aa r X i v : . [ m a t h . A P ] S e p HARNACK TYPE INEQUALITY FOR AN ELLIPTIC EQUATION.
SAMY SKANDER BAHOURAA
BSTRACT . We give a sup × inf inequality for an elliptic equation.
1. I
NTRODUCTION AND M AIN R ESULTS
We are on Riemannian manifold ( M, g ) of dimension n ≥ . In this paper we denote ∆ g = −∇ j ( ∇ j ) the Laplace-Beltrami operator and N = nn − .We consider the following equation(1) ∆ g u = V u N − + u α , u > . Where V is a function and α ∈ ] nn − , n +2 n − [ .For a, b, A > , we consider a sequence ( u i , V i ) i of solutions of the previous equation withthe following conditions: < a ≤ V i ≤ b < + ∞ , ||∇ V i || ∞ ≤ A. Here we study some properties of this nonlinear elliptic equation. We try to find some esti-mates of type sup × inf . We denote by S g the scalar curvature.There are many existence and compactness results which concern this type of equations, seefor example [1-21]. In particluar in [1], we can find some results about the Yamabe equation andthe Prescribed scalar curvature equation. Many methods where used to solve these problems, asa variationnal approach and some other topological methods. Note that the problems come fromthe nonlinearity of the critical Sobolev exponent. We can find in [1] some uniform estimates forvarious equations on the unit sphere or for the Monge-Ampere equation. Note that Tian and Siuproved uniform upper and lower bounds for the sup + inf for the Monge-Ampere equation undersome condition on the Chern class, see [1]. In the case of the Scalar curvature equation and in di-mension 2 Shafrir used the isoperimetric inequality of Alexandrov to prove an inequality of type sup + inf with only L ∞ assumption on the prescribed curvature, see [21]. The result of Shafriris an extention of a result of Brezis and Merle, see [4] and later, Brezis-Li-Shafrir proved a sharp sup + inf inequality for the same equation with Lipschitzian assumption on the prescribed scalarcurvature, see [3]. Li in[17] extend the previous last result to compact Riemannian surfaces. Inthe higher dimensional case, we can find in [15] a proof of the sup × inf inequality in the con-stant case for the scalar curvature equation on open set of R n . We have various estimates in [2]when we consider the nonconstant case. To prove our result, we use a blow-up analysis and themoving-plane method, based on the maximum principle and the Hopf Lemma as showed in [2,3, 15, 17], and a condition on the scalar curvature is sufficient to prove the estimate.Our main result is: Theorem 1.1.
Assume S g > on M , then, for every compact K of M , there exist a positiveconstant c = c ( α, a, b, A, K, M, n, g ) such that: sup K u i × inf M u i ≤ c. Remark: in the case where ( M, g ) = (Ω ⊂ R n , δ ) an open set of the euclidean space withthe flat metric, we have the same inequality on compact sets of Ω in this case the scalar curvature S δ ≡ , see [2]. f we consider the Green function G of the Laplacian with Dirichlet condition on small ballsof M , we can have a positive lower bound for G and we have the following corollary: Corollary 1.2.
Assume S g > on M , then, for every compact K of M , there exist a positiveconstant c ′ = c ′ ( α, a, b, A, K, M, n, g ) such that: Z K u nn − i dv g ≤ c ′ .
2. P
ROOF OF THE THEOREM .Let us consider x ∈ M , by a conformal change of the metric ˜ g = ϕ / ( n − g with ϕ > wecan consider an equation of type:(2) ∆ ˜ g u + R ˜ g u = V u N − + ϕ α +1 − N u α + R g ϕ − N u, u > . with, Ricci ˜ g ( x ) = 0 . Here; R g = n − n − S g and R ˜ g = n − n − S ˜ g It is clear see the computations in a previous paper [2], it is sufficient to consider an equationof type:(3) ∆ ˜ g u = V u N − + u α + µu, u > . with, Ricci ≡ and µ > . Part I: The metric in polar coordinates.
Let ( M, g ) a Riemannian manifold. We note g x,ij the local expression of the metric g in theexponential map centered in x .We are concerning by the polar coordinates expression of the metric. Using Gauss lemma, wecan write: g = ds = dt + g kij ( r, θ ) dθ i dθ j = dt + r ˜ g kij ( r, θ ) dθ i dθ j = g x,ij dx i dx j , in a polar chart with origin x ”, ]0 , ǫ [ × U k , with ( U k , ψ ) a chart of S n − . We can write theelement volume: dV g = r n − q | ˜ g k | drdθ . . . dθ n − = q [ det ( g x,ij )] dx . . . dx n , then, dV g = r n − q [ det ( g x,ij )][exp x ( rθ )] α k ( θ ) drdθ . . . dθ n − , where, α k is such that, dσ S n − = α k ( θ ) dθ . . . dθ n − . (Riemannian volume element of thesphere in the chart ( U k , ψ ) ).Then, q | ˜ g k | = α k ( θ ) q [ det ( g x,ij )] . Clearly, we have the following proposition: roposition 2.1. Let x ∈ M , there exist ǫ > and if we reduce U k , we have: | ∂ r ˜ g kij ( x, r, θ ) | + | ∂ r ∂ θ m ˜ g kij ( x, r, θ ) | ≤ Cr, ∀ x ∈ B ( x , ǫ ) ∀ r ∈ [0 , ǫ ] , ∀ θ ∈ U k . and, | ∂ r | ˜ g k | ( x, r, θ ) | + ∂ r ∂ θ m | ˜ g k | ( x, r, θ ) ≤ Cr, ∀ x ∈ B ( x , ǫ ) ∀ r ∈ [0 , ǫ ] , ∀ θ ∈ U k . Remark: ∂ r [log p | ˜ g k | ] is a local function of θ , and the restriction of the global function on the sphere S n − , ∂ r [log p det ( g x,ij )] . We will note, J ( x, r, θ ) = p det ( g x,ij ) . Part II: The laplacian in polar coordinates
Let’s write the laplacian in [0 , ǫ ] × U k , − ∆ = ∂ rr + n − r ∂ r + ∂ r [log q | ˜ g k | ] ∂ r + 1 r p | ˜ g k | ∂ θ i (˜ g θ i θ j q | ˜ g k | ∂ θ j ) . We have, − ∆ = ∂ rr + n − r ∂ r + ∂ r log J ( x, r, θ ) ∂ r + 1 r p | ˜ g k | ∂ θ i (˜ g θ i θ j q | ˜ g k | ∂ θ j ) . We write the laplacian ( radial and angular decomposition), − ∆ = ∂ rr + n − r ∂ r + ∂ r [log J ( x, r, θ )] ∂ r − ∆ S r ( x ) , where ∆ S r ( x ) is the laplacian on the sphere S r ( x ) .We set L θ ( x, r )( ... ) = r ∆ S r ( x ) ( ... )[exp x ( rθ )] , clearly, this operator is a laplacian on S n − for particular metric. We write, L θ ( x, r ) = ∆ g x,r, S n − , and, ∆ = ∂ rr + n − r ∂ r + ∂ r [ J ( x, r, θ )] ∂ r − r L θ ( x, r ) . If, u is function on M , then, ¯ u ( r, θ ) = u [exp x ( rθ )] is the corresponding function in polarcoordinates centered in x . We have,(4) − ∆ u = ∂ rr ¯ u + n − r ∂ r ¯ u + ∂ r [ J ( x, r, θ )] ∂ r ¯ u − r L θ ( x, r )¯ u. Part III: ”Blow-up” and ”Moving-plane” methodsThe ”blow-up” analysis
Let, ( u i ) i a sequence of functions on M such that,(5) ∆ ˜ g u i + R ˜ g u i = V i u N − i + ϕ α − N u αi + R g ϕ − N u i , u i > , N = 2 nn − , It is sufficient to consider an equation of type:(6) ∆ ˜ g u = V u N − + u α + µu, u > . with Ricci ≡ and µ > .We argue by contradiction and we suppose that sup × inf is not bounded.We assume that: ∀ c, R > ∃ u c,R solution of ( E ) such that: n − sup B ( x ,R ) u c,R × inf M u c,R ≥ c. ( H ) Proposition 2.2.
There exist a sequence of points ( y i ) i , y i → x and two sequences of positivereal number ( l i ) i , ( L i ) i , l i → , L i → + ∞ , such that if we consider v i ( y ) = u i [exp y i ( y )] u i ( y i ) , wehave: i ) 0 < v i ( y ) ≤ β i ≤ ( n − / , β i → .ii ) v i ( y ) → (cid:18)
11 + | y | (cid:19) ( n − / , uniformly on every compact set of R n .iii ) l ( n − / i [ u i ( y i )] × inf M u i → + ∞ Proof:
Without loss of generality, we can assume that: V ( x ) = n ( n − . We use the hypothesis ( H ) . We can take two sequences R i > , R i → and c i → + ∞ , suchthat, R i ( n − sup B ( x ,R i ) u i × inf M u i ≥ c i → + ∞ . Let, x i ∈ B ( x , R i ) , such that sup B ( x ,R i ) u i = u i ( x i ) and s i ( x ) = [ R i − d ( x, x i )] ( n − / u i ( x ) , x ∈ B ( x i , R i ) . Then, x i → x .We have, max B ( x i ,R i ) s i ( x ) = s i ( y i ) ≥ s i ( x i ) = R i ( n − / u i ( x i ) ≥ √ c i → + ∞ . Set : l i = R i − d ( y i , x i ) , ¯ u i ( y ) = u i [exp y i ( y )] , v i ( z ) = u i [exp y i (cid:0) z/ [ u i ( y i )] / ( n − (cid:1) ] u i ( y i ) . Clearly, y i → x . We obtain: L i = l i ( c i ) / n − [ u i ( y i )] / ( n − = [ s i ( y i )] / ( n − c / n − i ≥ c / ( n − i c / n − i = c / n − i → + ∞ . If | z | ≤ L i , then y = exp y i [ z/ [ u i ( y i )] / ( n − ] ∈ B ( y i , δ i l i ) with δ i = 1( c i ) / n − and d ( y, y i ) < R i − d ( y i , x i ) , thus, d ( y, x i ) < R i and, s i ( y ) ≤ s i ( y i ) , we can write, u i ( y )[ R i − d ( y, y i )] ( n − / ≤ u i ( y i )( l i ) ( n − / . But, d ( y, y i ) ≤ δ i l i , R i > l i and R i − d ( y, y i ) ≥ R i − δ i l i > l i − δ i l i = l i (1 − δ i ) , we obtain, < v i ( z ) = u i ( y ) u i ( y i ) ≤ (cid:20) l i l i (1 − δ i ) (cid:21) ( n − / ≤ ( n − / . We set, β i = (cid:18) − δ i (cid:19) ( n − / , clearly β i → .The function v i is solution of: g jk [exp y i ( y )] ∂ jk v i − ∂ k h g jk p | g | i [exp y i ( y )] ∂ j v i = 1[ u i ( y i )] N − − α v αi + µ [ u i ( y i )] / ( n − v i + V i v iN − , By elliptic estimates and Ascoli, Ladyzenskaya theorems, ( v i ) i converge uniformely on eachcompact to the function v solution on R n of,(7) ∆ v = n ( n − v N − , v (0) = 1 , ≤ v ≤ ≤ ( n − / , By using maximum principle, we have v > on R n , the result of Caffarelli-Gidas-Spruck (see [6]) give, v ( y ) = (cid:18)
11 + | y | (cid:19) ( n − / . We have the same properties for v i in the previouspaper [2]. Polar coordinates and ”moving-plane” method
Let, w i ( t, θ ) = e ( n − / ¯ u i ( e t , θ ) = e ( n − t/ u i o exp y i ( e t θ ) , et a ( y i , t, θ ) = log J ( y i , e t , θ ) . We set δ = ( n + 2) − ( n − α . Lemma 2.3.
The function w i is solution of: (8) − ∂ tt w i − ∂ t a∂ t w i − L θ ( y i , e t ) + cw i = V i w N − i + e δt w αi + µe t w i , with, c = c ( y i , t, θ ) = (cid:18) n − (cid:19) + n − ∂ t a. Proof:
We write: ∂ t w i = e nt/ ∂ r ¯ u i + n − w i , ∂ tt w i = e ( n +2) t/ (cid:20) ∂ rr ¯ u i + n − e t ∂ r ¯ u i (cid:21) + (cid:18) n − (cid:19) w i .∂ t a = e t ∂ r log J ( y i , e t , θ ) , ∂ t a∂ t w i = e ( n +2) t/ [ ∂ r log J∂ r ¯ u i ] + n − ∂ t aw i . the lemma is proved.Now we have, ∂ t a = ∂ t b b , b ( y i , t, θ ) = J ( y i , e t , θ ) > ,We can write, − √ b ∂ tt ( p b w i ) − L θ ( y i , e t ) w i + [ c ( t ) + b − / b ( t, θ )] w i = V i w N − i + e δt w αi + µe t w i , where, b ( t, θ ) = ∂ tt ( √ b ) = 12 √ b ∂ tt b − b ) / ( ∂ t b ) . Let, ˜ w i = p b w i , emma 2.4. The function ˜ w i is solution of: − ∂ tt ˜ w i + ∆ g yi,et, S n − ( ˜ w i ) + 2 ∇ θ ( ˜ w i ) . ∇ θ log( p b ) + ( c + b − / b − c ) ˜ w i = (9) = V i (cid:18) b (cid:19) ( N − / ˜ w N − i + e δt (cid:18) b (cid:19) ( α − / ˜ w αi + µe t ˜ w i , where, c = [ 1 √ b ∆ g yi,et, S n − ( √ b ) + |∇ θ log( √ b ) | ] . Proof:
We have: − ∂ tt ˜ w i − p b ∆ g yi,et, S n − w i + ( c + b ) ˜ w i = V i (cid:18) b (cid:19) ( N − / ˜ w N − i ++ e δt (cid:18) b (cid:19) ( α − / ˜ w αi + µe t ˜ w i , But, ∆ g yi,et, S n − ( p b w i ) = p b ∆ g yi,et, S n − w i − ∇ θ w i . ∇ θ p b + w i ∆ g yi,et, S n − ( p b ) , and, ∇ θ ( p b w i ) = w i ∇ θ p b + p b ∇ θ w i , we deduce than, p b ∆ g yi,et, S n − w i = ∆ g yi,et, S n − ( ˜ w i ) + 2 ∇ θ ( ˜ w i ) . ∇ θ log( p b ) − c ˜ w i , with c = [ 1 √ b ∆ g yi,et, S n − ( √ b ) + |∇ θ log( √ b ) | ] . The lemma is proved.
The ”moving-plane” method:
Let ξ i a real number, and suppose ξ i ≤ t . We set t ξ i = 2 ξ i − t and ˜ w ξ i i ( t, θ ) = ˜ w i ( t ξ i , θ ) .We have, − ∂ tt ˜ w ξ i i +∆ g yi,etξi S n − ( ˜ w i )+2 ∇ θ ( ˜ w ξ i i ) . ∇ θ log( p b ) ˜ w ξ i i +[ c ( t ξ i )+ b − / ( t ξ i , . ) b ( t ξ i ) − c ξ i ] ˜ w ξ i i == V i b ξ i ! ( N − / ( ˜ w ξ i i ) N − ++ e δt (cid:18) b (cid:19) ( α − / ˜ w αi + µe t ˜ w i , By using the same arguments than in [2], we have:
Proposition 2.5.
We have:
1) ˜ w i ( λ i , θ ) − ˜ w i ( λ i + 4 , θ ) ≥ ˜ k > , ∀ θ ∈ S n − . For all β > , there exist c β > such that:
2) 1 c β e ( n − t/ ≤ ˜ w i ( λ i + t, θ ) ≤ c β e ( n − t/ , ∀ t ≤ β, ∀ θ ∈ S n − . e set, ¯ Z i = − ∂ tt ( ... ) + ∆ g yi,et, S n − ( ... ) + 2 ∇ θ ( ... ) . ∇ θ log( p b ) + ( c + b − / b − c )( ... ) Remark:
In the operator ¯ Z i , by using the proposition 3, the coeficient c + b − / b − c satisfies: c + b − / b − c ≥ k ′ > , pour t << , it is fundamental if we want to apply Hopf maximum principle.We set δ = ( n + 2) − ( n − α . Goal:
Like in [2], we have elliptic second order operator. Here it is ¯ Z i , the goal is to use the ”moving-plane” method to have a contradiction. For this, we must have: ¯ Z i ( ˜ w ξ i i − ˜ w i ) ≤ , if ˜ w ξ i i − ˜ w i ≤ . We write: ¯ Z i ( ˜ w ξ i i − ˜ w i ) = (∆ g yi,etξi , S n − − ∆ g yi,et, S n − )( ˜ w ξ i i )++2( ∇ θ,e tξi − ∇ θ,e t )( w ξ i i ) . ∇ θ,e tξi log( q b ξ i ) + 2 ∇ θ,e t ( ˜ w ξ i i ) . ∇ θ,e tξi [log( q b ξ i ) − log p b ]++2 ∇ θ,e t w ξ i i . ( ∇ θ,e tξi − ∇ θ,e t ) log p b − [( c + b − / b − c ) ξ i − ( c + b − / b − c )] ˜ w ξ i i ++ V ξ i i b ξ i ! ( N − / ( ˜ w ξ i i ) N − − V i (cid:18) b (cid:19) ( N − / ˜ w N − i ++ e δt ξi b ξ i − α ) / ( ˜ w ξ i i ) α − e δt b (1 − α ) / ( ˜ w i ) α + µ ( e t ξi ˜ w ξ i i − e t ˜ w i ) ( ∗ ∗ ∗ Clearly, we have:
Lemma 2.6. b ( y i , t, θ ) = 1 − Ricci y i ( θ, θ ) e t + . . . ,R g ( e t θ ) = R g ( y i )+ < ∇ R g ( y i ) | θ > e t + . . . . According to proposition 1 and lemma 3,
Proposition 2.7. ¯ Z i ( ˜ w ξ i i − ˜ w i ) ≤ ˜ A ( e t − e t ξ )( ˜ w ξ i i ) N − ) + (1 / e δt ξi − e δt )( ˜ w ξ i i ) α ++ C | e t − e t ξi | h |∇ θ ˜ w ξ i i | + |∇ θ ( ˜ w ξ i i ) | + o (1)[ ˜ w ξ i i + ( ˜ w ξ i i ) N − + ( ˜ w ξ i i ) α ] + µ ˜ w ξ i i i . Proof:
We use proposition 1, we have: a ( y i , t, θ ) = log J ( y i , e t , θ ) = log b , | ∂ t b ( t ) | + | ∂ tt b ( t ) | + | ∂ tt a ( t ) | ≤ Ce t , and, | ∂ θ j b | + | ∂ θ j ,θ k b | + ∂ t,θ j b | + | ∂ t,θ j ,θ k b | ≤ Ce t , then, ∂ t b ( t ξ i ) − ∂ t b ( t ) | ≤ C ′ | e t − e t ξi | , on ] − ∞ , log ǫ ] × S n − , ∀ x ∈ B ( x , ǫ ) Locally, ∆ g yi,et, S n − = L θ ( y i , e t ) = − p | ˜ g k ( e t , θ ) | ∂ θ l [˜ g θ l θ j ( e t , θ ) q | ˜ g k ( e t , θ ) | ∂ θ j ] . Thus, in [0 , ǫ ] × U k , we have, A i = " p | ˜ g k | ∂ θ l (˜ g θ l θ j q | ˜ g k | ∂ θ j ) ξ i − p | ˜ g k | ∂ θ l (˜ g θ l θ j q | ˜ g k | ∂ θ j ) ( ˜ w ξ i i ) then, A i = B i + D i with, B i = h ˜ g θ l θ j ( e t ξi , θ ) − ˜ g θ l θ j ( e t , θ ) i ∂ θ l θ j ˜ w ξ i i ( t, θ ) , and, D i = " p | ˜ g k | ( e t ξi , θ ) ∂ θ l [˜ g θ l θ j ( e t ξi , θ ) q | ˜ g k | ( e t ξi , θ )] − p | ˜ g k | ( e t , θ ) ∂ θ l [˜ g θ l θ j ( e t , θ ) q | ˜ g k | ( e t , θ )] ∂ θ j ˜ w ξ i i ( t, θ ) , we deduce, A i ≤ C k | e t − e t ξi | h |∇ θ ˜ w ξ i i | + |∇ θ ( ˜ w ξ i i ) | i , We take C = max { C i , ≤ i ≤ q } and if we use ( ∗ ∗ ∗ , we obtain proposition 4.We have: ∂ θ j w λi ( t, θ ) w λi = e ( n − λ − λ i )+( ξ i − t )] / e [( λ − λ i )+( ξ i − t )] ( ∂ θ j v i )( e [( λ − λ i )+( λ − t )] θ ) e ( n − λ − λ i )+( λ − t )] / v i [ e ( λ − λ i )+( λ − t ) θ ] ≤ ¯ C i , Also: ∂ θ j ,θ l w λi ( t, θ ) w λi = e ( n − λ − λ i )+( ξ i − t )] / e λ − λ i )+( ξ i − t )] ( ∂ θ j ,θ l v i )( e [( λ − λ i )+( λ − t )] θ ) e ( n − λ − λ i )+( λ − t )] / v i [ e ( λ − λ i )+( λ − t ) θ ] ≤ ¯ C i . where ¯ C i tends to and does not depend on λ .We have, c ( y i , t, θ ) = (cid:18) n − (cid:19) + n − ∂ t a + R g e t , ( α ) b ( t, θ ) = ∂ tt ( p b ) = 12 √ b ∂ tt b − b ) / ( ∂ t b ) , ( α ) c = [ 1 √ b ∆ g yi,et, S n − ( p b ) + |∇ θ log( p b ) | ] , ( α ) Then, ∂ t c ( y i , t, θ ) = ( n − ∂ tt a, by proposition 1, | ∂ t c | + | ∂ t b | + | ∂ t b | + | ∂ t c | ≤ K e t . We have: w i (2 ξ i − t, θ ) = w i [( ξ i − t + ξ i − λ i −
2) + ( λ i + 2)] , Thus, w i (2 ξ i − t, θ ) = e [( n − ξ i − t + ξ i − λ i − / e n − v i [ θe e ( ξ i − t )+( ξ i − λ i − ] ≤ ( n − / e n − = ¯ c. e set δ = ( n + 2) − ( n − α .The left right side are denoted Z et Z , we can write: Z = ( ¯ V ξ i i − ¯ V i )( ˜ w ξ i i ) N − + ¯ V i [( ˜ w ξ i i ) N − − ˜ w N − i ] , and, Z = e δt [( ˜ w ξ i i ) α − ( ˜ w i ) α ] + ( ˜ w ξ i i ) α ( e δt ξi − e δt ) . We can write the part with nonlinear terms as: ( ˜ w ξ i i ) α [( A w ξ i i N − − α + B ) ( e t − e t ξi ) + c ( e δt ξi − e δt )] . Because w ξ i i ≤ ¯ c , we have: − ¯ Z i (( ˜ w i ) ξ i − ( ˜ w i )) ≤ ( ˜ w ξ i i ) α [( A ¯ c N − − α + B ) ( e t − e t ξi )+ c ( e δt ξi − e δt )]+( ˜ w i ) ξ i ( µ/ e t ξi − e t )] But α ∈ ] nn − , n + 2 n − , δ = n + 2 − ( n − α ∈ ]0 , .We obtain for t ≤ t < : e t ≤ e (1 − δ ) t e δt , pour tout t ≤ t . and, t ξ i ≤ t ( ξ i ≤ t ) , we integrate: ( e δt ξi − e δt ) ≤ δe (1 − δ ) t ( e t ξi − e t ) . Finaly: − ¯ Z i (( ˜ w i ) ξ i − ( ˜ w i )) ≤ ( ˜ w ξ i i ) α [ − δ ce (1 − δ ) t + A ¯ c N − − α + B ]( e t − e t ξi )+( ˜ w i ) ξ i ( µ/ e t ξi − e t ) We apply proposition 3. We take t i = log √ l i with l i like in proposition 2. The fact √ l i [ u i ( y i )] / ( n − → + ∞ ( see proposition 2), implies t i = log √ l i > n − u i ( y i ) + 2 = λ i + 2 . Finaly, we can work on ] − ∞ , t i ] .We define ξ i by: ξ i = sup { λ ≤ λ i + 2 , ˜ w i (2 λ − t, θ ) − ˜ w i ( t, θ ) ≤ λ, t i ] × S n − } . If we use proposition 4 and the similar technics that in [B2] we can deduce by Hopf maximumprinciple, min S n − ˜ w i ( t i , θ ) ≤ max S n − ˜ w i (2 ξ i − t i , θ ) , which implies, l i ( n − / u i ( y i ) × min M u i ≤ c. It is in contradiction with proposition 2.Then we have, sup K u × inf M u ≤ c = c ( α, a, b, A, K, M, g, n ) . EFERENCES[1] T. Aubin. Some Nonlinear Problems in Riemannian Geometry. Springer-Verlag 1998[2] S.S Bahoura. Majorations du type sup u × inf u ≤ c pour l’´equation de la courbure scalaire sur un ouvert de R n , n ≥ . J. Math. Pures. Appl.(9) 83 2004 no, 9, 1109-1150.[3] H. Brezis, YY. Li Y-Y, I. Shafrir. A sup+inf inequality for some nonlinear elliptic equations involving exponentialnonlinearities. J.Funct.Anal.115 (1993) 344-358.[4] H.Brezis and F.Merle, Uniform estimates and blow-up bihavior for solutions of − ∆ u = V e u in two dimensions,Commun Partial Differential Equations 16 (1991), 1223-1253.[5] M-F. Bidaut-Veron, L. Veron. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics odEmden equations. Invent.Math. 106 (1991), no3, 489-539.[6] L. Caffarelli, B. Gidas, J. Spruck. Asymptotic symmetry and local behavior of semilinear elliptic equations withcritical Sobolev growth. Comm. Pure Appl. Math. 37 (1984) 369-402.[7] A sharp sup+inf inequality for a nonlinear elliptic equation in R . Commun. Anal. Geom. 6, No.1, 1-19 (1998).[8] C-C.Chen, C-S. Lin. Estimates of the conformal scalar curvature equation via the method of moving planes. Comm.Pure Appl. Math. L(1997) 0971-1017.[9] O. Druet, E. Hebey, F.Robert, Blow-up theory in Riemannian Geometry, Princeton University Press 2004.[10] B. Gidas, W-M. Ni, L. Nirenberg. Symmetry and Related Properties via the Maximum Principle. Commun. Math.Phys. 68, 209-243 (1979).[11] Gidas, J. Spruck. Global and Local Behavior of Positive Solutions of Nonlinear Elliptic Equations. Comm. Pure.Appl. Math. 34 (1981), no 4, 525-598.[12] D. Gilbarg, N.S. Trudinger. Elliptic Partial Differential Equations of Second order, Berlin Springer-Verlag, Secondedition, Grundlehern Math. Wiss.,224, 1983.[13] E. Hebey, Analyse non lineaire sur les Vari´et´es, Editions Diderot.[14] E. Hebey, M. Vaugon. The best constant problem in the Sobolev embedding theorem for complete Riemannianmanifolds. Duke Math. J. 79 (1995), no. 1, 235–279.[15] N. Korevaar, F. Pacard, R. Mazzeo, R. Schoen. Refined asymptotics for constant scalar curvature metrics withisolated singularities. Invent. Math. 135 (1999), no. 2, 233–272.[16] J.M. Lee, T.H. Parker. The Yamabe problem. Bull.Amer.Math.Soc (N.S) 17 (1987), no.1, 37 -91.[17] YY. Li. Harnack Type Inequality: the Method of Moving Planes. Commun. Math. Phys. 200,421-444 (1999).[18] YY. Li. Prescribing scalar curvature on S n and related Problems. C.R. Acad. Sci. Paris 317 (1993) 159-164. PartI: J. Differ. Equations 120 (1995) 319-410. Part II: Existence and compactness. Comm. Pure Appl.Math.49 (1996)541-597.[19] YY. Li, L. Zhang. A Harnack type inequality for the Yamabe equation in low dimensions. Calc. Var. Partial Differ-ential Equations 20 (2004), no. 2, 133–151.[20] F.C. Marques. A Priori Estimates for the Yamabe Problem in the non-locally conformally flat case. J. Diff. Geom.71 (2005) 315-346.[21] I. Shafrir. A sup+inf inequality for the equation − ∆ u = V e u . C. R. Acad.Sci. Paris S´er. I Math. 315 (1992), no.2, 159-164.D EPARTMENT OF M ATHEMATICS , P
IERRE ET M ARIE C URIE U NIVERSITY , 75005 P
ARIS F RANCE . E-mail address : [email protected], [email protected]@yahoo.fr, [email protected]