Nuclear Magnetic Resonance-Based Dissection of a

Global and Local Behavior of Positive Solutions of Nonlinear Elliptic Equations B. GIDAS Institutefor Advanced Study and Courant Institute

AND

J. SPRUCK Brooklyn College

Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic Identities and Estimates-A Special Case . . . . . . . . . . 3. Local Results and Classification of Isolated SingularitiesA Special Case . . . . . . . . . . . . . . . . . . . . . . . . 4. Global Results-A Special Case . . . . . . . . . . . . . . . . 5. TheVector FieldandBasic Bounds-The General Case . . . . . . 6. Global and Local Results-The General Case. . . . . . . . . . . Appendix A. Example 1: Solutions of the Equation Au + Ixl'u" = 0 in R", n > 0 . . . . . . Example 2: Solutions of the Equation Au + b(xj/Ixl2)uj+ U"= 0 in R", n>2,bEk.. . . . . . . . . . . . . Appendix B. Global Results on Compact Manifolds . . . . . . . . .

525

. 529 539

. 561 . 567 . 577 . 589 .592

. 593

1. Introduction

In this paper we study the local and global behavior of positive solutions of nonlinear elliptic equations on R", n > 2, and, more generally, on n-dimensional Riemannian manifolds. More specifically, we investigate the behavior of positive solutions near an isolated singularity, and derive some global results of Liouville type. Our present work grew out of a study of specific equations relevant to physics and geometry. Our simplest global result is

THEOREM 1.1. Let u ( x ) be a non-negative C 2solution of (1.1)

Au+u"=O

Communications on Pure and Applied Mathematics, Vol. XXXIV, 525-598 (1981) CCC 0010-3640/8 I /040525-74$07.40 8 1981 John Wiley & Sons, Inc.

526

B. GIDAS AND I. SPRUCK

in R", n .> 2, with l S a < -n+2 n-2'

(13

Then u ( x ) = 0. An interesting feature of Theorem 1.1 is that nothing about the behavior of u ( x ) at infinity is assumed. Thus infinity could be a singular point. Theorem 1.1 is a special case of Theorem 4.1 which in turn is contained in Theorem 6.1. For a S n / ( n - 2), Theorem 1.1 has an elementary proof (cf. [4]).For n = 3, equation (1.1) is relevant to the stellar structure in astrophysics (cf. [2]).The case a = ( n 2 ) / ( n - 2) (which is relevant to Yang-Mills equations ( n = 4), and to differential geometry ( n > 2)) will appear elsewhere [l]. In this case, u ( x ) is not identically equal to zero, but it is given explicitly (cf. [7], [4]). Another simple global result is

+

THEOREM1.2. Let M be an n-dimensional, n > 2, complete Riemannian manijold with non-negative Ricci tensor. Then every non-negative C 2 solution of Au uu = 0, 1 5 a < ( n + 2 ) / ( n - 2), on M is identically zero.

+

The work of Yau [16] on harmonic functions on complete Riemannian manifolds shows that the condition on the Kcci tensor in Theorem 1.2 is natural. Our simplest local result is

THEOREM 1.3. Let u ( x ) be a non-negative C2solution of(l.l) in 0 < 1x1 5 R. Assume "2,

in R",

in Sections 2, 3 and 4. In Sections 5 and 6 we deal with the more general equations n >2,

(1.10) with linear part strictly elliptic, and (1.1 1)

Au

+ bu +f(x,u)

=0

on a complete n-dimensional, n > 2, Riemannian manifold M In (1. I),A is the Laplace-Beltrami operator on M",and b is a vector field on M. For our local results we could also allow some dependence off on Vu. Our global results require that the metric gv of M", or the metric gg = ag((x) defined on R" by the coefficients av, has non-negative Ricci tensor. This condition is not needed for our local results. For the local results we also allow some singularities in the ug (or the Ricci tensor). In general, our global results require more restrictive conditions on the coefficients ug and bj than our local results. But this is to be expected, since this is the case 5ven for linear equations (cf. [5], [14]). The precise conditions we impose on the bj (b in case of M"), and the nonlinearity f will be given in the text. The important feature of our results is 00. Previous that we allowf(x,u) to grow like u", a < (n 2)/(n - 2), as u+ techniques (cf. [12], [13]) apply only in the "classical" regions a < n/(n - 2). The flavor of our local results is as follows. For positive solutions of (1.9) (or (1.10)) in 0 < 1x1 5 R, with a possible singularity at the origin, we derive an a priori upper bound, and establish a special form of Harnack's inequality. These results are basic in classifying the singularities. The crucial technical L,, bounds used in the proofs are obtained through the use of a special vector field constructed from the solution. This vector field is probably the most important new device in this paper. Our proofs (once the appropriate Lp bounds are established) are based on the standard De Giorgi-Nash-Moser bootstrap arguments.

+

+

NONLINEAR ELLIFTIC EQUATIONS

5 29

The basic form of our vector field appeared in a geometric result of Obata [ 101 concerning conformal deformations of the usual metric on S". In Sections 2 and 5 we introduce the vector field for equations (1.9) and (1.1 l), respectively. The vector field leads to some identities (see (2.7) and (2.13) for equation (1.9), and (5.16), (5.18) for equation (1.1 1)) which are basic for much of our work. The organization of the paper is shown in the Table of Contents.

2. Basic Identities and EStimates--A

Special Case

In this section we introduce a new device for obtaining estimatesa geometric vector field associated with positive solutions of equation (1.9), which written in the weak form reads (2.1)

S,r v t

v u - ?J(bj9+ hu")} dx= 0,

where Q is a bounded open domain in R" and 9 E CT(Q). We introduce a new function

which satisfies

We define a vector field V with components

In terms of u(x),

Here subscripts represent partial derivatives and repeated indices are summed.

(2.4a)

-

4(n- 1) n(n - 212

(2.4b)

hx,( X ) U

-2 / ( n

- l)(n + 2) + 4(nn(n bju - q3 -

4(n- 1) n(n - 212

-

-2L i

*An- $' 1

V u12

u -2/(n-2hj(b,y)i,

where n We note that

For a 5 (n + 2)/(n - 2), the second term in (2.4) is also non-negative. The identity in Proposition 2.1 below is basic for much of the work in this and the next two sections. All integrals in this and the next two sections will be with respect to the Lebesgue measure over R".


53 1

PRo~osmoN2.1. Let u be a non-negative C 2 solution of (2.1), and Cl a bounded open domain in R", n > 2. Let q ( x ) E CT(Cl),and let y be a real number. Then -

b

+(?I 2)2

qU-rJ(x)

Remark. If q ( x ) h 0, and y E ( 0 , l ) is sufficiently small, then each term on the left-hand side of (2.7) is non-negative (in this paper we always assume a < ( n + 2 ) / ( n - 2)). Having in mind later applications of (2.7), we shall refer to the terms that contain derivatives of q as "surface" terms, and to the others as "volume" terms.

Proof of Proposition 2. I : By the divergence theorem, we have

532

B. GIDAS AND J. SPRUCK

Using (2.4b), equation (1,9),and the divergence theorem we get

t(n-2 ) 2 i q u - y ~ - ~

where

(2.10)

J = + ( n- 2)21qu-7J(x). n

Using (2.3b), equation (1.9), and the divergence theorem, we see that

(2.12)

Combining (2.8)-(2.12) we quickly obtain (2.7). We shall also need the following alternative form of Proposition 2.1.

533

NONLINEAR ELLIPTIC EQUATIONS

2.2. Let Q, and 7 be as in Proposition 2.1; furthermore, let Y # a + ( n - 4 ) / ( n - 2), Y # - 2 / ( n - 2). Then, with J given by (2.10), FkoPOSmoN

2(n - l ) ( n + 1 ) - (2n + I ) y ] i v ( b i h i bj,jh)~a+("-4)/(n-2)-y X[ n-2

+

+

where bj, i ( x ) = a $ / a x i .

( n - 1)n

PI2

534

B. GlDAS AND J. SPRUCK

The following lemma will be needed in the proof of Proposition 2.2.

LEMMA2.1. Let 52 and y be as in Proposition 2.1, and let Z E C,"(52) be a vector field with components Z Then

'.

(2.14)

Proof: By equation (2.l), (2.15)

n

[ - ( Z i ~ - y - 2 / ( nh-i2p j+ ziU-'-'/'"-''ui(bjuj + hu")) = 0.

We observe that

and 1

ziu-~-2/(n-2)uhu== i

a

+ ( n - 4 ) / ( n - 2) - y

Combining (2.15)-(2.17) one quickly obtains (2.14).

535


Remark. Equation (2.14) and its proof are similar to the well-known Pokhazaev identity (cf. [ 111).

Proof of Proposition 2.2: Identity (2.13) is obtained from (2.7) by modifying the appropriate terms on the right-hand side of (2.7) as follows:

(2.19) =-

1

a+(n-4)/(n-2)-y

$(q.hb- + qhjbj + qhbj,j ) ~ 4 + ( " - 4 ) / ( n - 2 ) - Y . n J J

The terms

and

are transformed by using (2.14) with 2' = qi and 2' = qbi, respectively. Inserting the above modifications into (2.7) we obtain (2.14). Next we derive our basic bound for the local problem. In the rest of the paper cw ill denote a strictly positive constant which may vary from line to line.

PROPOSITION 2.3. Let St be an open bounded domain in Rn, n > 2. Let (2.22) (a) For n h 4, assume

2 1.

x01/2= IxI~31xo1/2

For n 1 4 , we may takep = 2a - 2 / ( n - 2) - y , since 2a y sufficiently small. Then, by Lemma 3.1 and (3.la),

- 2(n - 2) - y > 1 for

Hence u(xo) s

proving (3.4) for n L 4. For n = 3, the choice p = 2a - 2 (see (3.6b)) presents a slight difficulty since p > 1 requires a > i. For this reason we proceed as follows: Since

AU

+ $9 = - h(x)u" 5 0,

any positive solution of (1.9) is a supersolution of a linear elliptic operator. Then

544


u satisfies the weak Harnack inequality (cf. [a], [15])

(3.18)

inf u ( x ) Z c R”/p

BR/~(XO)

(JBR(XO)

for any

n-2

> p > 0.

From this we estimate any Lp,0 < p < n / ( n - 2), norm in terms of any Lq estimate, e.g., q = 2a - 2, as follows: Clearly,

and by (3.18)

Choose q = 2a - 2, and 1 < p

< n / ( n - 2) = 3. Then, by Lemma 3.1,

This together with (3.14) yields (3.4) for n = 3, completing part (i) of the theorem. Hamark‘s inequality (3.5) is obtained by combining (3.14) and (3.18) with 1 < p = q < n / ( n - 2) and R = 4 lxolsufficiently small, and applying a standard covering argument for the annulus Q 5 1x1 5 (1 + e)c, with small enough e. By (3.15) and (3.16), the constant c is uniformly bounded. This proves the theorem. COROLLARY 3.2. = 0.

Under the assumption of Theorem 3.1, if a

< - 2 , then

u(x)

Proof: By (3.4), (3.19)

u(x)+O

as x-0;

since

A u + $ ~ = -huaSO and u ( x ) 2 0, by the maximum principle, u ( x ) cannot vanish in the interior of 1x1 d R. Thus (3.19) implies u ( x ) = 0. In order to characterize the “order” of an isolated singularity we need the following sufficient condition for removability of isolated singularities.

THEOREM 3.2. Let u ( x ) be a positive C2 solution of (1.9) in 0 < 1x1 S R with 1 < ( n + a ) / ( n - 2) < a < ( n 2 ) / ( n - 2). Suppose that h ( x ) and $(x) sari@

+

545


the conditions of Theorem 3.1 and also

(3.20)

b, E Ln/(l-~)(lxI 5 R ) for some 0 < 6 < 1.

Then, if

(3.21) with c independent of E, the singularity at x = 0 is removable, i.e., u ( x ) can be extended to a continuous HI weak solution in the entire ball 1x1 5 R .

Proof: The proof uses Theorem 3.1 and a double application of the De Giorgi-Nash-Moser bootstrap arguments. We first note that

(3.22)

I

( h u a - 1 ) n / 2 d x 5C

ESlXI 0, for O < U SI, ’(‘1 = - [q1q-qouqo + (qo - q ) ~ ]for I s u,

1“;

40

and

G(U ) = F( u)F’(U ) - 4. Clearly, F is a C I function of u and G is a piecewise smooth function of u with a comer at u = 1. Moreover since po > 0 (implied by ( n u ) / ( n - 2) < a), F, G satisfy

+

(3.23a)

1 G [ 5 FF’,

(3.23b)

(3.23~)

G’ B

1 -pFI2

for u < I,

-poF”

for u > 1.

(I


546

Now let ~ ( x ) i,j ( x ) be non-negative C" functions, q having compact support in B, = { x E R" : 1x1 S R }, and i j vanishing in some neighborhood of the origin. Inserting (M)2G(~)as a test function into (2.1) we find

(3.24)

d,SR{

+ 2qFjG(u)V(M) V U (qij)2G'(~)lV~/2 - ( M ) 2 G ( ~ )bpi [ + hu"]} dx=

0.

Using (3.23) and simplifying we obtain from (3.14)

(3.25)

Next we bound by Holder and Sobolev inequalities

where C,, is the Sobolev constant. Hence if we choose R small enough (depending on q) so that

(where C(q) is the constant in (3.25)) we obtain from (3.25), with a new constant C(q),

Similarly,


547

Again for R small enough so that (3.28)

11 IIl,,(4 )

'

11 11 I,,/,-a,( I

B,)

< ( c,".( 4 ) )- 'I2

we obtain from (3.27)

and

According to Lemma 8 of [12] there exists a sequence ijk tending to 1 almost everywhere, with llVfjkl[L(BR) +O as k+ +go.' Therefore, (3.3 1) Since, by (3.21),

we have from (3.21)

~~

I Here is an example of a sequence of Lipschitz functions which satisfy these conditions: Let c > 0 and define

Then ij'(x)+ 1 almost everywhere as c + O and


548

This together with (3.29), (3.30) yields

Now we can let I +

+

00.

Since F(u)+ uq = u ( ~ + ' ) / ' ,

(3.32a) (3.32b) Inequality (3.32) can be iterated a finite number of times to show that u E H , ( B R )n Lq(BR)for any q > 0, implying that u extends to an HI weak solution in B R and also that (3.33)

d ( x ) = hua-'

Ln/2+GI(BR)

for some 6 , > 0 depending on u. This suffices to apply again the bootstrap arguments (cf. [12]). By a standard result (cf. [5]), if u ( x ) is an H,weak solution of (1.9) in 0 < 1x1 S R and satisfies (3.33), then u ( x ) is equivalent to a continuous function in 1x1 5 R. The proof of the theorem is thus completed. Remarks. 1. Example 1 in Appendix A shows that Theorem 3.2 is not in general true for 1 < a < ( n u ) / ( n - 2). 2. We do not know whether it is possible to replace condition (3.20) by 9 E Ln(lxl S R ) . Example 2 of Appendix A shows that if bj(x)= O ( l / l x l ) , then Theorem 3.2 does not in general hold. 3. If u = -2, and 1 < a < ( n 2 ) / ( n - 2), then by Theorem 3.1, u ( x ) is bounded at the possible singularity x = 0. By Semn's results [ 121, the singularity is, therefore, removable.

+

+

COROLLARY 3.3. Assume the conditions of Theorem 3.1 hold and, in addition, (3.20) is satisfied and 1 < ( n + u ) / ( n - 2) < a < ( n 2 ) / ( n - 2). Then either the origin is a removable singulariy or u ( x )+ + 00 as x +0.

+

Proof: By Theorem 3.2, if the origin is not a removable singularity, then there exists a sequence of points ( x i } , xj+O, such that u(x,)+ + 00. Set

-

inf u ( x ) .

= (XI

lx,l

549


By Harnack's inequality, part (ii) of Theorem 3.1, 6 + - hu" 5 0, the maximum principle implies that

+

00.

Since A u

+ bjy =

u ( x ) h min( b, b+,) in Ixj+ll< 1x1 < Ixjl. Hence u(x)+

+ 00 as x+O.

THEOREM 3.3. Let u ( x ) be apositive C 2 solution of(1.9) in 0 < 1x1 5 R with ( n u ) / ( n - 2) < a < ( n 2 ) / ( n - 2), -2 < u < 2. Suppose that h ( x ) , bj(x) satisfv (3.1)-(3.3), and in addition bj satisfies (3.20). Then either the singularity at the origin is removable, or else there exist positive constants c I ,c2 such that

+

+

Furthermore, if (3.35) for all

ID'$(x)l 5 c ~ x ~ - ~ -ID'h(x)l ~ ' ~ , S ~lxl"-1~1near x = 0 171

5 k - 1, then

(3.36) Proof: By Theorem 3.1,

u(x)5

I f (3.34) does not hold, then

It follows from part (ii) o f Theorem 3.1 that

We shall show that the singularity at the origin is removable. By Theorem 2.2, it suffices to establish (3.21). To this end, we bound using (3.4):

5 50


Let

Then A'=

n-2 --(a a-1

-

n

+u

2

+ u1 ,.-(an-(n+2+a))/(a-1)

Let S(x) E C,"(c 5 1x1 S R) satisfy (3.7). Using q = Sv(r) as a test function in (2.1), and the divergence theorem we obtain

) =J

(cpA{+2V{*Vcp)u

c -2. Suppose that the bj satisfl (3.2), (3.3) near x = 00. Then, (i)

(3.53)

u ( x )5

C I x ) ( 2 + a ) / ( a - 1)

near x = 00.

+

(ii) I f 1 < a < ( n u ) / ( n - 2), then necessarib u ( x ) = 0 in 1x1 2 R. (iii) If in addition to the above conditions we ussume Ibj(x)l E ~ 5 , , / ( ~ - ~ )2 ( JRx )l for some 0 < 6

(3.54)

< 1,

+

and ( n u)/(n - 2) < a < (n + 2)/(n - 2), then either the singulariy removable, or there exist positive constants clrc2 such that

(3.55a)

CI Ix1(2+")/("-1) 5 u ( x ) s

c2 Ix 1(2+a)/(u- 1)

at

00

IYl

--

2 n-3

'

Then u ( x ) = 0. Remur.-r. 1. Condition ( 3b) is not needed in three dimensions. We suspect that it should not be needed for n 2 4 either. 2. Theorem 4.1 is a special case of Theorem 6.1. Theorem 1.1 is a corollary of Theorem 4.1. 3. Example 1 of Appendix A and Example 1 below show that Theorem 4.1 does not in general hold without a condition such as (3.2). Also, Example 2 of

562


Appendix A and Example 2 below show that Theorem 4.1 is not in general true with a $ ( x ) u j ( x ) term in the equation. 4. Condition (4.3a) could be replaced by an integral condition. Proof of Theorem 4.1: We consider identity (2.13) with bj = 0, and make the following modification: First, by the divergence theorem,

and next, using Lemma 2.2 with cp = qh,

First we assume n L 4,and a > 1. Then by choosing y sufficiently small we have - 2/(n - 2) - y > 0. Thus inserting (4.4) and (4.5) into (2.13), we obtain

u

+ y(l = -

-

- y ) J qnu - ~ - 2 ( n - ~ ) / ( n - 2 ) 3

(u

+ ( n - 4)/(n - 2) - y ) ( a - 2/(n - 2) - y )

2(n - 3 ) / ( n - 2) - 3y

Y + 2/(n

- 2)

pqu

- 7 - 2 A n - 2 ) lVUl2

ivUi4


563

For n = 3 and a > 2, we use (2.13), (4.4) and (4.5) to obtain

J (4.7)

+ $ ( 5 - a)iqh%2u-2 =

- (a - 1)(a - 2 ) I qn A h u " - '

The proof of the theorem will be based on (4.6) for n h 4, a > 1, and on (4.7) for n = 3 and a > 2. The remaining cases ( n h 4, a = 1, and n = 3, 1 Ia I2) are elementary and will be considered last. Since Ah B 0, the first term (a "volume" term) on the right-hand side of both (4.6) and (4.7) is non-negative ( y sufficiently small). The other terms are estimated as in Proposition 2.3. Thus for n h 4, a > 1 we have with a J satisfying (2.22)

5 eJ-(lDPI

(2u - 2/(n - 2 ) - 7 ) / ( a - I )

+ IVJI IVloghl)

and for n = 3, a > 2 (withp > 2) 4 (5 - a)iJph%2a-2

To complete the proof of the theorem we choose Q = { x E Wn : 1x1 S R } and (4.1Oa)

J(x) = 1 for

1x1 S t R ,

564


This together with (4.3) and (4.8) yields p - h + p -2(n-2)

-Y

JxldR

(4.1 1)

C 9 R (2/(a-1))(2~-2/(n-2)-7)+(5/(~-1))(-~+2(n-3)/(n-2))-n

which goes to zero as R + + ao, for u > - 2/(n - 3). Next we consider the case n = 3 (2 < a < 5). Estimate (4.9) together with (2.22) and (4.10) to get

We shall show that the right-hand side of (4.12) goes to zero as R + equation (4.1) we have

This gives, with 0

> 0,

(4.14)

For 8 sufficiently small we have (4.15) This and (4.13) yields (4.16)

+ 00. From

565


Combining (4.12)with (4.15)and (4.16)we see that the right-hand side of (4.12) goes to zero as R + + 00. This implies that u = 0. Next we consider the case n = 3, 1 < a 5 2. From equation (4.1),and using (2.27),we get

Choosing 8 small enough, and combining (4.17) and (4.10) we see that the right-hand side of (4.17)goes to zero like (R-'-'"), giving again u = 0. Finally, we consider the case n 2 3, (Y = 1. In this case we have

Taking R very large, the right-hand side of (4.18)may be absorbed into the left-hand side, giving

which implies that u = 0. This completes the proof of the theorem. As we mentioned before, Example 1 of Appendix A shows that Theorem 4.1 is not in general true without a condition such as (4.2). Here is another example which bears on the necessity of (4.2).

EXAMPLE1. Let

(4.19)

w(x) =

[

1

(n-2)/2

y2

+ Ix -

a12

,

y

> O,a E R",n > 2.

566


Let w ( x ) satisfy (1.1) with a = (n + 2)/(n - 2). Let a < (n + 2)/(n - 2), and

(4.20)

h( x ) =

+2)/(n- 2)-

a

Then the equation

Au

+ h(x)ua=0

has the obvious solution u ( x ) = w ( x ) . The function h ( x ) defined by (4.20) satisfies

Thus Ah 5 0 when a > 4/(n - 2). The following example as well as Example 2 of Appendix A show that Theorem 4.1 is not generally true with a $uj term in the equation.

EWLE

with p

2. Consider the equation

> 0 and 1 < a < (n + 2)/(n - 2). It has the explicit solution

(4.22) Thus Theorem 4.1 does not hold. It is interesting to note that if a > n / ( n - 2), (4.22) decays slower than 1/1x1n-2 at infinity. Equation (4.21) is a special case of the following equation:

AU + (/3 - l)(n - 2)

xi

p2

+ 1x12 ui


567

with /3 # 1. It has the explicit solution u(x) =(w(x))B

where w ( x ) is defined by (4.19). 5. The Vector Field and Basic B o m d s - ~ General Case

In this section we introduce the geometric vector field and derive the basic identities and bounds for equation (1.1 1). We shall see that the study of equation (1.10) can be reduced to the study of (1.1 1). Let gg be the metric on M",and (gg) the inverse matrix of (gg). With respect to a local coordinate system x = (xl, ,xn)E M",we denote by V i , hV, R = Riigg, and A, the Christoffel symbols, the covariant derivative, the Ricci tensor, the scalar curvature, and the Laplace-Beltrami operator, respectively, corresponding to the metric gg. We record some well-known formulas. In terms of the metric g, and for a (scalar) function f,

---

(5.1)

r: = 4 gkl(aiGl+ ajg, - algV),

(5.3)

where g = det( gg). Let Zi be the components of a vector field Z and (5.4)

z' = pzj.

Then

vizJ= aizj + rjkzk, (5.5)

568


Now we reduce equation (1.10) to (1.11). Without loss of generality we may assume that uq = a''. We shall assume that the linear part of (1.10) is uniformly elliptic, i.e., for some constant A > 0 and for all n-tuples El, * * ,&,, (5.8)

Setting (5.9)

where g = det(ag(x)), and (aq) is the inverse matrix of (a"), one easily sees that (1.10) is equivalent to (1.11). In this section we shall work only with (1.11). The volume element of M" is (5.10)

dV=Gdx, A

*

-

A

dx,,

where, as usual, " A " means exterior multiplication. All the integrals which appear in this section will be with respect to the volume element (5.10). The use of covariant derivatives makes the developments in this and the next sections parallel to the developments of Sections 2, 3, and 4. The part of the analysis which is the same as before will not be repeated, but we shall emphasize the new aspects such as the role of curvature of M",and the dependence of f ( x , u ) on u (some dependence off on V u could also be allowed). As in Section 2, we introduce a new function (5.1 1)

u ( x ) = w(x)-(n-2)/2

which satisfies

where ldwI2 = w i w i = g q w i y is the square of the gradient of w . Our basic vector field V has components (compare with (2.3)) (5.13a)

(5.13b)

1 w'Aw), V' = w - ( " - l ) wJVjw' - n


569

Using the notationf(x, t ) we have

(5.14a)

(n - l)(n + 2) w-"/tf(x,u)lmyI'+ n(n-2)

n-1 y

w - ( n - 1% (x, u )Im v l 2

1) w - ( n-2) /2w yx,(x, u ) + n(n -- 2) 2(n

where

The main difference between (5.14) and (2.4) is the appearance of the Ricci tensor RV.It entered through (5.6). The following proposition generalizes Proposition 2.1.

B.GIDAS AND J. SPRUCK

570

Let Q be an open bounded domain in M". Let q ( x ) E C F ( M ) , and y E R. Then,

PROPOSITION5.1.

=

-2bqR,(x)u

+ 2( 1 - y)J

- y -2/(n-2)Uiuj

Viqu-7-n/(n-2)Ui n

2(n - 1)

- 3Y]

Proof: The proof of (5.16) is similar to the proof of (2.7). We leave details to the reader. As in Section 2, we need an alternative form of (5.16) for the global problem in all dimensions, and for the local problem in three dimensions. This alternative form is contained in the next proposition. First we introduce the function

PROPOSITION 5.2. Let Q and q be as in Proposition 5.1. Then, with J given by

57 1


(2.10) (covuriunt derivatives implied), we have

+ n(r + 2/(n - 2))

+ n(y + l / ( n - 2))

2(n - l ) ( n + 1 ) n-2

b([ a

-

- (2n + 1 ) y

] bib-'

2

Before we give the proof of (5.18), we establish the analogue of Lemma 2.1.

572


LEMMA5.1. Let 52 and y be 0s in Proposition 5.1. Let Z be a vector field with components Zi E C,"(S2). Then p

i

-

( 1( - 2 / ( n - 2 ) - Y )ldUl2 = LViZiu- 2 / ( n 2, - 7 ldu12-

-2

vjziu-2/(n-2)-yu;uj

V'Zi@(X,u) - 2 ZiW(X,u)

I,

(5.19)

I,

2

I,

and use

and

Proof of Proposition 5.2: As in the proof of Proposition 2.2, we make the following substitutions in (5.16):

2(n - 1) Q j U - 2 / ( n - 2 ) - r (5.22)

= -

(5.24)

uj f ( x ,

n

=

-

2 ( n - 1) n

u)

I,{

(qibi + qV'bi)cP + qb,'bxl},

-2 k q 8

573


and

(5.25)

Inserting (5.22)-(5.25) into (5.16), we obtain (5.18). The next proposition generalizes Proposition 2.3. It contains the main technical conditions which will be imposed on the nonlinearity f ( x , u ) . Notice that bound (5.31) below is formulated differently from its analogue (2.25) in Section 2.

F'ROPOS~ON 5.3. Let P be an open bounded domain in M",n > 2. Let (5.26) Suppose that f ( x , u ) is a non-negative function satisbins (i)

- 2 { t - ( ( " + 2 ) / ( n - 2 ) - 8 ) f ( x , t ) }2 0

(5.27)

at

for some 6

> 0, and all t 2 0, (ii)

(5.28)

Ivi

[ log f ( x ,U ) ] I 5j ( x )

for some positive function j ( x ) independent of u. Then: (a) For n 2 4 and y positive but sufficient!y small, there exist positive constants c and a large p such that (with J given by (2.10))

J

(5.3 1)

+ cI,~ P f 2 ( ~ , ~ ) ~ - 2 / ( " - 2 ) z-pyU - 2 ( n - l ) / ( n - 2 ) - ~

vui4

574


where

(5.32)

IR12 = R,RU.

(b) For n = 3, there exist positive constants c and some p

> 2 such that

Proof: From (5.27) we have

(5.34)

UJ(X,U)S(

n-2 n + 2 -S)f(x,u).

Using this fact we bound the second terms on the left-hand side of (5.16) and (5.18):

(5.35)

For a given S

> 0, we choose y so that

(5.36) We now proceed as in the proof of Proposition 2.3. First we consider the case

NONLINEAR ELLIPT'lC EQUATIONS

n 2 4, and employ (5.16) and (5.35). With 1) = { p , p 2(n - 1) n

(5.37)

(5.38)

Here is the analogue of Lemma 2.2.

575

> 4, and 0 sufficiently small

Proof: Consider I q f ( x , U ) U - ~ - ' / ( " - ~ )( AU w

+ b.uj + f ) = 0 J

and apply the divergence theorem; (5.41b) is obtained from (5.41a) by using (5.34). Combining (5.14b) with (5.40) and applying (5.37) once more, we obtain idUi4

(5.42)

Using (2.27) with p = q = 2 and small enough 0, we get

hSJ'-2[ l&12 + (5.43)

lj(x)I2

+ Ibl']f(x,u)~("-~)/("-~)-~

s ce21,SPf2(x,u)U-2/(n-2) +f e

l{Pp4[

-7

+ l j ( x ) I 2 + 1b12]2~2(n-3)/(n-2)-Y.

n

Combining (5.42) and (5.43) we obtain (5.32).

577


The case n = 3 is similar to the one in Proposition 2.3. We leave the details to the reader. 6. Global and Local Results-Tbe

General Case

In this section we establish global and local results for equations (1.10) and (1.1 1). Some global results on compact manifolds are treated in Appendix B, 6.1. We present first our global results. Theorem 6.1 generalizes Theorem 4.1.

THEOREM 6.1. Let M",n > 2, be an n-dimensional complete Riemannian manifold with non-negative Ricci tensor Rii. Let A be the Laplace-Beltrami operator on M ,and u ( x ) be a non-negative C2 solution of (6.1)

Au

+f(x,u)=O

in M",

n>2.

Assume that the nonlinearity f satisfies: (i) For some ( n + 2)/(n - 2) > 6 > 0 and all t B 0,

(6-2)

- - [ dI

f ( x , t ) ] 20.

-((n+2)/(n-2)-~)

dt

(ii) For each fixed t 2 0, A, f ( x , t ) 2 0.

(6.3)

(iii) Let r = r ( x ) be the distance function on M from a fixed point P. Suppose that, for all t 2 0, (6.4)

as r +

ldxlogf(x,t)l 5

+ 00.

(iv) For n 2 4, assume (6.5a)

ch(x)uaS f ( x , u ) ,

c

> 0,

c

> 0,

where (6.5b)

n+2 1 2, be an n-dimensional complete Riemannian manifold with non-negative Ricci tensor. Every positive C 2 harmonic function on M" is constant. Remark. This corollary has also been proven by Yau [16] using totally different ideas. One may obtain other global results by combining (5.35) and (5.41b) to bound the left-hand side of (5.18). Using (5.21), we get, after a Straightforward computation,

J+

2(n n

1

1 2(n 1) n / ( n 2) -6-y n+2

- 2(n - 1) (6.16)

n

1

2(n n +-21) n / ( n - 2)

-6- y

58 1


n(n - 2)

n

with constants ci which are not necessarily positive. The first four terms on the right-hand side of (6.16) are “volume” terms, the others are “surface” terms. The surface terms are controlled as before. A global result is obtained whenever the volume terms are non-positive.

THEOREM 6.2. Let M ” be an n-dimensional, n > 2, complete Riemannian manifold with non-negative Ricci tensor RU.Suppose that u ( x ) is a non-negative C 2 solution of (6.17)

Au+bu=O

n > 2,

in M“,

where b ( x ) is a vector field. Let r = r ( x ) be the distance function on M from a fixed point Po, and assume

(6.18)

I~(X)~S;

as r + + m

for n B 3. For n = 3, assume in addition that l V b l S c / r 2 as r + tensor field with local coordinates

(6.19)

- 1 (Vkbk)g/

+ VJb,-

is positive definite, we have u = constant.

n

bib’

+ 00. Then, if

the


582

Proof: We use (6.16) withf 3 0, 51 being the geodesic ball of radius R.Since M" has non-negative Ricci tensor we may use Proposition 6.1. This together with (6.18) suffices to show (as in Theorem 6.1) that the "surface" terms go to zero as

R + + 00. Then (6.19) together with Rg.5 0 implies that the "volume" terms on the right-hand side of (6.14) are nonpositive. This yields the theorem.

Remark. We know of no other Louiville type theorem concerning the linear equation (6.5) on a complete Riemannian manifold. 6.2. In this subsection we present our local results for equations (1.10) and (1.1 1). The results (and their proofs) for these equations are identical. In fact, the Riemannian manifold M for equation (1.1 1) need not be complete. We shall state the results only for equation (1.10). The proofs for the most part are identical to those of Section 3. Therefore we shall point out only those parts of the proofs which differ from those of Section 3. The main difference occurs in the characterization of the order of the singularity in Theorem 6.4 below. The following theorem generalizes Theorem 3.1. THEOREM 6.3. Let u ( x ) be u non-negative C 2 solution of (1.10) in 0 < 1x1 S R. Suppose that the coefficients ag are C2(0< 1x1 S R ) n L" and satis& (5.8). Also suppose that the Ricci curvature (of the metric gg = ag) satisfies

(6.20)

IR~(x)~

5 C near x = 0. lx12

We assume that the nonlinearity f ( x , u ) satisfies: (i) For some S

>0

(6.21) (ii) For all t 2 0, (6.22) (iii) For n 2 4 (this condition on f is not needed in three dimensions),

for large u and

(6.24)

1 1.

€

This suffices to prove Harnack's inequality (combine (3.14) and (3.18) as in the proof of Theorem 3.1). In order to prove (6.30), we note that, by (6.29), (6.31), and (6.25), we have U2a-2 (n u ) / ( n - 2) > 1, these solutions are singular both at the origin and at infinity.

+

+

+ +

(ii) ( n a ) / ( n - 2) < a < (n 2 2a)/(n - 2): In this case, (A.2) (for functions depending only on r ) has no C 2 solutions. Besides (A.8), there exists another (one-parameter family) solution which is singular at the origin and regular at infinity, i.e., (A.ll)

IXlrn-2 where C, is given by (A.5b), and X

near r =

+00,

> 0.

(iii) (n + 2 + 2a)/(n - 2) < a: In this case, we have solution (A.8), and also another (one-parameter family) solution u ( r ) which is C 2 , and at infinity (A.12)

i.e., u(r) is singular at

00.

The above results illustrate the following remarks which bear on our theorems: 1. By Theorem A.2, Theorem 3.5 cannot hold with a < (n + u ) / ( n - 2). 2. For 1 < a < (n + a ) / ( n - 2), the singular solutions of Theorem A.2 satisfy (A.6). Thus, Theorem 4.3 cannot hold if a < (n + u ) / ( n - 2). 3. If - 2 < IJ < 0, then (n + 2 + 2a)/(n - 2) < (n + 2 ) / ( n - 2). The existence of solutions (A.lO) shows that (3.45) of Theorem 3.4 cannot hold in general. Also the existence of solutions (A.l) and (A.12) shows that the global result, Theorem 3.1, cannot hold without the assumption Ah h 0.

EXAMPLE 2. Consider the equation (A. 13)

X.

Au+b'u.+u"=O r2

'

in W",

n

> 2,

where b is a real number and a > 1. Equation (A.13) can be transformed into equations (A.3) and (AS) provided that we set IJ = 0 and n + n + b in these equations. Thus we have the analogue of Theorem A. 1.


593

THEOREM A.3. Let u ( x ) be a non-negative C 2 solution of (A.13) in 1x1 2 R. If 1 < a 5 ( n + b ) / ( n b - 2), then u ( x ) = 0.

+

A local analysis of the spherically symmetric solutions (A.13) (see [7] for the equation with b = 0) shows that a positive spherically symmetric solution of (A.13) with an isolated singularity at x = 0 satisfies Cl rn+b-2

(A.14)

+

5 u(r)S

c2 ,.n+b-2

near r =o,

+

for 1 < a < ( n b ) / ( n b - 2). We believe that (A.14) holds even for nonspherical symmetric solutions. For (A. 15)

the positive spherically symmetric solutions of (A.13) are obtained from the positive spherically symmetric solution of (A.l) by setting u = 0, and replacing n by n + b. We shall use a prime, e.g., (A. )’, to denote the corresponding solutions. These results illustrate the following remarks bearing on our theorems: 1. If -(n - 2) < b < 0, then n / ( n - 2) < ( n + b ) / ( n + n - 2). Thus (A.14), with n / ( n - 2) < a < (n b ) / ( n + b - 2), shows that Theorems 3.2 and 3.3 cannot hold in general when lb.(x)l = O(l/lxl). 2. If b > 0, then ( n b i ) / ( n + b - 2) < ( n + 2 ) / ( n - 2). Thus by (A.9)’ and (A.I2)’, no global theorem such as Theorem 4.1 is expected to hold without some additional conditions on the bj.

+ + +

Appendix B

In this appendix we derive two simple global theorems on compact Riemannian manifolds in n B 2 dimensions. They imply that the only positive solution of equation (3.67) is cp(8) = C,. First we consider the two-dimensional case. THEOREMB.l. Let M be a two-dimensional compact Riemannian manifold without boundaries and Gauss curvature K. Let u ( x ) be a positive C 2 ( M )solution of (B.1)

Au-Au+pu“=O

on M

with 1 < a < + GO. Here A and p are positive constants and A is the LaplaceBeltrami operator on M. Suppose that

(B.2a)

A 2y l +

$)s K


594

for some /3

> 0 such that

(B.2b)

Proof: We introduce the two-dimensional analogues of the function w in (5.1 l), and of the vector field V in (5.13): For any /3 > 0, define u-w-8

and (B.3a)

V' = w - ( 8+1) &'V wi - tw'Aw} {

Then

av

-

j

ViVi

where f ( x , u ) = -hu

+ pa

and J ( x ) = w-('+'){ V,Vjw - tAwgv} { V'ViU

- +Awgg} L 0.

For any two-dimensional manifold M we have Rv = Kgv. As in the proof of (5.16), we obtain the identity

595


The theorem is proven by choosing 2 ( B + 1)

(B-5)

B

= a if a > 2

and any

if

1

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