Abstract. We prove a gradient estimate and Liouville type theorems for the solutions of the Poisson equation on a complete manifold whose Ricci curvature is.
Tόhoku Math. J. 47 (1995), 509-519
GRADIENT BOUNDS AND LIOUVILLE'S TYPE THEOREMS FOR THE POISSON EQUATION ON COMPLETE RIEMANNIAN MANIFOLDS ANDREA RATTO AND MARCO RIGOLI
(Received May 9, 1994, revised November 18, 1994) Abstract. We prove a gradient estimate and Liouville type theorems for the solutions of the Poisson equation on a complete manifold whose Ricci curvature is suitably restricted.
1. Introduction and results. Throughout this paper M will denote a complete, connected, non-compact Riemannian manifold of dimension m>2. Our main aim is to establish various a priori estimates for the gradient of solutions to the Poisson equation Au = f(u) on M under suitable assumptions on the Ricci curvature (unless otherwise specified, the function / will be assumed to be of class C 1 ). Our first result is: THEOREM 1. (1)
Let FeC2 (i)
(R) be a function such that inf^=0
(ii)
F(u) = f(u).
R
Let u be a bounded solution of (2)
ΔW = /(M)
on
M
and assume that Ricci(M)>0. Then \Vu\2(x)A: That leads us to
514
A. RATTO AND M. RIGOLI 2
4d2"'
U>
a2ε2
16N2(a)
aN(a)ε
Now we choose d so small as to have the last term in (28) greater than zero and let a tend to +00. Then it is easy to see that (28) contradicts (5), so ending the proof of Theorem 2. • Step 3 (End of the proof of Proposition 1). Let N= Sup{| u |} on M. We proceed as in Step 2 above, with g(u) = [3N-u]~d and a=l: Then (28) takes the form 0
>
2
( 2 C ( 1 + ^ ) + 24 + 2A + 2fl) ε2
4d2 Nε
16N2
where R = Sup{\f'(u)\} on M and dis small as above. We observe the (29) must hold at any point peM at which \Vu\2(p)>ε2; but, if ε2 is large, then (29) does not hold. This is a contradiction unless | Vw | is bounded. • Step 4 (End of the proof of Proposition 2). Let u be a solution of Au = f(u) on M. Assuming that / is of class C 2 , we introduce the following two functions: (30)
W(a) = Inf{-f"(u)f(u)}
(3D
^
)=
Inf{/>)/(M)j I I/'Ml J
on Ba(p)
on Ba(p).
In particular, if f(u) = [uq — λu] we have (32)
(33)
Λ
»^(έi) = Inf{^-l)i/ ( «- 1 ) [-iί ( «" 1 ) + A]}
on
( f l ) =i n f | ^ - l ) i | t o - i > y - i _ ϊ y ^ j
on Ba(p).
Now Proposition 2 follows immediately if we take f(ύ) = uq — λu in the following more general result: THEOREM A. Suppose that Ricci(M)>0 and let u be a solution of Au = f(u) on M, where f is of class C2. Let W{a\ R(a) be as in (30), (31) and assume that W(a)>0for all a>0. Then either
Iiminf^(φ2 0 such that | Vu\2(p)>ε2/g(p). If qeBa{p) is a maximum of the function G in (10), we have Wu\2(q)>-
(36)
a4ε2 g(q)(a2-r2(q))2
'
2
Next, we apply (13) with g(u)=l/f (u) and A=0; we also divide both sides of the inequality (13) by | Vu\\q) and use (36). That leads us to
(37)
0>-
ε/W(φ.
It is now easy—using the definition of Wand R—to conclude that, if both (34) and (35) are false, then we contradict (37), so ending Theorem A and Step 4. • PROOF OF THEOREMS 1 AND 3. These two theorems are special cases of the following more general result: THEOREM B. Suppose that Ricci(M)> —A, A>0. Let u be a solution ofAu = f(u) on M such that (6) holds and assume that there exists a function Q such that (38)
(i)
(ii)
Q(u\Q\u) are bounded
Inf{β(κ)} = 0 . M
(iii)
[β'(iι)-2/(i/)]β'(n)>0
(iv)
|Vw| 2 2\Hess(u)\2-f(u)Q'(u), where the last inequality is due to the assumption (38) (iv). Next, we observe that, since η'(r) η(r)
PVr+Q'(u)Vu
n v) Now, by the Schwartz inequality as in (19), (47)
21 Hess(w) |21 Vw \2>— 2
[ η (r) 2
P2+2
n(r)
Using (47) into (46) we get (48)
I Wu
2
η
-(Q')2(ιή-f(u)Q'(u)}\Vu
η(ή η\r)
2
η\r)
η(r)
where the last inequality follows from (38) (iii). Now we put (48) into (45) to get r)Ar +
(49)
η \ r ) 2 ^ η{r) )
1 {η')\r) 2
η(r)
2
If I Vw| (x)
