Influence of an L^p-perturbation on Hardy-Sobolev inequality with singularity a curve
aa r X i v : . [ m a t h . A P ] F e b INFLUENCE OF AN L p -PERTURBATION ON HARDY-SOBOLEV INEQUALITYWITH SINGULARITY A CURVE IDOWU ESTHER IJAODORO AND EL HADJI ABDOULAYE THIAM
Abstract.
We consider a bounded domain Ω of R N , N ≥ h and b continuous functions on Ω. Let Γbe a closed curve contained in Ω. We study existence of positive solutions u ∈ H (Ω) to the perturbedHardy-Sobolev equation: − ∆ u + hu + bu δ = ρ − σ Γ u ∗ σ − in Ω , where 2 ∗ σ := N − σ ) N − is the critical Hardy-Sobolev exponent, σ ∈ [0 , < δ < N − and ρ Γ is thedistance function to Γ. We show that the existence of minimizers does not depend on the local geometryof Γ nor on the potential h . For N = 3, the existence of ground-state solution may depends on the traceof the regular part of the Green function of − ∆ + h and or on b . This is due to the perturbative term oforder 1 + δ . AMS Mathematics Subject Classification:
Key words : Hardy-Sobolev inequality; Positive minimizers; Parametrized curve; Mass; Green function.1.
Introduction
Hardy-Sobolev inequality with a cylindrical weight states, for N ≥
3, 0 ≤ k ≤ N − σ ∈ [0 , Z R N |∇ v | dx ≥ C (cid:18)Z R N | z | − σ | v | ∗ σ dx (cid:19) / ∗ σ for all v ∈ D , ( R N ), (1.1)where x = ( t, z ) ∈ R k × R N − k , C = C ( N, σ, k ) >
0, 2 ∗ σ := N − σ ) N − is the critical Hardy-Sobolev exponentand D , ( R N ) is the completion of C ∞ c ( R N ) with respect to the norm v (cid:18)Z R N |∇ v | dx (cid:19) / . Inequality (1.1) can be obtained by interpolating between Hardy (which corresponds to the case σ = 2and k = N −
2) and Sobolev (which is the case σ = 0) inequalities. This inequality is invariant byscaling on R N and by translations in the t -direction.When σ = 2 and k = N −
2, the best constant is (cid:0) N − k − (cid:1) but it is never achieved. For σ ∈ [0 , C in (1.1) is given by S N,σ := min (cid:26) Z R N |∇ v | dx − ∗ σ Z R N | z | − σ | v | ∗ σ dx, v ∈ D , ( R N ) (cid:27) . (1.2)In the case σ ∈ [0 ,
2) and k = 0, S N,σ is achieved by the standard bubble c N,σ (1 + | x | − σ ) − N − σ , seefor instance Aubin [19], Talenti [1] and Lieb [16]. When k = N −
1, the support of the minimizer iscontained in a half space, see Musina [17]. For 1 ≤ k ≤ N − σ ∈ (0 , w for (1.2). They were motivated by questions from astrophysics.Later Mancini, Fabbri and Sandeep used the moving plane method to prove that w ( t, z ) = θ ( | t | , | z | ), for some positive function θ . An interesting classification result was also derived in [7] when σ = 1,that every minimizer is of the form c N,k ((1 + | z | ) + | t | ) − N , up to scaling in R N and translations inthe t -direction.Since in this paper we are interested with Hardy-Sobolev inequality with weight singular at a givencurve, our asymptotic energy level is given by S N,σ with k = 1 and σ ∈ [0 , R N , N ≥ h and b continuous function on Ω. Let Γ ⊂ Ω be a smoothclosed curve. In this paper, we are concerned with the existence of minimizers for the infinimum µ σ (Ω , Γ , h, b ) := inf u ∈ H (Ω) \{ } Z Ω |∇ u | dx + 12 Z Ω hu dx + 12 + δ Z Ω bu δ dx − ∗ σ Z Ω ρ − σ Γ | u | ∗ σ dx, (1.3)where σ ∈ [0 , ∗ σ := 2( N − σ ) N − < δ < N − and ρ Γ ( x ) := dist( x, Γ) is the distance function to Γ.Here and in the following, we assume that − ∆ + h defines a coercive bilinear form on H (Ω) and that b ≤
0. We are interested with the effect of b and/or the location of the curve Γ on the existence ofminimmizer for µ σ (Ω , Γ , h, b ).When there is no perturbation, and σ = 0, problem (1.3) reduces to the famous Brezis-Nirenbergproblem [3]. In this case, for N ≥ h ( y ) < N = 3,the existence of minimizers is guaranteed by the positiveness of a certain mass, see Druet [6].Here, we deal with the case σ ∈ [0 , N ≥
4. Then we have
Theorem 1.1.
Let N ≥ , σ ∈ [0 , and Ω be a bounded domain of R N . Consider Γ a smooth closedcurve contained in Ω . Let h and b be continuous function such that the linear operator − ∆ + h iscoercive and b ≤ . We assume that b ( y ) < , (1.4) for some y ∈ Γ . Then µ (Ω , Γ , h, b ) is achieved by a positive function u ∈ H (Ω) . In contrast, to the result of the second author and Fall [9], inequality (1.4) in Theorem 1.1 showsthat there is no influence of the curvature of Γ nor the potential h . This is due to the influence of theadded perturbation term in (1.3).For N = 3, we let G ( x, y ) be the Dirichlet Green function of the operator − ∆ + h , with zero Dirichletdata. It satisfies − ∆ x G ( x, y ) + h ( x ) G ( x, y ) = 0 for every x ∈ Ω \ { y } G ( x, y ) = 0 for every x ∈ ∂ Ω. (1.5)In addition, there exists a continuous function m : Ω → R and a positive constant c > G ( x, y ) = c | x − y | + c m ( y ) + o (1) as x → y. (1.6)This function m : Ω → R is the mass of − ∆ + h in Ω. Our second main result is the following Theorem 1.2.
Let σ ∈ [0 , and Ω be a bounded domain of R . Consider Γ a smooth closed curvecontained in Ω . Let h and b be continuous functions such that the linear operator − ∆ + h is coerciveand b ≤ . We assume that b ( y ) < for < δ < m ( y ) > cb ( y ) for δ = 2 m ( y ) > for < δ < , (1.7) NFLUENCE OF AN L p -PERTURBATION ON HARDY-SOBOLEV INEQUALITY WITH SINGULARITY A CURVE 3 for some positive constant c and y ∈ Γ . Then µ σ (Ω , Γ , h, b ) is achieved by a positive function u ∈ H (Ω) . The literature about Hardy-Sobolev inequalities on domains with various singularities is very hudge.The existence of minimizers depends on the curvatures at a point of the singularity. For more details,we refer to Ghoussoub-Kang [12], Ghoussoub-Robert [10, 11], Demyanov-Nazarov [5], Chern-Lin [4],Lin-Li [15], Fall-Thiam [9], Fall-Minlend-Thiam in [8] and the references there in. We refer also toJaber [13, 14] and Thiam [20–22] and references therein, for Hardy-Sobolev inequalities on Riemannianmanifold. Here also the impact of the scalar curvature at the point singularity plays an important rolefor the existence of minimizers in higher dimensions N ≥
4. The paper [13] contains also existenceresult under positive mass condition for N = 3.The proof of Theorem 1.1 and Theorem 1.2 rely on test function methods. Namely to build appro-priate test functions allowing to compare µ σ (Ω , Γ , h, b ) and S N,σ . We find a continuous family of testfunctions ( u ε ) ε> concentrating at a point y ∈ Γ which yields µ (Ω , Γ , h, b ) < S N,σ , as ε →
0, provided(1.7) holds. In Section 4, we consider the case N = 3. Due to the fact that the ground-state w for S ,σ , σ ∈ (0 ,
2) is not known explicitly, it is not radially symmetric, it is not smooth and S ,σ is onlyinvariant under translations in the t − direction; we could only construct a discrete family of test func-tions (Ψ ε n ) n ∈ N that leads to the inequality µ σ (Ω , Γ , h, b ) < S ,σ . These are similar to the test functions( u ε n ) n ∈ N in dimension N ≥ y , but away from it is substituted with theregular part of the Green function G ( x, y ), which makes appear the mass m ( y ) and/or b ( y ) in itsfirst order Taylor expansion, see (1.6).The paper is organized as follows: In Section 2, we recall some geometric and analytic preliminariesresults relating to the local geometry of the curve Γ and the decay estimates of the ground state w of S N,σ . In Section 3 and Section 4, we construct a test function for µ σ (Ω , Γ , h, b ) in order to proveTheorem 1.1 and Theorem 1.2. Their proof is completed in Section 5.2. Preliminaries Results
Let Γ ⊂ R N be a smooth closed curve. Let ( E ; . . . ; E N ) be an orthonormal basis of R N . For y ∈ Γand r > γ : ( − r, r ) → Γ, parameterized by arclength such that γ (0) = y .Up to a translation and a rotation, we may assume that γ ′ (0) = E . We choose a smooth orthonormalframe field ( E ( t ); ... ; E N ( t )) on the normal bundle of Γ such that ( γ ′ ( t ); E ( t ); ... ; E N ( t )) is an orientedbasis of R N for every t ∈ ( − r, r ), with E i (0) = E i .We fix the following notation, that will be used a lot in the paper, Q r := ( − r, r ) × B R N − (0 , r ) , where B R N − (0 , r ) denotes the ball in R N − with radius r centered at the origin. Provided r > F y : Q r → Ω, given by ( t, z ) F y ( t, z ) := γ ( t ) + N X i =2 z i E i ( t ) , is smooth and parameterizes a neighborhood of y = F y (0 , ρ Γ : Γ → R the distancefunction to the curve given by ρ Γ ( y ) = min y ∈ R N | y − y | . In the above coordinates, we have ρ Γ ( F y ( x )) = | z | for every x = ( t, z ) ∈ Q r . (2.1) IDOWU ESTHER IJAODORO AND EL HADJI ABDOULAYE THIAM
Clearly, for every t ∈ ( − r, r ) and i = 2 , . . . N , there are real numbers κ i ( t ) and τ ji ( t ) such that E ′ i ( t ) = κ i ( t ) γ ′ ( t ) + N X j =2 τ ji ( t ) E j ( t ) . (2.2)The quantity κ i ( t ) is the curvature in the E i ( t )-direction while τ ji ( t ) is the torsion from the osculatingplane spanned by { γ ′ ( t ); E j ( t ) } in the direction E i . We note that provided r > κ i and τ ji aresmooth functions on ( − r, r ). Moreover, it is easy to see that τ ji ( t ) = − τ ij ( t ) for i, j = 2 , . . . , N . (2.3)The curvature vector is κ : Γ → R N is defined as κ ( γ ( t )) := P Ni =2 κ i ( t ) E i ( t ) and its norm is given by | κγ ( t ) | := qP Ni =2 κ i ( t ). Next, we derive the expansion of the metric induced by the parameterization F y defined above. For x = ( t, z ) ∈ Q r , we define g ( x ) = ∂ t F y ( x ) · ∂ t F y ( x ) , g i ( x ) = ∂ t F y ( x ) · ∂ z i F y ( x ) , g ij ( x ) = ∂ z j F y ( x ) · ∂ z i F y ( x ) . We have the following result.
Lemma 2.1.
There exits r > , only depending on Γ and N , such that for ever x = ( t, z ) ∈ Q r g ( x ) = 1 + 2 N X i =2 z i κ i (0) + 2 t N X i =2 z i κ ′ i (0) + N X ij =2 z i z j κ i (0) κ j (0) + N X ij =2 z i z j β ij (0) + O (cid:0) | x | (cid:1) g i ( x ) = N X j =2 z j τ ij (0) + t N X j =2 z j (cid:0) τ ij (cid:1) ′ (0) + O (cid:0) | x | (cid:1) g ij ( x ) = δ ij , (2.4) where β ij ( t ) := P Nl =2 τ li ( t ) τ lj ( t ) . Proof.
To alleviate the notations, we will write F = F y . We have ∂ t F ( x ) = γ ′ ( t ) + N X j =2 z j E ′ j ( t ) and ∂ z i F ( x ) = E i ( t ) . (2.5)Therefore g ij ( x ) = E i ( t ) · E j ( t ) = δ ij . (2.6)By (2.2) and (2.5), we have g i ( x ) = N X l =2 z l E ′ l ( t ) · E i ( t ) = N X j =2 z j τ ij ( t ) (2.7)and g ( x ) = ∂ t F ( x ) · ∂ t F ( x ) = 1 + 2 N X i =2 z i κ i ( t ) + N X ij =2 z i z j κ i ( t ) κ j ( t ) + N X ij =2 z i z j N X l =2 τ li ( t ) τ lj ( t ) ! . (2.8)By Taylor expansions, we get κ i ( t ) = κ i (0) + tκ ′ i (0) + O (cid:0) t (cid:1) and τ ki ( t ) = τ ki (0) + t (cid:0) τ ki (cid:1) ′ (0) + O (cid:0) t (cid:1) . Using these identities in (2.8) and (2.7), we get (2.4), thanks to (2.6). This ends the proof. (cid:3)
As a consequence we have the following result.
NFLUENCE OF AN L p -PERTURBATION ON HARDY-SOBOLEV INEQUALITY WITH SINGULARITY A CURVE 5 Lemma 2.2.
There exists r > only depending on Γ and N , such that for every x ∈ Q r , we have p | g | ( x ) = 1 + N X i =2 z i κ i (0) + t N X i =2 z i κ ′ i (0) + 12 N X ij =2 z i z j κ i (0) κ j (0) + O (cid:0) | x | (cid:1) , (2.9) where | g | stands for the determinant of g . Moreover g − ( x ) , the matrix inverse of g ( x ) , has componentsgiven by g ( x ) = 1 − N X i =2 z i κ i (0) − t N X i =2 z i κ ′ i (0) + 3 N X ij =2 z i z j κ i (0) κ j (0) + O (cid:0) | x | (cid:1) g i ( x ) = − N X j =2 z j τ ij (0) − t N X j =2 z j (cid:0) τ ij (cid:1) ′ (0) + 2 N X j =2 z l z j κ l (0) τ ij (0) + O (cid:0) | x | (cid:1) g ij ( x ) = δ ij + N X lm =2 z l z m τ jl (0) τ im (0) + O (cid:0) | x | (cid:1) . (2.10) Proof.
We write g ( x ) = id + H ( x ) , where id denotes the identity matrix on R N and H is a symmetric matrix with components H αβ , for α, β = 1 , . . . , N , given by H ( x ) = 2 N X i =2 z i κ i (0) + 2 t N X i =2 z i κ ′ i (0) + N X ij =2 z i z j κ i (0) κ j (0) + N X ij =2 z i z j β ij (0) + O (cid:0) | x | (cid:1) H i ( x ) = N X j =2 z i τ ij (0) + O (cid:0) | x | (cid:1) H ij ( x ) = 0 . (2.11)We recall that as | H | → p | g | = p det ( I + H ) = 1 + tr H H ) − tr ( H )4 + O (cid:0) | H | (cid:1) . (2.12)Now by (2.11), as | x | →
0, we havetr H N X i =2 z i κ i (0) + t N X i =2 z i κ ′ i (0) + 12 N X ij =2 z i z j κ i (0) κ j (0) + 12 N X ij =2 z i z j β ij (0) + O (cid:0) | x | (cid:1) , (2.13)so that ( tr H ) N X ij =2 z i z j κ i (0) κ j (0) + O (cid:0) | x | (cid:1) . (2.14)Moreover, from (2.11), we deduce thattr ( H )( x ) = N X α =1 (cid:0) H ( x ) (cid:1) αα = N X αβ =1 H αβ ( x ) H βα ( x ) = N X αβ =1 H αβ ( x ) = H ( x ) + 2 N X i =2 H i ( x ) , IDOWU ESTHER IJAODORO AND EL HADJI ABDOULAYE THIAM so that − tr ( H )4 = − N X ij =2 z i z j κ i (0) κ j (0) − N X ijl =2 z i z j τ li (0) τ lj (0) + O (cid:0) | x | (cid:1) . (2.15)Therefore plugging the expression from (2.13), (2.14) and (2.15) in (2.12), we get p | g | ( x ) = 1 + N X i =2 z i κ i (0) + t N X i =2 z i κ ′ i (0) + 12 N X ij =2 z i z j κ i (0) κ j (0) + O (cid:0) | x | (cid:1) . The proof of (2.9) is thus finished.By Lemma 2.1 we can write g ( x ) = id + A ( x ) + B ( x ) + O (cid:0) | x | (cid:1) , where A and B are symmetric matrix with components ( A αβ ) and ( A αβ ), α, β = 1 , . . . , N , givenrespectively by A ( x ) = 2 N X i =2 z i κ i (0) , A i ( x ) = N X j =2 z j τ ij (0) and A ij ( x ) = 0 (2.16)and B ( x ) = 2 t N X i =2 z i κ ′ (0) + N X i =2 z i z j κ i (0) κ j (0) + N X ij =2 z i z j β ij (0) B i ( x ) = t X j =2 z j (cid:0) τ ij (cid:1) ′ (0) and B ij ( x ) = 0 . (2.17)We observe that, as | x | →
0, we have g − ( x ) = id − A ( x ) − B ( x ) + A ( x ) + O ( | x | ) . We then deducefrom (2.16) and (2.17) that g ( x ) = 1 − A ( x ) − B ( x ) + A ( x ) + N X i =1 A i ( x ) + O (cid:0) | x | (cid:1) = 1 − N X i =2 z i κ i (0) − t N X i =2 z i κ ′ (0) + 3 N X i =2 z i z j κ i (0) κ j (0) + 3 N X ij =2 z i z j β ij (0) + O (cid:0) | x | (cid:1) ,g i ( x ) = − A i ( x ) − B i ( x ) + N X α =1 A iα A α + O (cid:0) | x | (cid:1) = − A i ( x ) − B i ( x ) + A i ( x ) A ( x ) + N X j =2 A ij ( x ) A j ( x ) + O (cid:0) | x | (cid:1) = − N X j =2 z j τ ij (0) − t X j =2 z j (cid:0) τ ij (cid:1) ′ (0) + 2 N X jl =2 z l z j κ l (0) τ ij (0) NFLUENCE OF AN L p -PERTURBATION ON HARDY-SOBOLEV INEQUALITY WITH SINGULARITY A CURVE 7 and g ij ( x ) = δ ij − A ij ( x ) − B ij ( x ) + (cid:0) A (cid:1) ij ( x ) + O (cid:0) | x | (cid:1) = δ ij − A ij ( x ) − B ij ( x ) + A i A j + N X l =2 A il ( x ) A jl ( x ) + O (cid:0) | x | (cid:1) = δ ij + N X lm =2 z l z m τ im (0) τ jl (0) + O (cid:0) | x | (cid:1) . This ends the proof. (cid:3)
We recall that the best constant for the cylindrical Hardy-Sobolev inequality is given by S N,σ = min (cid:26) Z R N |∇ w | dx − ∗ σ Z R N | z | − σ | w | ∗ σ dx : w ∈ D , ( R N ) , (cid:27) . Further it is attained by a positive function w ∈ D , ( R N ), that satisfies the Euler-Lagrange equation − ∆ w = | z | − σ w ∗ σ − in R N , (2.18)see e.g. [2]. By [7], we have the last result of this section. Lemma 2.3.
For N ≥ , we have w ( x ) = w ( t, z ) = θ ( | t | , | z | ) for a function θ : R + × R + → R + . (2.19) Moreover, there exists two constants < C < C , such that C | x | N − ≤ w ( x ) ≤ C | x | N − in R N . (2.20)3. Existence of minimzers for µ (Ω , Γ , h, b ) in dimension N ≥ R N , N ≥ ⊂ Ω be a smooth closed curve. For u ∈ H (Ω) \ { } , we define the functional J ( u ) := 12 Z Ω |∇ u | dy + 12 Z Ω hu dy + 12 + δ Z Ω bu δ dy − ∗ σ Z Ω ρ − σ Γ | u | ∗ σ dy. (3.1)We let η ∈ C ∞ c ( F y ( Q r )) be such that0 ≤ η ≤ η ≡ Q r . For ε >
0, we consider u ε : Ω → R given by u ε ( y ) := ε − N η ( F − y ( y )) w (cid:0) ε − F − y ( y ) (cid:1) . (3.2)In particular, for every x = ( t, z ) ∈ R × R N − , we have u ε ( F y ( x )) := ε − N η ( x ) θ ( | t | /ε, | z | /ε ) . (3.3)It is clear that u ε ∈ H (Ω) . Then we have the following
Proposition 3.1.
For all N ≥ , we have J ( u ε ) = S N,σ + ε − δ ( N − b ( y ) Z R N w δ +2 dx + o (cid:16) ε − δ ( N − (cid:17) , (3.4) as ε → . IDOWU ESTHER IJAODORO AND EL HADJI ABDOULAYE THIAM
The proof of Proposition 3.1 is divided in two parts, Lemma 3.2 and Lemma 3.3 below. For that weset J ( u ) := 12 Z Ω |∇ u | dx + 12 Z Ω hu dx − ∗ σ Z Ω ρ − σ Γ | u | ∗ σ dx, the following is due to the second author and Fall [9]. Lemma 3.2.
We have J ( u ε ) = S N,σ + O ( ε ) for all N ≥ O ( ε | log( ε ) | ) for all N = 4 . (3.5)We finish the proof by the following Lemma 3.3.
We have Z Ω bu δε dx = ε − δ ( N − b ( y ) Z R N w δ +2 dx + O (cid:0) ε (cid:1) for N ≥ Z Ω bu δε dx = ε − δ b ( y ) Z Q r/ε w δ +2 dx + O (cid:0) ε (cid:1) for N = 3 and δ ≤ Z Ω bu δε dx = ε − δ b ( y ) Z R N w δ +2 dx + O (cid:16) ε δ (cid:17) for N = 3 and δ > as ε → .Proof. We have Z Ω b ( x ) u δε dx = Z F y ( Q r ) b ( x ) u δε dx + Z F y ( Q r ) \ F y ( Q r ) b ( x ) u δε dx. Since b is continuous and r is small, then by the change of variable formula y = F ( x ) ε , we have Z Ω b ( x ) u δε dx = b ( y ) ε − δ ( N − Z Q r/ε w δ dx + O ε − δ ( N − Z Q r/ε | x | w δ dx + ε − δ ( N − Z Q r/ε \ Q r/ε w δ dx ! = b ( y ) ε − δ ( N − Z Q r/ε w δ dx + O ε − δ ( N − Z Q r/ε | x | w δ dx + ε − δ ( N − Z Q r/ε \ Q r/ε w δ dx ! Thanks to (2.20), we have ε − δ ( N − Z Q r/ε | x | w δ dx + ε − δ ( N − Z Q r/ε \ Q r/ε w δ dx = O ( ε ) for all N ≥ ε − δ ( N − Z Q r/ε \ Q r/ε w δ dx = O ( ε ) for all N ≥ . NFLUENCE OF AN L p -PERTURBATION ON HARDY-SOBOLEV INEQUALITY WITH SINGULARITY A CURVE 9 We finish by noticing that, for N = 3, we have Z R N \ Q r/ε w δ dx = O ( ε δ − ) . This then ends the proof of the Lemma. (cid:3) Existence of minimizer for µ h (Ω , Γ , h, b ) in dimension three We consider the function R : R \ { } → R , x
7→ R ( x ) = 1 | x | which satisfies − ∆ R = 0 in R \ { } . (4.1)We denote by G the solution to the equation (cid:26) − ∆ x G ( y, · ) + hG ( y, · ) = 0 in Ω \ { y } . G ( y, · ) = 0 on ∂ Ω, (4.2)and satisfying G ( x, y ) = R ( x − y ) + O (1) for x, y ∈ Ω and x = y . (4.3)We note that G is proportional to the Green function of − ∆ + h with zero Dirichlet data.We let χ ∈ C ∞ c ( − ,
2) with χ ≡ − ,
1) and 0 ≤ χ <
1. For r >
0, we consider the cylindricalsymmetric cut-off function η r ( t, z ) = χ (cid:18) | t | + | z | r (cid:19) for every ( t, z ) ∈ R × R . (4.4)It is clear that η r ≡ Q r , η r ∈ H ( Q r ) , |∇ η r | ≤ Cr in R . For y ∈ Ω, we let r ∈ (0 ,
1) such that y + Q r ⊂ Ω . (4.5)We define the function M y : Q r → R given by M y ( x ) := G ( y , x + y ) − η r ( x ) 1 | x | for every x ∈ Q r . (4.6)It follows from (4.3) that M y ∈ L ∞ ( Q r ). By (4.2) and (4.1), | − ∆ M y ( x ) + h ( x ) M y ( x ) | ≤ C | x | = C R ( x ) for every x ∈ Q r , whereas R ∈ L p ( Q r ) for every p ∈ (1 , M y ∈ W ,p ( Q r / ) forevery p ∈ (1 , k M y k C ,̺ ( Q r / ) ≤ C for every ̺ ∈ (0 , − ∆ + h in Ω at the point y ∈ Ω is given by m ( y ) = M y (0) . (4.8)We recall that the positive ground state solution w satisfies − ∆ w = | z | − σ w ∗ σ − in R , (4.9) where x = ( t, z ) ∈ R × R . In addition by (2.20), we have C | x | ≤ w ( x ) ≤ C | x | in R . (4.10)The following result will be crucial in the sequel. Lemma 4.1.
Consider the function v ε : R \ { } → R given by v ε ( x ) = ε − w (cid:16) xε (cid:17) . Then there exists a constant c > and a sequence ( ε n ) n ∈ N (still denoted by ε ) such that v ε ( x ) → c | x | for all most every x ∈ R and v ε ( x ) → c | x | for every x ∈ R \ { z = 0 } . (4.11)For a proof, see for instance [ [9], Lemma 5.1]Next, given y ∈ Γ ⊂ Ω ⊂ R , we let r as defined in (4.5). For r ∈ (0 , r / F y : Q r → Ω (see Section 2) parameterizing a neighborhood of y in Ω, with the property that F y (0) = y . For ε >
0, we consider u ε : Ω → R given by u ε ( y ) := ε − / η r ( F − y ( y )) w (cid:18) F − y ( y ) ε (cid:19) . We can now define the test function Ψ ε : Ω → R byΨ ε ( y ) = u ε ( y ) + ε / c η r ( F − y ( y )) M y ( F − y ( y )) . (4.12)It is plain that Ψ ε ∈ H (Ω) andΨ ε ( F y ( x )) = ε − / η r ( x ) w (cid:16) xε (cid:17) + ε / c η r ( x ) M y ( x ) for every x ∈ R N .The main result of this section is contained in the following result. Proposition 4.2.
Let ( ε n ) n ∈ N and c be the sequence and the number given by Lemma 4.1. Then thereexists r , n > such that for every r ∈ (0 , r ) and n ≥ n J (Ψ ε ) = S ,σ − ε n π m ( y ) c + ε − δ n δ Z Q r/ε w δ dx + O r ( ε n ) for δ ≤ J (Ψ ε ) = S ,σ − ε n π m ( y ) c + ε − δ n δ Z R w δ dx + O r ( ε n ) for δ > , for some numbers O r ( ε n ) satisfying lim r → lim n →∞ ε − n O r ( ε n ) = 0 . The proof of this proposition will be separated into two steps given by Lemma 4.3 and Lemma 4.4below. To alleviate the notations, we will write ε instead of ε n and we will remove the subscript y , bywriting M and F in the place of M y and F y respectively. We define e η r ( y ) := η r ( F − ( y )) , V ε ( y ) := v ε ( F − ( y )) and f M r ( y ) := η r ( F − ( y )) M ( F − ( y )) , NFLUENCE OF AN L p -PERTURBATION ON HARDY-SOBOLEV INEQUALITY WITH SINGULARITY A CURVE 11 where v ε ( x ) = ε − w (cid:0) xε (cid:1) . With these notations, (4.12) becomesΨ ε ( y ) = u ε ( y ) + ε c f M r ( y ) = ε V ε ( y ) + ε c f M r ( y ) . (4.13)We first consider the numerator in (4.2). Lemma 4.3.
We have J (Ψ ε ) = S ,σ − επ c m ( y ) + O r ( ε ) , as ε → . For a proof, see for instance [ [9], Proposition 5.3]. The following result together with the previouslemma provides the proof of Proposition4.2.
Lemma 4.4.
We have Z Ω | Ψ ε | δ dy = ε − δ b ( y ) Z Q r/ε w δ dx + o (cid:16) ε − δ (cid:17) , as ε → .Proof. Since δ >
0, by Taylor expansion we have Z Ω | Ψ ε | δ dy = Z Ω | u ε + ε / f M r | δ dy = Z Ω | u ε | δ dy + O (cid:18) ε / Z Ω | u ε | δ | f M r | dy + Z Ω | u ε | δ | f M r | dy + Z Ω | f M r | δ dy (cid:19) . (4.14)Using H¨older’s inequality and (2.9), we have Z F ( Q r ) | ηu ε | δ (cid:16) ε / f M r (cid:17) dy ≤ ε k u ε k δL δ ( F ( Q r )) k f M r k L δ ( F ( Q r )) = ε − δ k w k δL δ ( Q r ; √ | g | ) k f M r k L δ ( F ( Q r )) ≤ ε − δ k f M r k L δ ( F ( Q r )) = o ( ε ) , (4.15)Since δ >
0, by (4.7), we easily get Z F ( Q r ) | ε / f M r | δ dy = O ( ε δ ) = o ( ε ) . (4.16)By (4.14), (4.16), (4.15) and Lemma 3.3, it results Z Ω | Ψ ε | δ dy = Z F ( Q r ) | u ε | δ dy + O (cid:18) ε / Z F ( Q r ) | u ε | δ f M r dy (cid:19) + o ( ε )= ε − δ b ( y ) Z Q r/ε w δ +2 dx + O (cid:18) ε / Z F ( Q r ) | u ε | δ f M r dy (cid:19) + o ( ε ) . We define B ε ( x ) := M ( εx ) p | g ε | ( x ) = M ( εx ) p | g | ( εx ). Then by the change of variable y = F ( x ) ε in theabove identity and recalling (2.9), then by oddness, we have ε / Z Ω | u ε | δ | f M r | dy = O ε − δ/ Z Q r/ε | w | δ dx ! = O (cid:0) ε − δ/ (cid:1) . Therefore Z Ω | Ψ ε | δ dy = ε − δ b ( y ) Z Q r/ε w δ dx + o (cid:16) ε − δ (cid:17) , as ε →
0. This then ends the proof. (cid:3) Proofs of Theorem 1.1 and Theorem 1.2
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