A decomposition result for a singular elliptic equation on compact Riemannian manifolds
aa r X i v : . [ m a t h . A P ] J a n A DECOMPOSITION RESULT FOR A SINGULAR ELLIPTICEQUATION ON COMPACT RIEMANNIAN MANIFOLDS.
Y. MALIKI ∗ AND F.Z. TERKI
Abstract.
On compact Riemannian manifolds, we prove a decompositiontheorem for arbitrarily bounded energy sequence of solutions of a singularelliptic equation. Introduction
Let (
M, g ) be an ( n ≥ − dimensional Riemannian manifold. In this paper,we are interested in studying on ( M, g ) the asymptotic behaviour of a sequence ofsolutions u α , when α → ∞ , of the following singular elliptic equation:( E α ) ∆ g u − h α ρ p ( x ) u = f ( x ) | u | ∗ − u, where 2 ∗ = nn − , h α and f are functions on M , p is a fixed point of M and ρ p ( x ) = dist g ( p, x ) is the distance function on M based at p ( see definition (2.1)).Certainly, if the singular term h α ρ p ( x ) is replaced by n − n − Scal g , then equation E α becomes the famous prescribed scalar curvature equation which is very known inthe literature. When f is constant and the function ρ p is of power 0 < γ <
2, equa-tion ( E α ) can be seen as a case of equations that arise in the study of conformaldeformation to constant scalar curvature of metrics which are smooth only in someball B p ( δ ) ( see [4]).Equations of type ( E α ) have been the subject of interest especially on the Eu-clidean space IR n . A famous result has been obtained in [7] and it consists of theclassification of positive solutions of the equation( E ) ∆ u − λ | x | = u n +2 n − , where 0 < λ < ( n − , into the family of functions u λ ( x ) = C λ | x | a − | x | a ! n − . where c λ is some constant and a = q − λ ( n − . In terms of decomposition of Palais-Smale sequences of functional energy, this fam-ily of solutions was employed in [5] in constructing singularity bubbles, B ε α ,y α λ = ε − n α u λ ( x − y α ε α ) , with | y α | ε α → , which, together with the classical bubbles caused by the existence of critical expo-nent B ε α ,y α = ε − n α u ( x − y α ε α ) , with | y α | ε α → ∞ , where u being the solution of the non perturbed equation ∆ u = u n +2 n − give a wholepicture of the decomposition of the Palaise-Smale sequences. This decompositionresult has been proved in [5] and was the key component for the obtention of in-teresting existence results for equation ( E ) with a function K get involved in thenonlinear term. Similar decomposition result has been obtained in [1] for equation( E ) with small perturbation, the authors described asymptotically the associatedPalais-Smale sequences of bounded energy.The compactness result obtained in this paper can be seen as an extension to Rie-mannian context of those obtained in [5] and [1] in the Euclidean context, thedifficulties when working in the Riemannian setting reside mainly in the construc-tion of bubbles.Historically, a famous compactness result for elliptic value problems on domains of R n has been obtained by M.Struwe in [6]. Struwe’s result has been extended laterby O.Druet et al. in [2] to elliptic equations on Riemannian manifolds like∆ g u + h α u = u ∗ − . Many results have been obtained by the authors describing the asymptotic be-haviour of Palais-Smale sequences. The authors gave a detailed construction ofbubbles by means of a re-scaling process via the exponential map at some points,supposed to be the centers of bubbles. The author in [3] followed the same pro-cedure to prove a decomposition result on compact Riemannian manifolds for aSobolev-Poincar´e equation.For our case, we will use, when necessary, ideas from [2] to prove a decompositiontheorem for equation ( E α ). More explicitly, after determining conditions underwhich solutions of ( E α ) exist, we prove as in [5] and [1] that, under some con-ditions on the sequence h α and the function f , a sequence of solutions of ( E α ) ofarbitrarily bounded energy decomposes into the sum of a solution of the the limitingequation( E ∞ ) ∆ g u − h ∞ ( p ) ρ p ( x ) u = f ( p ) | u | ∗ − u, where h ∞ is the uniform limit of h α , and two kinds of bubbles, namely the classicaland the singular ones due to the presence respectively of the critical exponent andthe singular term. 2. Notations and preliminaries
In this section, we introduce some notations and materials necessary in ourstudy. Let H ( M ) be the Sobolev space consisting of the completion of C ∞ ( M )with respect to the norm || u || H ( M ) = Z M ( |∇ u | + u ) dv g .M being compact, H ( M ) is then embedded in L q ( M ) compactly for q < ∗ = nn − and continuously for q = 2 ∗ . DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 3
Let K ( n,
2) denote the best constant in Sobolev inequality that asserts that thereexists a constant
B > u ∈ H ( M ),(2.0) || u || L ∗ ( M ) ≤ K ( n, ||∇ u || L ( M ) + B || u || L ( M ) . Throughout the paper, we will denote by B ( a, r ) a ball of center a and radius r > a will be specified either in M or in IR n , and B ( r ) is a ball in IR n of center0 and radius r > δ g the injectivity radius of M . Let p ∈ M be a fixed point, as in [4] wedefine the function ρ p on M by(2.1) ρ p ( x ) = (cid:26) dist g ( p, x ) , dist g ( p, x ) < δ g ,δ g , dist g ( p, x ) ≥ δ g For q ≥
1, we denote by L q ( M, ρ θp ) the space of functions u such that uρ θp is inte-grable. This space is endowed with norm k u k qq,ρ θp = R M | u | q ρ θp dv g .In [4], the following Hardy inequality has been proven on any compact manifold M ,for every ε > A ( ε ) such that for any u ∈ H ( M ),(2.2) Z M u ρ p dv g ≤ ( K ( n, , −
2) + ε ) Z M |∇ u | dv g + A ( ε ) Z M u dv g , with K ( n, , −
2) being the best constant in the Euclidean Hardy inequality Z R n u | x | dx ≤ K ( n, , − Z R n |∇ u | dx, u ∈ C ∞ o ( R n ) . If u is supported in a ball B ( p, δ ) , < δ < δ g , then Z B ( p,δ ) u ρ p dv g ≤ K δ ( n, , − Z B ( p,δ ) |∇ u | dv g , with K δ ( n, , −
2) goes to K ( n, , −
2) when δ goes to 0.Concerning the existence of solutions of equations ( E α ), the author in [4] provedthrough the classical variational techniques an existence result with f a constantfunction. By following the same procedure, though the presence of the non constantfunction f adds further technical difficulties, we can prove the existence of a nontrivial weak solution of ( E α ). This existence result is formulated in the followingtheorem and due to the very familiarity of the techniques used, we omit the proof.For u ∈ H ( M ), set µ = inf u ∈ H ( M ) ,u =0 R M ( |∇ u | − hρ p u ) dv g ( R M f | u | ∗ dv g ) ∗ . The following theorem ensures conditions under which a weak solution u α of ( E α )exists. Theorem 2.1.
Let ( M, g ) be a compact n ( n ≥ − dimensional Riemannian man-ifold and f, h α ( α ∈ [0 , ∞ [) be continuous functions on M . Under the followingconditions : (1) 0 < h α ( p ) < K ( n, , − (2) f ( x ) > , ∀ x ∈ M and µ < − h α ( p ) K ( n, , − M f ) n − n K ( n, , equation ( E α ) admits a nontrivial weak solution u α ∈ H ( M ) . Y. MALIKI, F.Z. TERKI Decomposition theorem
Let J α be the functional defined on H ( M ) by J α ( u ) = 12 Z M ( |∇ u | − h α ρ u ) dv g − ∗ Z M f | u | ∗ dv g . Traditionally, we define a Palais-Smale sequence v α of J α at a level β as to be thesequence that satisfies J α ( v α ) → β and DJ α ( v α ) ϕ → , ∀ ϕ ∈ H ( M ).Define the following limiting functionals J ∞ ( u ) = 12 ( Z M ( |∇ u | − h ∞ ρ u ) dv g − ∗ Z M f | u | ∗ dv g , u ∈ H ( M ) G ( u ) = 12 Z IR n |∇ u | dx − ∗ Z IR n | u | ∗ dx, u ∈ D , ( IR n ) , and G ∞ ( u ) = 12 Z IR n |∇ u | dx − h ∞ ( p )2 Z M u | x | dx − f ( p )2 ∗ Z IR n | u | ∗ dx, u ∈ D , ( IR n )For α ∈ [0 , ∞ [, let h α be a sequence of continuous functions on M such that( H ) a- | h α ( x ) | ≤ C, for some constant C > , ∀ x ∈ M and ∀ α ∈ [0 , ∞ [ . b- There exists a function such that sup M | h α − h ∞ | → , c- 0 < h α ( p ) < K ( n, , − , for all α, ≤ α ≤ ∞ . Now, we state our main result
Theorem 3.1.
Let ( M, g ) be a Riemannian manifold with dim ( M ) = n ≥ , h α be a sequence of continuous functions on M satisfying ( H ) , f be a positivecontinuous function on M that satisfies with h α the conditions of theorem 2.1. Let u α be a sequence of weak solutions of ( E α ) such that R M f | u α | ∗ dv g ≤ C, ∀ α > .Then, there exist m ∈ IN , sequences R iα > , R iα → α →∞ , k ∈ IN n sequences τ jα > , τ jα → α →∞ , converging sequences x jα → x jo = p in M , a solution u o ∈ H ( M ) of ( E ∞ ) , solutions v i ∈ D , ( IR n ) of (3.11) and nontrivial solutions ν j ∈ D , ( IR n ) of (3.17) such that up to a subsequence u α = u o + k X i =1 ( R iα ) − nn η δ (exp − p ( x )) v i (( R iα ) − exp − p ( x ))+ l X j =1 ( r iα ) − nn f ( x o ) − n η δ (exp − x jα ( x )) ν j (( r jα ) − exp − x jα ( x )) + W α , with W α → in H ( M ) , and J α ( u α ) = J ∞ ( u o ) + k X i =1 G ∞ ( v i ) + l X j =1 f ( x jo ) − n G ( ν j ) + o (1) . In order to prove this theorem, we prove some useful lemmas. In all what follows, h α is supposed to satisfy conditions ( H ). Lemma 3.2.
Let u α be a Palais-Smale sequence for J α at level β that converges toa function u weakly in H ( M ) and L ( M, ρ p ) , strongly in L q ( M ) , ≤ q < ∗ and DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 5 almost everywhere to a function u . Then, the sequence v α = u α − u is sequence ofPalais-Smale for J α and J α ( v α ) = β − J ∞ ( u ) + o (1) . Proof.
First, in view of the fact that u α is a Palais-Smale sequence for J α , u α isbounded in H ( M ). In fact, DJ α ( u α ) u α = o ( || u || H ( M ) ) implies that J α ( u α ) = 1 n Z M f | u α | ∗ dv g = β + o (1) + o ( || u || H ( M ) ) . Since f >
0, this implies in turn that u α is bounded in L ∗ ( M ) and then in L ( M ).Furthermore, we have Z M |∇ u α | dv g = nJ α ( u α ) + Z h α ρ p u α dv g + o ( || u || H ( M ) )By continuity of h α on p , we have that for all ǫ > δ > R M |∇ u α | dv g ≤ nβ + ( ε + h α ( p )) R B ( p,δ ) h α ρ p u α dv g + δ − R M \ B ( p,δ ) h α u α dv g + o ( || u || H ( M ) ) + o (1) , then, by applying Hardy inequality (2.2) that for every ε > A ( ε ) such that R M |∇ u α | dv g ≤ nβ + ( ε + h α ( p ))( ε + K ( n, , − R M |∇ u α | dv g + A ( ε ) R M u α dv g + o ( || u || H ( M ) ) + o (1)since 0 < h α ( p ) < K ( n, , − , we can find ε > − ( ε + h α ( p ))( ε + K ( n, , − > R M |∇ u α | dv g is bounded. Thus, u α boundedin H ( M ).Now, for two functions ϕ, φ ∈ H ( M ), H¨older and Hardy inequalities give(3.3) Z M | h α − h ∞ ρ p φϕ | dv g ≤ C || ϕ || H ( M ) || φ || H ( M ) sup M | h α − h ∞ | , writing Z M h α ρ p φϕdv g = Z M h α − h ∞ ρ p φϕdv g + Z M h ∞ ρ p φϕdv g , we get by the assumption made on the sequence h α that(3.4) Z M h α ρ p φϕdv g = Z M h ∞ ρ p φϕdv g + o (1) . Then, since the sequence u α is bounded in H ( M ), by taking φ = u α , we get from(3.3) together with the weak convergence of u α to u in L ( M, ρ − ) that(3.5) Z M h α ρ p u α ϕdv g = Z M h ∞ ρ p uϕdv g + o (1) , thus, applying the last identity to ϕ = u , we get by the weak convergence in H ( M )that J α ( v α ) = J α ( u α ) − J ∞ ( u ) + Φ( u α ) + o (1) , with Φ α ( u α ) = 12 ∗ Z M f ( | u α | ∗ − | u | ∗ − | v α | ∗ ) dv g , Y. MALIKI, F.Z. TERKI which by the Brezis-Leib convergence Lemma equals to o (1), hence we obtain J α ( v α ) = β − J ∞ ( u ) + o (1) . Moreover, for ϕ ∈ H ( M ), by taking φ = u in (3.4), we can write DJ α ( v α ) ϕ = DJ α ( u α ) ϕ − DJ ∞ ( u ) ϕ + Φ( v α ) ϕ + o (1) , with Φ( v α ) ϕ = Z M f (cid:16) | v α + u | ∗ − ( v α + u ) − | v α | ∗ − v α − | u | ∗ − u (cid:17) ϕdv g . Knowing that there exists a positive constant C such that | | v α + u | ∗ − ( v α + u ) − | v α | ∗ − v α − | u | ∗ − u |≤ C ( | v α | ∗ − | u | + | u | ∗ − | v α | ) , we get, after applying H¨older inequality, that there exists a positive constant C such that | Φ( v α ) ϕ | ≤ C (cid:18) k| v α | ∗ − | u |k L ∗ ∗− ( M ) + k| u | ∗ − | v α |k L ∗ ∗− ( M ) (cid:19) k ϕ k L ∗ ( M ) , which gives that Φ( u α ) = o (1) since both ∗ (2 ∗ − ∗ − and ∗ ∗ − are smaller that 2 ∗ and the inclusion of H ( M ) in L q ( M ) is compact for q < ∗ .On the other hand, since the sequence u ∗ − α u α is bounded in L ∗ ∗− ( M ) andconverges almost everywhere to u ∗ − u , we get that u ∗ − α u α converges weaklyin L ∗ ∗− ( M ) to u ∗ − u . This, together with the weak convergence in H ( M )of u α to u and relation (3.5), imply that DJ ∞ ( u ) ϕ = 0 , ∀ ϕ ∈ H ( M ). Hence, DJ α ( v α ) ϕ → , ∀ ϕ ∈ H ( M ). (cid:3) Lemma 3.3.
Let v α be a Palais-Smale sequence of J α at level β that convergesweakly to in H ( M ) . If β < β ∗ = ( − h ∞ ( p ) K ( n, , − ) n n (sup M f ) n − K ( n, n , then v α convergesstrongly to in H ( M ) .Proof. If v α is a Palais-Smale sequence of J α at level β that converges to 0 weaklyin H ( M ), then R M u α dv g = o (1) and β = 1 n Z M ( |∇ v α | − h α ρ p v α ) dv g = 1 n Z M f | v α | ∗ dv g + o (1) . This implies that β ≥
0. Hence, on the one hand, by Hardy inequality (2.2) we getas in Lemma 3.2, that for small enough ε > Z M |∇ v α | dv g ≤ nβ − [( h α ( p ) + ε )( ε + K ( n, , − o (1) , and on the other hand, by Sobolev inequality (2.0), we also get(3.7) Z M |∇ v α | dv g ≥ (cid:18) nβ (sup M f ) K ∗ ( n, (cid:19) ∗ + o (1) . Now, suppose that β >
0, then the above inequalities (3.6) and (3.7) , for α bigenough, give β ≥ (cid:0) − ( h ∞ ( p ) + 2 ε )( K ( n, , −
2) + ε ) (cid:1) ) n n (sup M f ) n − K ( n, n , DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 7 that is β n ≥ β ∗ n − ε + ε ( h ∞ ( p ) + 2 εK ( n, , − n n (sup M f ) n − n K ( n, . By assumption β ∗ > β , by taking ε > − ε − ε ( h ∞ ( p ) − εK ( n, , − n n (sup M f ) n − n K ( n, ( β ∗ n − β n ) > , we get a contradiction. Thus β = 0 and (3.6) assures that Z M |∇ v α | dv g = o (1) , that is v α → H ( M ). (cid:3) In the following, for a given positive constant R , define a cut-off function η R ∈ C ∞ o ( IR n ) such that η R ( x ) = 1 , x ∈ B ( R ) and η R ( x ) = 0 , x ∈ IR n \ B (2 R ),0 ≤ η R ≤ |∇ η R | ≤ CR . Lemma 3.4.
Let v α be Palais-Smale sequence for J α at level β that weakly, butnot strongly, converges to in H ( M ) . Then, there exists a sequence of positivereals R α → such that, up to a subsequence, ˆ η α ˆ v α with ˆ v α ( x ) = R n − α v α (exp p ( R α x )) , and ˆ η α ( x ) = η δ ( R α x )) ( δ is some positive constant), converges weakly in D ( R n ) to a function v ∈ D ( R n ) such that, if v = 0 , v is weak solution of the Euclideanequation (3.8) ∆ v − h ∞ ( p ) | x | v = f ( p ) | v | ∗ − v. Proof.
Since the Palais-Smale sequence v α of J α at level β converges weakly andnot strongly in H ( M ) to 0, we get by Lemma 4.3 that β ≥ β ∗ .Write Z M ( |∇ v α | − h α ρ p v α ) dv g = Z M f | v α | ∗ dv g + o (1) = nβ + o (1) , since, up to a subsequence, v α converges strongly to 0 in L ( M ), we get by Hardyinequality (2.2) that for all ε > nβ ∗ + o (1) ≤ Z M |∇ v α | dv g ≤ nβ − ( h α ( p ) + ε )( K ( n, , −
2) + ε ) + o (1) . In other words,(3.9) c ≤ Z M |∇ v α | dv g ≤ c , for some positive constants c and c .Let ˆ δ a small positive constant such that(3.10) lim sup α →∞ Z M |∇ v α | > γ. Up to a subsequence, for each α > , we can find the smallest constant r α > Z B ( p,r α ) |∇ v α | dv g = γ. Y. MALIKI, F.Z. TERKI
For a sequence of positive constants R α and x ∈ B ( R − α δ g ) ⊂ R n , defineˆ v α ( x ) = R n − α v α (exp p ( R α x )) , and ˆ g α ( x ) = (exp ∗ p g )( R α x )) . We follow the same arguments as in [2]. Let z ∈ R n be such that | z | + r < δ g R − α ,then we have Z B ( z,r ) |∇ ˆ v α | dv ˆ g = Z exp p ( R α B ( z,r )) |∇ v α | dv g . Let 0 < r o < δ g be such that for any x, y ∈ B ( r o ) ⊂ R n , the following inequalityholds(3.11) dist g (exp p ( x ) , exp p ( y )) ≤ C o | x − y | , for some positive constant C o . Also, for r ∈ (0 , r o ), take R α be such that c o rR α = r α , then we get exp p ( R α B ( C o r ))) = B ( p, C o rR α )and then(3.12) Z B ( C o r ) |∇ ˆ v α | dv ˆ g = Z B ( p,r α ) |∇ v α | dv g = γ. Take δ such that 0 < δ ≤ min( C o r, δ g ), there exists a positive constant such that,for all u ∈ D , ( R n ) with Supp ( u ) ∈ B ( δR − α ), the following inequalities hold1 C Z R n |∇ u | dx ≤ Z R n |∇ u | dv ˆ g ≤ C Z R n |∇ u | dx, and (3.13) 1 C Z R n | u | dx ≤ Z R n | u | dv ˆ g ≤ C Z R n | u | dx (3.14)Define a sequence of cut-off functions ˆ η α by ˆ η α ( x ) = η δ ( R α x ). Then, it follows from(3.12), (3.13) and (3.14) that the sequence ˜ v α = ˆ η α ˆ v α is bounded in D , ( IR n ).Consequently, up to a subsequence, ˜ v α converges weakly to some function v ∈ D , ( IR n ).Suppose that v = 0, since v α converges weakly to 0, it follows that R α → v is a weak solution on D , ( IR n ) to (3.8). For this task, welet ϕ ∈ C ∞ o ( R n ) be a function with compact support included in the ball B ( δ ). For α large, define on M the sequence ϕ α as ϕ α ( x ) = R − n α ϕ ( R − α (exp − p ( x ))) . Then, we have Z M ∇ v α ∇ ϕ α dv g = Z R n ∇ ˜ v α ∇ ϕdv ˆ g α , Z M h α ρ p v α ϕ α dv g = R α Z R n h α (exp p ( R α x )) dist g α (0 , R α x ) ˜ v α ϕdv ˆ g α , and Z M f | v α | ∗ − v α ϕ α dv g = Z R n f (exp p ( R α x )) | ˜ v α | ∗ − ˜ v α ϕdv ˆ g α . When tending α to ∞ , ˆ g α tends smoothly to the Euclidean metric on IR n , then bypassing to the limit when α → ∞ and since v α is a Palais-Smale sequence of J α ,we get that v is weak solution of (3.8). (cid:3) DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 9
Lemma 3.5.
Let v be the solution of (3.8) given by Lemma 3.4, then up to asubsequence, w α = v α − R − n α η δ (exp − ( x )) v ( R − α exp − p ( x )) , where < δ < δ g , is a Palais-Sequence for J α at level β − G ∞ ( v ) that weaklyconverges to in H ( M ) . Proof.
For 0 < δ < δ g , define B α ( x ) = R − n α η δ (exp − p ( x )) v ( R − α exp − p ( x )) , x ∈ M and put w α = v α − B α . We begin proving that w α converges weakly to 0 in H ( M ), it suffices to prove that B α does. Take a function ϕ ∈ C ∞ ( M ), then we have R B ( p, δ ) ( ∇B α ∇ ϕ + B α ϕ ) dv g = R n α R B (2 δR − α ) [ R α v ( x )( ∇ η δ )( R α x ) + η δ ( R α x ) ∇ v ] ∇ ϕ (exp p ( R α x )) dv ˆ g α + R n +22 α R B (2 δR − α ) vη δ ( R α x ) ϕ (exp p ( R α x )) dv ˆ g α , then, for a positive constant C ′ such that dv ˆ g α ≤ C ′ dx , it follows that R B ( p, δ ) ( ∇B α ∇ ϕ + B α ϕ ) dv g ≤ C ′ R n α [sup M |∇ ϕ | R IR n ( |∇ v | + | v | Cδ − ) dx + R α sup M | ϕ | R IR n | v | ) dx ] . Thus, when tending α → ∞ , we ge that B α → H ( M ).Now, let us evaluate J α ( w α ). First, we have Z M |∇ w α | dv g = Z M \ B ( p, δ ) |∇ v α | dv g + Z B ( p, δ ) |∇ ( v α − B α ) | dv g , and of course Z B ( p, δ ) |∇ ( v α − B α ) | dv g = Z B ( p, δ ) |∇ v α | dv g − Z B ( p, δ ) ∇ v α ∇B α dv g + Z B ( p, δ ) |∇B α | dv g . Direct calculation gives R B ( p, δ ) |∇B α | dv g = R B (2 δR − α ) η δ ( R α x ) |∇ v | dv ˆ g α + R α R B (2 δR − α ) v |∇ η δ | ( R α x ) dv ˆ g α + 2 R α ∇ η δ ( R α x ) ∇ vdv ˆ g α . It can be easily seen that the second term of right-hand side member of the aboveequality tends to 0 as α → ∞ . Furthermore, for R >
0, a positive constant, wewrite Z B (2 δR − α ) η δ ( R α x ) |∇ v | dv ˆ g α = Z B ( R ) η δ ( R α x ) |∇ v | dv ˆ g α + Z R n \ B ( R ) η δ ( R α x ) |∇ v | dv ˆ g α . with Z R n \ B ( R ) η δ ( R α x ) |∇ v | dv ˆ g α ≤ C Z R n \ B ( R ) |∇ v | dx = ε R , where ε R is a function in R such that ε R → R → ∞ .Noting that, that ˆ g α goes locally in C to the Euclidean metric ξ , we get then(3.15) Z B ( p, δ ) |∇B α | dv g = Z R n |∇ v | dx + o (1) + ε R . Moreover, we have R B ( p, δ ) ∇ v α ∇B α dv g = R B (2 δR − α ) ∇ ( η δ ( R α x )ˆ v α ) ∇ vdv ˆ g α (3.16) + R α R B (2 δR − α ) ( v ∇ ˆ v α − ˆ v α ∇ v ) ∇ η δ ( R α x ) dv ˆ g α with | Z B (2 δR − α ) ∇ η δ ( R α x )( v ∇ ˆ v α − ˆ v α ∇ v ) dv ˆ g α |≤ cδ − [ Z B (2 δR − α ) |∇ ˆ v α | dv ˆ g α ) ( Z B (2 δR − α ) v dx ) + ( Z B (2 δR − α ) ˆ v α dv ˆ g α ) ( Z B (2 δR − α )) |∇ v | dx ) ] . Since v α is bounded in H ( M ), the quantities R B (2 δR − α ) |∇ ˆ v α | dv ˆ g α and R B (2 δR − α ) | ˆ v α | dv ˆ g α are bounded and hence the second term of the right-hand side member of (3.16) is o (1). Thus, by using the weak convergence of ˆ η α ˆ v α to v in D , ( IR n ) that Z B ( p,δ ) ∇ v α ∇B α dv g = Z R n |∇ v | dx + o (1) . so that Z M |∇ w α | dv g = Z M |∇ v α | dv g − Z R n |∇ v | dx + o (1) + ε R . In the same vain, for R a positive constant and α large, we write Z B ( p, δ ) h α ρ p B α dv g = Z B ( p,RR α ) h α ρ p B α dv g + Z B ( p, δ ) \ B ( p,RR α ) h α ρ p B α dv g with Z B ( p, δ ) \ B ( p,RR α ) h α ρ p B α dv g ≤ C ( RR α ) − Z B ( p, δ ) \ B ( p,RR α ) B α dv g then, by a direct calculations, we get Z B ( p, δ ) \ B ( p,RR α ) h α ρ p B α dv g ≤ CR − Z IR n \ B ( R ) v dx = ε R . Hence, Z B ( p, δ ) h α ρ p B α = R α Z B ( R ) h α (exp p ( R α x ))( dist ˆ g α (0 , R α x ) η α ( R α x ) v dv ˆ g α + ε R = h ∞ ( p ) Z R n v | x | dx + o (1) + ε R . Also, in similar way, since v α is bounded in H ( M ), after using H¨older and Hardyinequalities, we can easily have Z B ( p, δ ) \ B ( p,RR α ) h α ρ p v α B α dv g ≤ CR − Z IR n \ B ( R ) v dv g = ε R , DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 11 which yields Z B ( p,δ ) h α ρ p v α B α dv g = R α Z B ( R ) h α (exp p ( R α x ))( dist ˆ g α (0 , R α x )) ( η ( R α x )ˆ v α ) vdv ˆ g α + ε R = h ∞ ( p ) Z R n v | x | dx + o (1) + ε R . so that in the end we obtain Z M h α ρ p w α dv g = Z M h α ρ p v α dv g − h ∞ ( p ) Z R n v | x | dx + o (1) + ε R . In similar way, we can prove that Z M | w α | ∗ dv g = Z M | v α | ∗ dv g − f ( p ) Z M | v | ∗ dv g + o (1) + ε R , Finally, since R is arbitrary, when summing up we obtain J α ( w α ) = J α ( u α ) − G ∞ ( v ) + o (1) = β − G ∞ ( v ) + o (1) . It remains to prove that DJ α ( B α ) → H ( M ) ′ . Let ϕ ∈ H ( M ), for x ∈ B ( δR − α ) put ϕ α ( x ) = R n − α ϕ (exp p ( R α x )) and ϕ α ( x ) = η δ ( R α x )) ϕ α ( x ), then wehave Z B ( p, δ ) ∇B α ∇ ϕdv g = Z B (2 δR − α ) ∇ v ∇ ϕ α dv ˆ g α + R α Z B (2 δR − α ) ∇ η δ ( R α x )( v ∇ ϕ α − ϕ α ∇ v ) dv ˆ g α . Knowing that R B ( p, δ ) |∇ ϕ | dv g = R B (2 δR − α ) |∇ ϕ α | dv ˆ g α , we get that Z B (2 δR − α ) |∇ η δ ( R α x )( v ∇ ϕ α − ϕ α ∇ v ) | dv ˆ g α ≤ C || ϕ || H ( M ) , which gives that Z B ( p, δ ) ∇B α ∇ ϕdv g = Z B (2 δR − α ) ∇ v ∇ ϕ α dv ˆ g α + o ( || ϕ || H ( M ) ) . Next, for
R > Z B (2 δR − α ) ∇ v ∇ ϕ α dv ˆ g α = Z B ( R ) ∇ v ∇ ϕ α dv ˆ g α + Z B (2 δR − α ) \ B ( R ) ∇ v ∇ ϕ α dv ˆ g α , note that Z B (2 δR − α ) \ B ( R ) ∇ v ∇ ϕ α dv ˆ g α ≤ C || ϕ || H ( M ) ( Z B (2 δR − α ) \ B ( R ) |∇ v | dx ) = O ( || ϕ || H ( M ) ) ε ( R ) , where ε R → R → ∞ . Since the sequence of metrics ˆ g α tends locally in C when α → ∞ to the Euclidean metric, we obtain Z B ( p, δ ) ∇B α ∇ ϕdv g = Z IR n ∇ v ∇ ϕ α dx + o ( || ϕ || H ( M ) ) + O ( || ϕ || H ( M ) ) ε ( R ) . Moreover, for a given
R >
0, we have for α large, Z B ( p, δ ) h α ρ p B α ϕdv g = Z B ( p,RR α ) h α ρ p B α ϕdv g + Z B ( p, δ ) \ B ( p,RR α ) h α ρ p B α ϕdv g . On the one hand, we have Z B ( p, δ ) \ B p ( RR α ) h α ρ p B α ϕdv g ≤ C ( RR α ) || ϕ || H ( M ) Z B ( p, δ ) \ B ( p,RR α ) B α dv g , and a straightforward computation shows that Z B ( p, δ ) \ B ( p,RR α ) |B α | dv g ≤ CR α Z B (2 δR − α ) \ B ( R ) v dx, which implies that Z B ( p, δ ) \ B ( p,RR α ) h α ρ p B α ϕdv g = O ( || ϕ || H ( M ) ) ε R with ε R → R → ∞ .On the other hand, we have Z B ( p,RR α ) h α ρ p B α ϕdv g = R α Z B ( R ) h α (exp p R α x )( dist ˆ g α (0 , R α x )) vϕdv ˆ g . which leads to Z B p ( RR α ) h α ρ p B α ϕdv g = Z B ( R ) h ∞ ( p ) | x | vϕdx + o ( || ϕ || H ( M ) )= Z R n h ∞ ( p ) | x | vϕdx − Z R n \ B ( R ) h ∞ ( p ) | x | vϕdx + o ( || ϕ || H ( M ) ) , with Z R n \ B ( R ) h ∞ ( p ) | x | vϕdx ≤ CR || ϕ || H ( M ) = O ( || ϕ || H ( M ) ) ε R . so that Z B ( p, δ ) h α ρ p B α ϕdv g = Z R n h ∞ ( p ) | x | vϕdx + o ( || ϕ || H ( M ) ) + O ( || ϕ || H ( M ) ) ε R . In the same way, we can also have Z B ( p, δ ) f |B α | n − B α ϕdv g = f ( p ) Z R n | v | n − vϕ α dx + o ( || ϕ || H ( M ) )+ O ( || ϕ || H ( M ) ) ε R . Summing up, we obtain R B ( p, δ ) ( ∇B α ∇ ϕdv g + h α ρ p B α ϕ ) dv g − R B ( p, δ ) f |B α | n − B α ϕdv g = R IR n ( ∇ v ∇ ϕ α dx + h ∞ ( p ) | x | vϕ α ) dx − f ( p ) R R n | v | n − vϕ α dx + o ( || ϕ || H ( M ) ) + O ( || ϕ || H ( M ) ) ε R , and since v is weak solution of ( E ∞ ), we get the desired result. (cid:3) Keeping the notations adapted above, we prove the following lemma
Lemma 3.6.
Let v α a Palais-Smale sequence for J α at level β . Suppose that thesequence ˜ v = ˆ η α ˆ v α of the above lemma converges weakly to in D , ( IR n ) . Then,there exist a sequence of positive numbers { τ α } , τ α → and a sequence of points DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 13 x i ∈ M, x i → x o ∈ M \ { p } such that up to a subsequence, the sequence η δ ( τ α x ) ν α ,with δ is some constant and ν α = τ n − α v α (exp x i ( τ α x )) converges weakly to a nontrivial weak solution ν of the Euclidean equation (3.17) ∆ ν = f ( x o ) | ν | n − ν and the sequence W α = v α − τ − n α η δ (exp − x i ( x )) ν ( τ − α exp − x i ( x )) is a Palais-Smale sequence for J α at level β − f ( x o ) n − G ( ν ) that converges weaklyto in H ( M ) .Proof. Suppose that the sequence ˜ v α = ˆ η α ˆ v α converges weakly to 0 in D , ( IR n ) .Take a function ϕ ∈ C ∞ o ( B ( C o r )) and put ϕ α ( x ) = ϕ ( R − α exp − p ( x )). As in [5] and[1], by the strong convergence of ˜ v α to 0 in L loc ( IR n ), we have for α large Z IR n |∇ (˜ v α ϕ ) | dv ˆ g α = Z IR n ∇ ˜ v α ∇ (˜ v α ϕ ) dv ˆ g α + o (1)= Z M ∇ v α ∇ ( v α ϕ α ) dv g + o (1)= kD J α kk v α ϕ α k + Z M h α ρ p ( v α ϕ α ) dv g + Z M f | v α | n − ( v α ϕ α ) dv g + o (1)(3.18) ≤ ( h α ( p ) + ε )( K ( n, , −
2) + ε ) Z IR n |∇ (˜ v α ϕ ) | dv ˆ g α +sup M f K ∗ ( n, Z B ( C o r ) |∇ ˜ v α | dv ˆ g α ) n − Z IR n |∇ (˜ v α ϕ ) | dv ˆ g α + o (1) . Thus, for γ chosen small enough, we get that for each t, < t < C o r, (3.19) Z B ( p,tR α ) |∇ v α | dv g = Z B ( t ) |∇ ˜ v α | dv ˆ g → , as α → ∞ . Now, for t > t −→ F ( t ) = max x ∈ M Z B ( x,t ) |∇ v α | dv g . Since F is continuous, under (3.9) and (3.10), it follows that for any λ ∈ (0 , γ ),there exist t α > x α ∈ M such that Z B ( x α ,t α ) |∇ v α | dv g = λ. Since M is compact, up to a subsequence, we may assume that x α converges tosome point x o ∈ M .Note first that for all α ≥ t α < r α = C o rR α , otherwise if there exists α o ≥ t α o < r α o , we get a contradiction due to the fact that λ = Z B ( x αo ,t αo ) |∇ v α o | dv g ≥ Z B ( p,t αo ) |∇ v α o | dv g ≥ Z B ( p,r αo ) |∇ v α o | dv g = γ. Now, suppose that for all ε >
0, there exists α ε > dist g ( x α , p ) ≤ ε for all α ≥ α ε . Choose r ′ α such that, t α < r ′ α < r α and take ε ′ = r ′ α − t α , we get thatfor some α ε ′ > α ≥ α ε ′ B ( x α , t α ) ⊂ B ( p, r ′ α ) , which, by virtue of (3.19), is impossible. We deduce then that x o = p .Now, let 0 < τ α <
1, for x ∈ B ( τ − α δ g ) ⊂ R n consider the sequences ν α ( x ) = τ n − α v α (exp x α ( τ α x )) , and˜ g α ( x ) = exp ∗ x α g ( τ α x )) . Take τ α such that C o rτ α = t α . As in the above lemma, we can easily check thatthere is a subsequence of ˆ ν α = η δ ( τ α x ) ν α where δ is as in the above lemma, thatweakly converges in D , ( IR n ) to some function ν , a weak solution on D , ( IR n )to (3.17). Note that this time the singular term disappears because x o = p andbecause of course t α → ν = 0. For this purpose, take a point a ∈ IR n and aconstant r > | a | + r < r o τ − α , where r o ∈ (0 , δ g ) is a constant such thatinequality (3.11) is satisfied. Then, we haveexp x α ( τ α B ( a, r )) ⊂ B (exp xα ( τ α a ) , C o rτ α ) , and exp x α ( τ α B ( C o r )) = B ( x α , C o rτ α ) C o , here, is the constant appearing in inequality (3.11). Since we have Z B ( a,r ) |∇ ν α | dv ˜ g α = Z exp xα ( τ α B ( a,r )) |∇ v α | dv g , we get by construction of x α that for such a and r , Z B ( a,r ) |∇ ν α | dv ˜ g ≤ λ Suppose now that ν ≡
0. Take any function h ∈ D , ( IR n ) with support includedin a ball B ( a, r ) ⊂ IR n , with a and r as above. Then, by taking λ small enough, weget by the same calculation done in (3.18) that R B ( a,r ) ∇ ˆ ν α dv ˜ g converges to 0 forall a ∈ IR n and r > | a | + r < r o τ − α . In particular, Z B ( x α ,t α ) |∇ v α | dv g = Z B ( C o r ) |∇ ν α | dv ˜ g → , which makes a contradiction. Thus ν = 0.The proof of the remaining statements of the lemma goes in the same way as inlemma 3.5. (cid:3) Proof of Theorem 4.1.
First, it is worthy to mention that the value G ∞ ( v ) takenon a nontrivial weak solution v of the Euclidean equation (3.11) is greater or equalto the constant β ∗ . In fact, if v is solution of (3.11),then by Hardy and Sobolevinequalities we have(3.20) Z IR n ( |∇ v | − h ∞ ( p ) v | x | ) dx = f ( p ) Z IR n | v | ∗ dx ≤ f ( p ) K ∗ ( n, Z IR n |∇ v | dx ) ∗ , DECOMPOSITION RESULT FOR A SINGULAR ELLIPTIC EQUATION. 15 and(3.21) Z IR n ( |∇ v | − h ∞ ( p ) v | x | ) dx ≥ (1 − h ∞ ( p ) K ( n, − , Z IR n |∇ v | dx, then by (3.20) and (3.21) we get G ∞ ( v ) = 1 n Z IR n ( |∇ v | − h ∞ ( p ) v | x | ) dx ≥ (1 − h ∞ ( p ) K ( n, − , n nf ( p ) n − K n ( n,
2) = β ∗ . (3.22)Now, let u α a sequence of solutions of ( E α ) such that R M f | u α | ∗ dv g ≤ C , u α isthen a bounded Palais-Smale sequence of J α at some level β . Up to a subsequence,we may assume that u α converges weakly in H ( M ) and almost everywhere in M to a solution u of ( E ∞ ). Set v α = u α − u , then by Lemma 3.2, v α is a Palaissequence of J α at level β = β − J ∞ ( u ) + o (1). If v α → H ( M ), thenthe theorem is proved with k = l = 0. If v α → H ( M ), thenwe apply Lemmas 3.4, 3.4 and 3.6 to get a new Palais-Smale sequence v α at level β ≤ β − β ∗ + o (1). So, either β < β ∗ and then v α converges strongly to 0, or β ≥ β ∗ and in this case we repeat the procedure for v α to obtain again a newPalais -Smale sequence at smaller level. By induction, after a number of iterations,we obtain a Plais-Smale sequence at a level smaller than β ∗ . (cid:3) Corollary 3.7.
Suppose that the sequence u α of weak solutions of ( E α ) is suchthat E ( u α ) = Z M f | u α | ∗ dv g ≤ c ≤ (cid:0) − h ∞ ( p ) K ( n, , − (cid:1) n (sup f ) M n − K n ( n, . Then, up to a subsequence, u α converges strongly in H ( M ) to a nontrivial weaksolution u of ( E ∞ ) .Proof. By theorem 4.1, there is a weak solution u of ( E ∞ ) such that, up to asubsequence of u α , we have u α = u + k X i =1 ( R iα ) − nn η δ (exp − p ( x )) v i (( R iα ) − exp − p ( x ))+ l X j =1 f ( x jo ) − n ( r iα ) − nn η δ (exp − x jα ( x )) ν j (( r jα ) − exp − x jα ( x )) + W α , with W α → H ( M ) , and c ≥ E ( u α ) = nJ α ( u α )= nJ ∞ ( u ) + n k X i =1 G ∞ ( v i ) + n l X j =1 f ( x jo ) − n G ( ν j ) + o (1) . Suppose that u ≡
0, if there exists i, ≤ i ≤ k such that v i = 0, then by (3.22) weget c ≥ (cid:0) − h ∞ ( p ) K ( n, , − (cid:1) n (sup f ) M n − K n ( n, , thus, v i ≡ , ∀ i, ≤ i ≤ k , case in which Lemma 4.5 applies, that is, there exists ν j = 0 such that c ≥ f ( x jo ) − n K n ( n, > (cid:0) − h ∞ ( p ) K ( n, , − (cid:1) n (sup f ) M n − K n ( n, . Hence, u = 0. Furthermore, J ∞ ( u ) > k = l = 0.In particular, u α converges strongly in H ( M ) to u . (cid:3) References [1] D.Cao and S.Peng, A global compactness result for singular elliptic problemsinvolving critcal Sobolev exponent. Transcation of AMS, Volume 131, numbe6 (2002), 1857-1966.[2] O. Druet, E. Hebbey and F. Robert, Blow-up theory for elliptic PDEs inRiemannian geometry, Princeton University Press, 2004.[3] M. Dellinger, Etude asymptotique et multiplicit´e pour l’´equation de Sobolevpoincar´e. Thesis, Universit´e Paris VI (2007).[4] F. Madani, Le probl`eme de Yamabe avec singularit´es et la conjecture de Hebey-Vaugon. Th´esis, Universit´e Pi`erre et Marie Curie( 2009).[5] D. Smet, Nonlinear Schr¨odinger equations with Hardy potential and criticalnonlinearaties. Transactions of AMS, Volume 357, number 7 (2004), 2909-2938.[6] M. Struwe, A global compactnes result for elliptic boudary value problemsinvolving limiting nonlinearities. math.Z. 187,(1987) 511-517.[7] S.Terracini, On positive solutions to a class equations with a singular coefficientand critical exponent, Adv.Diff.Equats.,2(1996),241-264.
Y. Maliki, F.Z. TerkiD´epartement de Math´ematiques, Universit´e Abou Bakr Belka¨ıd, Tlemcen,Tlemcen 13000, Algeria.
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