Unique continuation property and local asymptotics of solutions to fractional elliptic equations
aa r X i v : . [ m a t h . A P ] J u l UNIQUE CONTINUATION PROPERTY AND LOCAL ASYMPTOTICS OFSOLUTIONS TO FRACTIONAL ELLIPTIC EQUATIONS
MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI
Abstract.
Asymptotics of solutions to fractional elliptic equations with Hardy type potentialsis studied in this paper. By using an Almgren type monotonicity formula, separation of variables,and blow-up arguments, we describe the exact behavior near the singularity of solutions to linearand semilinear fractional elliptic equations with a homogeneous singular potential related to thefractional Hardy inequality. As a consequence we obtain unique continuation properties forfractional elliptic equations. Introduction
The purpose of the present paper is to describe the asymptotic behavior of solutions to thefollowing class of fractional elliptic semilinear equations with singular homogeneous potentials(1) ( − ∆) s u ( x ) − λ | x | s u ( x ) = h ( x ) u ( x ) + f ( x, u ( x )) , in Ω , where u ∈ D s, ( R N ) (see definition below) and Ω ⊂ R N is a bounded domain containing the origin, N > s, s ∈ (0 , , λ < Λ N,s := 2 s Γ (cid:0) N +2 s (cid:1) Γ (cid:0) N − s (cid:1) , (2) h ∈ C (Ω \ { } ) , | h ( x ) | + | x · ∇ h ( x ) | C h | x | − s + ε as | x | → , (3) ( f ∈ C (Ω × R ) , t F ( x, t ) ∈ C (Ω × R ) , | f ( x, t ) t | + | f ′ t ( x, t ) t | + |∇ x F ( x, t ) · x | C f | t | p for a.e. x ∈ Ω and all t ∈ R , (4)where 2 < p ∗ ( s ) = NN − s , F ( x, t ) = R t f ( x, r ) dr , C f , C h , ε > x ∈ Ω and t ∈ R , ∇ x F denotes the gradient of F with respect to the x variable, and f ′ t ( x, t ) = ∂f∂t ( x, t ).We recall that for any ϕ ∈ C ∞ c ( R N ) and s ∈ (0 , − ∆) s ϕ is definedas(5) ( − ∆) s ϕ ( x ) = C ( N, s ) P . V . Z R N ϕ ( x ) − ϕ ( y ) | x − y | N +2 s dy = C ( N, s ) lim ρ → + Z | x − y | >ρ ϕ ( x ) − ϕ ( y ) | x − y | N +2 s dy where P.V. indicates that the integral is meant in the principal value sense and C ( N, s ) = π − N s Γ (cid:0) N +2 s (cid:1) Γ(2 − s ) s (1 − s ) . The Dirichlet form associated to ( − ∆) s on C ∞ c ( R N ) is given by(6) ( u, v ) D s, ( R N ) = C ( N, s )2 Z R N ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | N +2 s dx dy = Z R N | ξ | s b v ( ξ ) b u ( ξ ) dξ, Date : Revised version, June 27, 2013.M. M. Fall is supported by the Alexander von Humboldt foundation. V. Felli was partially supported by thePRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”.2010
Mathematics Subject Classification.
Keywords.
Fractional elliptic equations, Caffarelli-Silvestre extension, Hardy inequality, Unique continuationproperty. where b u denotes the unitary Fourier transform of u . It defines a scalar product thanks to (8) or(9) below. From now on we define D s, ( R N ) as the completion on C ∞ c ( R N ) with respect to thenorm induced by the scalar product (6).By a weak solution to (1) we mean a function u ∈ D s, ( R N ) such that(7) ( u, ϕ ) D s, ( R N ) = Z Ω (cid:18) λ | x | s u ( x ) + h ( x ) u ( x ) + f ( x, u ( x )) (cid:19) ϕ ( x ) dx, for all ϕ ∈ C ∞ c (Ω) . We notice that the right hand side of (7) is well defined in view of assumptions (3)–(4), theHardy-Littlewood-Sobolev inequality(8) S N,s k u k L ∗ ( s ) ( R N ) k u k D s, ( R N ) , and the following Hardy inequality, due to Herbst in [19] (see also [27]),(9) Λ N,s Z R N u ( x ) | x | s dx Z R N | ξ | s | b u ( ξ ) | dξ = k u k D s, ( R N ) , for all u ∈ D s, ( R N ) . It should also be remarked that, in (7), we allow p = 2 ∗ ( s ) and that u is not prescribed outside Ω.One of the aim of this paper is to give the precise behaviour of a solution u to (1). The rate andthe shape of u are given by the the eigenvalues and the eigenfunctions of the following eigenvalueproblem − div S N ( θ − s ∇ S N ψ ) = µ θ − s ψ, in S N + , − lim θ → + θ − s ∇ S N ψ · e = κ s λψ, on ∂ S N + , (10)where(11) κ s = Γ(1 − s )2 s − Γ( s )and S N + = { ( θ , θ , . . . , θ N +1 ) ∈ S N : θ > } = n z | z | : z ∈ R N +1 , z · e > o , with e = (1 , , . . . , µ ( λ ) µ ( λ ) · · · µ k ( λ ) · · · Moreover µ ( λ ) > − (cid:0) N − s (cid:1) , see Lemma 2.2. We notice that if λ = 0 then µ k (0) > k .Our first main result is the following theorem. Theorem 1.1.
Let u ∈ D s, ( R N ) be a nontrivial solution to (1) in a bounded domain Ω ⊂ R N containing the origin as in (7) with s, λ, h and f satisfying assumptions (2) , (3) and (4) . Thenthere exists an eigenvalue µ k ( λ ) of (10) and an eigenfunction ψ associated to µ k ( λ ) such that (12) τ − s − N − q ( s − N ) + µ k ( λ ) u ( τ x ) → | x | − N − s + q ( s − N ) + µ k ( λ ) ψ (cid:16) , x | x | (cid:17) as τ → + , in C ,α loc ( B ′ \ { } ) for some α ∈ (0 , , where B ′ := { x ∈ R N : | x | < } , and, in particular, (13) τ − s − N − q ( s − N ) + µ k ( λ ) u ( τ θ ′ ) → ψ (0 , θ ′ ) in C ,α ( S N − ) as τ → + , where S N − = ∂ S N + . We should mention that the above result is stated in such form for the sake of simplicity. In fact,for the eigenfunction ψ in (13), we obtain precisely its components in any basis of the eigenspacecorresponding to µ k ( λ ), see Theorem 4.1 (below). We also remark that µ ( λ ) < λ > s = 1) and Hardy-type potentials for different kinds of problems: in [12, 14]for Schr¨odinger equations with electromagnetic potentials and in [13] for Schr¨odinger equationswith inverse square many-particle potentials.As a particular case of Theorem 1.1, if λ = 0 we obtain that the convergence stated in 12 holdsin C ,α ( B ′ ). RACTIONAL ELLIPTIC EQUATIONS 3
Corollary 1.2.
Let u ∈ D s, ( R N ) be a nontrivial weak solution to (14) ( − ∆) s u ( x ) = h ( x ) u ( x ) + f ( x, u ( x )) , in Ω , in a bounded domain Ω ⊂ R N with s ∈ (0 , , h ∈ C (Ω) , and f satisfying (4) . Then, for every x ∈ Ω , there exists an eigenvalue µ k = µ k (0) of problem (10) with λ = 0 and an eigenfunction ψ associated to µ k such that (15) τ − s − N − q ( s − N ) + µ k u ( x + τ ( x − x )) → | x − x | − N − s + q ( s − N ) + µ k ψ (cid:16) , x − x | x − x | (cid:17) as τ → + , in C ,α ( { x ∈ R N : x − x ∈ B ′ } ) . Our next result contains the so called strong unique continuation property which is a directconsequence of Theorem 1.1.
Theorem 1.3.
Suppose that all the assumptions of Theorem 1.1 hold true. Let u be a solution to (1) in a bounded domain Ω ⊂ R N containing the origin. If u ( x ) = o ( | x | n ) = o (1) | x | n as | x | → for all n ∈ N , then u ≡ in Ω . From Theorem 1.2 a unique continuation principle related to sets of positive measures follows.
Theorem 1.4.
Let u ∈ D s, ( R N ) be a weak solution to (14) in a bounded domain Ω ⊂ R N with s ∈ (0 , , h ∈ C (Ω) , and f satisfying (4) . If u ≡ on a set E ⊂ Ω of positive measure, then u ≡ in Ω . An interesting application of Theorem 1.4 is that the nodal sets of eigenfunctions for the frac-tional laplacian operator have zero Lebesgue measure. We should point out that this was a keyassumption in [15], where the authors studied the existence of weak solutions to some non-localequations.Recent research in the field of second order elliptic equations has devoted a great attention tothe problem of unique continuation property in the presence of singular lower order terms, see e.g.[11, 20, 22]. Two different kinds of approach have been developed to treat unique continuation:a first one is due to Carleman [5] and is based on weighted priori inequalities, whereas a secondone is due to Garofalo and Lin [17] and is based on local doubling properties proved by Almgrenmonotonicity formula. In the present paper we will follow the latter approach. Furthermore, in thespirit of [12, 13, 14], the combination of monotonicity methods with blow-up analysis will enableus to prove not only unique continuation but also the precise asymptotics of solutions stated inTheorem 1.1.As far as unique continuation from sets of positive measures is concerned, we mention [6] whereit was proved for second order elliptic operators by combining strong unique continuation propertywith the De Giorgi inequality. Since the validity of a the De Giorgi type inequality for the fractionalproblem (or even its extension, see (17)) seems to be hard to prove, we will base the proof ofTheorem 1.4 directly on the asymptotic of solutions proved in Theorem 1.1.To explain our argument of proving the asymptotic behavior, let us we write (7) as(16) ( − ∆) s u = G ( x, u ) in Ω . The proof of our results is based on the study of the
Almgren frequency function at the origin 0:“ratio of the local energy over mass near the origin”. Due to the non-locality of the Dirichlet formassociated to ( − ∆) s , it is not clear how to set up an Almgren’s type frequency function using thisenergy as in the local case s = 1. A way out for this difficulty is to use the Caffarelli-Silvestreextension [4] which can be seen as a local version of (16). The Caffarelli-Silvestre extension of asolution u to (16) is a function w defined on R N +1+ = { z = ( t, x ) : t ∈ (0 , + ∞ ) , x ∈ R N } satisfying w = u on Ω and solving in some weak sense (see Section 2 for more details) the boundaryvalue problem(17) ( div( t − s ∇ w ) = 0 , in R N +1+ , − lim t → + t − s ∂w∂t = κ s G ( x, w ) , on Ω . MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI
Here and in the following, we write z = ( t, x ) ∈ R N +1+ with x ∈ R N and t >
0, and we identify R N with ∂ R N +1+ , so that Ω is contained in ∂ R N +1+ .We then consider the Almgren’s frequency function N ( r ) = D ( r ) H ( r )where D ( r ) = 1 r N − s (cid:20) Z B + r t − s |∇ w | dt dx − κ s Z B ′ r G ( x, w ) w dx (cid:21) , H ( r ) = 1 r N +1 − s Z S + r t − s w dS, being B + r = { z = ( t, x ) ∈ R N +1+ : | z | < r } , B ′ r := { x ∈ R N : | x | < r } ,S + r = { z = ( t, x ) ∈ R N +1+ : | z | = r } , and dS denoting the volume element on N -dimensional spheres. We mention that an Almgren’sfrequency function for degenerate elliptic equations of the form (17) was first formulated in [4,Section 6].As a first but nontrivial step, we prove that lim r → N ( r ) := γ exists and it is finite, seeLemma 3.15. Next, we make a blow-up analysis by zooming around the origin the solution w normalized also by √ H . More precisely, setting w τ ( z ) = w ( τz ) √ H ( τ ) , we have that w τ converges, (insome H¨older and Sobolev spaces) to e w solving the limiting equation ( div( t − s ∇ e w ) = 0 , in B +1 , − lim t → + t − s ∂ e w∂t = κ s λ | z | s e w, on B ′ . To obtain this, the fact that h is negligible with respect to the Hardy potential and f is at mostcritical with respect to the Sobolev exponent (see assumptions (3) and (4)) plays a crucial role; werefer to Lemma 4.2 for more details.The main point is that the Almgren’s frequency for e w is e N ( r ) = lim τ → N ( τ r ) = γ , i.e. e N is constant; hence e w and ∇ e w · z | z | are proportional on L ( S + r ; t − s ). As a consequence we ob-tain e w ( z ) = ϕ ( | z | ) ψ (cid:0) z | z | (cid:1) . By separating variables in polar coordinates, we obtain that ψ is aneigenfunction of (10) for some eigenvalue µ k ( λ ). By the method of variation of constants andthe fact that e w has finite energy near the origin, we then prove that ϕ ( r ) is proportional to r γ and that γ = s − N + q(cid:0) s − N (cid:1) + µ k ( λ ). We finally complete the proof by showing thatlim r → r − γ H ( r ) >
0, see Lemma 4.5.We should mention that in the recent literature, a great attention has been addressed to non-local fractional diffusion and many papers have been devoted to the study of existence, non-existence, regularity and qualitative properties of solutions to elliptic equations associated tofractional Laplace type operators, see e.g. [2, 3, 4, 7, 9, 10, 24, 25] and references therein. Inparticular, semilinear fractional elliptic equations involving the Hardy potential were treated in[9], where some existence and nonexistence results were obtained from lower bounds of positivesolutions. Adapting the ideas in [12] in the nonlocal case of the present paper, we provide thebehavior (therefore regularity) of solutions at the singularity. In particular, the asymptotics wehave here show that the lower bound of positive solutions proved in [9] is sharp.2.
Preliminaries and notations
Notation.
We list below some notation used throughout the paper.- S N = { z ∈ R N +1 : | z | = 1 } is the unit N -dimensional sphere.- S N + = { ( θ , θ , . . . , θ N +1 ) ∈ S N : θ > } := S N ∩ R N +1+ .- dS denotes the volume element on N -dimensional spheres.- dS ′ denotes the volume element on ( N − RACTIONAL ELLIPTIC EQUATIONS 5
Let D , ( R N +1+ ; t − s ) be the completion of C ∞ c ( R N +1+ ) with respect to the norm k w k D , ( R N +1+ ; t − s ) = (cid:18) Z R N +1+ t − s |∇ w ( t, x ) | dt dx (cid:19) / . We recall that there exists a well defined continuous trace map Tr : D , ( R N +1+ ; t − s ) → D s, ( R N ),see e.g. [2].For every u ∈ D s, ( R N ), let H ( u ) ∈ D , ( R N +1+ ; t − s ) be the unique solution to the minimiza-tion problem(18) Z R N +1+ t − s |∇H ( u ) | dt dx = min (cid:26) Z R N +1+ t − s |∇ v | dt dx : v ∈ D , ( R N +1+ ; t − s ) , Tr( v ) = u (cid:27) . By Caffarelli and Silvestre [4] we have that(19) Z R N +1+ t − s ∇H ( u ) · ∇ e ϕ dt dx = κ s ( u, Tr e ϕ ) D s, ( R N ) for all e ϕ ∈ D , ( R N +1+ ; t − s ) , where κ s is defined in (11).We notice that combining (9), (18), (19) and (8), we obtain the following Hardy-trace inequality κ s Λ N,s Z R N (Tr v ) ( x ) | x | s dx Z R N +1+ t − s |∇ v | dt dx, for all v ∈ D , ( R N +1+ ; t − s )(20)and also the Sobolev-trace inequality κ s S N,s k Tr v k L ∗ ( s ) ( R N ) Z R N +1+ t − s |∇ v | dt dx, for all v ∈ D , ( R N +1+ ; t − s ) . (21)2.1. Separation of variables in the extension operator.
For every
R >
0, we define the space H ( B + R ; t − s ) as the completion of C ∞ ( B + R ) with respect to the norm k w k H ( B + R ; t − s ) = (cid:18) Z B + R t − s (cid:16) |∇ w ( t, x ) | + w ( t, x ) (cid:17) dt dx (cid:19) / . By direct calculations, we can obtain the following lemma concerning separation of variables inthe extension operator.
Lemma 2.1. If v ∈ H ( B + R ; t − s ) is such that v ( z ) = f ( r ) ψ ( θ ) for a.e. z = ( t, x ) ∈ R N +1+ , with r = | z | < R and θ = z | z | ∈ S N , then div( t − s ∇ v ( z )) = 1 r N (cid:0) r N +1 − s f ′ (cid:1) ′ θ − s ψ ( θ ) + r − − s f ( r ) div S N ( θ − s ∇ S N ψ ( θ )) in the distributional sense, where θ = tr = θ · e with e = (1 , , . . . , , div S N (respectively ∇ S N )denotes the Riemannian divergence (respectively gradient) on the unit sphere S N endowed with thestandard metric. Let us define H ( S N + ; θ − s ) as the completion of C ∞ ( S N + ) with respect to the norm k ψ k H ( S N + ; θ − s ) = (cid:18) Z S N + θ − s (cid:0) |∇ S N ψ ( θ ) | + ψ ( θ ) (cid:1) dS (cid:19) / . We also denote L ( S N + ; θ − s ) := n ψ : S N + → R measurable such that R S N + θ − s ψ ( θ ) dS < + ∞ o . The following trace inequality on the unit half-sphere S N + holds. MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI
Lemma 2.2.
There exists a well defined continuous trace operator H ( S N + ; θ − s ) → L ( ∂ S N + ) = L ( S N − ) . Moreover for every ψ ∈ H ( S N + ; θ − s ) κ s Λ N,s (cid:18) Z S N − | ψ ( θ ′ ) | dS ′ (cid:19) (cid:16) N − s (cid:17) Z S N + θ − s | ψ ( θ ) | dS + Z S N + θ − s |∇ S N ψ ( θ ) | dS. where dS ′ denotes the volume element on the sphere S N − = ∂ S N + = { ( θ , θ ′ ) ∈ S N + : θ = 0 } . Proof . Let ψ ∈ C ∞ ( S N + ) and f ∈ C ∞ c (0 , + ∞ ) with f = 0. Rewriting (20) for v ( z ) = f ( r ) ψ ( θ ), r = | z | , θ = z | z | , we obtain that κ s Λ N,s (cid:18) Z + ∞ r N − − s f ( r ) dr (cid:19)(cid:18) Z S N − | ψ (0 , θ ′ ) | dS ′ (cid:19) (cid:18) Z + ∞ r N +1 − s | f ′ ( r ) | dr (cid:19)(cid:18) Z S N + θ − s | ψ ( θ ) | dS (cid:19) + (cid:18) Z + ∞ r N − − s f ( r ) dr (cid:19)(cid:18) Z S N + θ − s |∇ S N + ψ ( θ ) | dS (cid:19) , and hence, by optimality of the classical Hardy constant, see [18, Theorem 330], κ s Λ N,s (cid:18) Z S N − | ψ (0 , θ ′ ) | dS ′ (cid:19) (cid:18) Z S N + θ − s | ψ ( θ ) | dS (cid:19) inf f ∈ C ∞ c (0 , + ∞ ) R + ∞ r N +1 − s | f ′ ( r ) | dr R + ∞ r N − − s f ( r ) dr + Z S N + θ − s |∇ S N + ψ ( θ ) | dS = (cid:16) N − s (cid:17) Z S N + θ − s | ψ ( θ ) | dS + Z S N + θ − s |∇ S N + ψ ( θ ) | dS. By density of C ∞ ( S N + ) in H ( S N + ; θ − s ), we obtain the conclusion.In view of Lemma 2.1, in order to construct an orthonormal basis of L ( S N + ; θ − s ) for expandingsolutions to (17) in Fourier series, we are naturally lead to consider the eigenvalue problem (10),which admits the following variational formulation: we say that µ ∈ R is an eigenvalue of problem(10) if there exists ψ ∈ H ( S N + ; θ − s ) \ { } (called eigenfunction) such that Q ( ψ, υ ) = µ Z S N + θ − s ψ ( θ ) υ ( θ ) dS, for all υ ∈ H ( S N + ; θ − s ) , where Q : H ( S N + ; θ − s ) × H ( S N + ; θ − s ) → R ,Q ( ψ, υ ) = Z S N + θ − s ∇ S N ψ ( θ ) · ∇ υ ( θ ) dS − λκ s Z S N − T ψ ( θ ′ ) T υ ( θ ′ ) dS ′ . By Lemma 2.2 the bilinear form Q is continuous and weakly coercive on H ( S N + ; θ − s ). Moreoverthe belonging of the weight t − s to the second Muckenhoupt class ensures that the embedding H ( S N + ; θ − s ) ֒ → ֒ → L ( S N + ; θ − s ) is compact (see [8] for weighted embeddings with Muckenhoupt A weights). Then, from classical spectral theory (see e.g. [23, Theorem 6.16]), problem (10) admitsa diverging sequence of real eigenvalues with finite multiplicity µ ( λ ) µ ( λ ) · · · µ k ( λ ) · · · the first of which admits the variational characterization(22) µ ( λ ) = min ψ ∈ H ( S N + ; θ − s ) \{ } Q ( ψ, ψ ) R S N + θ − s ψ ( θ ) dS . Furthermore, in view of Lemma 2.2, we have that(23) µ ( λ ) > − (cid:16) N − s (cid:17) . RACTIONAL ELLIPTIC EQUATIONS 7
To each k >
1, we associate an L ( S N + ; θ − s )-normalized eigenfunction ψ k ∈ H ( S N + ; θ − s ) \ { } corresponding to the k -th eigenvalue µ k ( λ ), i.e. satisfying(24) Q ( ψ k , υ ) = µ k ( λ ) Z S N + θ − s ψ k ( θ ) υ ( θ ) dS, for all υ ∈ H ( S N + ; θ − s ) . In the enumeration µ ( λ ) µ ( λ ) · · · µ k ( λ ) · · · we repeat each eigenvalue as many timesas its multiplicity; thus exactly one eigenfunction ψ k corresponds to each index k ∈ N , k >
1. Wecan choose the functions ψ k in such a way that they form an orthonormal basis of L ( S N + ; θ − s ).We can also determine µ ( λ ) for λ ∈ (0 , Λ N,s ), where Λ
N,s is the fractional Hardy constantdefined in (2).
Proposition 2.3.
For every α ∈ (cid:0) , N − s (cid:1) , we define λ ( α ) = 2 s Γ (cid:0) N +2 s +2 α (cid:1) Γ (cid:0) N − s − α (cid:1) Γ (cid:0) N +2 s − α (cid:1) Γ (cid:0) N − s +2 α (cid:1) . Then the mapping α λ ( α ) is continuous and decreasing. In addition we have that µ ( λ ( α )) = α − (cid:18) N − s (cid:19) for all α ∈ (cid:18) , N − s (cid:19) . Proof . It was proved in [9, Lemma 3.1] that, for every α ∈ (cid:0) , N − s (cid:1) , there exists a positivecontinuous function Φ α : R N +1+ → R such that(25) div( t − s ∇ Φ α ) = 0 in R N +1+ Φ α = | x | s − N + α on R N \ { }− t − s ∂ Φ α ∂t = κ s λ ( α ) | x | − s Φ α on R N \ { } , Moreover Φ α ∈ H ( B + R ; t − s ) for every R > α is scale invariant, i.e.Φ α ( τ z ) = τ s − N + α Φ α ( z ) , for all τ > , thus implying that, for all z ∈ R N +1+ ,(26) c | z | s − N + α Φ α ( z ) c | z | s − N + α , for some positive constants c , c . It is also known (see for instance [16]) that the map α λ ( α )is continuous and monotone decreasing.We write Φ α ( z ) = ∞ X k =1 Φ kα ( | z | ) ψ k ( z/ | z | ) , Φ kα ( r ) = Z S N + ψ k ( θ )Φ α ( rθ ) dS. In particular, since ψ >
0, by (26) we have, for every r ∈ (0 , R ),(27) c ′ | r | s − N + α Φ α ( r ) c ′ | r | s − N + α , for some positive constants c ′ , c ′ . Using (25), we have, weakly, for every k > r ∈ (0 , R ), ( r N (cid:0) r N +1 − s (Φ kα ) ′ (cid:1) ′ θ − s ψ k ( θ ) + r − − s Φ kα div S N ( θ − s ∇ S N ψ k ( θ )) = 0 , − Φ kα lim θ → + θ − s ∇ S N ψ k ( θ ) · e = κ s λ ( α ) ψ k (0 , θ ′ )Φ kα . Testing the above equation with ψ > α >
0, we obtain(Φ α ) ′′ + N + 1 − sr (Φ α ) ′ − µ ( λ ( α )) r Φ α = 0and hence Φ α ( r ) is of the form Φ α ( r ) = c r σ + + c r σ − for some c , c ∈ R , where σ + = − N − s s(cid:18) N − s (cid:19) + µ ( λ ( α )) and σ − = − N − s − s(cid:18) N − s (cid:19) + µ ( λ ( α )) . MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI
Since the function | z | σ − k ψ ( z | z | ) / ∈ H ( B +1 ; t − s ), we deduce that c = 0 and thus for every r ∈ (0 , R ) Φ α ( r ) = c r σ + . This together with (27) implies that µ ( λ ( α )) = α − (cid:18) N − s (cid:19) for all α ∈ (cid:18) , N − s (cid:19) , as claimed.2.2. Hardy type inequalities.
From well-known weighted embedding inequalities and the factthat the weight t − s belongs to the second Muckenhoupt class (see e.g. [8]), the embedding H ( B + r ; t − s ) ֒ → L ( B + r ; t − s ) is compact. It can be also proved that both the trace operators(28) H ( B + r ; t − s ) ֒ → ֒ → L ( S + r ; θ − s ) , (29) H ( B + r ; t − s ) ֒ → ֒ → L ( B ′ r )are well defined and compact.For sake of simplicity, in the following of this paper, we will often denote the trace of a functionwith the same letter as the function itself.The following Hardy type inequality with boundary terms holds. Lemma 2.4.
For all r > and w ∈ H ( B + r ; t − s ) , the following inequality holds (cid:18) N − s (cid:19) Z B + r t − s w ( z ) | z | dz Z B + r t − s (cid:18) ∇ w ( z ) · z | z | (cid:19) dz + (cid:18) N − s r (cid:19) Z S + r t − s w dS. Proof . By scaling, it is enough to prove the stated inequality for r = 1. Let V ( z ) = | z | s − N , z ∈ R N +1+ \ { } . We notice that V satisfies(30) − div( t − s ∇ V ) = (cid:18) N − s (cid:19) t − s | z | − V ( z ) in R N +1+ \ { } . Hence, letting w ∈ C ∞ ( B +1 ), multiplying (30) with w V , and integrating over B +1 \ B + δ with δ ∈ (0 , (cid:18) N − s (cid:19) Z B +1 \ B + δ t − s w ( z ) | z | dz = Z B +1 \ B + δ t − s ∇ V ( z ) · ∇ (cid:16) w V (cid:17) ( z ) dz − Z S +1 t − s ( ∇ V · ν ) w V dS + Z S + δ t − s ( ∇ V · ν ) w V dS = − ( N − s ) Z B +1 \ B + δ t − s w | z | (cid:16) ∇ w · z | z | (cid:17) dz − (cid:18) N − s (cid:19) Z B +1 \ B + δ t − s w | z | dz + N − s Z S +1 t − s w dS − N − s δ Z S + δ t − s w dS where ν ( z ) = z | z | . Since, by Schwarz’s inequality(31) − ( N − s ) Z B +1 \ B + δ t − s w | z | (cid:16) ∇ w · z | z | (cid:17) dz (cid:18) N − s (cid:19) Z B +1 \ B + δ t − s w | z | dz + Z B +1 \ B + δ t − s (cid:16) ∇ w · z | z | (cid:17) dz and 1 δ Z S + δ t − s w dS = O ( δ N − s ) = o (1) for δ → + , RACTIONAL ELLIPTIC EQUATIONS 9 letting δ → r = 1 and w ∈ C ∞ ( B +1 ). The conclusion followsby density of C ∞ ( B +1 ) in H ( B +1 ; t − s ). Lemma 2.5.
For every r > and w ∈ H ( B + r ; t − s ) , the following inequality holds κ s Λ N,s Z B ′ r w | x | s dx (cid:18) N − s r (cid:19) Z S + r t − s w dS + Z B + r t − s |∇ w | dz. Proof . Let w ∈ C ∞ ( B + r ). Then, passing to polar coordinates and using Lemmas 2.2 and 2.4,we obtain κ s Λ N,s Z B ′ r w (0 , x ) | x | s dx = κ s Λ N,s Z r ρ N − − s (cid:18) Z S N − w (0 , ρθ ′ ) dS ′ (cid:19) dρ Z r ρ N − − s (cid:18)(cid:16) N − s (cid:17) Z S N + θ − s | w ( ρθ ) | dS + Z S N + θ − s |∇ S N w ( ρθ ) | dS (cid:19) dρ = (cid:16) N − s (cid:17) Z B + r t − s w | z | dz + Z r ρ N − − s (cid:18) Z S N + θ − s |∇ S N w ( ρθ ) | dS (cid:19) dρ (cid:18) N − s r (cid:19) Z S + r t − s w dS + Z r ρ N +1 − s (cid:18) Z S N + θ − s (cid:18) ρ |∇ S N w ( ρθ ) | + (cid:12)(cid:12)(cid:12) ∂w∂ρ ( ρθ ) (cid:12)(cid:12)(cid:12) (cid:19) dS (cid:19) dρ = (cid:18) N − s r (cid:19) Z S + r t − s w dS + Z B + r t − s |∇ w | dz. The conclusion follows by density of C ∞ ( B + r ) in H ( B + r ; t − s ).The following Sobolev type inequality with boundary terms holds. Lemma 2.6.
There exists e S N,s > such that, for all r > and w ∈ H ( B + r ; t − s ) , (cid:18) Z B ′ r | w | ∗ ( s ) dx (cid:19) ∗ ( s ) e S N,S (cid:20) N − s r Z S + r t − s w dS + Z B + r t − s |∇ w | dz (cid:21) . Proof . By scaling, it is enough to prove the statement for r = 1. Let w ∈ C ∞ ( B +1 ) and denoteby e w its fractional Kelvin transform defined as e w ( z ) = | z | − ( N − s ) w (cid:0) z | z | (cid:1) . Some computations (seealso [10]) show that Z B +1 t − s |∇ w | dz + ( N − s ) Z S +1 t − s w dS = Z R N +1+ \ B +1 t − s |∇ e w | dz, (32) Z B ′ | w | ∗ ( s ) dx = Z R N \ B ′ | e w | ∗ dx. (33)In particular the function v ( z ) = ( w ( z ) , if z ∈ B +1 , e w ( z ) , if z ∈ R N +1+ \ B +1 belongs to D , ( R N +1+ ; t − s ), so that, by (21) we have(34) (cid:18) Z R N | v | ∗ ( s ) dx (cid:19) ∗ ( s ) S N,s κ s Z R N +1+ t − s |∇ v | dz. From (32), (33), and (34), it follows that (cid:18) Z B ′ | w | ∗ ( s ) dx (cid:19) ∗ ( s ) S N,s κ s (cid:20) Z B +1 t − s |∇ w | dz + N − s Z S +1 t − s w dS (cid:21) which, by density, yields the conclusion.Combining Lemma 2.5 and Lemma 2.6 the following corollary follows. Corollary 2.7.
For all r > and w ∈ H ( B + r ; t − s ) , the following inequalities hold (35) Z B + r t − s |∇ w | dz − κ s λ Z B ′ r w | x | s dx + N − s r Z S + r t − s w dS > κ s (Λ N,s − λ ) Z B ′ r w | x | s dx and (36) Z B + r t − s |∇ w | dz − κ s λ Z B ′ r w | x | s dx + N − s r Z S + r t − s w dS > Λ N,s − λ (1 + Λ N,s ) e S N,s (cid:18) Z B ′ r | w | ∗ ( s ) dx (cid:19) ∗ ( s ) . The Almgren type frequency function
Let
R > B ′ R ⊂⊂ Ω and w ∈ H ( B + R ; t − s ) be a nontrivial solution to(37) ( div( t − s ∇ w ) = 0 , in B + R , − lim t → + t − s ∂w∂t ( t, x ) = κ s (cid:16) λ | x | s w + hw + f ( x, w ) (cid:17) , on B ′ R , in a weak sense, i.e., for all ϕ ∈ C ∞ c ( B + R ∪ B ′ R ), we have that(38) Z R N +1+ t − s ∇ w · ∇ ϕ dt dx = κ s Z B ′ R (cid:18) λ | x | s w + hw + f ( x, w ) (cid:19) ϕ dx, with s, λ, h, f as in assumptions (2), (3), and (4).For every r ∈ (0 , R ] we define(39) D ( r ) = 1 r N − s (cid:20) Z B + r t − s |∇ w | dt dx − κ s Z B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) (cid:19) dx (cid:21) and(40) H ( r ) = 1 r N +1 − s Z S + r t − s w dS = Z S N + θ − s w ( rθ ) dS. The main result of this section is the existence of the limit as r → + of the Almgren’s frequency function (see [17] and [1]) associated to w (41) N ( r ) = D ( r ) H ( r ) = r "Z B + r t − s |∇ w | dt dx − κ s Z B ′ r (cid:18) λ | x | s w + h ( x ) w + f ( x, w ) w (cid:19) dx S + r t − s w dS . We notice that, by Lemma 2.5, w (0 , · ) ∈ L ( B ′ R ; | x | − s ) and so the L (0 , R )-function r Z S + r t − s |∇ w | dS, respectively r Z ∂B ′ r w | x | s dS ′ , is the weak derivative of the W , (0 , R )-function r → Z B + r t − s |∇ w | dz, respectively r Z B ′ r w | x | s dx. In particular, for a.e. r ∈ (0 , R ), ∂w∂ν ∈ L ( S + r ; t − s ), where ν = ν ( z ) = z | z | .Next we observe the following integration by parts. Lemma 3.1.
For a.e. r ∈ (0 , R ) and every e ϕ ∈ C ∞ ( B + r ) Z B + r t − s ∇ w · ∇ e ϕ dz = Z S + r t − s ∂w∂ν e ϕ dS + κ s Z B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) (cid:19) e ϕ dx. RACTIONAL ELLIPTIC EQUATIONS 11
Proof . It follows by testing (38) with e ϕ ( z ) η n ( | z | ) where η n ( ρ ) = 1 if ρ < r − n , η n ( r ) = 0 if ρ > r , η ( ρ ) = n ( r − ρ ) if r − n ρ r , passing to the limit, and noticing that a.e. r ∈ (0 , R ) is aLebesgue point for the L (0 , R )- function r R S + r t − s ∂w∂ν e ϕ dS .3.1. Regularity estimates and Pohozaev-type identity.
In this section, we will prove localregularity estimates for a general class of fractional elliptic equations in Lemma 3.3 below. Thisestimate will be useful for the blow-up analysis and also for establishing the Pohozaev identity inTheorem 3.7 below which is crucial in this paper. The proof of Lemma 3.3 uses mainly a result ofJin, Li and Xiong in [21] that we state here for sake of completeness.
Proposition 3.2 ([21] Proposition 2.4) . Let v ∈ H ( B +1 ; t − s ) be a weak solution to ( div( t − s ∇ v ) = 0 , in B +1 − lim t → + t − s v t = c ( x ) v + b ( x ) , on B ′ , with c, b ∈ L p ( B ′ ) for some p > N s . Then v ∈ L ∞ loc (cid:0) B +1 ∪ B ′ (cid:1) and there exists C > dependingonly on N, s, p, k c k L p ( B ′ ) such that k v k L ∞ (cid:16) [0 , / × B ′ / (cid:17) C (cid:16) k v k H ( B +1 ; t − s ) + k b k L p ( B ′ ) (cid:17) . Also v ∈ C ,α loc ( B + r ∪ B ′ r )) for some α ∈ (0 , depending only on N, s, p, k c k L p ( B ′ ) . In addition wehave k v k C ,α (cid:16) [0 , / × B ′ / (cid:17) C (cid:16) k v k L ∞ ( [0 , × B ′ ) + k b k L p ( B ′ ) (cid:17) . We now state the following technical but crucial result.
Lemma 3.3. (i)
Let r > and V ∈ L q ( B ′ r ) for some q > N s . For every t , r > such that [0 , t ) × B ′ r ⋐ B + r ∪ B ′ r there exist positive constants A > , α ∈ (0 , depending on t , r , r, k V k L q ( B ′ r ) such that for every v ∈ H ( B + r ; t − s ) solving ( div( t − s ∇ v ) = 0 , in B + r − lim t → + t − s v t = V ( x ) v, on B ′ r , we have that v ∈ C ,α ([0 , t ) × B ′ r ) and (42) k v k C ,α ([0 ,t ) × B ′ r ) A k v k H ( B + r ; t − s ) . (ii) Let v ∈ H ( B + r ; t − s ) ∩ C ,α ( B + r ) for some α ∈ (0 , be a function satisfying ( div( t − s ∇ v ) = 0 , in B + r − lim t → + t − s v t = g ( x, v ) , on B ′ r , where g ∈ C ( B ′ r × R ) and | g ( x, ρ ) | c ( | ρ | + | ρ | p − ) for some < p ∗ ( s ) = NN − s , c > ,and every x ∈ B ′ r and ρ ∈ R . Let t , r > such that [0 , t ) × B ′ r ⋐ B + r . Then thereexist positive constants A , β depending only on N , p , s , c , r , r , t , k v k H ( B + r ; t − s ) and k g k C ( B ′ r × [0 ,A ]) where A = k v k C ,α ([0 ,t ) × B ′ r ) , with β ∈ (0 , , such that (43) k∇ x v k C ,β ([0 ,t ) × B ′ r ) A , (44) k t − s v t k C ,β ([0 ,t ) × B ′ r ) A . Remark 3.4.
The dependence of the constant A in Lemma 3.3 on k v k H ( B + r ; t − s ) is continuous;in particular we can take the same A for a family of solutions which are uniformly bounded in H ( B + r ; t − s ) ∩ C ,α ( B + r ). Proof . Part (i) and (42) follows from Proposition 3.2.To prove (ii), for h ∈ R N with | h | <<
1, we set v h ( t, x ) = v ( t,x + h ) − v ( t,x ) | h | for every x ∈ B ′ r / .Then we have ( div( t − s ∇ v h ) = 0 , in B + r / − lim t → + t − s v ht = c h ( x ) v h + b h , on B ′ r / , where c h ( x ) = g ( x, v (0 , x + h )) − g ( x, v (0 , x )) v (0 , x + h ) − v (0 , x ) χ { v (0 ,x + h ) = v (0 ,x ) } ( x )with χ A being the characteristic function of a set A and b h ( x ) = g ( x + h, v (0 , x + h )) − g ( x, v (0 , x + h )) | h | . Let A = k v k C ,α ([0 ,t ) × B ′ r ) . Then we have k c h k L ∞ ( B ′ r / ) + k b h k L ∞ ( B ′ r / ) k g ρ k L ∞ ( B ′ r / × [0 ,A ]) + k∇ x g k L ∞ ( B ′ r / × [0 ,A ]) , for every small h . Applying once again Proposition 3.2, k v h k C ,α ([0 ,t / × B ′ r / ) C ( k v h k L ∞ ([0 ,t / × B ′ r / ) + k b h k L ∞ ( B ′ r / ) ) C k v h k L ([0 ,t / × B ′ r / ; t − s ) + C k∇ g k L ∞ ( B ′ r / × [0 ,A ]) C k∇ v k L ([0 ,t ) × B ′ r ; t − s ) + C k∇ g k L ∞ ( B ′ r / × [0 ,A ]) A for every small h . By the Arzel`a-Ascoli Theorem, passing to the limit as | h | →
0, we conclude that z
7→ ∇ x v ( z ) ∈ C ([0 , t / × B ′ r / ) and estimate (43) holds for all β ∈ (0 , α ).Since the map x g ( x, v (0 , x )) ∈ C ,β ( B ′ r / ), estimate (44) follows from [[3], Lemma 4.5].Because of the presence of the Hardy potential | x | − s , we cannot expect the solution w of (37)to be L q loc near the origin for every q , but we can expect w to be in a space better than L ∗ ( s )loc .This is what we will prove in the next result. Lemma 3.5.
Let w be a solution to (37) in the sense of (38) . Then there exist p > ∗ ( s ) and R ∈ (0 , R ) such that w ∈ L p ( B + R ) . Proof . By (3), there are δ > R δ ∈ (0 , R ) such that(45) ( λ + | x | s | h ( x ) | ) λ + δ < Λ N,s , for all x ∈ B ′ R δ . Let β >
1. For all
L >
0, we define F L ( τ ) = | τ | β if | τ | < L and F L ( τ ) = βL β − | τ | + (1 − β ) L β if | τ | > L . Put G = 1 β F L F ′ L . It is easy to verify that, for all τ ∈ R ,(46) τ G L ( τ ) τ G ′ L ( τ ) , τ G L ( τ ) ( F L ( τ )) , ( F ′ L ( τ )) βG ′ L ( τ ) . Let η ∈ C ∞ c ( B + R δ ∪ B ′ R δ ) be a radial cut-off function such that η ≡ B + R δ / . It is clear that ζ := η G L ( w ) ∈ H ( B + R ; t − s ) and F L ( w ) ∈ H ( B + R ; t − s ). Using ζ as a test in (37), from (45)and integration by parts, we have that Z B + Rδ t − s η |∇ w | G ′ L ( w ) dtdx − κ s ( λ + δ ) Z B ′ Rδ | x | − s η wG L ( w ) dx − Z B + Rδ t − s η ∇ w · ∇ ηG L ( w ) dtdx + c Z B ′ Rδ η wG L ( w ) dx + c Z B ′ Rδ η | w | ∗ ( s ) − wG L ( w ) dx, for some positive c > C f , s, p, N . By Young’s inequality and (46), we havethat, for every σ > (cid:12)(cid:12)(cid:12)(cid:12) η ∇ w ) · ( w ∇ η ) G L ( w ) w (cid:12)(cid:12)(cid:12)(cid:12) σ η |∇ w | G ′ L ( w ) + 2 σ |∇ η | ( F L ( w )) , hence we obtain that (cid:16) − σ (cid:17) Z B + Rδ t − s η |∇ w | G ′ L ( w ) dt dx − κ s ( λ + δ ) Z B ′ Rδ | x | − s η wG L ( w ) dx c Z B ′ Rδ η ( | w | ∗ ( s ) − + 1) w G L ( w ) dx + 2 σ Z B + Rδ t − s |∇ η | ψ dt dx, RACTIONAL ELLIPTIC EQUATIONS 13 where we have set ψ = F L ( w ). Therefore by (46)1 β (cid:16) − σ (cid:17) Z B + Rδ t − s η |∇ ψ | dt dx − κ s ( λ + δ ) Z B ′ Rδ | x | − s ( ηψ ) dx c Z B ′ Rδ ( | w | ∗ ( s ) − + 1)( ηψ ) dx + 2 σ Z B + Rδ t − s |∇ η | ψ dt dx. Since |∇ ( ηψ ) | (1 + σ ) η |∇ ψ | + (1 + σ ) ψ |∇ η | , we have that1 β (1 + σ ) (cid:16) − σ (cid:17) Z B + Rδ t − s |∇ ( ηψ ) | dt dx − κ s ( λ + δ ) Z B ′ Rδ | x | − s ( ηψ ) dx c Z B ′ Rδ ( | w | ∗ ( s ) − + 1)( ηψ ) dx + C ( c, β, σ, R δ ) Z B + Rδ t − s ψ dt dx for some positive C ( c, β, σ, R δ ) > c , σ , R δ , and β . By H¨older inequality andLemma 2.6, we have that Z B ′ Rδ ( | w | ∗ ( s ) − + 1)( ηψ ) dx e S N,s (cid:18) Z B ′ Rδ | w | ∗ ( s ) dx (cid:19) ∗ ( s ) − ∗ ( s ) + | B ′ R δ | sN Z B + Rδ t − s |∇ ( ηψ ) | dt dx. We deduce that(47) A Z B + Rδ t − s |∇ ( ηψ ) | dt dx − κ s ( λ + δ ) Z B ′ Rδ | x | − s ( ηψ ) dx const Z B + Rδ t − s ψ dtdx, for some positive const > C f , s, p, N, β, σ, R δ , where A = 1(1 + σ ) β (cid:16) − σ (cid:17) − c e S N,s (cid:18) Z B ′ Rδ | w | ∗ ( s ) dx (cid:19) ∗ ( s ) − ∗ ( s ) + | B ′ R δ | sN . From Hardy inequality (20), we have that A Z B + Rδ t − s |∇ ( ηψ ) | dt dx − κ s ( λ + δ ) Z B ′ Rδ | x | − s ( ηψ ) dx > (cid:18) A − λ + δ Λ N,s (cid:19) Z B + Rδ t − s |∇ ( ηψ ) | dt dx, and, by (45), we can choose β sufficiently close to 1 and σ, R δ sufficiently small such that A − λ + δ Λ N,s > . Hence we have that C Z B + Rδ t − s |∇ ( ηψ ) | dt dx Z B + Rδ t − s ψ dtdx, for some constant C > f, h, s, p, N, β, ε, w, λ, δ . From Lemma 2.6 it follows that C e S − N,s (cid:18) Z B ′ Rδ | ηF L ( w ) | ∗ ( s ) dx (cid:19) ∗ ( s ) Z B + Rδ t − s | w | β dtdx for all L > . Hence by taking the limit as L → + ∞ (48) C e S − N,s (cid:18) Z B ′ Rδ/ | w | β ∗ ( s ) dx (cid:19) ∗ ( s ) Z B + Rδ t − s | w | β dtdx. The conclusion follows since β > H ( B + R ; t − s ) ֒ → L q ( B + R ; t − s ) for some q >
2, see forinstance [[8], Theorem 1.2].
Remark 3.6.
From Lemma 3.5, we deduce that, if w ∈ H ( B + R ; t − s ) is a weak solution to (37),then λ | x | s + h + f ( x,w ) w ∈ L q loc ( B ′ R \ { } ) for some N s < q p p − and hence, from Lemma 3.3, weconclude that w ∈ C ,α loc ( B + r \ { } ), ∇ x v ∈ C ,β loc ( B + r \ { } ), and t − s v t ∈ C ,β loc ( B + r \ { } ) for all r ∈ (0 , R ) and some 0 < β < α < D ′ (see Lemma3.9 below) and therefore N ′ . Theorem 3.7.
Let w solves (38) . Then for a.e. r ∈ (0 , R ) there holds (49) − N − s (cid:20) Z B + r t − s |∇ w | dz − κ s λ Z B ′ r w | x | s dx (cid:21) + r (cid:20) Z S + r t − s |∇ w | dS − κ s λ Z ∂B ′ r w | x | s dS ′ (cid:21) = r Z S + r t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS − κ s Z B ′ r ( N h + ∇ h · x ) w dx + rκ s Z ∂B ′ r hw dS ′ + rκ s Z ∂B ′ r F ( x, w ) dS ′ − κ s Z B ′ r [ ∇ x F ( x, w ) · x + N F ( x, w )] dx and (50) Z B + r t − s |∇ w | dz − κ s λ Z B ′ r w | x | s dx = Z S + r t − s ∂w∂ν w dS + κ s Z B ′ r (cid:18) hw + f ( x, w ) w (cid:19) dx. Proof . We write our problem in the form ( div( t − s ∇ w ) = 0 , in B + R , − lim t → + t − s w t = G ( x, w ) , on B ′ R , where G ∈ C ( B ′ R \ { } × R ), G ( x, ̺ ) = κ s (cid:0) λ | x | s ̺ + h ( x ) ̺ + f ( x, ̺ ) (cid:1) .We have, on B + R , the formula(51) div (cid:18) t − s |∇ w | z − t − s ( z · ∇ w ) ∇ w (cid:19) = N − s t − s |∇ w | − ( z · ∇ w ) div( t − s ∇ w ) . Let ρ < r < R . Now we integrate by parts over the set O δ := ( B + r \ B + ρ ) ∩ { ( t, x ) , t > δ } with δ >
0. We have N − s Z O δ t − s |∇ w ( z ) | dz = − δ − s Z B ′ √ r − δ \ B ′ √ ρ − δ |∇ w | ( δ, x ) dx + δ − s Z B ′ √ r − δ \ B ′ √ ρ − δ | w t | ( δ, x ) dx + r Z S + r ∩{ t>δ } t − s |∇ w | dS − r Z S + r ∩{ t>δ } t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS − ρ Z S + ρ ∩{ t>δ } t − s |∇ w | dS + ρ Z S + ρ ∩{ t>δ } t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS + Z B ′ √ r − δ \ B ′ √ ρ − δ ( x · ∇ x w ( δ, x )) δ − s w t ( δ, x ) dx. We now claim that there exists a sequence δ n → n →∞ " δ − sn Z B ′ r |∇ w | ( δ n , x ) dx + δ − sn Z B ′ r | w t | ( δ n , x ) dx = 0 . RACTIONAL ELLIPTIC EQUATIONS 15
If no such sequence exists, we would havelim inf δ → " δ − s Z B ′ r |∇ w | ( δ, x ) dx + δ − s Z B ′ r | w t | ( δ, x ) dx > C > δ > δ − s Z B ′ r |∇ w | ( δ, x ) dx + δ − s Z B ′ r | w t | ( δ, x ) dx > C δ ∈ (0 , δ ) . It follows that12 δ − s Z B ′ r |∇ w | ( δ, x ) dx + δ − s Z B ′ r | w t | ( δ, x ) dx > C δ for all δ ∈ (0 , δ )and so integrating the above inequality on (0 , δ ) we contradict the fact that w ∈ H ( B + R ; t − s ).Next, from the Dominated Convergence Theorem, Lemma 3.3 and Remark 3.6, we have thatlim δ → Z B ′ √ r − δ \ B ′ √ ρ − δ ( x · ∇ x w ( δ, x )) δ − s w t ( δ, x ) dx = − Z B ′ r \ B ′ ρ ( x · ∇ x w ) G ( x, w ) dx. We conclude that (replacing O δ with O δ n , for a sequence δ n →
0) that(52) N − s Z B + r \ B + ρ t − s |∇ w ( z ) | dz = r Z S + r t − s |∇ w | dS − r Z S + r t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS − ρ Z S + ρ t − s |∇ w | dS − ρ Z S + ρ t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS − Z B ′ r \ B ′ ρ ( x · ∇ x w ) G ( x, w ) dx. Furthermore, integration by parts yields Z B ′ r \ B ′ ρ ( x · ∇ x w ) G ( x, w ) dx = − N − s κ s λ Z B ′ r \ B ′ ρ w | x | s dx (53) − κ s Z B ′ r \ B ′ ρ ( N h ( x ) + ∇ h ( x ) · x ) w dx + κ s λ r Z ∂B ′ r w | x | s dS ′ + rκ s Z ∂B ′ r h ( x ) w dS ′ − κ s λ ρ Z ∂B ′ ρ w | x | s dS ′ − ρκ s Z ∂B ′ ρ h ( x ) w dS ′ − κ s Z B ′ r \ B ′ ρ [ ∇ x F ( x, w ) · x + N F ( x, w )] dx + rκ s Z ∂B ′ r F ( x, w ) dS ′ − ρκ s Z ∂B ′ ρ F ( x, w ) dS ′ . Since w ∈ H ( B + R ; t − s ), in view of Lemma 2.5 and (8), there exists a sequence ρ n → n →∞ ρ n (cid:20) Z S + ρn t − s |∇ w | dS + Z ∂B ′ ρn w | x | s dS ′ + Z ∂B ′ ρn | F ( x, w ) | dS ′ (cid:21) = 0 . Hence, taking ρ = ρ n and letting n → ∞ in (52) and (53), we obtain (49).(50) follows from Lemma 3.1 and density of C ∞ ( B + r ) in H ( B + r ; t − s ).3.2. On the Almgren type frequency N . In this section, we shall study the differentiabilityof N , it’s limit at 0 and provide estimates of N ′ . Lemma 3.8. H ∈ C (0 , R ) and H ′ ( r ) = 2 r N +1 − s Z S + r t − s w ∂w∂ν dS, for every r ∈ (0 , R ) , (54) H ′ ( r ) = 2 r D ( r ) , for every r ∈ (0 , R ) . (55) Proof . Fix r ∈ (0 , R ) and consider the limit(56) lim r → r H ( r ) − H ( r ) r − r = lim r → r Z S N + θ − s | w ( rθ ) | − | w ( r θ ) | r − r dS. Since w ∈ C ( R N +1+ ), then, for every θ ∈ S N + ,(57) lim r → r | w ( rθ ) | − | w ( r θ ) | r − r = 2 ∂w∂ν ( r θ ) w ( r θ ) . On the other hand, for any r ∈ ( r / , R ) and θ ∈ S N + we have (cid:12)(cid:12)(cid:12)(cid:12) | w ( rθ ) | − | w ( r θ ) | r − r (cid:12)(cid:12)(cid:12)(cid:12) B + R \ B + r | w |· sup B + R \ B + r (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) B + R \ B + r | w |· sup B + R \ B + r (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t | z | w t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ x w · x | z | (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) and hence, by (56), (57), Lemma 3.3 and the Dominated Convergence Theorem, we obtain that H ′ ( r ) = 2 Z S N + θ − s ∂w∂ν ( r θ ) w ( r θ ) dS ( θ ) = 2 r N +1 − s Z S + r t − s w ∂w∂ν dS. The continuity of H ′ on the interval (0 , R ) follows by the representation of H ′ given above, Lemma3.3, and the Dominated Convergence Theorem.Finally, (55) follows from (54), (39), and (50).The regularity of the function D is established in the following lemma. Lemma 3.9.
The function D defined in (39) belongs to W , (0 , R ) and D ′ ( r ) = 2 r N +1 − s (cid:20) r Z S + r t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS − κ s Z B ′ r (cid:16) sh + 12 ( ∇ h · x ) (cid:17) w dx (cid:21) (58) + κ s r N +1 − s Z B ′ r (cid:0) ( N − s ) f ( x, w ) w − N F ( x, w ) − ∇ x F ( x, w ) · x (cid:1) dx + κ s r N − s Z ∂B ′ r (cid:0) F ( x, w ) − f ( x, w ) w (cid:1) dS ′ in a distributional sense and for a.e. r ∈ (0 , R ) . Proof . For any r ∈ (0 , r ) let I ( r ) = Z B + r t − s |∇ w | dt dx − κ s Z B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) w (cid:19) dx. (59)From the fact that w ∈ H ( B + R ; t − s ), Lemma 2.5, and (8), we deduce that I ∈ W , (0 , R ) and(60) I ′ ( r ) = Z S + r t − s |∇ w | dS − κ s Z ∂B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) w (cid:19) dS ′ for a.e. r ∈ (0 , R ) and in the distributional sense. Therefore D ∈ W , (0 , R ) and, using (49), (59),and (60) into D ′ ( r ) = r s − − N [ − ( N − s ) I ( r ) + rI ′ ( r )] , we obtain (58) for a.e. r ∈ (0 , R ) and in the distributional sense.Before going on, we recall that w is nontrivial and satisfies (38). We prove now that, if w H ( r ) does not vanish for r sufficiently small. Lemma 3.10.
There exists R ∈ (0 , R ) such that H ( r ) > for any r ∈ (0 , R ) , where H is definedby (40). Proof . Clearly from assumption (2), there exists R ∈ (0 , R ) such that λ Λ N,s + C h R ε Λ N,s + C f S − N,s (cid:0) ω N − N (cid:1) ∗ ( s ) − p ∗ ( s ) R N (2 ∗ ( s ) − p )2 ∗ ( s ) k w k p − L ∗ ( s ) ( B ′ R ) < , (61)where ω N − denotes the volume of the unit sphere S N − , i.e. ω N − = R S N − dS . RACTIONAL ELLIPTIC EQUATIONS 17
Next suppose by contradiction that there exists r ∈ (0 , R ) such that H ( r ) = 0. Then w = 0a.e. on S + r . From (50) it follows that Z B + r t − s |∇ w | dz − κ s λ Z B ′ r w | x | s dx − κ s Z B ′ r (cid:18) h ( x ) w + f ( x, w ) w (cid:19) dx = 0 . From Lemma 2.5, assumptions (3)-(4), H¨older’s inequality, and (8), it follows that0 = Z B + r t − s |∇ w | dz − κ s λ Z B ′ r w | x | s dx − κ s Z B ′ r (cid:18) h ( x ) w + f ( x, w ) w (cid:19) dx > (cid:20) − λ Λ N,s − C h R ε Λ N,s − C f S − N,s (cid:0) ω N − N (cid:1) ∗ ( s ) − p ∗ ( s ) R N (2 ∗ ( s ) − p )2 ∗ ( s ) k w k p − L ∗ ( s ) ( B ′ R ) (cid:21) Z B + r t − s |∇ w | dz, which, together with (61), implies w ≡ B + r by Lemma 2.4. Classical unique continuationprinciples for second order elliptic equations with locally bounded coefficients (see e.g. [26]) allowto conclude that w = 0 a.e. in B + R , a contradiction.Letting R be as in Lemma 3.10 and recalling (41), the Almgren type frequency function(62) N ( r ) = D ( r ) H ( r )is well defined in (0 , R ). Using Lemmas 3.8 and 3.9, we can now compute the derivative of N . Lemma 3.11.
The function N defined in (62) belongs to W , (0 , R ) and N ′ ( r ) = ν ( r ) + ν ( r )(63) in a distributional sense and for a.e. r ∈ (0 , R ) , where ν ( r ) = 2 r h (cid:16)R S + r t − s (cid:12)(cid:12) ∂w∂ν (cid:12)(cid:12) dS (cid:17) · (cid:16)R S + r t − s w dS (cid:17) − (cid:16)R S + r t − s w ∂w∂ν dS (cid:17) i(cid:16)R S + r t − s w dS (cid:17) , (64) ν > , and ν ( r ) = − κ s R B ′ r (2 sh + ∇ h · x ) | w | dx R S + r t − s w dS + κ s r R ∂B ′ r (cid:0) F ( x, w ) − f ( x, w ) w (cid:1) dS ′ R S + r t − s w dS (65) + κ s R B ′ r (cid:0) ( N − s ) f ( x, w ) w − N F ( x, w ) − ∇ x F ( x, w ) · x (cid:1) dx R S + r t − s w dS . Proof . From Lemmas 3.8, 3.10, and 3.9, it follows that
N ∈ W , (0 , R ). From (55) it followsthat N ′ ( r ) = D ′ ( r ) H ( r ) − D ( r ) H ′ ( r )( H ( r )) = D ′ ( r ) H ( r ) − r ( H ′ ( r )) ( H ( r )) and the proof of the lemma easily follows from (54) and (58). Now it is easy to see that ν > N ( r ) admits a finite limit as r → + . To this aim, the following estimate playsa crucial role. Lemma 3.12.
Let N be the function defined in (62). There exist ˜ R ∈ (0 , R ) and a constant C > such that (66) Z B + r t − s |∇ w | dt dx − κ s Z B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) w (cid:19) dx > − (cid:18) N − s r (cid:19) Z S + r t − s w dS + C Z B ′ r w | x | s dx + (cid:18) Z B ′ r | w | ∗ ( s ) dx (cid:19) ∗ ( s ) ! , (67) Z B + r t − s |∇ w | dt dx − κ s Z B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) w (cid:19) dx > − (cid:18) N − s r (cid:19) Z S + r t − s w dS + C Z B + r t − s |∇ w | dt dx, and (68) N ( r ) > − N − s for every r ∈ (0 , ˜ R ) . Proof . From Corollary 2.7, (3), and (4), it follows that Z B + r t − s |∇ w | dt dx − κ s Z B ′ r (cid:18) λ | x | s w + hw + f ( x, w ) w (cid:19) dx + (cid:18) N − s r (cid:19) Z S + r t − s w dS > (cid:18) κ s (Λ N,s − λ )2 − C h κ s r ε (cid:19) Z B ′ r w | x | s dx + (cid:18) Λ N,s − λ N,s ) e S N,s − C f κ s | B ′ r | ∗ ( s ) − p ∗ ( s ) k w k p − L ∗ ( s ) ( B ′ r ) (cid:19)(cid:18) Z B ′ r | w | ∗ ( s ) dx (cid:19) ∗ ( s ) for every r ∈ (0 , R ). Since λ < Λ N,s , from the above estimate it follows that we can choose˜ R ∈ (0 , R ) sufficiently small such that estimate (66) holds for r ∈ (0 , ˜ R ) for some positive constant C >
0. The proof of (67) can be performed in a similar way, using Lemmas 2.5 and 2.6. Estimate(66), together with (39) and (40), yields (68).
Lemma 3.13.
Let ˜ R be as in Lemma 3.12 and ν as in (65). Then there exist a positive constant C > and a function g ∈ L (0 , ˜ R ) , g > a.e. in (0 , ˜ R ) , such that | ν ( r ) | C (cid:20) N ( r ) + N − s (cid:21) (cid:16) r − ε + r − s ( p − ∗ ( s )) p + g ( r ) (cid:17) for a.e. r ∈ (0 , ˜ R ) and Z r g ( ρ ) dρ − α k w k p (1 − α ) L p ( B ′ ˜ R ) r N (cid:0) α ∗ ( s ) − ∗ ( s ) − pα − p (cid:1) for all r ∈ (0 , ˜ R ) with some α satisfying p < α < . Proof . From (3) and (66) we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ′ r (2 sh ( x ) + ∇ h ( x ) · x ) | w | dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C h r ε Z B ′ r | w | | x | s dx C h C − r ε + N − s (cid:2) D ( r ) + N − s H ( r ) (cid:3) , and, therefore, for any r ∈ (0 , ˜ R ), we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R B ′ r (2 sh ( x ) + ∇ h ( x ) · x ) | w | dx R S + r t − s w dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C h C − r − ε D ( r ) + N − s H ( r ) H ( r )(69) = 2 C h C − r − ε (cid:20) N ( r ) + N − s (cid:21) . RACTIONAL ELLIPTIC EQUATIONS 19
By (4), H¨older’s inequality, and (66), for some constant const = const (
N, s, C f ) > N, s, C f , and for all r ∈ (0 , ˜ R ), there holds (cid:12)(cid:12)(cid:12)(cid:12) Z B ′ r (cid:0) ( N − s ) f ( x, w ) w − N F ( x, w ) − ∇ x F ( x, w ) · x (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) const Z B ′ r ( | w | + | w | ∗ ( s ) ) dx const (cid:18)(cid:16) ω N − N (cid:17) sN r s + k w k ∗ ( s ) − L ∗ ( s ) ( B ′ ˜ R ) (cid:19)(cid:18) Z B ′ r | w | ∗ ( s ) dx (cid:19) ∗ ( s ) const C (cid:18)(cid:16) ω N − N (cid:17) sN r s + (cid:16) ω N − N (cid:17) s ( p − ∗ ( s )) Np r s ( p − ∗ ( s )) p k w k ∗ ( s ) − L p ( B ′ ˜ R ) (cid:19) r N − s (cid:2) D ( r ) + N − s H ( r ) (cid:3) and hence(70) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R B ′ r (cid:0) ( N − s ) f ( x, w ) w − N F ( x, w ) − ∇ x F ( x, w ) · x (cid:1) dx R S + r t − s w dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) const C (cid:18)(cid:16) ω N − N (cid:17) sN r s ∗ ( s ) p + (cid:16) ω N − N (cid:17) s ( p − ∗ ( s )) Np k w k ∗ ( s ) − L p ( B ′ ˜ R ) (cid:19) r − s ( p − ∗ ( s )) p (cid:2) N ( r ) + N − s (cid:3) . Let us fix p < α <
1. By H¨older’s inequality and (66), (cid:18) Z B ′ r | w | p dx (cid:19) α = (cid:18) Z B ′ r | w | p − α | w | α dx (cid:19) α (71) (cid:18) Z B ′ r | w | ∗ ( s ) pα − ∗ ( s ) α − dx (cid:19) α ∗ ( s ) − ∗ ( s ) (cid:18) Z B ′ r | w | ∗ ( s ) dx (cid:19) ∗ ( s ) (cid:16) ω N − N (cid:17) α ∗ ( s ) − ∗ ( s ) − pα − p r N (cid:0) α ∗ ( s ) − ∗ ( s ) − pα − p (cid:1) k w k pα − L p ( B ′ ˜ R ) r N − s C (cid:2) D ( r ) + N − s H ( r ) (cid:3) = C − (cid:16) ω N − N (cid:17) βN r − β (cid:20) N ( r ) + N − s (cid:21) (cid:18) Z S + r t − s w dS (cid:19) for all r ∈ (0 , ˜ R ), where β = N (cid:0) α ∗ ( s ) − ∗ ( s ) − pα − p (cid:1) >
0. From (4), (71), and (68), there exists someconst = const (
N, s, C f ) > N, s, C f such that, for all r ∈ (0 , ˜ R ),(72) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r R ∂B ′ r (cid:0) F ( x, w ) − f ( x, w ) w (cid:1) dS ′ R S + r t − s w dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) const r R ∂B ′ r | w | p dS ′ R S + r t − s w dS const C (cid:16) ω N − N (cid:17) βN (cid:20) N ( r ) + N − s (cid:21) r β R ∂B ′ r | w | p dS ′ (cid:16) R B ′ r | w | p dx (cid:17) α . By a direct calculation, we have that(73) r β R ∂B ′ r | w | p dS ′ (cid:16) R B ′ r | w | p dx (cid:17) α = 11 − α " ddr (cid:18) r β (cid:18) Z B ′ r | w | p dx (cid:19) − α (cid:19) − β r − β (cid:18) Z B ′ r | w | p dx (cid:19) − α in the distributional sense and for a.e. r ∈ (0 , ˜ R ). Sincelim r → + r β (cid:18) Z B ′ r | w | p dx (cid:19) − α = 0we deduce that the function r ddr (cid:18) r β (cid:18) Z B ′ r | w | p dx (cid:19) − α (cid:19) is integrable over (0 , ˜ R ). Being r − β (cid:18) Z B ′ r | w | p dx (cid:19) − α = o ( r − β )as r → + , we have that also the function r r − β (cid:18) Z B ′ r | w | p dx (cid:19) p − p is integrable over (0 , ˜ R ). Therefore, by (73), we deduce that(74) g ( r ) := r β R ∂B ′ r | w | p dS ′ (cid:16) R B ′ r | w | p dx (cid:17) α ∈ L (0 , ˜ R )and 0 Z r g ( ρ ) dρ − α k w k p (1 − α ) L p ( B ′ r ) r β (75)for all r ∈ (0 , ˜ R ). Collecting (69), (70), (72), (74), and (75), we obtain the stated estimate. Lemma 3.14.
Let ˜ R be as in Lemma 3.12 and N as in (62). Then there exist a positive constant C > such that (76) N ( r ) C for all r ∈ (0 , ˜ R ) . Proof . By Lemma 3.11 and Lemma 3.13, we obtain(77) (cid:18) N + N − s (cid:19) ′ ( r ) > ν ( r ) > − C (cid:20) N ( r ) + N − s (cid:21) (cid:16) r − ε + r − s ( p − ∗ ( s )) p + g ( r ) (cid:17) for a.e. r ∈ (0 , ˜ R ). Integration over ( r, ˜ R ) yields N ( r ) − N − s (cid:18) N ( ˜ R ) + N − s (cid:19) exp C (cid:18) ˜ R ε ε + p ˜ R s ( p − ∗ ( s )) p s ( p − ∗ ( s )) + Z ˜ R g ( ρ ) dρ (cid:19) for any r ∈ (0 , ˜ R ), thus proving estimate (76). Lemma 3.15.
The limit γ := lim r → + N ( r ) exists and is finite. Proof . By Lemmas 3.13 and 3.14, the function ν defined in (65) belongs to L (0 , ˜ R ). Hence,by Lemma 3.11, N ′ is the sum of a nonnegative function and of a L -function on (0 , ˜ R ). Therefore N ( r ) = N ( ˜ R ) − Z ˜ Rr N ′ ( ρ ) dρ admits a limit as r → + which is necessarily finite in view of (68) and (76).The function H defined in (40) can be estimated as follows. Lemma 3.16.
Let γ := lim r → + N ( r ) be as in Lemma 3.15 and ˜ R as in Lemma 3.12. Then thereexists a constant K > such that (78) H ( r ) K r γ for all r ∈ (0 , ˜ R ) . Moreover, for any σ > there exists a constant K ( σ ) > depending on σ such that (79) H ( r ) > K ( σ ) r γ + σ for all r ∈ (0 , ˜ R ) . RACTIONAL ELLIPTIC EQUATIONS 21
Proof . By Lemma 3.15, N ′ ∈ L (0 , ˜ r ) and, by Lemma 3.14, N is bounded, then from (77) and(75) it follows that(80) N ( r ) − γ = Z r N ′ ( ρ ) dρ > − C r δ for some constant C > r ∈ (0 , ˜ R ), where(81) δ = min (cid:26) ε, s ( p − ∗ ( s )) p , N (cid:18) α ∗ ( s ) − ∗ ( s ) − pα − p (cid:19)(cid:27) > . Therefore by (55), (62), and (80) we deduce that, for all r ∈ (0 , ˜ R ), H ′ ( r ) H ( r ) = 2 N ( r ) r > γr − C r − δ , which, after integration over the interval ( r, ˜ R ), yields (78).Since γ = lim r → + N ( r ), for any σ > r σ > N ( r ) < γ + σ/ r ∈ (0 , r σ ) and hence H ′ ( r ) H ( r ) = 2 N ( r ) r < γ + σr for all r ∈ (0 , r σ ) . Integrating over the interval ( r, r σ ) and by continuity of H outside 0, we obtain (79) for someconstant K ( σ ) depending on σ . 4. The blow-up argument
The main result of this section, which also contains Theorem 1.1, is the following theorem.
Theorem 4.1.
Let w satisfy (38) , with s, λ, h, f as in assumptions (2) , (3) , and (4) . Then, letting N ( r ) as in (41), there there exists k ∈ N , k > , such that (82) lim r → + N ( r ) = − N − s s(cid:18) N − s (cid:19) + µ k ( λ ) . Furthermore, if γ denotes the limit in (82), m > is the multiplicity of the eigenvalue µ j ( λ ) = µ j +1 ( λ ) = · · · = µ j + m − ( λ ) and { ψ i } j + m − i = j ( j k j + m − ) is an L ( S N + ; θ − s ) -orthonormal basis for the eigenspace of problem (10) associated to µ k ( λ ) , then τ − γ w (0 , τ x ) → | x | γ j + m − X i = j β i ψ i (cid:16) , x | x | (cid:17) in C ,α loc ( B ′ \ { } ) as τ → + ,τ − γ w ( τ θ ) → j + m − X i = j β i ψ i ( θ ) in C ,α ( S N + ) as τ → + ,τ − γ w (0 , τ θ ′ ) → j + m − X i = j β i ψ i (0 , θ ′ ) in C ,α ( S N − ) as τ → + , and τ − γ ∇ x w (0 , τ θ ′ ) → j + m − X i = j β i (cid:16) γψ i (0 , θ ′ ) θ ′ + ∇ S N − ψ i (0 , · )( θ ′ ) (cid:17) in C ,α ( S N − ) as τ → + , for some α ∈ (0 , , where β i = R − γ Z S N + θ − s w ( R θ ) ψ i ( θ ) dS ( θ )+ κ s Z S N − "Z R h ( tθ ′ ) w (0 , tθ ′ ) + f ( tθ ′ , w (0 , tθ ′ ))2 γ + N − s (cid:18) t s − γ − − t γ + N − R γ + N − s (cid:19) dt ψ i (0 , θ ′ ) dS ( θ ′ ) , for all R > such that B ′ R = { x ∈ R N : | x | R } ⊂ Ω and ( β j , β j +1 , . . . , β j + m − ) = (0 , , . . . , . To prove Theorem 4.1, we start by determining the asymptotic profile of blowing up renormalizedsolutions to (37).
Lemma 4.2.
Let w as in Theorem 4.1. Let γ := lim r → + N ( r ) as in Lemma 3.15. Then (i) there exists k ∈ N , k > , such that γ = − N − s + q(cid:0) N − s (cid:1) + µ k ( λ ) ; (ii) for every sequence τ n → + , there exist a subsequence { τ n k } k ∈ N and an eigenfunction ψ ofproblem (10) associated to the eigenvalue µ k ( λ ) such that k ψ k L ( S N + ; θ − s ) = 1 and w ( τ n k z ) p H ( τ n k ) → | z | γ ψ (cid:16) z | z | (cid:17) strongly in H ( B + r ; t − s ) and in C ,α loc ( B + r \ { } ) for some α ∈ (0 , and all r ∈ (0 , and w (0 , τ n k x ) p H ( τ n k ) → | x | γ ψ (cid:16) , x | x | (cid:17) in C ,α loc ( B ′ \ { } ) . Proof . Let us set(83) w τ ( z ) = w ( τ z ) p H ( τ ) . We notice that R S +1 t − s | w τ | dS = 1. Moreover, by scaling and (76),(84) Z B +1 t − s |∇ w τ ( z ) | dz − κ s Z B ′ (cid:18) λ | x | s | w τ | + τ s h ( τ x ) | w τ | + τ s p H ( τ ) f (cid:16) τ x, p H ( τ ) w τ (cid:17) w τ (cid:19) dx = N ( τ ) C for every τ ∈ (0 , ˜ R ), whereas, from (67),(85) N ( τ ) > τ − N +2 s H ( τ ) (cid:18) − (cid:18) N − s τ (cid:19) Z S + τ t − s w dS + C Z B + τ t − s |∇ w | dt dx (cid:19) = − N − s C Z B +1 t − s |∇ w τ ( z ) | dz for every τ ∈ (0 , ˜ R ). From (84) and (85) we deduce that(86) { w τ } τ ∈ (0 , ˜ R ) is bounded in H ( B +1 ; t − s ) . Therefore, for any given sequence τ n → + , there exists a subsequence τ n k → + such that w τ nk ⇀ e w weakly in H ( B +1 ; t − s ) for some e w ∈ H ( B +1 ; t − s ). Due to compactness of the traceembedding (28), we obtain that R S +1 t − s | e w | dS = 1. In particular e w τ ∈ (0 , ˜ R ), w τ satisfies(87) div( t − s ∇ w τ ) = 0 , in B +1 , − lim t → + t − s ∂w τ ∂t = κ s (cid:16) λ | x | s w τ + τ s h ( τ x ) w τ + τ s √ H ( τ ) f ( τ x, p H ( τ ) w τ ) (cid:17) , on B ′ , in a weak sense, i.e.(88) Z B +1 t − s ∇ w τ · ∇ e ϕ dt dx = κ s Z B ′ (cid:18) λ | x | s w τ + τ s h ( τ x ) w τ + τ s p H ( τ ) f (cid:16) τ x, p H ( τ ) w τ (cid:17)(cid:19) e ϕ (0 , x ) dx RACTIONAL ELLIPTIC EQUATIONS 23 for all e ϕ ∈ H ( B +1 ; t − s ) s.t. e ϕ = 0 on S N + and, for such e ϕ , by (4) and H¨older’s inequality,(89) τ s p H ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) Z B ′ f (cid:16) τ x, p H ( τ ) w τ (0 , x ) (cid:17) e ϕ (0 , x ) dx (cid:12)(cid:12)(cid:12)(cid:12) C f τ s Z B ′ | w (0 , τ x ) | p − | w τ (0 , x ) || e ϕ (0 , x ) | dx C f k Tr e ϕ k L p ( B ′ ) k w τ k L p ( B ′ ) k w k p − L p ( B ′ τ ) τ ( N − s )(2 ∗ ( s ) − p ) p = o (1) as τ → + and, by (3) and Lemma 2.5,(90) τ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ′ h ( τ x ) w τ e ϕ (0 , x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C h τ ε κ s Λ N,s (cid:18) Z B +1 t − s |∇ w τ ( z ) | dz + N − s (cid:19) / (cid:18) Z B +1 t − s |∇ e ϕ ( z ) | dz (cid:19) / = o (1) as τ → + . From (89), (90), and weak convergence w τ nk ⇀ e w in H ( B +1 ; t − s ), we can pass to the limit in(87) along the sequence τ n k and obtain that e w weakly solves(91) ( div( t − s ∇ e w ) = 0 , in B +1 , − lim t → + t − s ∂ e w∂t = κ s λ | x | s e w, on B ′ . From (4), letting q = p p − > N s with p as in Lemma 3.5, we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ s p H ( τ ) f (cid:0) τ x, p H ( τ ) w τ ( x ) (cid:1) w τ ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( B ′ ) C f τ s − Nq (cid:18) Z B τ | w ( x ) | p dx (cid:19) /q (cid:19) = O (1)as τ → + . Therefore from Lemma 3.3 part (i) there holds(92) w τ nk → e w in C ,α loc ( B + r \ { } ) , while Lemma 3.3 part (ii) and Remark 3.4 imply that(93) ∇ x w τ nk → ∇ x e w, and t − s ∂w τ nk ∂t → t − s ∂ e w∂t in C ,α loc ( B + r \ { } )for some α ∈ (0 ,
1) and all r ∈ (0 , τ s p H ( τ ) Z B ′ f (cid:16) τ x, p H ( τ ) w τ (cid:17) w τ dx = o (1) as τ → + and(95) τ s Z B ′ h ( τ x ) | w τ | dx = o (1) as τ → + . Multiplying equation (87) with w τ , integrating in B + r , and using (93), (94), (95), we easily obtainthat k w τ nk k H ( B + r ; t − s ) → k e w k H ( B + r ; t − s ) for all r ∈ (0 , w τ nk → e w in H ( B + r ; t − s )for any r ∈ (0 , r ∈ (0 ,
1) and k ∈ N , let us define the functions D k ( r ) = 1 r N − s (cid:20) Z B + r t − s |∇ w τ nk | dt dx − κ s Z B ′ r (cid:18) λ | x | s | w τ nk | + τ sn k h ( τ n k x ) | w τ nk | + τ sn k p H ( τ n k ) f (cid:16) τ n k x, p H ( τ n k ) w τ nk (cid:17) w τ nk (cid:19) dx (cid:21) and H k ( r ) = 1 r N +1 − s Z S + r t − s | w τ nk | dS. Direct calculations yield(97) N k ( r ) := D k ( r ) H k ( r ) = D ( τ n k r ) H ( τ n k r ) = N ( τ n k r ) for all r ∈ (0 , . From (96), (94), and (95), it follows that, for any fixed r ∈ (0 , D k ( r ) → e D ( r ) , where(99) e D ( r ) = 1 r N − s (cid:20) Z B + r t − s |∇ e w | dt dx − κ s Z B ′ r λ | x | s e w dx (cid:21) for all r ∈ (0 , . On the other hand, by compactness of the trace embedding (28), we also have(100) H k ( r ) → e H ( r ) for any fixed r ∈ (0 , , where(101) e H ( r ) = 1 r N +1 − s Z S + r t − s e w dS. From (35) it follows that e D ( r ) > − N − s e H ( r ) for all r ∈ (0 , r ∈ (0 , e H ( r ) = 0 then e D ( r ) >
0, and passing to the limit in (97) should give a contradiction with Lemma3.15. Hence H w ( r ) > r ∈ (0 ,
1) and the function e N ( r ) := e D ( r ) e H ( r )is well defined for r ∈ (0 , e N ( r ) = lim k →∞ N ( τ n k r ) = γ for all r ∈ (0 , e N is constant in (0 ,
1) and hence e N ′ ( r ) = 0 for any r ∈ (0 , h ≡ f ≡
0, we obtain Z S + r t − s (cid:12)(cid:12)(cid:12)(cid:12) ∂ e w∂ν (cid:12)(cid:12)(cid:12)(cid:12) dS ! · (cid:18)Z S + r t − s e w dS (cid:19) − (cid:18)Z S + r t − s e w ∂ e w∂ν dS (cid:19) = 0 for all r ∈ (0 , , which implies that e w and ∂ e w∂ν have the same direction as vectors in L ( S + r ; t − s ) and hence thereexists a function η = η ( r ) such that ∂ e w∂ν ( r, θ ) = η ( r ) e w ( r, θ ) for all r ∈ (0 ,
1) and θ ∈ S N + . Afterintegration we obtain(103) e w ( r, θ ) = e R r η ( s ) ds e w (1 , θ ) = ϕ ( r ) ψ ( θ ) , r ∈ (0 , , θ ∈ S N + , where ϕ ( r ) = e R r η ( s ) ds and ψ ( θ ) = e w (1 , θ ). From (91), (103), and Lemma 2.1, it follows that,weakly, ( r N (cid:0) r N +1 − s ϕ ′ (cid:1) ′ θ − s ψ ( θ ) + r − − s ϕ ( r ) div S N ( θ − s ∇ S N ψ ( θ )) = 0 , − lim θ → + θ − s ∇ S N ψ ( θ ) · e = κ s λψ (0 , θ ′ ) . Taking r fixed we deduce that ψ is an eigenfunction of the eigenvalue problem (10). If µ k ( λ ) isthe corresponding eigenvalue then ϕ ( r ) solves the equation1 r N (cid:0) r N +1 − s ϕ ′ (cid:1) ′ − µ k ( λ ) r − − s ϕ ( r ) = 0i.e. ϕ ′′ ( r ) + N + 1 − sr ϕ ′ − µ k ( λ ) r ϕ ( r ) = 0and hence ϕ ( r ) is of the form ϕ ( r ) = c r σ + k + c r σ − k for some c , c ∈ R , where σ + k = − N − s s(cid:18) N − s (cid:19) + µ k ( λ ) and σ − k = − N − s − s(cid:18) N − s (cid:19) + µ k ( λ ) . RACTIONAL ELLIPTIC EQUATIONS 25
Since the function | x | σ − k ψ ( x | x | ) / ∈ L ( B ′ ; | x | − s ) and hence | z | σ − k ψ ( z | z | ) / ∈ H ( B +1 ; t − s ) in virtueof Lemma 2.5, we deduce that c = 0 and ϕ ( r ) = c r σ + k . Moreover, from ϕ (1) = 1, we obtain that c = 1 and then(104) e w ( r, θ ) = r σ + k ψ ( θ ) , for all r ∈ (0 ,
1) and θ ∈ S N + . It remains to prove part (i). From (104) and the fact that R S N + θ − s ψ ( θ ) dS = 1 it follows that e D ( r ) = 1 r N − s (cid:20) Z B + r t − s |∇ e w | dt dx − κ s Z B ′ r λ | x | s e w dx (cid:21) = r s − N ( σ + k ) Z r t N − − s +2 σ + k dt + r s − N (cid:18) Z r t N − − s +2 σ + k dt (cid:19)(cid:18) Z S N + θ − s |∇ S N ψ ( θ ) | dS − λκ s Z S N − |T ψ ( θ ′ ) | dS ′ (cid:19) = ( σ + k ) + µ k ( λ ) N − s + 2 σ + k r σ + k = σ + k r σ + k and e H ( r ) = Z S N + θ − s e w ( rθ ) dS = r σ + k , and hence from (102) it follows that γ = e N ( r ) = e D ( r ) e H ( r ) = σ + k . This completes the proof of thelemma.The following lemma describes the behavior of H ( r ) as r → + . Lemma 4.3.
Let w satisfy (37) , H be defined in (40) , and let γ := lim r → + N ( r ) as in Lemma3.15. Then the limit lim r → + r − γ H ( r ) exists and it is finite. Proof . In view of (78) it is sufficient to prove that the limit exists. By (40), (55), and Lemma 3.15we have ddr H ( r ) r γ = − γr − γ − H ( r ) + r − γ H ′ ( r ) = 2 r − γ − ( D ( r ) − γH ( r )) = 2 r − γ − H ( r ) Z r N ′ ( ρ ) dρ. Integration over ( r, ˜ R ) yields(105) H ( ˜ R )˜ R γ − H ( r ) r γ = Z ˜ Rr s − γ − H ( ρ ) (cid:18)Z ρ ν ( t ) dt (cid:19) dρ + Z ˜ Rr ρ − γ − H ( ρ ) (cid:18)Z ρ ν ( t ) dt (cid:19) dρ where ν and ν are as in (64) and (65). Since, by Schwarz’s inequality, ν >
0, we have thatlim r → + R ˜ Rr ρ − γ − H ( ρ ) (cid:0)R ρ ν ( t ) dt (cid:1) dρ exists. On the other hand, by (78), Lemma 3.13, and(76), we deduce that (cid:12)(cid:12)(cid:12)(cid:12) ρ − γ − H ( ρ ) (cid:18)Z ρ ν ( t ) dt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) K C (cid:16) C + N − s (cid:17) ρ − Z ρ (cid:16) t − ε + t − s ( p − ∗ ( s )) p + g ( t ) (cid:17) dt K C (cid:16) C + N − s (cid:17) ρ − (cid:18) ρ ε ε + p ρ s ( p − ∗ ( s )) p s ( p − ∗ ( s )) + 11 − α k w k p (1 − α ) L p ( B ′ ˜ R ) ρ N (cid:0) α ∗ ( s ) − ∗ ( s ) − pα − p (cid:1)(cid:19) for all ρ ∈ (0 , e R ), which proves that ρ − γ − H ( ρ ) (cid:0)R ρ ν ( t ) dt (cid:1) ∈ L (0 , e R ). Hence both terms at theright hand side of (105) admit a limit as r → + thus completing the proof.From Lemma 4.2, the following pointwise estimate for solutions to (1) and (38) can be derived. Lemma 4.4.
Let w satisfying (38) . Then there exist C , C > and ¯ r ∈ (0 , ˜ R ) such that (i) sup S + r | w | C r N +1 − s R S + r t − s | w ( z ) | dS for every < r < ¯ r , (ii) | w ( z ) | C | z | γ for all z ∈ B +¯ r and in particular | w (0 , x ) | C | x | γ for all x ∈ B ′ ¯ r , where γ := lim r → + N ( r ) is as in Lemma 3.15. Proof . We first notice that (ii) follows directly from (i) and (78). In order to prove (i), we argueby contradiction and assume that there exists a sequence τ n → + such thatsup θ ∈ S N + (cid:12)(cid:12)(cid:12) w (cid:16) τ n θ (cid:17)(cid:12)(cid:12)(cid:12) > nH (cid:16) τ n (cid:17) with H as in (40), i.e. sup x ∈ S +1 / | w ( τ n z ) | > N +1 − s n Z S +1 / t − s w ( τ n z ) dS, i.e., defining w τ as in (83)(106) sup x ∈ S +1 / | w τ n ( z ) | > N +1 − s n Z S +1 / t − s | w τ n ( z ) | dS. From Lemma 4.2, along a subsequence τ n k we have that w τ nk → | z | γ ψ (cid:0) z | z | (cid:1) in C ,α loc ( S +1 / ), for some ψ eigenfunction of problem (10), hence passing to the limit in (106) gives rise to a contradiction.We will now prove that lim r → + r − γ H ( r ) is strictly positive. Lemma 4.5.
Under the same assumption as in Lemmas 4.3 and 4.4, we have lim r → + r − γ H ( r ) > . Proof . For all k >
1, let ψ k be as in (24), i.e. ψ k is a L ( S N + ; θ − s )-normalized eigenfunction ofproblem (10) associated to the eigenvalue µ k ( λ ) and { ψ k } k is an orthonormal basis of L ( S N + ; θ − s ).From Lemma 4.2 there exist j , m ∈ N , j , m > m is the multiplicity of the eigenvalue µ j ( λ ) = µ j +1 ( λ ) = · · · = µ j + m − ( λ ) and(107) γ = lim r → + N ( r ) = − N − s s(cid:18) N − s (cid:19) + µ i ( λ ) , i = j , . . . , j + m − . Let us expand w as w ( z ) = w ( τ θ ) = ∞ X k =1 ϕ k ( τ ) ψ k ( θ )where τ = | z | ∈ (0 , R ], θ = z/ | z | ∈ S N + , and(108) ϕ k ( τ ) = Z S N + θ − s w ( τ θ ) ψ k ( θ ) dS. The Parseval identity yields(109) H ( τ ) = Z S N + θ − s w ( τ θ ) dS = ∞ X k =1 ϕ k ( τ ) , for all 0 < τ R. In particular, from (78) and (109) it follows that, for all k > ϕ k ( τ ) = O ( τ γ ) as τ → + . Equations (38) and (24) imply that, for every k , − ϕ ′′ k ( τ ) − N + 1 − sτ ϕ ′ k ( τ ) + µ k ( λ ) τ ϕ k ( τ ) = ζ k ( τ ) , in (0 , R ) , where(111) ζ k ( τ ) = κ s τ − s Z S N − (cid:0) h ( τ θ ′ ) w (0 , τ θ ′ ) + f ( τ θ ′ , u ( τ θ ′ )) (cid:1) ψ k (0 , θ ′ ) dS ′ . A direct calculation shows that, for some c k , c k ∈ R ,(112) ϕ k ( τ ) = τ σ + k (cid:18) c k + Z Rτ t − σ + k +1 σ + k − σ − k ζ k ( t ) dt (cid:19) + τ σ − k (cid:18) c k + Z Rτ t − σ − k +1 σ − k − σ + k ζ k ( t ) dt (cid:19) , RACTIONAL ELLIPTIC EQUATIONS 27 where(113) σ + k = − N − s s(cid:18) N − s (cid:19) + µ k ( λ ) and σ − k = − N − s − s(cid:18) N − s (cid:19) + µ k ( λ ) . From (3), (4), Lemma 4.4, (107), and the fact that, in view of (23),2 s + ( p − γ = ( N − s )(2 ∗ ( s ) − p )2 + ( p − s(cid:18) N − s (cid:19) + µ j ( λ ) > , we deduce that, for all i = j , . . . , j + m − ζ i ( τ ) = O ( τ − δ + σ + i ) as τ → + , with ˜ δ = min { ε, s + ( p − γ } >
0. Consequently, the functions t t − σ + i +1 σ + i − σ − i ζ i ( t ) and t t − σ − i +1 σ − i − σ + i ζ i ( t )belong to L (0 , R ). Hence τ σ + i (cid:18) c i + Z Rτ ρ − σ + i +1 σ + i − σ − i ζ i ( ρ ) dρ (cid:19) = o ( τ σ − i ) as τ → + , and then, by (110), there must be c i = − Z R t − σ − i +1 σ − i − σ + i ζ i ( t ) dt. Using (114), we then deduce that τ σ − i (cid:18) c i + Z Rτ t − σ − i +1 σ − i − σ + i ζ i ( t ) dt (cid:19) = τ σ − i (cid:18) Z τ t − σ − i +1 σ + i − σ − i ζ i ( t ) dt (cid:19) = O ( τ σ + i +˜ δ )(115)as τ → + . From (112) and (115), we obtain that, for all i = j , . . . , j + m − ϕ i ( τ ) = τ σ + i (cid:18) c i + Z Rτ t − σ + i +1 σ + i − σ − i ζ i ( t ) dt + O ( τ ˜ δ ) (cid:19) as τ → + . Let us assume by contradiction that lim λ → + λ − γ H ( λ ) = 0. Then, for all i ∈ { j , . . . , j + m − } ,(107) and (109) would imply that lim τ → + τ − σ + i ϕ i ( τ ) = 0 . Hence, in view of (116), c i + Z R t − σ + i +1 σ + i − σ − i ζ i ( t ) dt = 0 , which, together with (114), implies τ σ + i (cid:18) c i + Z Rτ t − σ + i +1 σ + i − σ − i ζ i ( t ) dt (cid:19) = τ σ + i Z τ t − σ + i +1 σ − i − σ + i ζ i ( t ) dt = O ( τ σ + i +˜ δ )(117)as τ → + . Collecting (112), (115), and (117), we conclude that ϕ i ( τ ) = O ( τ σ + i +˜ δ ) as τ → + for every i ∈ { j , . . . , j + m − } , namely, p H ( τ ) ( w τ , ψ ) L ( S N + ; θ − s ) = O ( τ γ +˜ δ ) as τ → + for every ψ ∈ A = span { ψ i } j + m − i = j , where A is the eigenspace of problem (10) associated tothe eigenvalue µ j ( λ ) = µ j +1 ( λ ) = · · · = µ j + m − ( λ ). From (79), there exists C (˜ δ ) > p H ( τ ) > C (˜ δ ) τ γ + δ for τ small, and therefore(118) ( w τ , ψ ) L ( S N + ; θ − s ) = O ( τ ˜ δ ) as τ → +8 MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI for every ψ ∈ A . From Lemma 4.2, for every sequence τ n → + , there exist a subsequence { τ n k } k ∈ N and an eigenfunction e ψ ∈ A (119) Z S N + θ − s e ψ ( θ ) dS = 1 and w τ nk → e ψ in L ( S N + ; θ − s ) . From (118) and (119), we infer that0 = lim k → + ∞ ( w τ nk , e ψ ) L ( S N − ) = k e ψ k L ( S N + ; θ − s ) = 1 , thus reaching a contradiction.We can now completely describe the behavior of solutions to (38) near the singularity, hence provingTheorem 4.1. Proof of Theorem 4.1.
Identity (82) follows from part (i) of Lemma 4.2, thus there exists k ∈ N , k >
1, such that γ = lim r → + N ( r ) = − N − s + q(cid:0) N − s (cid:1) + µ k ( λ ). Let us denote as m the multiplicity of µ j ( λ ) so that, for some j ∈ N , j > j k j + m − µ j ( λ ) = µ j +1 ( λ ) = · · · = µ j + m − ( λ ) and let { ψ i } j + m − i = j be an L ( S N + ; θ − s )-orthonormal basis for theeigenspace associated to µ k ( λ ).Let { τ n } n ∈ N ⊂ (0 , + ∞ ) such that lim n → + ∞ τ n = 0. Then, from part (ii) of Lemma 4.2 andLemmas 4.3 and 4.5, there exist a subsequence { τ n k } k ∈ N and m real numbers β j , . . . , β j + m − ∈ R such that ( β j , β j +1 , . . . , β j + m − ) = (0 , , . . . ,
0) and τ − γn k w (0 , τ n k x ) → | x | γ j + m − X i = j β i ψ i (cid:16) , x | x | (cid:17) in C ,α loc ( B ′ \ { } ) as k → + ∞ , (120) τ − γn k w ( τ n k θ ) → j + m − X i = j β i ψ i ( θ ) in C ,α ( S N + ) as k → + ∞ , (121) τ − γn k w (0 , τ n k θ ′ ) → j + m − X i = j β i ψ i (0 , θ ′ ) in C ,α ( S N − ) as k → + ∞ , (122)and(123) τ − γn k ∇ x w (0 , τ n k θ ′ ) → j + m − X i = j β i ( γψ i (0 , θ ′ ) θ ′ + ∇ S N − ψ i (0 , θ ′ )) in C ,α ( S N − ) as k → + ∞ for some α ∈ (0 , β i ’s depend neither on the sequence { τ n } n ∈ N nor on its subsequence { τ n k } k ∈ N .Defining ϕ i and ζ i as in (108) and (111), from (121) it follows that, for any i = j , . . . , j + m − τ − γn k ϕ i ( τ n k ) = Z S N + θ − s w ( τ n k θ ) τ γn k ψ i ( θ ) dS → j + m − X j = j β j Z S N + θ − s ψ j ( θ ) ψ i ( θ ) dS = β i as k → + ∞ . As deduced in the proof of Lemma 4.5, for any i = j , . . . , j + m − τ ∈ (0 , R ]there holds ϕ i ( τ ) = τ σ + i (cid:18) c i + Z Rτ t − σ + i +1 σ + i − σ − i ζ i ( t ) dt (cid:19) + τ σ − i (cid:18) Z τ t − σ − i +1 σ + i − σ − i ζ i ( t ) dt (cid:19) (125) = τ σ + i (cid:18) c i + Z Rτ t − σ + i +1 σ + i − σ − i ζ i ( t ) dt + O ( τ ˜ δ ) (cid:19) as τ → + , for some c i ∈ R , where σ ± i are defined in (113). Choosing τ = R in the first line of (125), weobtain c i = R − σ + i ϕ i ( R ) − R σ − i − σ + i Z R s − σ − i +1 σ + i − σ − i ζ i ( s ) ds. RACTIONAL ELLIPTIC EQUATIONS 29
Hence (125) yields τ − γ ϕ i ( τ ) → R − σ + i ϕ i ( R ) − R σ − i − σ + i Z R t − σ − i +1 σ + i − σ − i ζ i ( t ) dt + Z R t − σ + i +1 σ + i − σ − i ζ i ( t ) dt as τ → + , and therefore from (124) we deduce that β i = R − γ Z S N + θ − s w ( R θ ) ψ i ( θ ) dS − R − γ − N +2 s Z R κ s ρ γ + N − γ + N − s (cid:18) Z S N − (cid:0) h ( ρθ ′ ) w (0 , ρθ ′ ) + f ( ρθ ′ , w (0 , ρθ ′ )) (cid:1) ψ i (0 , θ ′ ) dS ′ (cid:19) dρ + Z R κ s ρ s − γ − γ + N − s (cid:18) Z S N − (cid:0) h ( ρθ ′ ) w (0 , ρθ ′ ) + f ( ρθ ′ , w (0 , ρθ ′ )) (cid:1) ψ i (0 , θ ′ ) dS ′ (cid:19) dρ. In particular the β i ’s depend neither on the sequence { τ n } n ∈ N nor on its subsequence { τ n k } k ∈ N ,thus implying that the convergences in (120), (121), (122), and (123) actually hold as τ → + andproving the theorem. (cid:3) Proof of Theorem 1.1
Let D s, (Ω) denote the completion on C ∞ c (Ω) with respect to the norm k · k D s, ( R N ) . Simpledensity arguments show that (7) is equivalent to(126) ( u, ϕ ) D s, ( R N ) = Z Ω (cid:18) λ | x | s u ( x ) + h ( x ) u ( x ) + f ( x, u ( x )) (cid:19) ϕ ( x ) dx, for all ϕ ∈ D s, (Ω) . Since u ∈ D s, ( R N ), we can let H ( u ) ∈ D , ( R N +1+ ; t − s ) such that Z R N +1+ t − s ∇H ( u ) · ∇ ϕ dt dx = 0 , for all ϕ ∈ C ∞ c ( R N +1+ ) , and H ( u ) = u on R N identified with ∂ R N +1+ , i.e. H ( u ) weakly satisfies ( div( t − s ∇H ( u )) = 0 , in R N +1+ , H ( u ) = u, on ∂ R N +1+ = { } × R N . From [4] we have that Z R N +1+ t − s ∇H ( u ) · ∇ e ϕ dt dx = κ s ( u, e ϕ ) D s, ( R N ) for all e ϕ ∈ D , ( R N +1+ ; t − s ) , where κ s = Γ(1 − s )2 s − Γ( s ) , i.e. − lim t → + t − s ∂ H ( u ) ∂t = κ s ( − ∆) s u ( x ) , in a weak sense. Therefore u ∈ D s, ( R N ) weakly solves (1) in Ω in the sense of (7) if and only ifits extension w = H ( u ) satisfies(127) div( t − s ∇ w ) = 0 , in R N +1+ ,w = u, on R N , − lim t → + t − s ∂w∂t ( t, x ) = κ s (cid:16) λ | x | s w + hw + f ( x, w ) (cid:17) , on Ω , in a weak sense, i.e. if for all e ϕ ∈ D , ( R N +1+ ; t − s ) such that x e ϕ (0 , x ) ∈ D s, (Ω) we have(128) Z R N +1+ t − s ∇ w · ∇ e ϕ dt dx = κ s Z Ω (cid:18) λ | x | s w + hw + f ( x, w ) (cid:19) e ϕ dx. Since D , ( R N +1+ ; t − s ) ֒ → H ( B + R ; t − s ) for all R >
0, the result follows from Theorem 4.1. (cid:3)
Proof of Theorem 1.2.
The proof follows from the proof of Theorem 1.1, observing that, if λ = 0, since no singularity occurs in the coefficients of the equation, the convergences in (92) and(93) hold in C ,α ( B + r ) by virtue of Lemma 3.3. (cid:3) Proof of Theorem 1.3.
The proof follows directly from Theorem 1.1. Indeed, let u be asolution to (1) satisfying u ( x ) = o ( | x | n ) = o (1) | x | n as | x | → n ∈ N ; if, by contradiction, u u ( x ) = o ( | x | n ) if n > − N − s + q(cid:0) s − N (cid:1) + µ k ( λ ). (cid:3) Proof of Theorem 1.4
Let u ∈ D s, ( R N ) be a weak solution to (14) in Ω such that u ≡ E ⊂ Ω with L ( E ) > L denotes the N -dimensional Lebesgue measure. By Lebesgue’s density Theorem, for a.e.point x ∈ E there holdslim r → + L ( E ∩ B ( x, r )) L ( B ( x, r )) = 1 and lim r → + L (( R N \ E ) ∩ B ( x, r )) L ( B ( x, r )) = 0 , where B ( x, r ) = { y ∈ R N : | y − x | < r } , i.e. a.e. point of E is a density point of E . Let x be adensity point of E ; then for all ε > r = r ( ε ) ∈ (0 ,
1) such that, for all r ∈ (0 , r ),(129) L (( R N \ E ) ∩ B ( x , r )) L ( B ( x , r )) < ε. Assume by contradiction that u r − γ u ( x + r ( x − x )) → | x − x | γ ψ (cid:18) , x − x | x − x | (cid:19) as r → + , in C ,α ( B ( x , γ = − N − s + q ( N − s ) + µ k and µ k = µ k (0) > λ = 0 and ψ is a corresponding eigenfunction. Since u ≡ E , by (129) wehave Z B ( x ,r ) u ( x ) dx = Z ( R N \ E ) ∩ B ( x ,r ) u ( x ) dx (cid:18) Z ( R N \ E ) ∩ B ( x ,r ) | u ( x ) | ∗ ( s ) dx (cid:19) / ∗ ( s ) |L (( R N \ E ) ∩ B ( x , r )) | ∗ ( s ) − ∗ ( s ) < ε ∗ ( s ) − ∗ ( s ) |L ( B ( x , r )) | ∗ ( s ) − ∗ ( s ) (cid:18) Z ( R N \ E ) ∩ B ( x ,r ) | u ( x ) | ∗ ( s ) dx (cid:19) / ∗ ( s ) for all r ∈ (0 , r ), i.e. letting u r ( x ) := r − γ u ( x + r ( x − x )), Z B ( x , | u r ( x ) | dx < (cid:16) ω N − N (cid:17) ∗ ( s ) − ∗ ( s ) ε ∗ ( s ) − ∗ ( s ) (cid:18) Z B ( x , | u r ( x ) | ∗ ( s ) dx (cid:19) / ∗ ( s ) for all r ∈ (0 , r ). Letting r → + , from (130) we have that Z B ( x , | x − x | γ ψ (cid:0) , x − x | x − x | (cid:1) dx (cid:16) ω N − N (cid:17) ∗ ( s ) − ∗ ( s ) ε ∗ ( s ) − ∗ ( s ) (cid:18) Z B ( x , | x − x | ∗ ( s ) γ ψ ∗ ( s ) (cid:0) , x − x | x − x | (cid:1) dx (cid:19) / ∗ ( s ) which yields a contradiction as ε → + . (cid:3) Acknowledgements.
The authors would like to thank Prof. Enrico Valdinoci for his interestand for taking their attention to reference [6] and the problem of unique continuation for sets ofpositive measure.
RACTIONAL ELLIPTIC EQUATIONS 31
References [1] F. J. Jr. Almgren, Q valued functions minimizing Dirichlet’s integral and the regularity of area minimizingrectifiable currents up to codimension two , Bull. Amer. Math. Soc., 8 (1983), no. 2, 327–328.[2] C. Br¨andle, E. Colorado, A. de Pablo, U. S´anchez, A concave-convex elliptic problem involving the fractionalLaplacian , Proc. Roy. Soc. Edinburgh, 143A (2013), 39–71.[3] X. Cabr´e and Y. Sire,
Nonlinear equations for fractional Laplacians I: Regularity, maximum principles,and Hamiltonian estimates , Annales de l’Institut Henri Poincar´e (C) Non Linear Analysis, to appear, http://arxiv.org/abs/1012.0867 .[4] L. Caffarelli, L. Silvestre,
An extension problem related to the fractional Laplacian , Comm. Partial DifferentialEquations 32 (2007), no. 7–9, 1245–1260.[5] T. Carleman,
Sur un probl`eme d’unicit´e pur les syst`emes d’´equations aux d´eriv´ees partielles `a deux variablesind´ependantes , Ark. Mat., Astr. Fys. 26 (1939), no. 17, 9 pp.[6] D. G. de Figueiredo, J.-P. Gossez,
Strict monotonicity of eigenvalues and unique continuation , Comm. PartialDifferential Equations 17 (1992), no. 1–2, 339–346.[7] E. Di Nezza, G. Palatucci, E. Valdinoci,
Hitchhiker’s guide to the fractional Sobolev spaces , Bull. Sci. math.136 (2012), no. 5, 521–573.[8] E. Fabes, C. Kenig, R. P. Serapioni,
The local regularity of solutions of degenerate elliptic equations , Comm.Partial Differential Equations 7 (1982), no. 1, 77–116.[9] M. M. Fall,
On a semilinear elliptic equation with fractional Laplacian and Hardy potential , http://arxiv.org/abs/1109.5530 .[10] M. M. Fall, T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems , J. Funct.Anal., Vol. 263, no. 8 (2012) 2205–2227 .[11] E. Fabes, N. Garofalo, F.-H. Lin,
A partial answer to a conjecture of B. Simon concerning unique continuation ,J. Funct. Anal. 88 (1990), no. 1, 194–210.[12] V. Felli, A. Ferrero, S. Terracini,
Asymptotic behavior of solutions to Schr¨odinger equations near an isolatedsingularity of the electromagnetic potential,
Journal of the European Mathematical Society 13 (2011), 119–174.[13] V. Felli, A. Ferrero, S. Terracini,
On the behavior at collisions of solutions to Schr¨odinger equations withmany-particle and cylindrical potentials,
Discrete Contin. Dynam. Systems. 32 (2012), 3895–3956.[14] V. Felli, A. Ferrero, S. Terracini,
A note on local asymptotics of solutions to singular elliptic equations viamonotonicity methods,
Milan J. Math 80 (2012), no. 1, 203–226.[15] A. Fiscella, R. Servadei, E. Valdinoci,
Asymptotically linear problems driven by the fractional , Preprint 2012.[16] R. Frank, E. H. Lieb, R. Seiringer,
Hardy-Lieb-Thirring inequalities for fractional Schr¨odinger operators , J.Amer. Math. Soc. 21 (2008), no. 4, 925-950.[17] N. Garofalo, F.-H. Lin,
Monotonicity properties of variational integrals, A p weights and unique continuation ,Indiana Univ. Math. J. 35 (1986), no. 2, 245–268.[18] G. H. Hardy, J. E. Littlewood, G. P´olya, Inequalities , 2d ed. Cambridge, at the University Press, 1952.[19] I. W. Herbst,
Spectral theory of the operator ( p + m ) / − Ze /r , Comm. Math. Phys. 53 (1977), no. 3,285–294.[20] D. Jerison, C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schr¨odinger operators ,Ann. of Math. (2) 121 (1985), no. 3, 463–494.[21] T. Jin, Y.Y. Li, J. Xiong,
On a fractional Nirenberg problem, part I: blow up analysis and compactness ofsolutions , J. Eur. Math. Soc. (JEMS), to appear, arXiv:1111.1332v1 .[22] K. Kurata,
A unique continuation theorem for uniformly elliptic equations with strongly singular potentials ,Comm. Partial Differential Equations 18 (1993), no. 7-8, 1161–1189.[23] S. Salsa,
Partial differential equations in action. From modelling to theory , Universitext. Springer–Verlag Italia,Milan, 2008.[24] L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator , Comm. PureAppl. Math. 60 (2007), 67–112.[25] Y. Sire, E. Valdinoci,
Fractional Laplacian phase transitions and boundary reactions: a geometric inequalityand a symmetry result , Journal of Functional Analysis 256 (2009), no. 6, 1842–1864.[26] T. H. Wolff,
A property of measures in R N and an application to unique continuation , Geom. Funct. Anal., 2(1992), no. 2, 225–284.[27] D. Yafaev, Sharp constants in the Hardy-Rellich inequalities , J. Funct. Anal. 168 (1999), no. 1, 121–144.
M.M. FallAfrican Institute for Mathematical Sciences (A.I.M.S.) of Senegal,KM 2, Route de Joal, AIMS-SenegalB.P. 1418. Mbour, S´en´egal.
E-mail address: [email protected].
V. FelliUniversit`a di Milano Bicocca,Dipartimento di Matematica e Applicazioni,Via Cozzi 53, 20125 Milano, Italy.