2 and assume that f Ž x, j . is strictly increasing in j for all x g V. If Ži. the matrix Ž u1 x x . is positi¨ e definite in V, i j Žii. M w u1 xŽ x . G f Ž x, u1Ž x .., ; x g V, Žiii. M w u 2 xŽ x . F f Ž x, u 2 Ž x .., ; x g V, Živ. u1Ž x . F u 2 Ž x ., ; x g V,then u1 Ž x . F u 2 Ž x . ,
; x g V.
Proof. Assuming the contrary, there exists x 0 g V such that u1Ž x 0 . ) u 2 Ž x 0 . and u1 y u 2 assumed its maximum on V at x 0 . Therefore, the matrix
Ž u 2 x x Ž x 0 . . y Ž u1 x x Ž x 0 . . i j
i j
MONGE ] AMPERE ´ OPERATOR
345
is positive semidefinite at x 0 . From the variational characterization of the eigenvalues of a symmetric matrix w2, p. 115x it follows that if a 1 G a 2 G ??? G a N are the eigenvalues of Ž u 2 x i x jŽ x 0 .. and b 1 G b 2 G ??? G bN are the eigenvalues of Ž u1 x i x jŽ x 0 .., then a k G b k , 1 F k F N. Since the matrix Ž u1 x i x jŽ x 0 .. is positive definite, b k ) 0 for k s 1, . . . , N. Therefore, we conclude that f Ž x 0 , u2 Ž x 0 . . G M w u2 x Ž x 0 . s a 1 ??? a N G b 1 . . . bN s M w u1 x Ž x 0 . G f Ž x 0 , u1 Ž x 0 . . which is a contradiction to the assumption that f Ž x, j . is strictly increasing in j . This proves the lemma. We shall also need the following known result concerning interior estimates for derivatives of smooth solutions of semilinear Monge]Ampere ´ equations. LEMMA 2.2. Let V be a bounded domain in R N, N G 2, with V g C`. Let f g C`Ž V = Ž0, `.. with f Ž x, j . ) 0 for Ž x, j . g V = Ž0, `.. Let u g C`Ž V . be a solution of the Dirichlet problem
Ž D.
½
M w u x Ž x . s f Ž x, u . u Ž x . s c s constant
xgV x g V,
with 0 - uŽ x . - c in V. Let V9 be a subdomain of V with V9 ; V and assume that a F uŽ x . F b for x g V9 and let k G 1 be an integer. There exists a constant C* which depends only on k, a, b, bounds for the deri¨ ati¨ es of f Ž x, j . for Ž x, j . g V9 = w a, b x, and distŽ V9, V . such that 5 u 5 C k Ž V 9. F C*. This is only a slight variation of Proposition 2.4Žii. of w18x. If g Ž x, j . ' f Ž x, c y j . for 0 F j F c y m, where m - c is the minimum of u on V, g is extended smoothly to V = w0, `., and we set uŽ x . s c q ¨ Ž x ., then ¨ Ž x . F 0 on V and M w ¨ xŽ x . s g Ž x, y¨ Ž x ... A C k-estimate of ¨ on w a y c, b y c x = V9 gives a C k-estimate of u on w a, b x = V9. Therefore, Lemma 2.2 is a consequence of Proposition 2.4Žii. of w18x. Proposition 2.4Žii. of w18x follows from a result of Pogorelov w17x. Finally, we make use of the following known existence result. LEMMA 2.3. Let V be a strictly con¨ ex, bounded domain in R N, N G 2, with V g C` . Let f Ž x, j . be a positi¨ e C` function on Ž0, c x = V, where
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LAZER AND MCKENNA
c ) 0 is a constant. If there exists a C 2-function u#, which is con¨ ex on V, such that
½
M w u# x Ž x . G f Ž x, u# Ž x . . u# Ž x . s c
on V , on V ,
then there exists a solution u of Ž D . with u g C`Ž V . and u strictly con¨ ex. Moreo¨ er, uŽ x . G u#Ž x . on V. This is a special case of Theorem 7.1 of w4x in which f may depend on =u and u may be equal to a smooth function on V. PROPOSITION 2.4. Let V ; R N, N G 2, be a smooth, bounded con¨ ex domain. Assume c g C`Ž V ., c Ž x . ) 0 on V, c N V s 0, and Žy1. N M w c xŽ x . ) 0 on V. Let p g C`Ž V . with pŽ x . ) 0 on V. If g ) N,
as
Nq1
gyN
¨ Ž x. s c Ž x.
ya
,
Ž 2.1. x g V,
,
Ž 2.2.
then there exist constants c1 and c 2 with 0 - c1 - c 2 such that if w k Ž x . s c k¨ Ž x ., then M w w1 x Ž x . ) p Ž x . w1Ž x .
g
on V ,
Ž 2.3.
M w w2 x Ž x . - p Ž x . w2 Ž x .
g
on V .
Ž 2.4.
Proof. The jth column of the matrix Ž ¨ x i x j . is the sum of two columns. The first of these two has entries yacyŽ aq1.c x i x j
Ž i s 1, . . . , N .
and the second has entries
a Ž a q 1 . cyŽ aq2.c x i c x j
Ž i s 1, . . . , N . .
Since the determinant of a matrix is linear in each of its columns, M w ¨ x s detŽ ¨ x i x j . can be expressed as a sum of 2 N determinants where each summand has as its jth column one of the two types given above. Since for j / k the two columns col Ž c x 1 c x j , c x 2 c x j , . . . , c x N c x j . and col Ž c x 1 c x k , c x 2 c x k , . . . , c x N c x k .
MONGE ] AMPERE ´ OPERATOR
347
are proportional, any of the 2 N summands which have two different columns of the second type are zero. Therefore, M w ¨ x s det Ž ¨ x i x j . s D q
N
Dj ,
Ý js1
where D is the determinant whose Ž i, j .th entry is yacyŽ aq1.c x i x j and Dj , j s 1, . . . , N, is the determinant obtained from D by replacement of the jth column of D by the column with entries
a Ž a q 1 . cyŽ aq2.c x i c x j ,
i s 1, . . . , N.
Therefore, if we denote the cofactor of the Ž i, j .th entry of the matrix Ž c x x . by Ci j Ž c ., then i j M w ¨ x s Ž y1 . a NcyNŽ aq1. M w c x N
q Ž y1 .
Ny 1
a N Ž a q 1 . cyNŽ aq1.y1
N
N
Ý Ý
Ci j Ž c . c x i c x j . Ž 2.5.
js1 is1
Since the matrix Ž c x i x j . is symmetric, if we write =c s col Ž c x 1 , c x 2 , . . . , c x N . , then, by the formula for the inverse of a matrix, N
N
Ý Ý
Ci j Ž c . c x i c x j s M w c x Ž =c . B Ž c . =c T
Ž 2.6.
is1 js1
where Ž =c .T is the row matrix
cx1 , cx 2 , . . . , cx N and B Ž c . is the inverse of the matrix Ž c x i x j .. We claim that the matrix B Ž c . is negative definite on V. In fact, since Žy1. N M w c x ) 0 on V, no eigenvalue of Ž c x x . can be zero at any point of i j V. Therefore, by continuity of the eigenvalues of Ž c x i x j ., they all have the same sign throughout V and, since the Hessian matrix of c is negative semidefinite at the point in V where c assumes its maximum on V, the
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LAZER AND MCKENNA
eigenvalues of Ž c x i x j . are negative throughout V. Hence, the same is true of the inverse matrix B Ž c ., and the claim is established. From the above we infer the existence of a number d1 ) 0 such that T Ž =c . B Ž c . =c F yd1 5 =c 5 2
on V .
Ž 2.7.
Since Ž c x i x j .Ž x . is negative definite on V, trace Ž c x i x j . s c x 1 x 1 q ??? q c x N x N s D c - 0. Therefore, since DŽyc . ) 0 on V and yc attains its maximum on V at each point of V, it follows from the maximum principle that there exists an open set U containing V such that 5 =c Ž x . 5 G d 2 ) 0,
;x g U.
Ž 2.8.
Let p1 and p 2 be constants such that p1 F pŽ x . F p 2 ; x g V. From Ž2.1. ag s N Ž a q 1. q 1, so if c ) 0 it follows from Ž2.5. and Ž2.6. that M w c¨ x s Ž y1 . c N M w c x a Ncy ag c y Ž a q 1 . Ž =c . B Ž c . =c . Ž 2.9. N
T
Since g ) N, it follows that if c is a large positive number, then for all xgV g
M w c¨ x Ž x . y p Ž x . Ž c¨ Ž x . . F M w c¨ x Ž x . y cg p1 c Ž x .
y ag
- 0. Ž 2.10.
Let c 2 ) 0 be so large that Ž2.4. holds if w 2 Ž x . s c 2 c Ž x .ya . From Ž2.7. and Ž2.9. with c s 1, we have that for x g V, M w ¨ x Ž x . G Ž y1 . M w c x Ž x . a Nc Ž x . N
ya g
c Ž x . q Ž a q 1 . d1 5 =c 5 2 .
Since, by Ž2.8., 5 =c Ž x .5 is bounded below by a positive constant in a neighborhood U of V and c is bounded below by a positive constant on V y U , we infer the existence of d 2 ) 0 such that M w ¨ x Ž x . G d2 c Ž x .
yg a
for all x g V. Therefore, since g ) N, if c ) 0 is small then g
M w c¨ x Ž x . y p Ž x . Ž c¨ Ž x . . G Ž d 2 c N y p 2 cg . c Ž x .
y ag
)0
for all x g V. Therefore, we can choose a constant c1 with 0 - c1 - c2 such that if w 1Ž x . s c1¨ Ž x ., then Ž2.3. holds. This proves Proposition 2.4. Until the final section we shall assume that V is a smooth, bounded, strictly, con¨ ex domain in R N , i.e., there exists w g C`Ž V . such that w s 0 on V, =w / 0 on V, and the matrix Ž wx i x j . is positive definite in V. It is
MONGE ] AMPERE ´ OPERATOR
349
clear that if c s yw, then Žy1. N M w c x ) 0 in V and c Ž x . ) 0 for x g V. THEOREM 2.1. Let V be a smooth, bounded, strictly con¨ ex domain in R N , N G 2, and let p g C`Ž V . with pŽ x . ) 0 for all x g V. If g ) N, then there exists a unique solution u of the boundary blowup problem
½
g
M w u x Ž x . s p Ž x . uŽ x . xgV u Ž x . ª ` as dist Ž x, V . ª 0,
Ž BB .
such that u g C`Ž V .. Moreo¨ er, there exist constants k 1 ) 0 and k 2 such that k 1 d Ž x, V .
ya
F u Ž x . F k 2 d Ž x, V .
ya
,
where a is as in Ž2.1.. Proof. From the remarks preceding the statement of the theorem there exists c satisfying the hypothesis of Proposition 2.4. Let w 1Ž x . and w 2 Ž x . be as in Proposition 2.4. Let sn4`1 be a strictly increasing sequence of positive numbers such that sn ª ` as n ª `, and let V n s x g V N w 1Ž x . - sn4 . Since any level surface of w 1 is a level surface of c , for each n G 1, V n is a strictly convex C`-submanifold of R N of dimension N y 1. Using Lemma 2.3, there exists u n g C`Ž V n . for n G 1 such that
½
M w u n x s p Ž x . ugn on V n < < u n V n s sn s w 1 V n
and u nŽ x . is strictly convex on V n and satisfies u n Ž x . G w1Ž x .
on V n .
Ž 2.11.
From the fact that ; x g Vn,
u n Ž x . s w1Ž x . F w 2 Ž x . , equation Ž2.4., and Lemma 2.1, we see that w2 Ž x . G un Ž x .
on V n .
Ž 2.12.
Clearly, V n ; V nq1
for n ) 1
Ž 2.13.
and Vs
`
D Vn.
ns1
Ž 2.14.
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We claim that u n Ž x . F u nq1 Ž x . ,
; x g Vn.
Ž 2.15.
Indeed, since u n and u nq1 are both positive solutions of M w u x s pŽ x . uŽ x .g on V n , u n is strictly convex in V n and, for x g V n ; V nq1 , u nq 1 Ž x . G w 1 Ž x . s u n Ž x . , the inequality Ž2.15. is a consequence of Lemma 2.1. Let x 0 g V be fixed. If m is so large that x 0 g V m , then for all n G m, we have w 1 Ž x 0 . F u n Ž x 0 . F u nq1 Ž x 0 . F w 2 Ž x 0 . . Therefore, lim n ª` u nŽ x 0 . exists. Since x 0 is arbitrary we see that for x g V, lim u n Ž x . s u Ž x .
nª`
exists and w1Ž x . F u Ž x . F w 2 Ž x . ,
; x g V.
To finish the proof we show that u g C`Ž V . and that M w u x s pŽ x . ug in V. Fix an integer m. For n ) m Vm ; Vn, M w u n x s pŽ x . ugn in V n , u nŽ x . s sn s constant on V n , and for x g V m , a F u nŽ x . F b, where a is the minimum of w 1Ž x . on V m and b is the maximum of w 2 Ž x . on V m . Moreover, for n ) m, 0 - dist Ž V m , V mq1 . F dist Ž V m , V n . - dist Ž V m , V . . Let j G 3 be an integer. Since u n is convex on V n , it follows from Lemma 2.2 that there exists a constant C* such that if n ) m, then < D a u n Ž x . < F C*,
; x g Vm ,
where D a u n is any partial derivative of u n of order F j. It follows from Ascoli’s lemma that there exists a subsequence u n jŽ x .4`1 of u nŽ x .4`mq1 such that if D g is any partial derivative operator of order F j y 1, then the
MONGE ] AMPERE ´ OPERATOR
351
sequence D g u n jŽ x .4`1 converges uniformly on V m . Hence u g C jy1 Ž V m . and for x g V m , M w u x Ž x . s lim M w u n j x Ž x . s lim p Ž x . u n jŽ x . jª`
g
jª`
g
s p Ž x . uŽ x . . Since j G 3 was arbitrary and m G 1 is arbitrary, this argument proves that u g C`Ž V . and M w u x s pŽ x . ug on V. To establish the estimates given by the last statement of the theorem it suffices to note that since c Ž x . ) 0 in V and =c Ž x . / 0 for x g V, there exist constants a1 ) 0 and a2 such that for all x g V, a1 dist Ž x, V . F c Ž x . F a2 dist Ž x, V . .
Ž 2.16.
Therefore, from the form of w 1 and w 2 given in Proposition 2.4 and the fact that w 1Ž x . F uŽ x . F w 2 Ž x . for all x ; V, we infer the existence of constants k 1 ) 0 and k 2 such that k 1 distŽ x, V .y a F uŽ x . F k 2 distŽ x, V .ya . Uniqueness of the solution of ŽBB. can be established using an argument analogous to the one used to prove uniqueness for a boundary blowup problem for the Laplacian in w12x. Given a constant k with 1 - k, let Vk s
½
1 k
5
xNxgV .
Suppose that u1Ž x . and u 2 Ž x . are both solutions of ŽBB.. Let k ) 1 and let w Ž x . s cu1Ž kx . for x g V k . We have that for x g V k , M w w x Ž x . s c N k 2 N p Ž kx . u1 Ž kx .
g g
s c Ny g k 2 N p Ž kx . w Ž x . . If we choose c s cŽ k . so that c Ny g s min xgV k
pŽ x . k
2N
p Ž kx .
,
then cŽ k . ª 1 as k ª 1 and M w w x Ž x . F pŽ x . w Ž x .
g
for all x g V k . We claim that u 2 Ž x . F w Ž x . for all x g V k . Assuming on the contrary that u 2 Ž x 0 . ) w Ž x 0 . for some x 0 g V k , it follows that, since
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LAZER AND MCKENNA
w Ž x . ª ` as x ª V k while u 2 Ž x . is bounded on V k , there exists a subdomain D of V k such that x 0 g D, D ; V k , u 2 Ž x . ) w Ž x . in D, and u 2 Ž x . s w Ž x . for x g D. However, since M w u 2 x s pŽ x . ug2 and M w w x F pŽ x . w g in V k , this gives a contradiction to Lemma 2.1. ŽThe matrix Ž u 2 x i x j . is obviously positive definite in V since its eigenvalues are never zero and must consequently always be positive since u 2 attains a minimum on V.. This contradiction shows that for any k ) 1, u 2 Ž x . F c Ž k . u1 Ž kx . for all x g V k . Since every x g V is contained in V k for k close enough to 1, it follows, by letting k ª 1, that u 2 Ž x . F u1Ž x . on V. Similarly u1Ž x . F u 2 Ž x . on V so solutions of ŽBB. are unique. This proves the theorem. We also have the following negative result. THEOREM 2.2. Let V be a bounded domain in R N, N G 2. Let p be continuous on V, with pŽ x . ) 0 on V. If 0 - g F N, then there cannot exist u g C 2 Ž V . such that uŽ x . G 0 on V, and
½
g
M w u x Ž x . s p Ž x . uŽ x . xgV u Ž x . ª ` as dist Ž x, V . ª 0.
Ž 2.17.
Proof. Let a s Ž a1 , . . . , a N . g V and let N
z Ž x . s exp
Ý k Ž x j y aj . 2 , js1
where k G 1 is a constant to be determined. We have for 1 F i, j F N, z x i x jŽ x . s 4 k 2 Ž x i y a i . Ž x j y a j . z Ž x . ,
if i / j,
and for 1 F i F N, z x i x iŽ x . s Ž 2 k q 4 k 2 Ž x i y a i .
2
. zŽ x. .
Therefore, if Ž z x i x j . denotes the Hessian matrix of z, then
Ž z x i x j . Ž x . s z Ž x . 2 kI q 4 k 2 Ž Ž x i y ai . Ž x j y a j . . , where I is the N = N identity matrix and ŽŽ x i y a i .Ž x j y a j .. is the matrix whose Ž i, j .th entry is Ž x i y a i .Ž x j y a j .. We evaluate M w z xŽ x . s detŽ z x i x j .Ž x . as the product of the eigenvalues of Ž z x i x j .Ž x .. Since any two different columns of the matrix ŽŽ x i y a i .Ž x j y a j .. are proportional, the
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353
rank of this matrix is at most 1. Therefore N y 1 of the eigenvalues of this matrix are equal to 0 and the other eigenvalue is equal to the trace of this matrix which is N
Ý Ž x j y aj . 2 . js1
It follows that N y 1 of the eigenvalues of Ž z x i x j .Ž x . are equal to 2 kz Ž x . and the other eigenvalue is equal to N
zŽ x. 2k q 4k2
Ý Ž x j y aj . 2
.
js1
Therefore, for x g R N, M w z x Ž x . G 2 N k Nz Ž x . . N
Assume, contrary to the assertion of the theorem, that there exists a solution u of the problem Ž2.17.. Let A G 1 be so large that A ) u Ž a. and let w Ž x . s Az Ž x .. We have for x g R N , M w w x Ž x . s A N M w z x Ž x . G 2 NA N k N z Ž x . . N
If p 2 is an upper bound for pŽ x . on V, then M w w x Ž x . G pŽ x . w Ž x .
g
on V ,
Ž 2.18.
provided that for x g V 2 NA N k N z Ž x . G p 2 Ag z Ž x . N
g
or 2 NA Ny g k N z Ž x .
Ny g
G p2
on V .
Since z Ž x . G 1 for all x and g F N, we can choose k so large that Ž2.18. holds. Since w Ž a. s Az Ž a. s A ) uŽ a. and uŽ x . ª ` as dist Ž x, V . ª 0, while w Ž x . is continuous on V, we infer the existence of an open set D such that a g D, D ; V, uŽ x . - w Ž x . for x g D, and uŽ x . s w Ž x . for x g D. However, since for x g V, the function f Ž x, j . s pŽ x . j g is strictly increasing in j on 0 F j - `, M w u xŽ x . s pŽ x . uŽ x .g in D, M w w xŽ x . G pŽ x . w Ž x .g in D, the matrix Ž wx i x j . is positive definite on D Žits
354
LAZER AND MCKENNA
eigenvalues are all positive on D ., and w s u on D, it follows from Lemma 2.1 that w Ž x . F uŽ x . in D. This contradiction shows that the problem Ž2.17. cannot have a solution and the theorem is proved. We conclude this section with a brief treatment of the generalized Bieberbach problem discussed in the introduction. THEOREM 2.3. Let V be a smooth, bounded strictly con¨ ex domain in R N , N G 2. If p g C`Ž V . and pŽ x . ) 0 for all x g V, then there exists a unique u g C`Ž V . such that
½
M w u x Ž x . s p Ž x . e uŽ x . uŽ x . ª `
xgV as dist Ž x, V . ª 0.
Ž GBP .
Moreo¨ er, if dŽ x . s distŽ x, V . then uŽ x . y lnŽ dŽ x .yŽ Nq1. . is bounded on V. Proof. As in the proof of Theorem 2.1, let c be a function in C`Ž V . such that c Ž x . ) 0 and Žy1. N M w c xŽ x . ) 0 for all x g V, and c s 0 and =c / 0 for all x g V. Let ¨ Ž x . s y Ž N q 1 . ln c Ž x . q k
where k is a constant. We have that for 1 F i, j F N, ¨ xi xj s y
Ž N q 1. c
cx i x j q
Ž N q 1. c2
cx i cx j .
Therefore the determinant M w ¨ x is the sum of 2 N determinants where in each summand the jth column has one of two forms, the first form having the entries yŽ N q 1. c x i x jrc , i s 1, . . . , N, and the second form having the entries Ž N q 1. c x i c x jrc 2 , i s 1, . . . , N. Since for k / j the column having entries Ž N q 1. c x i c x jrc 2 , i s 1, . . . , N, is proportional to the column having entries Ž N q 1. c x i x jrc 2 , i s 1, . . . , N, it follows from the argument used in the proof of Theorem 2.1 that M w ¨ x s Ž y1 . Ž N q 1 . cyN M w c x N
N
q M w c x Ž y1 .
Ny 1
N T Ž N q 1 . cyŽ Nq1. Ž =c . B Ž c . =c
where B Ž c . is the inverse of the matrix Ž c x i x j .. As in the proof of Theorem 2.1, B Ž c . is negati¨ e definite at each point of V. We can write M w ¨ x s F Ž c . cyŽ Nq1.
MONGE ] AMPERE ´ OPERATOR
355
where N
N
T
F Ž c . s Ž y1 . Ž N q 1 . M Ž c . c y Ž =c . B Ž c . =c . Since =c / 0 on V and c ) 0 on V, there exist constants b1 ) 0 and b 2 such that b1 F F Ž c . Ž x . F b 2 ,
; x g V.
As in the proof of Theorem 2.1, let p1 ) 0 and p 2 be constants such that p1 F pŽ x . F p 2 for all x g V. Let k 2 be a large positive constant satisfying p1 e k 2 G b 2 , and let k 1 - k 2 be a constant, possibly negative, satisfying p 2 e k 1 F b1 . If for j s 1, 2 wj Ž x . s ¨ Ž x . q k j s y Ž N q 1 . ln c q k j ,
Ž 2.19.
then M w w2 x Ž x . s M ¨ Ž x . s F Ž c . Ž x . c Ž x . F p1 e k 2c Ž x .
y Ž Nq1 .
y Ž Nq1 .
F b2 c Ž x .
y Ž Nq1 .
F p Ž x . ew 2Ž x . ,
and M w w 1 x Ž x . s M ¨ Ž x . G b1 c Ž x .
y Ž Nq1 .
G p Ž x . e w 1Ž x . .
Since w 1Ž x . - w 2 Ž x . on V, a repetition of the argument used in the Proof of Theorem 2.1 shows that there exists a function u g C`Ž V . such that w1Ž x . F u Ž x . F w 2 Ž x .
Ž 2.20.
for all x g V and M w u x Ž x . s p Ž x . e uŽ x . on V. To estimate how fast uŽ x . tends to infinity as dŽ x . ª 0, where dŽ x . s distŽ x, V ., let a1 and a2 be constants such that Ž2.16. holds. We have that for x g V, yln d Ž x . y ln a2 F yln c Ž x . F yln d Ž x . y ln a1 ,
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LAZER AND MCKENNA
and therefore, for x g V, it follows from Ž2.19. that w 2 Ž x . F ln Ž d Ž x .
y Ž Nq1 .
. q c2 , y Ž Nq1 . w 1 Ž x . G ln Ž d Ž x . . q c1 ,
where c1 and c 2 are constants. Hence, from Ž2.20., we see that the difference u Ž x . y ln Ž d Ž x .
y Ž Nq1 .
.
is bounded on V. To prove uniqueness we have an argument similar to the one used to prove the uniqueness part of Theorem 2.1. Again, without loss of generality, we may assume that 0 g V. Assume that u1 and u 2 are both solutions of ŽGBP., let k ) 1, let V k ; V be defined as in the proof of Theorem 2.1, and for x g V k let w Ž x . s u1 Ž kx . y ln c, where c is a positive constant to be determined. For x g V, M w w x Ž x . s k 2 N M w u1 x Ž kx . s k 2 N p Ž kx . e u1Ž k x . s ck 2 N p Ž kx . e wŽ x . . If we let c s c Ž k . s min xgV k
pŽ x . k
2N
p Ž kx .
,
then cŽ k . ª 1
as k ª 1
and M w w x Ž x . F p Ž x . e wŽ x . ,
; x g Vk .
Since w Ž x . ª ` as distŽ x, V k . ª 0 and u 2 Ž x . is bounded on V k , the same argument used in proving the uniqueness part of Theorem 2.1 shows that u 2 Ž x . F w Ž x . s u1 Ž kx . y ln c Ž k . for all x g V k . Fixing x 0 g V and letting k ª 1, we obtain u 2 Ž x 0 . F u1Ž x 0 .. Hence u 2 Ž x . F u1Ž x . for all x g V. similarly, u1Ž x . F u 2 Ž x . on V so u1 ' u 2 . This proves the theorem.
MONGE ] AMPERE ´ OPERATOR
357
3. ANOTHER SINGULAR PROBLEM In this section we apply the methods used to study boundary blowup problems to study the problem
¡M w u x Ž x . s p Ž x . Ž yuŽ x . . ~ uŽ x . - 0 ¢ uŽ x . s 0
yg
xgV in V , x g V,
Ž S.
where g ) 0. By a solution we mean a function u g C 2 Ž V . l C Ž V . such that M w u x s pŽ x .Žyu.g in V and u s 0 on V. THEOREM 3.1. Let V and p be as in Theorem 2.1. If g ) 1, then there exists a unique solution of ŽS.. Moreo¨ er, if
as
Nq1 Nqg
,
Ž 3.1.
then the solution uŽ x . of Ž S . satisfies a
c1 dist Ž x, V . F u Ž x . F c 2 dist Ž x, V .
a
Ž 3.2.
where c1 and c 2 are negati¨ e constants. Proof. Let w g C`Ž V . be a function such that w g C`Ž V ., w Ž x . - 0 in V, w Ž x . s 0 for x g V, and the matrix Ž wx i x j . is positive definite on V. Let a
¨ Ž x . s y Ž yw Ž x . . ,
where a is as in Ž3.1.. We have for 1 F i, j F N, ¨ x i x j s a Ž yw .
ay1
wx i x j y a Ž a y 1 . Ž yw .
a y2
wx i wx j .
Using the same reasoning as that in the proof of Theorem 2.1, we compute that M w ¨ x s a N Ž yw .
N Ž a y1 .
Mwwx
q a N Ž 1 y a . Ž yw .
N Ž a y1 .y1
N
N
Ý Ý
C i j Ž w . wx i wx j ,
is1 js1
where Ci j Ž w . is the cofactor of the i, j entry of M w w x. If Ew w x is the inverse of the matrix Ž wx i x j ., the M w ¨ x s M w w x Ž yw .
N Ž a y1 .y1
T
a N Ž yw . q Ž 1 y a . Ž =w . E Ž w . =w .
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LAZER AND MCKENNA
Let G Ž x . s M w w x a N Ž yw . q 1 Ž 1 y a . Ž =w . E Ž w . =w . T
Since Ž wx i x j . is positive definite on V, EŽ w . is positive definite on V. Therefore, since 1 y a ) 0, yw ) 0 in V, and =w / 0 on V, there exist constants b1 ) 0 and b 2 such that b1 F G Ž x . F b 2 ,
; x g V.
From Ž3.1. we compute that N Ž a y 1. y 1 s yag and, therefore, M w ¨ x s Ž yw .
ag
GŽ x . .
If 0 - p1 F pŽ x . F p 2 on V where p1 and p 2 are constant, then for a positive constant c, it follows from the above that for x g V, M w c¨ x Ž x . y p Ž x . Ž yc¨ Ž x . .
yg
F Ž yw Ž x . .
y ag
Ž b2 c N y p1 cy g .
ag
b1 c N y p 2 cy g .
and M w c¨ x Ž x . y p Ž x . Ž yc¨ Ž x . .
yg
G Ž yw Ž x . .
If c 2 is a small positive constant chosen so that g b 2 c 2N y p1 cy 2 - 0,
and c1 ) c 2 is a large positive constant chosen so that g b1 c1N y p 2 cy 1 ) 0,
and we set w k Ž x . s c k¨ Ž x . for k s 1, 2, then since ¨ Ž x . s y Ž yw Ž x . .
ya
-0
on V, we have that w1Ž x . - w 2 Ž x . - 0 M w w 1 x Ž x . G p Ž x . Ž yw 1 Ž x . .
on V , yg
on V ,
and M w w 2 x Ž x . F p Ž x . Ž yw 2 Ž x . .
yg
on V .
MONGE ] AMPERE ´ OPERATOR
359
Since the function f Ž x, j . s pŽ x .Žyj .yg is strictly increasing in j for y` - j - 0, the arguments used in the proof of Theorem 2.1 yield the existence of a function u defined on V such that w1Ž x . - u Ž x . - w 2 Ž x . ,
x g V,
and M w u x Ž x . s p Ž x . Ž yu Ž x . .
yg
,
x g V.
The estimates Ž3.2. follow from the fact that since w Ž x . - 0 in V and =w / 0 on V, there exist constants a1 ) 0 and a2 such that for x g V, a1 dist Ž x, V . F yw Ž x . F a2 dist Ž x, V . . Uniqueness of the solution of Ž3.1. follows easily from Lemma 2.1. If u1Ž x . and u 2 Ž x . are two solutions of Ž3.1. and there were a point x 0 in V such that u1Ž x 0 . - u 2 Ž x 0 ., then there would exist a domain D such that x 0 g D : V, u1Ž x . - u 2 Ž x . in D, and u1Ž x . s u 2 Ž x . for x g D. Since f Ž x, j . s pŽ x .Žyj .yg is strictly increasing on Žy`, 0., and since the facts that M w u1 x ) 0 on V and u attains its minimum on V at a point in V implies that matrix Ž u1 x i x j . is positive definite on V, we have a contradiction to Lemma 2.1. Therefore, u1 s u 2 and the proof of Theorem 3.1 is complete. 4. REMARKS ON RELAXATION OF REGULARITY OF V Theorems 2.1, 2.3, and 2.4 were proved under the assumption that there exists w g C`Ž V . such that w s 0 on V, =w / 0 on V, and the matrix Ž wx x . is positive definite in V. In this section we briefly indicate how a i j result of Lions w15x and ideas used by the authors in w13x can be used to establish the following improvement: Theorems 2.1, 2.3, and 2.4 remain true if it is only assumed that p g C`Ž V . l C Ž V ., and there are constants p1 ) 0 and p 2 such that p1 F pŽ x . F p 2 for all x g V, and V satisfies the following condition: Ž V .: The region V ; R N is bounded and there exists w g C 2 Ž V . such that w s 0 on V and the matrix Ž wx i x j . is positi¨ e definite on V. We indicate the modification in the proof of Theorem 2.1. First we note that there exist numbers R1 ) 0 and R 2 ) 0 such that if x 0 g V, then there exist balls B1 and B2 , having radii R1 and R 2 respectively, such that B1 ; V ; B2 ,
Ž 4.1.
B1 l V s B2 l V s x 0 4 .
Ž 4.2.
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LAZER AND MCKENNA
We observe that there exists a function u g C`Ž V . l C 1 Ž V . such that u Ž x . - 0 for x g V, u N V s 0, and the matrix Ž ux i x j . is positive definite in V. In fact, Lions w15x has proved the existence of such a u which is a ‘‘first eigenfunction of M ’’ in V, in the sense that M w u x s Žylu . N in V, where l ) 0. Let « n4`1 be a strictly decreasing sequence of positive numbers such that « n ª 0 as n ª ` and for each integer n let V n s x g V N u Ž x . F y« n4 . We have that V n ; V nq1 ; V and V is the union of the V n , for n s 1, 2, . . . . Since the hypotheses of Theorem 2.1 are satisfied for V n and pŽ x . for n G 1, there exists a u n g C`Ž V n . such that u ) 0 in V n , M w u n x s pŽ x . ugn in V n Žg ) N is fixed throughout this argument., and u nŽ x . ª ` as distŽ x, V n . ª 0. Using Lemma 2.1 and an argument used several times before, it follows that if n G 1, then u nq 1 Ž x . F u n Ž x . ,
; x g Vn.
Ž 4.3.
To obtain a lower bound for u nŽ x . for x g V n let x 1 g V n and let x 0 be a point on V closest to x 1. Let B2 be a ball of radius R 2 such that V ; B2 and V l B2 s x 0 4 . The hypotheses of Theorem 2.1 are satisfied for the region B2 and the constant function p 2 . Therefore, there exists z 2 g C`Ž B2 . such that z 2 Ž x . ) 0 in B2 , M w z 2 x s p 2 z 2g
in B2
Ž 4.4.
and z 2 Ž x . G s 2 dist Ž x, B2 .
ya
Ž 4.5.
where a s Ž N q 1.rŽg y N . and s 2 is a positive constant. Since the equation Ž4.4. is invariant under translation, the same constant s 2 will ser¨ e for any such ball B2 regardless of the particular point x 0 g V. For x g V n , M w z 2 x s p 2 z 2g G pŽ x . z 2g. Therefore, since z 2 is bounded on V n while u n becomes infinite on V n , it follows from Lemma 2.1 that u nŽ x . G z 2 Ž x . for x g V n . In particular, u n Ž x 1 . G z 2 Ž x 1 . G s 2 dist Ž x 1 , B2 .
ya
s s 2 dist Ž x 1 , V .
ya
.
Since x 1 g V n was arbitrary and s 2 is independent of x 0 , we see that u n Ž x . G s 2 dist Ž x, V .
ya
n G 1, x g V n .
,
Ž 4.6.
Since u nq 1Ž x . F u nŽ x . for x g V n , the argument used in the proof of Theorem 2.1 shows that there exists u g C`Ž V . such that lim nª` u nŽ x . s uŽ x . on V and g
M w u x Ž x . s p Ž x . uŽ x . ,
x g V.
Ž 4.7.
MONGE ] AMPERE ´ OPERATOR
361
Moreover, from Ž4.6. we have u Ž x . G s 2 dist Ž x, V .
ya
Ž 4.8.
for all x g V. Let x 1 g V and assume that distŽ x 1 , V . - R1. Let x 0 g V be a point in V closest to x 1 and let B1 be a ball of radius R1 such that B1 ; V and B1 l V s x 0 4 . Since the hypotheses of Theorem 2.1 are satisfied for B1 and the constant function p1 , there exists a function z1 , such that M w z1 x s p1 z1g in B1 , z N B1 s `, z1 ) 0 in B1 , and there exists s 1 ) 0 such that for x g B1 , z1 Ž x . F s 1 dist Ž x, B1 .
ya
.
Ž 4.9.
Since the P.D.E. satisfied by z1 is translation invariant, s 1 can be chosen independent of x 0 g V. Since M w z xŽ x . F pŽ x . z1Ž x .g on B1 , z1 N B1 s ` which u is finite on B1 y x 0 4 , and Ž4.7. holds, the argument used in w13, pp. 1003]1004x shows that uŽ x . F z1Ž x . in B1. In particular, u Ž x 1 . F z Ž x 1 . F s 1 dist Ž x 1 , B1 .
ya
s s 1 dist Ž x 1 , V .
ya
.
Since x 1 was any point in V with distŽ x 1 , V . - R1 and s 1 is independent of x 0 g V, it follows that uŽ x . F s 1 distŽ x, V 1 .ya if distŽ x, V . - R1. By making s 1 larger, if necessary, we can ensure that u Ž x . F s 1 dist Ž x, V .
ya
,
; x g V.
Ž 4.10.
Since the uniqueness part of the proof of Theorem 2.1 did not make use of the regularity of V and only required that p g C Ž V . and pŽ x . ) 0 on V, we see that under the hypotheses given at the beginning of this section, there exists a unique solution u of ŽBB. and the estimates Ž4.8. and Ž4.10. hold.
REFERENCES 1. C. Bandle and M. Marcus, ‘‘Large’’ solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J. Analyse Math. 58 Ž1992., 9]24. 2. L. Bieberbach, D u s e u und die automorphen Funktionen, Math. Ann. 77 Ž1916., 173]212. 3. R. Bellman, ‘‘Introduction to Matrix Analysis,’’ McGraw]Hill, New York, 1960. 4. L. Caffarelli, L. Nirenberg, and J. Spruk, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge]Ampere ´ equation, Comm. Pure Appl. Math. 37 Ž1984., 369]402.
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5. S. Y. Cheng and S. T. Yau, On the existence of a complete Kahler metric on non-compact ¨ complex manifolds and regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 Ž1980., 507]544. 6. S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampere ´ equation detŽ 2 ur x i x j . s F Ž x, u., Comm. Pure Appl. Math. 30 Ž1977., 41]68. 7. M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 Ž1977., 193]222. 8. G. Diaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Analysis T. M. A. 20 Ž1993., 97]125. 9. C. Fefferman, Monge]Ampere ´ equations, the Bergman Kernel and the geometry of pseudo convex domains, Ann. of Math. 103 Ž1976., 395]416. 10. V. A. Kondrat’ev and V. A. Nikishkin, Asymptotics near the boundary of a solution of a singular boundary-value problem for a semilinear elliptic equation, Differentsial’nye Ura¨ neniya 26 Ž1990., 465]468; English transl., Differential Equations 26 Ž1990., 345]348. 11. A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 Ž1991., 721]730. 12. A. C. Lazer and P. J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. T. M. A. 21 Ž1993., 327]335. 13. A. C. Lazer and P. J. McKenna, Asymptotic behaviour of solutions of boundary blowup problems, Differential Integral Equations 7 Ž1994., 1001]1019. 14. P. L. Lions, Sur les equations de Monge-Ampere, ´ Arch. Rational Mech. Anal. 89 Ž1985., 93]122. 15. P. L. Lions, Two remarks on Monge-Ampere ´ equations, Ann. Mat. Pura Appl. 142 Ž1985., 263]275. 16. C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in ‘‘Contributions to Analysis ŽA Collection of Papers Dedicated to Lipman Bers.,’’ pp. 245]272, Academic Press, New York, 1974. 17. A. V. Pogorelov, ‘‘The Multidimensional Minkowski Problem,’’ Wiley, New York, 1978. 18. K. Tso, On a real Monge-Ampere ´ functional, In¨ ent. Math. 101 Ž1990., 425]448.
197 341]362 Ž1996.
0024
On Singular Boundary Value Problems for the Monge]Ampere ´ Operator A. C. Lazer Department of Mathematics and Computer Science, Uni¨ ersity of Miami, Coral Gables, Florida 33124
and P. J. McKenna Department of Mathematics, Uni¨ ersity of Connecticut, Storrs, Connecticut 06269 Submitted by Jack K. Hale Received February 28, 1995
We consider different types of singular boundary value problems for the Monge]Ampere ´ operator. The approach is based on existing regularity theory and a subsolution]supersolution method. Nonexistence and uniqueness results are also given. Q 1996 Academic Press, Inc.
1. INTRODUCTION AND SUMMARY We consider singular boundary value problems for the P. D. E. M w u x Ž x . s f Ž x, u . ,
x g V,
Ž 1.1.
where V is a smooth, strictly convex domain in R N , N G 2, and M w u x s detŽ u x i x j ., 1 F i, j F N is the so-called Monge]Ampere ´ operator. Here x s Ž x 1 , . . . , x N . is a generic point in R N . We consider two types of singular problems for Ž1.. In the first type, which we call boundary blowup problems, we look for solutions which are smooth and satisfy Ž1.1. in V and tend to infinity as x approaches V, the boundary of V. Problems of this type in which the Monge]Ampere ´ operator is replaced by the Laplacian have been the subject of several investigations in recent years}see for example w1, 10, 12, 13, and 16x. The first paper concerning boundary 341 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
342
LAZER AND MCKENNA
blowup problems for the Laplace operator appears to be a 1916 paper by Bieberbach w2x. Bieberbach considers the problem Du s eu,
in V ,
Ž 1.2.
where V is a bounded domain in R 2 with C 2 boundary, and shows that there is a unique solution such that the difference u Ž x . y ln Ž d Ž x .
y2
.
is bounded in V, where d Ž x . s distance Ž x, V . . Extensions of this result are given in w12x and w13x. We shall show that if V is a smooth, strictly convex, bounded domain in R N, N G 2, and f Ž x, j . s p Ž x . e j , where p is a smooth function which is positive on V, then there exists a unique solution u of Ž1.1. in C`Ž V . such that uŽ x . ª ` as x ª `. Moreover, it is shown that the difference u Ž x . y ln Ž d Ž x .
y Ž Nq1 .
.
is bounded in V. In work related to a problem of Fefferman w9x, Cheng and Yau w5x considered a similar blowup problem in which M is replaced by the complex Monge]Ampere ´ operator. Another type of boundary blowup problem for the Laplacian which seems to be well understood is the problem
½
D u s p Ž x . ug uŽ x . ª `
in V as d Ž x . ª `.
Ž 1.3.
where g ) 1. It follows from w1, 10, and 13x that if V ; R N is sufficiently regular and p is positive and smooth on V, then there exists a unique solution of Ž1.3. and for x g V, c1 Ž d Ž x . .
y2r Ž g y1 .
F u Ž x . F c2 Ž d Ž x . .
y2r Ž g y1 .
,
where c1 and c 2 are positive constants. Results of this type are extended to the case where D is replaced by the p-Laplacian in w8x. We shall show that if V is a smooth, strictly convex, bounded domain in R N , N G 2, f Ž x, j . s pŽ x . j g , where g ) N, then there exists a unique
MONGE ] AMPERE ´ OPERATOR
343
solution of Ž1.1. in C`Ž V . such that uŽ x . ª ` as distŽ x, V . ª 0. Moreover, there exist positive constants k 1 and k 2 such that for x g V, k1 d Ž x .
ya
F uŽ x . F k2 dŽ x .
ya
where
as
Nq1
gyN
.
We also show that if 0 - g F N and f Ž x, j . s pŽ x . j g , where p is as above, then there does not exist a solution of Ž1.1. in V such that uŽ x . ª ` as dŽ x . ª 0. Our tools in studying the two above-mentioned boundary blowup problems consist of a simple comparison result for the Monge]Ampere ´ operator ŽLemma 2.1 below., a subsolution]supersolution method, motivated in part by work of Lions w14x, existence theory for smooth solutions for Monge]Ampere ´ equations developed by Caffarelli et al. in w4x, and regularity theory due to Pogorelov w17x in a form used by Tso in w18x. A second type of singular problem for Ž1.1. is concerned with finding a solution u of Ž1.1., which is continuous on V and smooth on V, such that u s 0 on V and f Ž x, j . becomes infinite as j ª 0. Problems of this type have also been considered when M w u x is replaced by D u. It is known Žsee w7x and w11x. that if pŽ x . is positive and smooth on V and if V ; R N , N ) 1, is bounded and sufficiently regular, then for g ) 1 there exists a unique u g C Ž V . l C 2 Ž V . such that
½
yg
Ž D u. Ž x . q p Ž x . uŽ x . s 0 uŽ x . s 0
in V for x g V .
Moreover, there are positive constants b1 and b 2 such that for x g V, b1 d Ž x .
2r Ž1q g .
F u Ž x . F b2 d Ž x .
2r Ž1q g .
.
In the final section we indicate briefly how the same subsolution] supersolution method used to study boundary blowup problems can be used to establish the existence of a unique function u g C`Ž V . l C Ž V . such that
¡M w u x Ž x . s p Ž x . Ž yuŽ x . . ~ uŽ x . - 0 ¢ uŽ x . s 0
yg
in V in V x g V
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LAZER AND MCKENNA
provided that g ) 1; V is smooth, strictly convex, and bounded; pŽ x . ) 0 on V; and p is smooth. We show that there are negative constants c1 and c 2 such that c1 dŽ x . a F uŽ x . F c 2 dŽ x . a in V where
as
Nq1 Nqg
.
We remark that a problem of this type has been considered by Cheng and Yau w6x. They show that if V in R N is convex and bounded but not necessarily strictly convex then there exists a unique u g C`Ž V . l C Ž V . which is negative on V such that
¡ 1 ~M w u x Ž x . s ž y uŽ x . / ¢ uŽ x . s 0
Nq 2
in V on V .
For clarity of presentation we first prove the existence of solutions of various singular problems under the assumption that V is a strictly convex bounded domain with V of class C`. In the final section using a result of Lions w15x and ideas of the authors w13x, we show that it is enough that V is of class C 2 .
2. BOUNDARY BLOWUP PROBLEMS Our first main tool in this section, whose proof is given for completeness, is the following elementary comparison result. LEMMA 2.1. Let V be a bounded domain in R N, N G 2, and let u k g C 2 Ž V . l C Ž V . for k s 1, 2. Let f Ž x, j . be defined for x g V and j in some inter¨ al containing the ranges of u1 and u 2 and assume that f Ž x, j . is strictly increasing in j for all x g V. If Ži. the matrix Ž u1 x x . is positi¨ e definite in V, i j Žii. M w u1 xŽ x . G f Ž x, u1Ž x .., ; x g V, Žiii. M w u 2 xŽ x . F f Ž x, u 2 Ž x .., ; x g V, Živ. u1Ž x . F u 2 Ž x ., ; x g V,then u1 Ž x . F u 2 Ž x . ,
; x g V.
Proof. Assuming the contrary, there exists x 0 g V such that u1Ž x 0 . ) u 2 Ž x 0 . and u1 y u 2 assumed its maximum on V at x 0 . Therefore, the matrix
Ž u 2 x x Ž x 0 . . y Ž u1 x x Ž x 0 . . i j
i j
MONGE ] AMPERE ´ OPERATOR
345
is positive semidefinite at x 0 . From the variational characterization of the eigenvalues of a symmetric matrix w2, p. 115x it follows that if a 1 G a 2 G ??? G a N are the eigenvalues of Ž u 2 x i x jŽ x 0 .. and b 1 G b 2 G ??? G bN are the eigenvalues of Ž u1 x i x jŽ x 0 .., then a k G b k , 1 F k F N. Since the matrix Ž u1 x i x jŽ x 0 .. is positive definite, b k ) 0 for k s 1, . . . , N. Therefore, we conclude that f Ž x 0 , u2 Ž x 0 . . G M w u2 x Ž x 0 . s a 1 ??? a N G b 1 . . . bN s M w u1 x Ž x 0 . G f Ž x 0 , u1 Ž x 0 . . which is a contradiction to the assumption that f Ž x, j . is strictly increasing in j . This proves the lemma. We shall also need the following known result concerning interior estimates for derivatives of smooth solutions of semilinear Monge]Ampere ´ equations. LEMMA 2.2. Let V be a bounded domain in R N, N G 2, with V g C`. Let f g C`Ž V = Ž0, `.. with f Ž x, j . ) 0 for Ž x, j . g V = Ž0, `.. Let u g C`Ž V . be a solution of the Dirichlet problem
Ž D.
½
M w u x Ž x . s f Ž x, u . u Ž x . s c s constant
xgV x g V,
with 0 - uŽ x . - c in V. Let V9 be a subdomain of V with V9 ; V and assume that a F uŽ x . F b for x g V9 and let k G 1 be an integer. There exists a constant C* which depends only on k, a, b, bounds for the deri¨ ati¨ es of f Ž x, j . for Ž x, j . g V9 = w a, b x, and distŽ V9, V . such that 5 u 5 C k Ž V 9. F C*. This is only a slight variation of Proposition 2.4Žii. of w18x. If g Ž x, j . ' f Ž x, c y j . for 0 F j F c y m, where m - c is the minimum of u on V, g is extended smoothly to V = w0, `., and we set uŽ x . s c q ¨ Ž x ., then ¨ Ž x . F 0 on V and M w ¨ xŽ x . s g Ž x, y¨ Ž x ... A C k-estimate of ¨ on w a y c, b y c x = V9 gives a C k-estimate of u on w a, b x = V9. Therefore, Lemma 2.2 is a consequence of Proposition 2.4Žii. of w18x. Proposition 2.4Žii. of w18x follows from a result of Pogorelov w17x. Finally, we make use of the following known existence result. LEMMA 2.3. Let V be a strictly con¨ ex, bounded domain in R N, N G 2, with V g C` . Let f Ž x, j . be a positi¨ e C` function on Ž0, c x = V, where
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LAZER AND MCKENNA
c ) 0 is a constant. If there exists a C 2-function u#, which is con¨ ex on V, such that
½
M w u# x Ž x . G f Ž x, u# Ž x . . u# Ž x . s c
on V , on V ,
then there exists a solution u of Ž D . with u g C`Ž V . and u strictly con¨ ex. Moreo¨ er, uŽ x . G u#Ž x . on V. This is a special case of Theorem 7.1 of w4x in which f may depend on =u and u may be equal to a smooth function on V. PROPOSITION 2.4. Let V ; R N, N G 2, be a smooth, bounded con¨ ex domain. Assume c g C`Ž V ., c Ž x . ) 0 on V, c N V s 0, and Žy1. N M w c xŽ x . ) 0 on V. Let p g C`Ž V . with pŽ x . ) 0 on V. If g ) N,
as
Nq1
gyN
¨ Ž x. s c Ž x.
ya
,
Ž 2.1. x g V,
,
Ž 2.2.
then there exist constants c1 and c 2 with 0 - c1 - c 2 such that if w k Ž x . s c k¨ Ž x ., then M w w1 x Ž x . ) p Ž x . w1Ž x .
g
on V ,
Ž 2.3.
M w w2 x Ž x . - p Ž x . w2 Ž x .
g
on V .
Ž 2.4.
Proof. The jth column of the matrix Ž ¨ x i x j . is the sum of two columns. The first of these two has entries yacyŽ aq1.c x i x j
Ž i s 1, . . . , N .
and the second has entries
a Ž a q 1 . cyŽ aq2.c x i c x j
Ž i s 1, . . . , N . .
Since the determinant of a matrix is linear in each of its columns, M w ¨ x s detŽ ¨ x i x j . can be expressed as a sum of 2 N determinants where each summand has as its jth column one of the two types given above. Since for j / k the two columns col Ž c x 1 c x j , c x 2 c x j , . . . , c x N c x j . and col Ž c x 1 c x k , c x 2 c x k , . . . , c x N c x k .
MONGE ] AMPERE ´ OPERATOR
347
are proportional, any of the 2 N summands which have two different columns of the second type are zero. Therefore, M w ¨ x s det Ž ¨ x i x j . s D q
N
Dj ,
Ý js1
where D is the determinant whose Ž i, j .th entry is yacyŽ aq1.c x i x j and Dj , j s 1, . . . , N, is the determinant obtained from D by replacement of the jth column of D by the column with entries
a Ž a q 1 . cyŽ aq2.c x i c x j ,
i s 1, . . . , N.
Therefore, if we denote the cofactor of the Ž i, j .th entry of the matrix Ž c x x . by Ci j Ž c ., then i j M w ¨ x s Ž y1 . a NcyNŽ aq1. M w c x N
q Ž y1 .
Ny 1
a N Ž a q 1 . cyNŽ aq1.y1
N
N
Ý Ý
Ci j Ž c . c x i c x j . Ž 2.5.
js1 is1
Since the matrix Ž c x i x j . is symmetric, if we write =c s col Ž c x 1 , c x 2 , . . . , c x N . , then, by the formula for the inverse of a matrix, N
N
Ý Ý
Ci j Ž c . c x i c x j s M w c x Ž =c . B Ž c . =c T
Ž 2.6.
is1 js1
where Ž =c .T is the row matrix
cx1 , cx 2 , . . . , cx N and B Ž c . is the inverse of the matrix Ž c x i x j .. We claim that the matrix B Ž c . is negative definite on V. In fact, since Žy1. N M w c x ) 0 on V, no eigenvalue of Ž c x x . can be zero at any point of i j V. Therefore, by continuity of the eigenvalues of Ž c x i x j ., they all have the same sign throughout V and, since the Hessian matrix of c is negative semidefinite at the point in V where c assumes its maximum on V, the
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LAZER AND MCKENNA
eigenvalues of Ž c x i x j . are negative throughout V. Hence, the same is true of the inverse matrix B Ž c ., and the claim is established. From the above we infer the existence of a number d1 ) 0 such that T Ž =c . B Ž c . =c F yd1 5 =c 5 2
on V .
Ž 2.7.
Since Ž c x i x j .Ž x . is negative definite on V, trace Ž c x i x j . s c x 1 x 1 q ??? q c x N x N s D c - 0. Therefore, since DŽyc . ) 0 on V and yc attains its maximum on V at each point of V, it follows from the maximum principle that there exists an open set U containing V such that 5 =c Ž x . 5 G d 2 ) 0,
;x g U.
Ž 2.8.
Let p1 and p 2 be constants such that p1 F pŽ x . F p 2 ; x g V. From Ž2.1. ag s N Ž a q 1. q 1, so if c ) 0 it follows from Ž2.5. and Ž2.6. that M w c¨ x s Ž y1 . c N M w c x a Ncy ag c y Ž a q 1 . Ž =c . B Ž c . =c . Ž 2.9. N
T
Since g ) N, it follows that if c is a large positive number, then for all xgV g
M w c¨ x Ž x . y p Ž x . Ž c¨ Ž x . . F M w c¨ x Ž x . y cg p1 c Ž x .
y ag
- 0. Ž 2.10.
Let c 2 ) 0 be so large that Ž2.4. holds if w 2 Ž x . s c 2 c Ž x .ya . From Ž2.7. and Ž2.9. with c s 1, we have that for x g V, M w ¨ x Ž x . G Ž y1 . M w c x Ž x . a Nc Ž x . N
ya g
c Ž x . q Ž a q 1 . d1 5 =c 5 2 .
Since, by Ž2.8., 5 =c Ž x .5 is bounded below by a positive constant in a neighborhood U of V and c is bounded below by a positive constant on V y U , we infer the existence of d 2 ) 0 such that M w ¨ x Ž x . G d2 c Ž x .
yg a
for all x g V. Therefore, since g ) N, if c ) 0 is small then g
M w c¨ x Ž x . y p Ž x . Ž c¨ Ž x . . G Ž d 2 c N y p 2 cg . c Ž x .
y ag
)0
for all x g V. Therefore, we can choose a constant c1 with 0 - c1 - c2 such that if w 1Ž x . s c1¨ Ž x ., then Ž2.3. holds. This proves Proposition 2.4. Until the final section we shall assume that V is a smooth, bounded, strictly, con¨ ex domain in R N , i.e., there exists w g C`Ž V . such that w s 0 on V, =w / 0 on V, and the matrix Ž wx i x j . is positive definite in V. It is
MONGE ] AMPERE ´ OPERATOR
349
clear that if c s yw, then Žy1. N M w c x ) 0 in V and c Ž x . ) 0 for x g V. THEOREM 2.1. Let V be a smooth, bounded, strictly con¨ ex domain in R N , N G 2, and let p g C`Ž V . with pŽ x . ) 0 for all x g V. If g ) N, then there exists a unique solution u of the boundary blowup problem
½
g
M w u x Ž x . s p Ž x . uŽ x . xgV u Ž x . ª ` as dist Ž x, V . ª 0,
Ž BB .
such that u g C`Ž V .. Moreo¨ er, there exist constants k 1 ) 0 and k 2 such that k 1 d Ž x, V .
ya
F u Ž x . F k 2 d Ž x, V .
ya
,
where a is as in Ž2.1.. Proof. From the remarks preceding the statement of the theorem there exists c satisfying the hypothesis of Proposition 2.4. Let w 1Ž x . and w 2 Ž x . be as in Proposition 2.4. Let sn4`1 be a strictly increasing sequence of positive numbers such that sn ª ` as n ª `, and let V n s x g V N w 1Ž x . - sn4 . Since any level surface of w 1 is a level surface of c , for each n G 1, V n is a strictly convex C`-submanifold of R N of dimension N y 1. Using Lemma 2.3, there exists u n g C`Ž V n . for n G 1 such that
½
M w u n x s p Ž x . ugn on V n < < u n V n s sn s w 1 V n
and u nŽ x . is strictly convex on V n and satisfies u n Ž x . G w1Ž x .
on V n .
Ž 2.11.
From the fact that ; x g Vn,
u n Ž x . s w1Ž x . F w 2 Ž x . , equation Ž2.4., and Lemma 2.1, we see that w2 Ž x . G un Ž x .
on V n .
Ž 2.12.
Clearly, V n ; V nq1
for n ) 1
Ž 2.13.
and Vs
`
D Vn.
ns1
Ž 2.14.
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LAZER AND MCKENNA
We claim that u n Ž x . F u nq1 Ž x . ,
; x g Vn.
Ž 2.15.
Indeed, since u n and u nq1 are both positive solutions of M w u x s pŽ x . uŽ x .g on V n , u n is strictly convex in V n and, for x g V n ; V nq1 , u nq 1 Ž x . G w 1 Ž x . s u n Ž x . , the inequality Ž2.15. is a consequence of Lemma 2.1. Let x 0 g V be fixed. If m is so large that x 0 g V m , then for all n G m, we have w 1 Ž x 0 . F u n Ž x 0 . F u nq1 Ž x 0 . F w 2 Ž x 0 . . Therefore, lim n ª` u nŽ x 0 . exists. Since x 0 is arbitrary we see that for x g V, lim u n Ž x . s u Ž x .
nª`
exists and w1Ž x . F u Ž x . F w 2 Ž x . ,
; x g V.
To finish the proof we show that u g C`Ž V . and that M w u x s pŽ x . ug in V. Fix an integer m. For n ) m Vm ; Vn, M w u n x s pŽ x . ugn in V n , u nŽ x . s sn s constant on V n , and for x g V m , a F u nŽ x . F b, where a is the minimum of w 1Ž x . on V m and b is the maximum of w 2 Ž x . on V m . Moreover, for n ) m, 0 - dist Ž V m , V mq1 . F dist Ž V m , V n . - dist Ž V m , V . . Let j G 3 be an integer. Since u n is convex on V n , it follows from Lemma 2.2 that there exists a constant C* such that if n ) m, then < D a u n Ž x . < F C*,
; x g Vm ,
where D a u n is any partial derivative of u n of order F j. It follows from Ascoli’s lemma that there exists a subsequence u n jŽ x .4`1 of u nŽ x .4`mq1 such that if D g is any partial derivative operator of order F j y 1, then the
MONGE ] AMPERE ´ OPERATOR
351
sequence D g u n jŽ x .4`1 converges uniformly on V m . Hence u g C jy1 Ž V m . and for x g V m , M w u x Ž x . s lim M w u n j x Ž x . s lim p Ž x . u n jŽ x . jª`
g
jª`
g
s p Ž x . uŽ x . . Since j G 3 was arbitrary and m G 1 is arbitrary, this argument proves that u g C`Ž V . and M w u x s pŽ x . ug on V. To establish the estimates given by the last statement of the theorem it suffices to note that since c Ž x . ) 0 in V and =c Ž x . / 0 for x g V, there exist constants a1 ) 0 and a2 such that for all x g V, a1 dist Ž x, V . F c Ž x . F a2 dist Ž x, V . .
Ž 2.16.
Therefore, from the form of w 1 and w 2 given in Proposition 2.4 and the fact that w 1Ž x . F uŽ x . F w 2 Ž x . for all x ; V, we infer the existence of constants k 1 ) 0 and k 2 such that k 1 distŽ x, V .y a F uŽ x . F k 2 distŽ x, V .ya . Uniqueness of the solution of ŽBB. can be established using an argument analogous to the one used to prove uniqueness for a boundary blowup problem for the Laplacian in w12x. Given a constant k with 1 - k, let Vk s
½
1 k
5
xNxgV .
Suppose that u1Ž x . and u 2 Ž x . are both solutions of ŽBB.. Let k ) 1 and let w Ž x . s cu1Ž kx . for x g V k . We have that for x g V k , M w w x Ž x . s c N k 2 N p Ž kx . u1 Ž kx .
g g
s c Ny g k 2 N p Ž kx . w Ž x . . If we choose c s cŽ k . so that c Ny g s min xgV k
pŽ x . k
2N
p Ž kx .
,
then cŽ k . ª 1 as k ª 1 and M w w x Ž x . F pŽ x . w Ž x .
g
for all x g V k . We claim that u 2 Ž x . F w Ž x . for all x g V k . Assuming on the contrary that u 2 Ž x 0 . ) w Ž x 0 . for some x 0 g V k , it follows that, since
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LAZER AND MCKENNA
w Ž x . ª ` as x ª V k while u 2 Ž x . is bounded on V k , there exists a subdomain D of V k such that x 0 g D, D ; V k , u 2 Ž x . ) w Ž x . in D, and u 2 Ž x . s w Ž x . for x g D. However, since M w u 2 x s pŽ x . ug2 and M w w x F pŽ x . w g in V k , this gives a contradiction to Lemma 2.1. ŽThe matrix Ž u 2 x i x j . is obviously positive definite in V since its eigenvalues are never zero and must consequently always be positive since u 2 attains a minimum on V.. This contradiction shows that for any k ) 1, u 2 Ž x . F c Ž k . u1 Ž kx . for all x g V k . Since every x g V is contained in V k for k close enough to 1, it follows, by letting k ª 1, that u 2 Ž x . F u1Ž x . on V. Similarly u1Ž x . F u 2 Ž x . on V so solutions of ŽBB. are unique. This proves the theorem. We also have the following negative result. THEOREM 2.2. Let V be a bounded domain in R N, N G 2. Let p be continuous on V, with pŽ x . ) 0 on V. If 0 - g F N, then there cannot exist u g C 2 Ž V . such that uŽ x . G 0 on V, and
½
g
M w u x Ž x . s p Ž x . uŽ x . xgV u Ž x . ª ` as dist Ž x, V . ª 0.
Ž 2.17.
Proof. Let a s Ž a1 , . . . , a N . g V and let N
z Ž x . s exp
Ý k Ž x j y aj . 2 , js1
where k G 1 is a constant to be determined. We have for 1 F i, j F N, z x i x jŽ x . s 4 k 2 Ž x i y a i . Ž x j y a j . z Ž x . ,
if i / j,
and for 1 F i F N, z x i x iŽ x . s Ž 2 k q 4 k 2 Ž x i y a i .
2
. zŽ x. .
Therefore, if Ž z x i x j . denotes the Hessian matrix of z, then
Ž z x i x j . Ž x . s z Ž x . 2 kI q 4 k 2 Ž Ž x i y ai . Ž x j y a j . . , where I is the N = N identity matrix and ŽŽ x i y a i .Ž x j y a j .. is the matrix whose Ž i, j .th entry is Ž x i y a i .Ž x j y a j .. We evaluate M w z xŽ x . s detŽ z x i x j .Ž x . as the product of the eigenvalues of Ž z x i x j .Ž x .. Since any two different columns of the matrix ŽŽ x i y a i .Ž x j y a j .. are proportional, the
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353
rank of this matrix is at most 1. Therefore N y 1 of the eigenvalues of this matrix are equal to 0 and the other eigenvalue is equal to the trace of this matrix which is N
Ý Ž x j y aj . 2 . js1
It follows that N y 1 of the eigenvalues of Ž z x i x j .Ž x . are equal to 2 kz Ž x . and the other eigenvalue is equal to N
zŽ x. 2k q 4k2
Ý Ž x j y aj . 2
.
js1
Therefore, for x g R N, M w z x Ž x . G 2 N k Nz Ž x . . N
Assume, contrary to the assertion of the theorem, that there exists a solution u of the problem Ž2.17.. Let A G 1 be so large that A ) u Ž a. and let w Ž x . s Az Ž x .. We have for x g R N , M w w x Ž x . s A N M w z x Ž x . G 2 NA N k N z Ž x . . N
If p 2 is an upper bound for pŽ x . on V, then M w w x Ž x . G pŽ x . w Ž x .
g
on V ,
Ž 2.18.
provided that for x g V 2 NA N k N z Ž x . G p 2 Ag z Ž x . N
g
or 2 NA Ny g k N z Ž x .
Ny g
G p2
on V .
Since z Ž x . G 1 for all x and g F N, we can choose k so large that Ž2.18. holds. Since w Ž a. s Az Ž a. s A ) uŽ a. and uŽ x . ª ` as dist Ž x, V . ª 0, while w Ž x . is continuous on V, we infer the existence of an open set D such that a g D, D ; V, uŽ x . - w Ž x . for x g D, and uŽ x . s w Ž x . for x g D. However, since for x g V, the function f Ž x, j . s pŽ x . j g is strictly increasing in j on 0 F j - `, M w u xŽ x . s pŽ x . uŽ x .g in D, M w w xŽ x . G pŽ x . w Ž x .g in D, the matrix Ž wx i x j . is positive definite on D Žits
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LAZER AND MCKENNA
eigenvalues are all positive on D ., and w s u on D, it follows from Lemma 2.1 that w Ž x . F uŽ x . in D. This contradiction shows that the problem Ž2.17. cannot have a solution and the theorem is proved. We conclude this section with a brief treatment of the generalized Bieberbach problem discussed in the introduction. THEOREM 2.3. Let V be a smooth, bounded strictly con¨ ex domain in R N , N G 2. If p g C`Ž V . and pŽ x . ) 0 for all x g V, then there exists a unique u g C`Ž V . such that
½
M w u x Ž x . s p Ž x . e uŽ x . uŽ x . ª `
xgV as dist Ž x, V . ª 0.
Ž GBP .
Moreo¨ er, if dŽ x . s distŽ x, V . then uŽ x . y lnŽ dŽ x .yŽ Nq1. . is bounded on V. Proof. As in the proof of Theorem 2.1, let c be a function in C`Ž V . such that c Ž x . ) 0 and Žy1. N M w c xŽ x . ) 0 for all x g V, and c s 0 and =c / 0 for all x g V. Let ¨ Ž x . s y Ž N q 1 . ln c Ž x . q k
where k is a constant. We have that for 1 F i, j F N, ¨ xi xj s y
Ž N q 1. c
cx i x j q
Ž N q 1. c2
cx i cx j .
Therefore the determinant M w ¨ x is the sum of 2 N determinants where in each summand the jth column has one of two forms, the first form having the entries yŽ N q 1. c x i x jrc , i s 1, . . . , N, and the second form having the entries Ž N q 1. c x i c x jrc 2 , i s 1, . . . , N. Since for k / j the column having entries Ž N q 1. c x i c x jrc 2 , i s 1, . . . , N, is proportional to the column having entries Ž N q 1. c x i x jrc 2 , i s 1, . . . , N, it follows from the argument used in the proof of Theorem 2.1 that M w ¨ x s Ž y1 . Ž N q 1 . cyN M w c x N
N
q M w c x Ž y1 .
Ny 1
N T Ž N q 1 . cyŽ Nq1. Ž =c . B Ž c . =c
where B Ž c . is the inverse of the matrix Ž c x i x j .. As in the proof of Theorem 2.1, B Ž c . is negati¨ e definite at each point of V. We can write M w ¨ x s F Ž c . cyŽ Nq1.
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355
where N
N
T
F Ž c . s Ž y1 . Ž N q 1 . M Ž c . c y Ž =c . B Ž c . =c . Since =c / 0 on V and c ) 0 on V, there exist constants b1 ) 0 and b 2 such that b1 F F Ž c . Ž x . F b 2 ,
; x g V.
As in the proof of Theorem 2.1, let p1 ) 0 and p 2 be constants such that p1 F pŽ x . F p 2 for all x g V. Let k 2 be a large positive constant satisfying p1 e k 2 G b 2 , and let k 1 - k 2 be a constant, possibly negative, satisfying p 2 e k 1 F b1 . If for j s 1, 2 wj Ž x . s ¨ Ž x . q k j s y Ž N q 1 . ln c q k j ,
Ž 2.19.
then M w w2 x Ž x . s M ¨ Ž x . s F Ž c . Ž x . c Ž x . F p1 e k 2c Ž x .
y Ž Nq1 .
y Ž Nq1 .
F b2 c Ž x .
y Ž Nq1 .
F p Ž x . ew 2Ž x . ,
and M w w 1 x Ž x . s M ¨ Ž x . G b1 c Ž x .
y Ž Nq1 .
G p Ž x . e w 1Ž x . .
Since w 1Ž x . - w 2 Ž x . on V, a repetition of the argument used in the Proof of Theorem 2.1 shows that there exists a function u g C`Ž V . such that w1Ž x . F u Ž x . F w 2 Ž x .
Ž 2.20.
for all x g V and M w u x Ž x . s p Ž x . e uŽ x . on V. To estimate how fast uŽ x . tends to infinity as dŽ x . ª 0, where dŽ x . s distŽ x, V ., let a1 and a2 be constants such that Ž2.16. holds. We have that for x g V, yln d Ž x . y ln a2 F yln c Ž x . F yln d Ž x . y ln a1 ,
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LAZER AND MCKENNA
and therefore, for x g V, it follows from Ž2.19. that w 2 Ž x . F ln Ž d Ž x .
y Ž Nq1 .
. q c2 , y Ž Nq1 . w 1 Ž x . G ln Ž d Ž x . . q c1 ,
where c1 and c 2 are constants. Hence, from Ž2.20., we see that the difference u Ž x . y ln Ž d Ž x .
y Ž Nq1 .
.
is bounded on V. To prove uniqueness we have an argument similar to the one used to prove the uniqueness part of Theorem 2.1. Again, without loss of generality, we may assume that 0 g V. Assume that u1 and u 2 are both solutions of ŽGBP., let k ) 1, let V k ; V be defined as in the proof of Theorem 2.1, and for x g V k let w Ž x . s u1 Ž kx . y ln c, where c is a positive constant to be determined. For x g V, M w w x Ž x . s k 2 N M w u1 x Ž kx . s k 2 N p Ž kx . e u1Ž k x . s ck 2 N p Ž kx . e wŽ x . . If we let c s c Ž k . s min xgV k
pŽ x . k
2N
p Ž kx .
,
then cŽ k . ª 1
as k ª 1
and M w w x Ž x . F p Ž x . e wŽ x . ,
; x g Vk .
Since w Ž x . ª ` as distŽ x, V k . ª 0 and u 2 Ž x . is bounded on V k , the same argument used in proving the uniqueness part of Theorem 2.1 shows that u 2 Ž x . F w Ž x . s u1 Ž kx . y ln c Ž k . for all x g V k . Fixing x 0 g V and letting k ª 1, we obtain u 2 Ž x 0 . F u1Ž x 0 .. Hence u 2 Ž x . F u1Ž x . for all x g V. similarly, u1Ž x . F u 2 Ž x . on V so u1 ' u 2 . This proves the theorem.
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357
3. ANOTHER SINGULAR PROBLEM In this section we apply the methods used to study boundary blowup problems to study the problem
¡M w u x Ž x . s p Ž x . Ž yuŽ x . . ~ uŽ x . - 0 ¢ uŽ x . s 0
yg
xgV in V , x g V,
Ž S.
where g ) 0. By a solution we mean a function u g C 2 Ž V . l C Ž V . such that M w u x s pŽ x .Žyu.g in V and u s 0 on V. THEOREM 3.1. Let V and p be as in Theorem 2.1. If g ) 1, then there exists a unique solution of ŽS.. Moreo¨ er, if
as
Nq1 Nqg
,
Ž 3.1.
then the solution uŽ x . of Ž S . satisfies a
c1 dist Ž x, V . F u Ž x . F c 2 dist Ž x, V .
a
Ž 3.2.
where c1 and c 2 are negati¨ e constants. Proof. Let w g C`Ž V . be a function such that w g C`Ž V ., w Ž x . - 0 in V, w Ž x . s 0 for x g V, and the matrix Ž wx i x j . is positive definite on V. Let a
¨ Ž x . s y Ž yw Ž x . . ,
where a is as in Ž3.1.. We have for 1 F i, j F N, ¨ x i x j s a Ž yw .
ay1
wx i x j y a Ž a y 1 . Ž yw .
a y2
wx i wx j .
Using the same reasoning as that in the proof of Theorem 2.1, we compute that M w ¨ x s a N Ž yw .
N Ž a y1 .
Mwwx
q a N Ž 1 y a . Ž yw .
N Ž a y1 .y1
N
N
Ý Ý
C i j Ž w . wx i wx j ,
is1 js1
where Ci j Ž w . is the cofactor of the i, j entry of M w w x. If Ew w x is the inverse of the matrix Ž wx i x j ., the M w ¨ x s M w w x Ž yw .
N Ž a y1 .y1
T
a N Ž yw . q Ž 1 y a . Ž =w . E Ž w . =w .
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LAZER AND MCKENNA
Let G Ž x . s M w w x a N Ž yw . q 1 Ž 1 y a . Ž =w . E Ž w . =w . T
Since Ž wx i x j . is positive definite on V, EŽ w . is positive definite on V. Therefore, since 1 y a ) 0, yw ) 0 in V, and =w / 0 on V, there exist constants b1 ) 0 and b 2 such that b1 F G Ž x . F b 2 ,
; x g V.
From Ž3.1. we compute that N Ž a y 1. y 1 s yag and, therefore, M w ¨ x s Ž yw .
ag
GŽ x . .
If 0 - p1 F pŽ x . F p 2 on V where p1 and p 2 are constant, then for a positive constant c, it follows from the above that for x g V, M w c¨ x Ž x . y p Ž x . Ž yc¨ Ž x . .
yg
F Ž yw Ž x . .
y ag
Ž b2 c N y p1 cy g .
ag
b1 c N y p 2 cy g .
and M w c¨ x Ž x . y p Ž x . Ž yc¨ Ž x . .
yg
G Ž yw Ž x . .
If c 2 is a small positive constant chosen so that g b 2 c 2N y p1 cy 2 - 0,
and c1 ) c 2 is a large positive constant chosen so that g b1 c1N y p 2 cy 1 ) 0,
and we set w k Ž x . s c k¨ Ž x . for k s 1, 2, then since ¨ Ž x . s y Ž yw Ž x . .
ya
-0
on V, we have that w1Ž x . - w 2 Ž x . - 0 M w w 1 x Ž x . G p Ž x . Ž yw 1 Ž x . .
on V , yg
on V ,
and M w w 2 x Ž x . F p Ž x . Ž yw 2 Ž x . .
yg
on V .
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359
Since the function f Ž x, j . s pŽ x .Žyj .yg is strictly increasing in j for y` - j - 0, the arguments used in the proof of Theorem 2.1 yield the existence of a function u defined on V such that w1Ž x . - u Ž x . - w 2 Ž x . ,
x g V,
and M w u x Ž x . s p Ž x . Ž yu Ž x . .
yg
,
x g V.
The estimates Ž3.2. follow from the fact that since w Ž x . - 0 in V and =w / 0 on V, there exist constants a1 ) 0 and a2 such that for x g V, a1 dist Ž x, V . F yw Ž x . F a2 dist Ž x, V . . Uniqueness of the solution of Ž3.1. follows easily from Lemma 2.1. If u1Ž x . and u 2 Ž x . are two solutions of Ž3.1. and there were a point x 0 in V such that u1Ž x 0 . - u 2 Ž x 0 ., then there would exist a domain D such that x 0 g D : V, u1Ž x . - u 2 Ž x . in D, and u1Ž x . s u 2 Ž x . for x g D. Since f Ž x, j . s pŽ x .Žyj .yg is strictly increasing on Žy`, 0., and since the facts that M w u1 x ) 0 on V and u attains its minimum on V at a point in V implies that matrix Ž u1 x i x j . is positive definite on V, we have a contradiction to Lemma 2.1. Therefore, u1 s u 2 and the proof of Theorem 3.1 is complete. 4. REMARKS ON RELAXATION OF REGULARITY OF V Theorems 2.1, 2.3, and 2.4 were proved under the assumption that there exists w g C`Ž V . such that w s 0 on V, =w / 0 on V, and the matrix Ž wx x . is positive definite in V. In this section we briefly indicate how a i j result of Lions w15x and ideas used by the authors in w13x can be used to establish the following improvement: Theorems 2.1, 2.3, and 2.4 remain true if it is only assumed that p g C`Ž V . l C Ž V ., and there are constants p1 ) 0 and p 2 such that p1 F pŽ x . F p 2 for all x g V, and V satisfies the following condition: Ž V .: The region V ; R N is bounded and there exists w g C 2 Ž V . such that w s 0 on V and the matrix Ž wx i x j . is positi¨ e definite on V. We indicate the modification in the proof of Theorem 2.1. First we note that there exist numbers R1 ) 0 and R 2 ) 0 such that if x 0 g V, then there exist balls B1 and B2 , having radii R1 and R 2 respectively, such that B1 ; V ; B2 ,
Ž 4.1.
B1 l V s B2 l V s x 0 4 .
Ž 4.2.
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LAZER AND MCKENNA
We observe that there exists a function u g C`Ž V . l C 1 Ž V . such that u Ž x . - 0 for x g V, u N V s 0, and the matrix Ž ux i x j . is positive definite in V. In fact, Lions w15x has proved the existence of such a u which is a ‘‘first eigenfunction of M ’’ in V, in the sense that M w u x s Žylu . N in V, where l ) 0. Let « n4`1 be a strictly decreasing sequence of positive numbers such that « n ª 0 as n ª ` and for each integer n let V n s x g V N u Ž x . F y« n4 . We have that V n ; V nq1 ; V and V is the union of the V n , for n s 1, 2, . . . . Since the hypotheses of Theorem 2.1 are satisfied for V n and pŽ x . for n G 1, there exists a u n g C`Ž V n . such that u ) 0 in V n , M w u n x s pŽ x . ugn in V n Žg ) N is fixed throughout this argument., and u nŽ x . ª ` as distŽ x, V n . ª 0. Using Lemma 2.1 and an argument used several times before, it follows that if n G 1, then u nq 1 Ž x . F u n Ž x . ,
; x g Vn.
Ž 4.3.
To obtain a lower bound for u nŽ x . for x g V n let x 1 g V n and let x 0 be a point on V closest to x 1. Let B2 be a ball of radius R 2 such that V ; B2 and V l B2 s x 0 4 . The hypotheses of Theorem 2.1 are satisfied for the region B2 and the constant function p 2 . Therefore, there exists z 2 g C`Ž B2 . such that z 2 Ž x . ) 0 in B2 , M w z 2 x s p 2 z 2g
in B2
Ž 4.4.
and z 2 Ž x . G s 2 dist Ž x, B2 .
ya
Ž 4.5.
where a s Ž N q 1.rŽg y N . and s 2 is a positive constant. Since the equation Ž4.4. is invariant under translation, the same constant s 2 will ser¨ e for any such ball B2 regardless of the particular point x 0 g V. For x g V n , M w z 2 x s p 2 z 2g G pŽ x . z 2g. Therefore, since z 2 is bounded on V n while u n becomes infinite on V n , it follows from Lemma 2.1 that u nŽ x . G z 2 Ž x . for x g V n . In particular, u n Ž x 1 . G z 2 Ž x 1 . G s 2 dist Ž x 1 , B2 .
ya
s s 2 dist Ž x 1 , V .
ya
.
Since x 1 g V n was arbitrary and s 2 is independent of x 0 , we see that u n Ž x . G s 2 dist Ž x, V .
ya
n G 1, x g V n .
,
Ž 4.6.
Since u nq 1Ž x . F u nŽ x . for x g V n , the argument used in the proof of Theorem 2.1 shows that there exists u g C`Ž V . such that lim nª` u nŽ x . s uŽ x . on V and g
M w u x Ž x . s p Ž x . uŽ x . ,
x g V.
Ž 4.7.
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361
Moreover, from Ž4.6. we have u Ž x . G s 2 dist Ž x, V .
ya
Ž 4.8.
for all x g V. Let x 1 g V and assume that distŽ x 1 , V . - R1. Let x 0 g V be a point in V closest to x 1 and let B1 be a ball of radius R1 such that B1 ; V and B1 l V s x 0 4 . Since the hypotheses of Theorem 2.1 are satisfied for B1 and the constant function p1 , there exists a function z1 , such that M w z1 x s p1 z1g in B1 , z N B1 s `, z1 ) 0 in B1 , and there exists s 1 ) 0 such that for x g B1 , z1 Ž x . F s 1 dist Ž x, B1 .
ya
.
Ž 4.9.
Since the P.D.E. satisfied by z1 is translation invariant, s 1 can be chosen independent of x 0 g V. Since M w z xŽ x . F pŽ x . z1Ž x .g on B1 , z1 N B1 s ` which u is finite on B1 y x 0 4 , and Ž4.7. holds, the argument used in w13, pp. 1003]1004x shows that uŽ x . F z1Ž x . in B1. In particular, u Ž x 1 . F z Ž x 1 . F s 1 dist Ž x 1 , B1 .
ya
s s 1 dist Ž x 1 , V .
ya
.
Since x 1 was any point in V with distŽ x 1 , V . - R1 and s 1 is independent of x 0 g V, it follows that uŽ x . F s 1 distŽ x, V 1 .ya if distŽ x, V . - R1. By making s 1 larger, if necessary, we can ensure that u Ž x . F s 1 dist Ž x, V .
ya
,
; x g V.
Ž 4.10.
Since the uniqueness part of the proof of Theorem 2.1 did not make use of the regularity of V and only required that p g C Ž V . and pŽ x . ) 0 on V, we see that under the hypotheses given at the beginning of this section, there exists a unique solution u of ŽBB. and the estimates Ž4.8. and Ž4.10. hold.
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