A compactness result for an elliptic equation in dimension 2
aa r X i v : . [ m a t h . A P ] O c t A COMPACTNESS RESULT FOR AN ELLIPTIC EQUATION IN DIMENSION 2.
SAMY SKANDER BAHOURAA
BSTRACT . We give a blow-up behavior for the solutions of an elliptic equation under some conditions. Wealso derive a compactness creterion for this equation.
Mathematics Subject Classification: 35J60 35B45 35B50
Keywords: blow-up, boundary, elliptic equation, a priori estimate, Lipschitz condition, starshaped domains.
1. I
NTRODUCTION AND M AIN R ESULTS
Let us consider the following operator: L ǫ := ∆ + ǫ ( x ∂ + x ∂ ) = div [ a ǫ ( x ) ∇ ] a ǫ ( x ) with a ǫ ( x ) = e ǫ | x | . We consider the following equation: ( P ǫ ) ( − ∆ u − ǫ ( x ∂ u + x ∂ u ) = − L ǫ u = V e u in Ω ⊂ R ,u = 0 in ∂ Ω . Here, we assume that: Ω starshaped , and, u ∈ W , (Ω) , e u ∈ L (Ω) , ≤ V ≤ b, ≥ ǫ ≥ . When ǫ = 0 the previous equation was studied by many authors with or without the boundary condition,also for Riemann surfaces see [1-20] where one can find some existence and compactness results. Also wehave a nice formulation in the sens of the distributions of this Problem in [7].Among other results, we can see in [6] the following important Theorem, Theorem A (Brezis-Merle [6]) If ( u i ) i and ( V i ) i are two sequences of functions relative to the problem ( P ) with ǫ = 0 and, < a ≤ V i ≤ b < + ∞ then it holds, sup K u i ≤ c, with c depending on a, b, K and Ω . We can find in [6] an interior estimate if we assume a = 0 but we need an assumption on the integral of e u i , namely: heorem B (Brezis-Merle [6]) . For ( u i ) i and ( V i ) i two sequences of functions relative to the problem ( P ) with, ≤ V i ≤ b < + ∞ and Z Ω e u i dy ≤ C, then it holds; sup K u i ≤ c, with c depending on b, C, K and Ω . The condition R Ω e u i dy ≤ C is a necessary condition in the Problem ( P ǫ ) as showed by the followingcounterexample for ǫ = 0 : Theorem C (Brezis-Merle [6]) . There are two sequences ( u i ) i and ( V i ) i of the problem ( P ) with; ≤ V i ≤ b < + ∞ , Z Ω e u i dy ≤ C, such that, sup Ω u i → + ∞ . To obtain the two first previous results (Theorems A and B) Brezis and Merle used an inequality (Theorem1 of [6]) obtained by an approximation argument and they used Fatou’s lemma and applied the maximumprinciple in W , (Ω) which arises from Kato’s inequality. Also this weak form of the maximum principleis used to prove the local uniform boundedness result by comparing a certain function and the Newtonianpotential. We refer to [5] for a topic about the weak form of the maximum principle.Note that for the problem ( P ) , by using the Pohozaev identity, we can prove that R Ω e u i is uniformlybounded when < a ≤ V i ≤ b < + ∞ and ||∇ V i || L ∞ ≤ A and Ω starshaped, when a = 0 and ∇ log V i isuniformly bounded, we can bound uniformly R Ω V i e u i . In [17] Ma-Wei have proved that those results staytrue for all open sets not necessarily starshaped in the case a > .In [8] Chen-Li have proved that if a = 0 and R Ω e u i is uniformly bounded and ∇ log V i is uniformlybounded then ( u i ) i is bounded near the boundary and we have directly the compactness result for the prob-lem ( P ) . Ma-Wei in [17] extend this result in the case where a > .When ǫ = 0 and if we assume V more regular we can have another type of estimates called sup + inf typeinequalities. It was proved by Shafrir see [19] that, if ( u i ) i , ( V i ) i are two sequences of functions solutionsof the Problem ( P ) without assumption on the boundary and < a ≤ V i ≤ b < + ∞ then it holds: C (cid:16) ab (cid:17) sup K u i + inf Ω u i ≤ c = c ( a, b, K, Ω) . We can see in [9] an explicit value of C (cid:16) ab (cid:17) = r ab . In his proof, Shafrir has used the blow-up function,the Stokes formula and an isoperimetric inequality see [2]. For Chen-Lin, they have used the blow-upanalysis combined with some geometric type inequality for the integral curvature see [9].Now, if we suppose ( V i ) i uniformly Lipschitzian with A its Lipschitz constant then C ( a/b ) = 1 and c = c ( a, b, A, K, Ω) see Brezis-Li-Shafrir [4]. This result was extended for H ¨olderian sequences ( V i ) i by hen-Lin see [9]. Also have in [15], an extension of the Brezis-Li-Shafrir result to compact Riemanniansurfaces without boundary. One can see in [16] explicit form, ( πm, m ∈ N ∗ exactly), for the numbers infront of the Dirac masses when the solutions blow-up. Here the notion of isolated blow-up point is used.Also one can see in [10] refined estimates near the isolated blow-up points and the bubbling behavior of theblow-up sequences.Here we give the behavior of the blow-up points on the boundary and a proof of a compactness result withLipschitz condition. Note that our problem is an extension of the Brezis-Merle Problem.The Brezis-Merle Problem (see [6]) is: Problem.
Suppose that V i → V in C ( ¯Ω) with ≤ V i . Also, we consider a sequence of solutions ( u i ) of ( P ) relative to ( V i ) such that, Z Ω e u i dx ≤ C, is it possible to have: || u i || L ∞ ≤ C = C ( b, C, V, Ω)?
Here we give blow-up analysis on the boundary when V (similar to the prescribed curvature when ǫ = 0 )are nonegative and bounded, and on the other hand, if we add the assumption that these functions (similar tothe prescribed cruvature) are uniformly Lipschitzian, we have a compactness of the solutions of the problem ( P ǫ ) for ǫ small enough. (In particular we can take a sequence of ǫ i tending to ):For the behavior of the blow-up points on the boundary, the following condition is sufficient, ≤ V i ≤ b, The condition V i → V in C ( ¯Ω) is not necessary. But for the compactness of the solutions we add thefollowing condition: ||∇ V i || L ∞ ≤ A. Our main results are:
Theorem 1.1.
Assume that max Ω u i → + ∞ , where ( u i ) are solutions of the probleme ( P ǫ i ) with: ≤ V i ≤ b, and Z Ω e u i dx ≤ C, ǫ i → , then, after passing to a subsequence, there is a finction u , there is a number N ∈ N and N points x , . . . , x N ∈ ∂ Ω , such that, ∂ ν u i → ∂ ν u + N X j =1 α j δ x j , α j ≥ π, in the sens of measures on ∂ Ω .u i → u in C loc ( ¯Ω − { x , . . . , x N } ) . Theorem 1.2.
Assume that ( u i ) are solutions of ( P ǫ i ) relative to ( V i ) with the following conditions: ≤ V i ≤ b, ||∇ V i || L ∞ ≤ A and Z Ω e u i ≤ C, ǫ i → . Then we have: || u i || L ∞ ≤ c ( b, A, C, Ω) , . P ROOF OF THE THEOREMS
Proof of theorem 1.1:
First remark that: ( − ∆ u i = ǫ i ( x ∂ u i + x ∂ u i ) + V i e u i ∈ L (Ω) in Ω ⊂ R ,u i = 0 in ∂ Ω . and, u i ∈ W , (Ω) . By the corollary 1 of Brezis-Merle see [6] we have e u i ∈ L k (Ω) for all k > and the elliptic estimates ofAgmon and the Sobolev embedding see [1] imply that: u i ∈ W ,k (Ω) ∩ C ,ǫ ( ¯Ω) . Also remark that, we have for two positive constants C q = C ( q, Ω) and C = C (Ω) (see [7]) : ||∇ u i || L q ≤ C q || ∆ u i || L ≤ ( C ′ q + ǫC ||∇ u i || L ) , ∀ i and 1 < q < . Thus, if ǫ > is small enough and by the Holder inequality, we have the following estimate: ||∇ u i || L q ≤ C ′′ q , ∀ i and 1 < q < . Step 1: interior estimate
First remark that, if we consider the following equation: ( − ∆ w i = ǫ i ( x ∂ u i + x ∂ u i ) ∈ L q , < q < in Ω ⊂ R ,w i = 0 in ∂ Ω . If we consider v i the Newtonnian potential of ǫ i ( x ∂ u i + x ∂ u i ) , we have: v i ∈ C ( ¯Ω) , ∆( w i − v i ) = 0 . By the maximum principle w i − v i ∈ C ( ¯Ω) and thus w i ∈ C ( ¯Ω) .Also we have by the elliptic estimates that w i ∈ W , ǫ ⊂ L ∞ , and we can write the equation of theProblem as: − ∆( u i − w i ) = ˜ V i e u i − w i in Ω ⊂ R ,u i − w i = 0 in ∂ Ω . with, ≤ ˜ V i = V i e w i ≤ ˜ b, Z Ω e u i − w i ≤ ˜ C. We apply the Brezis-Merle theorem to u i − w i to have: u i − w i ∈ L ∞ loc (Ω) , and, thus: u i ∈ L ∞ loc (Ω) . Step2: boundary estimate
Set ∂ ν u i the inner derivative of u i . By the maximum principle ∂ ν u i ≥ .We have: Z ∂ Ω ∂ ν u i dσ ≤ C. We have the existence of a nonnegative Radon measure µ such that, Z ∂ Ω ∂ ν u i ϕdσ → µ ( ϕ ) , ∀ ϕ ∈ C ( ∂ Ω) . We take an x ∈ ∂ Ω such that, µ ( x ) < π . Set B ( x , ǫ ) ∩ ∂ Ω := I ǫ . We choose a function η ǫ such that, η ǫ ≡ , on I ǫ , < ǫ < δ/ ,η ǫ ≡ , outside I ǫ , ≤ η ǫ ≤ , ||∇ η ǫ || L ∞ ( I ǫ ) ≤ C (Ω , x ) ǫ . We take a ˜ η ǫ such that, ( − ∆˜ η ǫ = 0 in Ω ⊂ R , ˜ η ǫ = η ǫ in ∂ Ω . emark: We use the following steps in the construction of ˜ η ǫ :We take a cutoff function η in B (0 , or B ( x , :1- We set η ǫ ( x ) = η ( | x − x | /ǫ ) in the case of the unit disk it is sufficient.2- Or, in the general case: we use a chart ( f, ˜Ω) with f (0) = x and we take µ ǫ ( x ) = η ( f ( | x | /ǫ )) to haveconnected sets I ǫ and we take η ǫ ( y ) = µ ǫ ( f − ( y )) . Because f, f − are Lipschitz, | f ( x ) − x | ≤ k | x | ≤ for | x | ≤ /k and | f ( x ) − x | ≥ k | x | ≥ for | x | ≥ /k > /k , the support of η is in I (2 /k ) ǫ . η ǫ ≡ , on f ( I (1 /k ) ǫ ) , < ǫ < δ/ ,η ǫ ≡ , outside f ( I (2 /k ) ǫ ) , ≤ η ǫ ≤ , ||∇ η ǫ || L ∞ ( I (2 /k ǫ ) ≤ C (Ω , x ) ǫ .
3- Also, we can take: µ ǫ ( x ) = η ( | x | /ǫ ) and η ǫ ( y ) = µ ǫ ( f − ( y )) , we extend it by outside f ( B (0)) .We have f ( B (0)) = D ( x ) , f ( B ǫ (0)) = D ǫ ( x ) and f ( B + ǫ ) = D + ǫ ( x ) with f and f − smooth diffeo-morphism. η ǫ ≡ , on a the connected set J ǫ = f ( I ǫ ) , < ǫ < δ/ ,η ǫ ≡ , outside J ′ ǫ = f ( I ǫ ) , ≤ η ǫ ≤ , ||∇ η ǫ || L ∞ ( J ′ ǫ ) ≤ C (Ω , x ) ǫ . And, H ( J ′ ǫ ) ≤ C H ( I ǫ ) = C ǫ , because f is Lipschitz. Here H is the Hausdorff measure.We solve the Dirichlet Problem: ( ∆¯ η ǫ = ∆ η ǫ in Ω ⊂ R , ¯ η ǫ = 0 in ∂ Ω . and finaly we set ˜ η ǫ = − ¯ η ǫ + η ǫ . Also, by the maximum principle and the elliptic estimates we have : ||∇ ˜ η ǫ || L ∞ ≤ C ( || η ǫ || L ∞ + ||∇ η ǫ || L ∞ + || ∆ η ǫ || L ∞ ) ≤ C ǫ , with C depends on Ω .As we said in the beguening, see also [3, 7, 13, 20], we have: ||∇ u i || L q ≤ C q , ∀ i and 1 < q < . e deduce from the last estimate that, ( u i ) converge weakly in W ,q (Ω) , almost everywhere to a function u ≥ and R Ω e u < + ∞ (by Fatou lemma). Also, V i weakly converge to a nonnegative function V in L ∞ .The function u is in W ,q (Ω) solution of : ( − ∆ u = V e u ∈ L (Ω) in Ω ⊂ R ,u = 0 in ∂ Ω . According to the corollary 1 of Brezis-Merle result, see [6], we have e ku ∈ L (Ω) , k > . By the ellipticestimates, we have u ∈ W ,k (Ω) ∩ C ,ǫ ( ¯Ω) .We denote by f · g the inner product of any two vectors f and g of R .We can write, − ∆(( u i − u )˜ η ǫ ) = ( V i e u i − V e u )˜ η ǫ − ∇ ( u i − u ) · ∇ ˜ η ǫ + ǫ i ( ∇ u i · x )˜ η ǫ . (1)We use the interior esimate of Brezis-Merle, see [6], Step 1:
Estimate of the integral of the first term of the right hand side of (1) .We use the Green formula between ˜ η ǫ and u , we obtain, Z Ω V e u ˜ η ǫ dx = Z ∂ Ω ∂ ν uη ǫ ≤ Cǫ = O ( ǫ ) (2)We have, ( − ∆ u i − ǫ i ∇ u i · x = V i e u i in Ω ⊂ R ,u = 0 in ∂ Ω . We use the Green formula between u i and ˜ η ǫ to have: Z Ω V i e u i ˜ η ǫ dx = Z ∂ Ω ∂ ν u i η ǫ dσ − ǫ i Z Ω ( ∇ u i · x )˜ η ǫ == Z ∂ Ω ∂ ν u i η ǫ dσ + o (1) → µ ( η ǫ ) ≤ µ ( J ′ ǫ ) ≤ π − ǫ , ǫ > (3)From (2) and (3) we have for all ǫ > there is i such that, for i ≥ i , Z Ω | ( V i e u i − V e u )˜ η ǫ | dx ≤ π − ǫ + Cǫ (4)Step 2.1: Estimate of integral of the second term of the right hand side of (1) . et Σ ǫ = { x ∈ Ω , d ( x, ∂ Ω) = ǫ } and Ω ǫ = { x ∈ Ω , d ( x, ∂ Ω) ≥ ǫ } , ǫ > . Then, for ǫ small enough, Σ ǫ is an hypersurface.The measure of Ω − Ω ǫ is k ǫ ≤ meas (Ω − Ω ǫ ) = µ L (Ω − Ω ǫ ) ≤ k ǫ . Remark : for the unit ball ¯ B (0 , , our new manifold is ¯ B (0 , − ǫ ) .(Proof of this fact; let’s consider d ( x, ∂ Ω) = d ( x, z ) , z ∈ ∂ Ω , this imply that ( d ( x, z )) ≤ ( d ( x, z )) for all z ∈ ∂ Ω which it is equivalent to ( z − z ) · (2 x − z − z ) ≤ for all z ∈ ∂ Ω , let’s consider a chartaround z and γ ( t ) a curve in ∂ Ω , we have; ( γ ( t ) − γ ( t ) · (2 x − γ ( t ) − γ ( t )) ≤ if we divide by ( t − t ) (with the sign and tend t to t ), we have γ ′ ( t ) · ( x − γ ( t )) = 0 , this imply that x = z − sν where ν is the outward normal of ∂ Ω at z ))With this fact, we can say that S = { x, d ( x, ∂ Ω) ≤ ǫ } = { x = z − sν z , z ∈ ∂ Ω , − ǫ ≤ s ≤ ǫ } . It issufficient to work on ∂ Ω . Let’s consider a charts ( z, D = B ( z, ǫ z ) , γ z ) with z ∈ ∂ Ω such that ∪ z B ( z, ǫ z ) is cover of ∂ Ω . One can extract a finite cover ( B ( z k , ǫ k )) , k = 1 , ..., m , by the area formula the measureof S ∩ B ( z k , ǫ k ) is less than a kǫ (a ǫ -rectangle). For the reverse inequality, it is sufficient to consider onechart around one point of the boundary).We write, Z Ω |∇ ( u i − u ) · ∇ ˜ η ǫ | dx = Z Ω ǫ |∇ ( u i − u ) · ∇ ˜ η ǫ | dx + Z Ω − Ω ǫ |∇ ( u i − u ) · ∇ ˜ η ǫ | dx. (5) Step 2.1.1:
Estimate of R Ω − Ω ǫ |∇ ( u i − u ) · ∇ ˜ η ǫ | dx .First, we know from the elliptic estimates that ||∇ ˜ η ǫ || L ∞ ≤ C /ǫ , C depends on Ω We know that ( |∇ u i | ) i is bounded in L q , < q < , we can extract from this sequence a subsequencewhich converge weakly to h ∈ L q . But, we know that we have locally the uniform convergence to |∇ u | (bythe Brezis-Merle’s theorem), then, h = |∇ u | a.e. Let q ′ be the conjugate of q .We have, ∀ f ∈ L q ′ (Ω) Z Ω |∇ u i | f dx → Z Ω |∇ u | f dx If we take f = 1 Ω − Ω ǫ , we have: for ǫ > ∃ i = i ( ǫ ) ∈ N , i ≥ i , Z Ω − Ω ǫ |∇ u i | ≤ Z Ω − Ω ǫ |∇ u | + ǫ . Then, for i ≥ i ( ǫ ) , Ω − Ω ǫ |∇ u i | ≤ meas (Ω − Ω ǫ ) ||∇ u || L ∞ + ǫ = ǫ ( k ||∇ u || L ∞ + 1) = O ( ǫ ) . Thus, we obtain, Z Ω − Ω ǫ |∇ ( u i − u ) · ∇ ˜ η ǫ | dx ≤ ǫC (2 k ||∇ u || L ∞ + 1) = O ( ǫ ) (6)The constant C does not depend on ǫ but on Ω . Step 2.1.2:
Estimate of R Ω ǫ |∇ ( u i − u ) · ∇ ˜ η ǫ | dx .We know that, Ω ǫ ⊂⊂ Ω , and ( because of Brezis-Merle’s interior estimates) u i → u in C (Ω ǫ ) . Wehave, ||∇ ( u i − u ) || L ∞ (Ω ǫ ) ≤ ǫ , for i ≥ i . We write, Z Ω ǫ |∇ ( u i − u ) · ∇ ˜ η ǫ | dx ≤ ||∇ ( u i − u ) || L ∞ (Ω ǫ ) ||∇ ˜ η ǫ || L ∞ = C ǫ = O ( ǫ ) for i ≥ i , For ǫ > , we have for i ∈ N , i ≥ i ′ , Z Ω |∇ ( u i − u ) · ∇ ˜ η ǫ | dx ≤ ǫC (2 k ||∇ u || L ∞ + 2) = O ( ǫ ) (7)From (4) and (7) , we have, for ǫ > , there is i ′′ such that, i ≥ i ′′ , Z Ω | ∆[( u i − u )˜ η ǫ ] | dx ≤ π − ǫ + ǫ C (2 k ||∇ u || L ∞ + 2 + C ) = 4 π − ǫ + O ( ǫ ) (8)We choose ǫ > small enough to have a good estimate of (1) .Indeed, we have: ( − ∆[( u i − u )˜ η ǫ ] = g i,ǫ in Ω ⊂ R , ( u i − u )˜ η ǫ = 0 in ∂ Ω . with || g i,ǫ || L (Ω) ≤ π − ǫ / . We can use Theorem 1 of [6] to conclude that there are q ≥ ˜ q > such that: V ǫ ( x ) e ˜ q | u i − u | dx ≤ Z Ω e q | u i − u | ˜ η ǫ dx ≤ C ( ǫ, Ω) . where, V ǫ ( x ) is a neighberhooh of x in ¯Ω . Here we have used that in a neighborhood of x by theelliptic estimates, − Cǫ ≤ ˜ η ǫ ≤ .Thus, for each x ∈ ∂ Ω − { ¯ x , . . . , ¯ x m } there is ǫ > , q > such that: Z B ( x ,ǫ ) e q u i dx ≤ C, ∀ i. By the elliptic estimate see [14] we have: || u i || C ,θ [ B ( x ,ǫ )] ≤ c ∀ i. We have proved that, there is a finite number of points ¯ x , . . . , ¯ x m such that the squence ( u i ) i is locallyuniformly bounded in C ,θ , ( θ > on ¯Ω − { ¯ x , . . . , ¯ x m } . Proof of theorem 1.2:
The Pohozaev identity gives : Z ∂ Ω
12 ( x · ν )( ∂ ν u i ) dσ + ǫ Z Ω ( x · ∇ u i ) dx + Z ∂ Ω ( x · ν ) V i e u i dσ = Z Ω ( x · ∇ V i + 2 V i ) e u i dx We use the boundary condition and the fact that Ω is starshaped and the fact that ǫ > to have that: Z ∂ Ω ( ∂ ν u i ) dx ≤ c ( b, A, C, Ω) . (9)Thus we can use the weak convergence in L ( ∂ Ω) to have a subsequence ∂ ν u i , such that: Z ∂ Ω ∂ ν u i ϕdx → Z ∂ Ω ∂ ν uϕdx, ∀ ϕ ∈ L ( ∂ Ω) , Thus, α j = 0 , j = 1 , . . . , N and ( u i ) is uniformly bounded. Remark 1:
Note that if we assume the open set bounded starshaped and V i uniformly Lipschitzian andbetween two positive constants we can bound, by using the inner normal derivative R Ω e u i . Remark 2:
If we assume the open set bounded starshaped and ∇ log V i uniformly bounded, by theprevious Pohozaev identity (we consider the inner normal derivative) one can bound R Ω V i e u i uniformly. emark 3: One can consider the problem on the unit ball and an ellipse. These two problems aredifferents, because:1) if we use a linear transformation, ( y , y ) = ( x /a, x /b ) , the Laplcian is not invariant under this map.2) If we use a conformal transformation, by a Riemann theorem, the quantity x · ∇ u is not invariant underthis map.We can not use, after using those transofmations, the Pohozaev identity.3. A COUNTEREXAMPLE
We start with the notation of the counterexample of Brezis and Merle.The domain Ω is the unit ball centered in (1 , .Lets consider z i (obtained by the variational method), such that: − ∆ z i − ǫ i x · ∇ z i = − L ǫ i ( z i ) = f ǫ i . With Dirichlet condition. By the regularity theorem we have z i ∈ C ( ¯Ω) .We have: || f ǫ i || = 4 πA. Thus by the duality theorem of Stampacchia or Brezis-Strauss, we have: ||∇ z i || q ≤ C q , ≤ q < . We solve: − ∆ w i = ǫ i x · ∇ z i , With Dirichlet condition.By the elliptic estimates, w i ∈ C ( ¯Ω) and w i ∈ C ( ¯Ω) uniformly.By the maximum principle we have: z i − w i ≡ u i . Where u i is the function of the counterexemple of Brezis Merle.We write: ∆ z i − ǫ i x · ∇ z i = f ǫ i = V i e z i . Thus, we have: Z Ω e z i ≤ C , and 0 ≤ V i ≤ C , and, z i ( a i ) ≥ u i ( a i ) − C → + ∞ , a i → O. R EFERENCES [1] T. Aubin. Some Nonlinear Problems in Riemannian Geometry. Springer-Verlag 1998[2] C. Bandle. Isoperimetric Inequalities and Applications. Pitman, 1980.[3] L. Boccardo, T. Gallouet. Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 no 1, (1989),149-169.[4] H. Brezis, YY. Li and I. Shafrir. A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlineari-ties. J.Funct.Anal.115 (1993) 344-358.[5] Brezis. H, Marcus. M, Ponce. A. C. Nonlinear elliptic equations with measures revisited. Mathematical aspects of nonlineardispersive equations, 55-109, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.[6] H. Brezis, F. Merle. Uniform estimates and Blow-up behavior for solutions of − ∆ u = V ( x ) e u in two dimension. Commun.in Partial Differential Equations, 16 (8 and 9), 1223-1253(1991).Comm.Part.Diff. Equations. 1991.[7] H. Brezis, W. A. Strauss. Semi-linear second-order elliptic equations in L1. J. Math. Soc. Japan 25 (1973), 565-590.[8] W. Chen, C. Li. A priori estimates for solutions to nonlinear elliptic equations. Arch. Rational. Mech. Anal. 122 (1993)145-157.[9] C-C. Chen, C-S. Lin. A sharp sup+inf inequality for a nonlinear elliptic equation in R . Commun. Anal. Geom. 6, No.1, 1-19(1998).[10] C-C.Chen, C-S. Lin. Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Comm. Pure Appl. Math.55 (2002), no. 6, 728-771.[11] Chang, Sun-Yung A, Gursky, Matthew J, Yang, Paul C. Scalar curvature equation on - and -spheres. Calc. Var. PartialDifferential Equations 1 (1993), no. 2, 205-229.[12] D.G. De Figueiredo, P.L. Lions, R.D. Nussbaum, A priori Estimates and Existence of Positive Solutions of Semilinear EllipticEquations, J. Math. Pures et Appl., vol 61, 1982, pp.41-63.[13] Ding.W, Jost. J, Li. J, Wang. G. The differential equation ∆ u = 8 π − πhe u on a compact Riemann surface. Asian J. Math.1 (1997), no. 2, 230-248.[14] D. Gilbarg, N. S, Trudinger. Elliptic Partial Differential Equations of Second order, Berlin Springer-Verlag.[15] YY. Li. Harnack Type Inequality: the method of moving planes. Commun. Math. Phys. 200,421-444 (1999).[16] YY. Li, I. Shafrir. Blow-up analysis for solutions of − ∆ u = V e u in dimension two. Indiana. Math. J. Vol 3, no 4. (1994).1255-1270.[17] L. Ma, J-C. Wei. Convergence for a Liouville equation. Comment. Math. Helv. 76 (2001) 506-514.[18] Nagasaki, K, Suzuki,T. Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominatednonlinearities. Asymptotic Anal. 3 (1990), no. 2, 173-188.[19] 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.[20] Tarantello, G. Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys. 37 (1996), no. 8, 3769-3796.D EPARTEMENT DE M ATHEMATIQUES , U
NIVERSITE P IERRE ET M ARIE C URIE , 2
PLACE J USSIEU , 75005, P
ARIS , F
RANCE . E-mail address : [email protected], [email protected]@yahoo.fr, [email protected]