On the behavior at collisions of solutions to Schr\"odinger equations with many-particle and cylindrical potentials

The asymptotic behavior of solutions to Schr\"odinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, point-wise upper estimate are derived.


Introduction
The purpose of the present paper is to describe the behavior of solutions to a class of Schrödinger equations with singular homogeneous potentials including cylindrical and quantum multi-body ones.
The interaction between M particles of coordinates y 1 , . . . , y M in R k is described in classical mechanics by potentials of the form where V j,m (y) → 0 as |y| → +∞, see [28]. From the mathematical point of view, a particular interest arises in the case of inverse square potentials V j,m (y) = λj λm |y| 2 , since they have the same order of homogeneity as the laplacian thus making the corresponding Schrödinger operator invariant by scaling. Schrödinger equations with the resulting M -body potential (1) V (y 1 , . . . , y M ) = M j,m=1 j<m λ j λ m |y j − y m | 2 , λ j , λ m ∈ R, have been studied by several authors; we mention in particular [27] where many-particle Hardy inequalities are proved and [12] where the existence of ground state solutions for semilinear Schrödinger equations with potentials of type (1) is investigated. It is worth pointing out that hamiltonians with singular potentials having the same homogeneity as the operator arise in relativistic quantum mechanics, see [31].
There is a natural relation between 2-particle potentials (1) and cylindrical potentials, whose singular set is some k-codimensional subspace of the configuration space. Indeed, in the special case M = 2, after the change of variables in R 2k (2) z 1 = 1 √ 2 (y 1 − y 2 ), z 2 = 1 √ 2 (y 1 + y 2 ), the potential V (y 1 , y 2 ) = λ1λ2 |y1−y2| 2 takes the form Elliptic equations with cylindrical inverse square potentials arise in several fields of applications, e.g. in the search for solitary waves with no vanishing angular momentum of nonlinear evolution equations of Schrödinger and Klein-Gordon type, see [3]. In the recent literature, many papers have been devoted to the study of semilinear elliptic equations with cylindrical potentials; we mention among others [3,4,5,32,36]. We point out that cylindrical type (and a fortiori many-particle) potentials give rise to substantially major difficulties with respect to the case of an isolated singularity, because in the cylindrical/many-particle case separation of variables (radial and angular) does not actually "eliminate" the singularity, being the angular part of the operator also singular. We consider both linear and semilinear Schrödinger equations with singular homogeneous potentials belonging to a class including as particular cases both (1)  where #J stands for the cardinality of J and J 1 < J 2 stands for the "alphabetic ordering" for multi-indices (see the list of notations at the end of this section).
In the sequel, for every x = (x 1 , x 2 , . . . , x N ) ∈ R N and J ∈ A k , we denote as x J the k-uple (x i ) i∈J so that |x J | 2 = i∈J x 2 i . In a similar way, for any x ∈ R N \ {0} and J ∈ A k we write θ J = xJ |x| . Moreover we denote Σ :={(θ 1 , . . . , θ N ) ∈ S N −1 : θ J = 0 for some J ∈ A k } (4) ∪ {(θ 1 , . . . , θ N ) ∈ S N −1 : θ J1 = θ J2 for some (J 1 , J 2 ) ∈ B k } and (5) The potentials we are going to consider are of the type where α J , α J1J2 ∈ R. We notice that B k is empty whenever k > N 2 ; in such a case we consider potentials V with only the cylindrical part, i.e. with only the first summation at right hand side of (6).
Letting, for all θ ∈ S N −1 \ Σ, we can write the potential V in (6) as a( x |x| ) |x| 2 and the associated hamiltonian as As a natural setting to study the properties of operators L a , we introduce the functional space D 1,2 (R N ) defined as the completion of C ∞ c (R N ) with respect to the Dirichlet norm The potential V in (6) satisfies a Hardy type inequality. Indeed, it was proved in [33] (see also [5] and [39]) that the following Hardy's inequality for cylindrically singular potentials holds: for all u ∈ D 1,2 (R N ) and J ∈ A k , being the constant k−2 2 2 optimal. Using a change of variables of type (2), from (8) it follows the "two-particle Hardy inequality": for all u ∈ D 1,2 (R N ) and (J 1 , J 2 ) ∈ B k , being the constant (k−2) 2 2 optimal. From (8) and (9) we deduce that the potential V in (6) satisfies the following "many-particle Hardy inequality": for all u ∈ D 1,2 (R N ), where α + J = max{α J , 0} and α + J1J2 = max{α J1J2 , 0}. We refer to [27] for a deep analysis of many-particle Hardy inequalities and related best constants.
The relation between the value Λ(a) and the first eigenvalue of the angular component of the operator on the unit (N − 1)-dimensional sphere S N −1 is discussed in Lemma 2.3. More precisely, Lemma 2.3 ensures that the quadratic form associated to L a is positive definite if and only if where µ 1 (a) is the first eigenvalue of the operator L a := −∆ S N −1 − a on the sphere S N −1 . The spectrum of the angular operator L a is discrete and consists in a nondecreasing sequence of eigenvalues µ 1 (a) µ 2 (a) · · · µ k (a) · · · diverging to +∞, see Lemma 2.2.
We study nonlinear equations obtained as perturbations of the operator L a in a bounded domain Ω ⊂ R N containing the origin. More precisely, we deal with semilinear equations of the type (13) L a u = h(x) u + f (x, u), in Ω .
We assume that the linear perturbing potential h is negligible with respect to the potential V near the collision singular set Σ defined in (5), in the sense that there exist C h > 0 and ε > 0 such that, for a.e. x ∈ Ω \ Σ, We notice that it is not restrictive to assume ε ∈ (0, 1) in (H). As far as the nonlinear perturbation is concerned, we assume that f satisfies (F) f ∈ C 0 (Ω × R), F ∈ C 1 (Ω × R), s → f (x, s) ∈ C 1 (R) for a.e. x ∈ Ω, |f (x, s)s| + |f ′ s (x, s)s 2 | + |∇ x F (x, s) · x| C f (|s| 2 + |s| 2 * ) for a.e. x ∈ Ω and all s ∈ R, where F (x, s) = s 0 f (x, t) dt, 2 * = 2N N −2 is the critical Sobolev exponent, C f > 0 is a constant independent of x ∈ Ω and s ∈ R, ∇ x F denotes the gradient of F with respect to the x variable, and f ′ s (x, s) = ∂f ∂s (x, s). We say that a function u ∈ H 1 (Ω) is a H 1 (Ω)-weak solution to (13) if, for all w ∈ H 1 0 (Ω), Schrödinger equations with inverse square homogeneous singular potentials can be regarded as critical from the mathematical point of view, as they do not belong to the Kato class. A rich literature deals with such critical equations, both in the case of one isolated pole, see e.g. [16,24,25,29,40,42], and in that of multiple singularities, see [7,14,15,19,23]. The analysis of fundamental spectral properties such as essential self-adjointness and positivity carried out in [19,21] for Schrödinger operators with isolated inverse square singularities, highlighted how the asymptotic behavior of solutions to associated elliptic equations near the singularity plays a crucial role. A precise evaluation of the asymptotics of solutions turned out to be an important tool also to establish existence of ground states for nonlinear Schrödinger equations with multi-singular Hardy potentials (see [23]) and of solutions to nonlinear systems of Schrödinger equations with Hardy potentials [1]. A first result about the study of the asymptotic behavior of solutions near isolated singularities is contained in [22], where Hölder continuity of solutions to degenerate elliptic equations with singular weights has been established thus allowing the evaluation of the exact asymptotic behavior of solutions to Schrödinger equations with Hardy potentials near the pole. An extension to the case of Schrödinger equations with dipole-type potentials (namely purely angular multiples of inverse square potentials) has been obtained in [20] by separation of variables and comparison principles, and later generalized to Schrödinger equations with singular homogeneous electromagnetic potentials of Aharonov-Bohm type [17] by the Almgren monotonicity formula. Comparison and maximum principles play a crucial role also in [37], where the existence of the limit at the singularity of any quotient of two positive solutions to Fuchsian type elliptic equations is proved. We mention that related asymptotic expansions near singularities were obtained in [34,35] for elliptic equations on manifolds with conical singularities by Mellin transform methods (see also [30]); we refer to [18] for a comparison between such results and asymptotics via Almgren monotonicity methods. It is also worth citing [9], where some asymptotic formulas are heuristically obtained for the three-body one-dimensional problem.
Due to the presence of multiple collisions, one should expect that solutions to equations (13) behave singularly at the origin: our purpose is to describe the rate and the shape of the singularity of solutions, by relating them to the eigenvalues and the eigenfunctions of the angular operator L a on the sphere S N −1 .
Due to the homogeneity of the potentials, Schrödinger operators L a are invariant by the Kelvin transform,ũ which is an isomorphism of D 1,2 (R N ). Indeed, if u ∈ H 1 (Ω) weakly solves (13) in a bounded open set Ω containing 0, then its Kelvin's transformũ weakly solves (13) with h replaced by |x| −4 h( x |x| 2 ) and f (x, ·) replaced by |x| −N −2 f x |x| 2 , |x| N −2 · in the external domain Ω = x ∈ R N : x/|x| 2 ∈ Ω . Therefore, under suitable decay conditions on h at ∞ and proper subcriticality assumptions on f , the asymptotic behavior at infinity of solutions to (13) in external domains can be easily deduced from Theorem 1.1 and the Kelvin transform (see [17,Theorems 1.4

and 1.6]).
A major breakthrough in the description of the singularity of solutions at zero can be done by evaluating the behavior of eigenfunctions ψ i ; indeed such eigenfunctions solve an elliptic equation on S N −1 exhibiting itself a potential which is singular on Σ. After a stereographic projection of S N −1 onto R N −1 , the equation satisfied by each ψ i takes a form which is similar to (13) in a lowered dimension with a potential whose singular set is (N −1−k)−dimensional and to which we can apply the above theorem to deduce a precise asymptotics in terms of eigenvalues and eigenfunctions of an operator on S N −2 ; the procedure can be iterated (N − k)−times until we come to an equation with a potential with isolated singularities whose corresponding angular operator is no more singular. A detailed analysis of the asymptotic behavior of eigenfunctions is performed in section 7.
A pointwise upper estimate on the behavior of solutions can be derived by a Brezis-Kato type iteration argument, see [8]. More precisely, we can estimate the solutions by terms of the first eigenvalue and eigenfunction of the angular potentialâ obtained by summing up only the positive contributions of a, i.e.
Under the assumption (19) Λ(â) = sup Theorem 1.2. Let u be a weak H 1 (Ω)-solution to (13) in a bounded open set Ω ⊂ R N containing 0, N k 3, with a satisfying (7) andâ as in (18) satisfying (19). If h satisfies (H) and f satisfies (F), then for any Ω ′ ⋐ Ω there exists C > 0 such that In particular, if all α J , α J1J2 are positive, thenâ ≡ a and the above theorem ensures that all solutions are pointwise bounded by |x| the other hand, if all α J , α J1J2 are negative, thenâ ≡ 0 and the above theorem implies that all solutions are bounded. The paper is organized as follows. In section 2 we prove some Hardy-type inequalities with singular potentials of type (6) and discuss the relation between the positivity of the quadratic form associated to L a and the first eigenvalue of the angular operator on the sphere S N −1 . In section 3 we derive a Pohozaev-type identity for solutions to (13) through a suitable approximating procedure which allows getting rid of the singularity of the angular potential. In Section 4 we deduce a Brezis-Kato estimate to prove an a-priori super-critical summability of solutions to (13) which allows us to include the critical growth case in the Almgren type monotonicity formula which is obtained in Section 5 and which is used in section 6 together with a blow-up method to prove Theorem 1.1. Section 7 is devoted to the study of the asymptotic behavior of the eigenfunctions of the angular operator. Section 8 contains some Brezis-Kato estimates in weighted Sobolev spaces which allow proving Theorem 1.2. A final appendix contains a Pohozaev-type identity for semilinear elliptic equations with an anisotropic inverse-square potential with a bounded angular coefficient.
Notation. We list below some notation used throughout the paper.
-For all r > 0, B r denotes the ball {x ∈ R N : |x| < r} in R N with center at 0 and radius r.
-For all r > 0, B r = {x ∈ R N : |x| r} denotes the closure of B r .
-dS denotes the volume element on the spheres ∂B r , r > 0.
Let us consider the following class of angular potentials . From Lemma 2.1 we have that, for every f ∈ F , the supremum (22) Λ(f ) := sup On the other hand, arguing as in the proof of [42, Lemma 1.1], we can easily verify that (23) Λ(f ) = sup Furthermore, it is easy to verify that Λ(f ) 0 and Λ(f ) = 0 if and only if f 0 a.e. in S N −1 . For every f ∈ F satisfying Λ(f ) < 1, we can perform a complete spectral analysis of the angular Schrödinger operator −∆ S N −1 − f on the sphere.
Then the spectrum of the operator real eigenvalues with finite multiplicity the first of which admits the variational characterization (24) µ 1 (f ) = min .
Moreover µ 1 (f ) is simple and its associated eigenfunctions do not change sign in S N −1 .
Proof. By Lemma 2.1 and assumption Λ(f ) < 1, the operator T : is well-defined, symmetric, and compact. The conclusion follows from classical spectral theory. In particular, we point out that the simplicity of the first eigenvalue follows from the fact that, since k > 1, the singular set Σ does not disconnect the sphere.
For all f ∈ F , let us consider the quadratic form associated to the Schrödinger operator L f , i.e.
The problem of positivity of Q f is solved in the following lemma. i) Q f is positive definite, i.e. inf (24).
Proof. The equivalence between i) and ii) follows from the definition of Λ(f ), see (23 Henceforward, we shall assume that (12) holds, so that the quadratic form associated to the operator L a is positive definite.
We extend to singular potentials of the form (6) the Hardy type inequality with boundary terms proved by Wang and Zhu in [43]. Lemma 2.6. Let a be as in (7) and assume that (12) holds. Then for all r > 0 and u ∈ H 1 (B r ).
Proof. By scaling, it is enough to prove the inequalities for r = 1. Let u ∈ C ∞ (B 1 ) ∩ H 1 (B 1 ) with 0 / ∈ supp u. Passing in polar coordinates we obtain By (22) and (12) we have Now, inequality (39) follows immediately from (35). From (39) and (37) we obtain for all J ∈ A k and for all u ∈ C ∞ (B 1 ) ∩ H 1 (B 1 ) with 0 / ∈ supp u. On the other hand by (39) and (38) for all (J 1 , J 2 ) ∈ B k and for all u ∈ C ∞ (B 1 ) ∩ H 1 (B 1 ) with 0 / ∈ supp u. By density the stated inequalities follow for any u ∈ H 1 (B 1 ).
From (33) and (39), we can derive a Hardy-Sobolev type inequality which takes into account the boundary terms; to this aim, the following lemma is needed.
Then, for every r > 0 and u ∈ H 1 (B r ), there holds Proof. Inequality (44) follows simply by scaling from the definition of S N .
The following boundary Hardy-Sobolev inequality holds true.
Corollary 2.10. Let a be as in (7) and assume that (12) holds. Then, for all r > 0 and u ∈ H 1 (B r ), there holds where S N is defined in (43).

A Pohozaev-type identity
In order to approximate L a := −∆ S N −1 − a with operators with bounded coefficients, for all λ ∈ R, we define in such a way that a λ ∈ L ∞ (S N −1 ) for any λ > 0. We notice that a λ ∈ F for any λ ∈ R.
Since we are interested in the asymptotics of solutions at 0, we focus our attention on a ball B r0 which is sufficiently small to ensure positivity of the quadratic forms associated to equation (13) and to some proper approximations of (13) in B r0 . Let u be a solution of (13), with the perturbation potential h satisfying (H) and the nonlinear term f satisfying (F). If condition (12) holds, there exists r 0 > 0 such that with a as (7), Λ(a) as in (22) and N −k k = 0 whenever N < 2k.
be a bounded open set such that 0 ∈ Ω, and let a satisfy (7) and (12). Suppose that h satisfies (H), f satisfies (F), u is a H 1 (Ω)-weak solution to (13) in Ω, and r 0 > 0 is as in (47). Then there existsλ > 0 such that, for every λ ∈ (0,λ), the Dirichlet boundary value problem The existence and uniqueness of such aṽ can be proven by introducing the continuous bilinear form Q : By (H), (8), (9), and (11), we have . By (50), (12) and (47) it follows that the bilinear form Q is coercive. The Lax-Milgram lemma yields existence and uniqueness of a solution v ∈ H 1 0 (B r0 ) of the variational problem Then the functionṽ := v + u is the unique solution of (49).
Let us now define the map Φ : By (7), (8), (9), (H) and (F), the function Φ is continuous and its first variation with respect to the w variable is an isomorphism. The first claim is an immediate consequence of the definition of u andṽ. Let us prove the second one. By (F), (11), and Hölder and Sobolev inequalities, for every w ∈ H 1 0 (B r0 ) we obtain The above estimate, together with (47), shows that the quadratic form w → Φ ′ w (0, u −ṽ)w, w is positive definite over H 1 0 (B r0 ). Then the Lax-Milgram lemma applied to the continuous and coercive bilinear form is an isomorphism and hence our second claim is proved. We are now in position to apply the Implicit Function Theorem to the map Φ, thus showing the existence ofλ > 0, ρ > 0, and of a continuous function . The function u λ := g(λ) +ṽ solves (48) for any λ ∈ (0,λ). Moreover, by the continuity of g over the interval (−λ,λ) and the fact that and hence the function s → ∂Bs |f | dS belongs to L 1 (0, r) and is the weak derivative of the In particular, for every u ∈ H 1 (Ω) and every J ∈ A k , Solutions to (13) satisfy the following Pohozaev-type identity.

A Brezis-Kato type estimate
Throughout this section, we let Ω ⊂ R N , N 3, be a bounded open set such that 0 ∈ Ω, a satisfy (7), (12), h satisfy (H), and V ∈ L 1 loc (Ω) satisfy the form-bounded condition see [33]. The above condition (which is in particular satisfied by L N/2 and L N/2,∞ functions, potentials of the form (6), etc.) in particular implies that for every u ∈ H 1 (Ω), V u ∈ H −1 (Ω). We assume that u ∈ H 1 (Ω) is a weak solution to In the spirit of [40, Theorem 2.3], we prove the following Brezis-Kato type result.
then for every 1 q < q lim there exists r q > 0 depending on q, N, k, a, h such that B rq ⊂ Ω and u ∈ L q (B rq ).

The Almgren type frequency function
Let u be a weak H 1 (Ω)-solution to equation (13) in a bounded domain Ω ⊂ R N containing the origin with a satisfying (7) and (12), h satisfying (H) and f satisfying (F).
By (F) and Sobolev embedding, we infer that the function belongs to L N/2 (Ω) and hence we may apply Proposition 4.1 to the function u. Therefore, throughout this section, we may fix (70) 2 * < q < q lim and r q as in Proposition 4.1 in such a way that u ∈ L q (B rq ). By Remark 3.2, the function It is also easy to verify that Further regularity of H is established in the following lemma.
be a bounded open set such that 0 ∈ Ω and let u be a weak H 1 (Ω)-solution to equation (13) in Ω with a satisfying (7) and (12), h satisfying (H), and in a distributional sense and for a.e. r ∈ (0, R).
Proof. Suppose by contradiction that there exists R ∈ (0, r 0 ) such that H(R) = 0. Then u = 0 a.e. on ∂B R and thus u ∈ H 1 0 (B R ). Multiplying both sides of (13) by u and integrating by parts over B R we obtain Proceeding as in (50) and using (F), Hölder and Sobolev inequalities, we obtain which, together with (47), implies u ≡ 0 in B R . Since u ≡ 0 in a neighborhood of the origin, we may apply, away from the singular set Σ (which has zero measure), classical unique continuation principles for second order elliptic equations with locally bounded coefficients (see e.g. [44]) to conclude that u = 0 a.e. in Ω, a contradiction.
We also consider the function D : (0, r 0 ) → R defined as where r 0 is defined in (47). The regularity of the function D is established in the following lemma.
be a bounded open set such that 0 ∈ Ω. Let a satisfy (7) and (12), and u be a weak H 1 (Ω)-solution to (13), with h satisfying (H) and f satisfying (F). Then the function D defined in (74) belongs to W 1,1 loc (0, r 0 ) and in a distributional sense and for a.e. r ∈ (0, r 0 ).
By virtue of Lemma 5.2, if u is a weak H 1 (Ω)-solution to (13), u ≡ 0, the Almgren type frequency function is well defined in (0, r 0 ). Collecting Lemmas 5.1 and 5.3, we compute the derivative of N .
We now prove that N (r) admits a finite limit as r → 0 + . To this aim, the following estimate plays a crucial role.
Lemma 5.5. Under the same assumptions as in Lemma 5.4, there existr ∈ (0, min{r 0 , r q }) and a positive constant C = C (N, k, a, h, f, u) > 0 depending on N , k, a, h, f , u but independent of r such that for every r ∈ (0,r).
Lemma 5.6. Under the same assumptions as in Lemma 5.4, letr be as in Lemma 5.5 and ν 2 as in (81). Then there exist a positive constant C 1 > 0 depending on N, q, C f , C h , C,r, u L q (Br ) and a function g ∈ L 1 (0,r), g 0 a.e. in (0,r), such that for a.e. r ∈ (0,r) and r 0 g(s) ds u for all r ∈ (0,r) and for some α satisfying 2 2 * < α < 1. Proof. From (H) and (83) we deduce that , and, therefore, for any r ∈ (0,r), we have that By (F), Hölder's inequality, and (83), for some constant const = const (N, C f ) > 0 depending on N, C f , and for all r ∈ (0,r), there holds Let us fix 2 2 * < α < 1. Then, by Hölder's inequality and (83), (87), and (84), there exists some const = const (N, q, C f ) > 0 depending on N, q, C f such that, for all r ∈ (0,r), By a direct calculation, we have that in the distributional sense and for a.e. r ∈ (0,r). Since we deduce that the function as r → 0 + , we have that also the function is integrable over (0,r). Therefore, by (89), we deduce that for all r ∈ (0,r).
Proof. By Lemma 5.4, Schwarz's inequality, and Lemma 5.6, we obtain for a.e. r ∈ (0,r). After integration over (r,r) it follows that for any r ∈ (0,r), thus proving estimate (92). Proof. By Lemmas 5.6 and 5.7, the function ν 2 defined in (81) belongs to L 1 (0,r). Hence, by Lemma 5.4 and Schwarz's inequality, N ′ is the sum of a nonnegative function and of a L 1 -function on (0,r). Therefore N (r) = N (r) − r r N ′ (s) ds admits a limit as r → 0 + which is necessarily finite in view of (84) and (92).
A first consequence of the above analysis on the Almgren's frequency function is the following estimate of H(r). On the other hand for any σ > 0 there exists a constant K 2 (σ) > 0 depending on σ such that (95) H(r) K 2 (σ) r 2γ+σ for all r ∈ (0,r).
In the next lemma we prove a doubling type result.
Suppose by contradiction that for any M > 0 there exists a sequence λ n → 0 + such that and, up to a subsequence still denoted by λ n , we may suppose that w λn ⇀ w in H 1 (B 1 ) for some w ∈ H 1 (B 1 ). Notice that, for any λ ∈ (0,r), ∂B1 |w λ | 2 dS = 1 and hence by compactness of the trace map from H 1 (B 1 ) into L 2 (∂B 1 ), it follows that ∂B1 |w| 2 dS = 1. Moreover, by weak lower semicontinuity and (109), we also have |∇w λn (x)| 2 dx = 0 from which it follows that w ≡ const in B 1 . On the other hand, for every λ ∈ (0,r), For every φ ∈ H 1 0 (B 1 ), by (F) and Hölder's inequality, = o(1) as λ → 0 + and, by (H) and Corollary 2.7, From (111), (112), and weak convergence w λn ⇀ w in H 1 (B 1 ), we can pass to the limit in (110) along the sequence λ n and obtain that w is a H 1 (B 1 )-weak solution to the equation Since w is constant in B 1 , this implies w ≡ 0 in B 1 which contradicts ∂B1 |w| 2 dS = 1. Lemma 6.4. Let w λ and R λ be as in the statement of Lemma 6.3. Then there exists M > 0 such that ∂B1 |∇w λR λ | 2 dS M for any 0 < λ < min λ 0 ,r 2 .
Consider now the sequence w λn k . Up to a further subsequence still denoted by w λn k , we may suppose that w λn k ⇀ w for some w ∈ H 1 (B 1 ) and that R λn k → R for some R ∈ [1,2].
Strong convergence of w λn k R λn k in H 1 (B 1 ) implies that, up to a subsequence, both w λn k R λn k and |∇w λn k R λn k | are dominated by a L 2 (B 1 )-function uniformly with respect to k. Moreover by (103), up to a subsequence we may assume that the limit exists and is finite. Then, by the Dominated Convergence Theorem, we have By a density argument, it follows that the previous convergence also holds for all v ∈ L 2 (B 1 ). This proves that w λn k ⇀ √ l w(·/R) in L 2 (B 1 ) (actually weakly in H 1 (B 1 )) and in particular w = √ l w(·/R). Moreover This shows that w λn k → w = √ l w(·/R) strongly in H 1 (B 1 ). Furthermore, by (128) and the fact that ∂B1 |w| 2 dS = ∂B1 |w| 2 dS = 1, we deduce that w = w.
Let us now describe the behavior of H(r) as r → 0 + . Lemma 6.6. Under the same assumptions as in Lemma 5.4 and letting γ := lim r→0 + N (r) ∈ R as in Lemma 5.8, the limit lim r→0 + r −2γ H(r) exists and it is finite.
Proof. In view of (94) it is sufficient to prove that the limit exists. By (71), (82), and Lemma 5.8 we have d dr Let ν 1 and ν 2 be as in (80) and (81). After integration over (r,r), for all s ∈ (0, r), which proves that s −2γ−1 H(s) s 0 ν 2 (t)dt ∈ L 1 (0, r). We may conclude that both terms in the right hand side of (129) admit a limit as r → 0 + thus completing the proof of the lemma. The next step of our asymptotic analysis relies on the proof that lim r→0 + r −2γ H(r) is indeed strictly positive. In the sequel we denote by ψ i a L 2 -normalized eigenfunction of the operator L a = −∆ S N −1 − a associated to the i-th eigenvalue µ i (a), i.e.
Moreover, we choose the ψ i 's in such a way that the set {ψ i } i∈N forms an orthonormal basis of L 2 (S N −1 ).
Combining Lemma 6.5 with Lemma 6.9, we can now prove Theorem 1.1.

Asymptotic behavior of eigenfunctions
We describe the asymptotic behavior of eigenfunctions of the operator L a = −∆ S N −1 − a near the singular set of the function a. Actually, for simplicity we study the asymptotic behavior of eigenfunctions near the south pole as an application of Theorem 1.1 after a stereographic projection of S N −1 onto R N −1 with respect to the "north pole".
Throughout this section we assume that 3 k N − 1 and that a satisfies (7) and (12). Note that if k = N then a is constant and hence the eigenfunctions of L a are smooth.
By Lemma 2.2 the spectrum of L a consists of a diverging sequence of eigenvalues µ 1 (a) < µ 2 (a) . . . µ n (a) . . . each of them having finite multiplicity.
Let µ i (a) be an eigenvalue of L a and let ψ ∈ H 1 (S N −1 ) be a corresponding eigenfunction, i.e.
Let Π : S N −1 \ {e N } → R N −1 be the standard stereographic projection with respect to the "north pole". Here by e N , we denote the vector (0, 0, . . . , 0, 1) ∈ R N . Let φ : R N −1 → R be the function given by If θ ∈ S N −1 \ {e N } and x, z ∈ T θ S N −1 (by T θ S N −1 we mean the tangent space to S N −1 at θ), then where the vector space T Π(θ) R N −1 is identified with R N −1 . In the following lemma the equation satisfied by the projection of ψ is deduced. (7) and (12), and let Π and φ be respectively the stereographic projection with respect to the north pole and the function defined in (153). Let µ i (a) be an eigenvalue of the operator L a and let ψ ∈ H 1 (S N −1 ) be a corresponding eigenfunction. Then the function belongs to D 1,2 (R N −1 ) and weakly solves where b and h are defined by for a.e. y ∈ R N −1 , where for any (J 1 , J 2 ) ∈ B k , n k = max J 1 , m k = max J 2 , and for any J = {n 1 , . . . , n k } ∈ A k , n 1 < n 2 < · · · < n k , we denote J ′ = J \ {n k } ∈ A k−1 .
Proof. The conformal laplacian on S N −1 is given by while, since R N −1 has zero scalar curvature, the conformal laplacian in R N −1 coincides with the usual Laplace operator. Then for any function for every θ ∈ S N −1 \ {e N }. For the definition of the conformal Laplacian and for a proof of (158) see [13, §3] or [6, (1.2.27)]. We claim that the function ψ defined in (154) belongs to D 1,2 (R N −1 ) and weakly solves i.e.
by the Dominated Convergence Theorem, and as n → +∞, we conclude that u n → u in D 1,2 (R N ), thus proving the density of C ∞ c (R N \{x J = 0}) in C ∞ c (R N ) and hence in D 1,2 (R N ). The density of C ∞ c (R N \ {x J1 = x J2 }) can be proven in a similar way. Lemma 8.3. Ifâ satisfy (18) and (19), then Proof. We first prove that (173) holds for all v ∈ C ∞ c (R N \ Σ). Indeed, by direct computation ρ solves Let v ∈ C ∞ c (R N \ Σ) and put u = ρv so that u ∈ C ∞ c (R N ) ⊂ D 1,2 (R N ). Then, testing (174) with ρv 2 we obtain By (175)-(177) we then have To prove (i), by density of C ∞ c (R N ) in D 1,2 ρ (R N ), it is enough to prove the density of C ∞ c (R N \ Σ) in C ∞ c (R N ) with respect to the norm · D 1,2 ρ (R N ) . Let v ∈ C ∞ c (R N ). It is easy to verify that u = ρv ∈ D 1,2 (R N ), hence, by Lemma 8.2, there exists a sequence {u n } n ⊂ C ∞ c (R N \ Σ) such that u n → u in D 1,2 (R N ). Letting v n = un ρ , we have that v n ∈ C ∞ c (R N \ Σ) and, by (178), Therefore, since u n is a Cauchy sequence in D 1,2 (R N ) and, by (19) and Lemma 2.3, (Qâ(u)) 1/2 is an equivalent norm in D 1,2 (R N ), we conclude that v n is a Cauchy sequence in D 1,2 ρ (R N ) and hence converges to someṽ ∈ D 1,2 ρ (R N ). Since v n → v a.e. in R N , we deduce thatṽ = v and then v n → v in D 1,2 ρ (R N ). The proof of (i) is thereby complete. To prove (ii-iii), let v ∈ D 1,2 ρ (R N ). By (i), there exists a sequence {v . Therefore, since v n is a Cauchy sequence in D 1,2 ρ (R N ) and (Qâ(u)) 1/2 is an equivalent norm in D 1,2 (R N ), we infer that u n is a Cauchy sequence in D 1,2 (R N ) and hence converges to some u in D 1,2 (R N ). Since u n = ρv n → ρv a.e. in R N , we deduce that ρv = u ∈ D 1,2 (R N ). Moreover, we can pass to the limit in (179), thus obtaining (173) and proving (iii). In a similar way, one can prove that if u ∈ D 1,2 (R N ) then u ρ ∈ D 1,2 ρ (R N ), thus completing the proof of (ii).
Thanks to Lemma 2.3, (19), and the standard Sobolev inequality, the number is strictly positive and provides the best constant in the following weighted Sobolev inequality.
The proof of the lemma then follows letting n → +∞.
Remark 8.7. The statement of Theorem 9.4 in our previous paper [17] should be corrected as in the statement of Theorem 1.2. The missing point in Theorem 9.4. as it was stated in [17] relies in the fact that the constant C ∞ such that |x| −σ u L ∞ (Ω ′ ) C ∞ u L 2 * (Ω) depends on u, more precisely on the distribution function of f (x, u)/u.
In a similar way, the statements of Theorems 9.3 and 10.4 should be corrected as in Theorem 8.6 above, i.e. the constant C s (respectively C s,2 ) appearing in the statement (ii) of Theorem 9.3 (respectively 10.4) depends on (ℜ(V )) + (more precisely on its distribution function) and not only on its L N/2 (ρ 2 * , Ω)-norm (respectively L s (ρ p , Ω)-norm) as incorrectly stated in [17].
Anyway, the proofs of Theorems 9.3 and 9.4 contained in [17] are correct and lead to analogous conclusion as those stated in Theorems 1.2 and 8.6 of the present paper. Moreover all the proofs and statements in the rest of the paper [17] are not affected by these corrections. f (x, u(x))(x · ∇u(x)) dx .