On the behavior of solutions to Schr\"odinger equations with dipole-type potentials near the singularity

Asymptotics of solutions to Schroedinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.


Introduction and statement of the main results
In nonrelativistic molecular physics, the interaction between an electric charge and the dipole moment D ∈ R N of a molecule is described by an inverse square potential with an anisotropic coupling strength. In particular the Schrödinger equation for the wave function of an electron interacting with a polar molecule (supposed to be point-like) can be written as ∆ + e x · D |x| 3 − E Ψ = 0, where e and m denote respectively the charge and the mass of the electron and D is the dipole moment of the molecule, see [11]. We aim to describe the asymptotic behavior near the singularity of solutions to equations associated to dipole-type Schrödinger operators of the form , being |D| the magnitude of the dipole moment D, and d = D/|D| denotes the orientation of D. A precise estimate of such a behavior is the first step in the analysis of fundamental properties of Schrödinger operators, such as positivity, essential self-adjointness, and spectral features. Such an analysis has been carried out in [7] for the case of Schrödinger operators with multipolar Hardy potentials based on the regularity results proved in [8] for degenerate elliptic equations. In a forthcoming paper, the authors will apply Theorem 1.1 to establish spectral properties of multi-singular dipole type operators.
We emphasize that, from the mathematical point of view, potentials of the form λ (x·d) |x| 3 have the same order of homogeneity as inverse square potentials and consequently share many features with them, such as invariance by scaling and Kelvin transform, as well as no inclusion in the Kato class. We mention that Schrödinger equations with Hardy-type singular potentials have been largely studied, see e.g. [1,6,9,10,17,18] and references therein.
More precisely, in this paper we deal with a more general class of Schrödinger operators including those with dipole-type potentials, namely with operators whose potentials are purely angular multiples of radial inverse-square potentials: in R N , where N 3 and a ∈ L ∞ (S N −1 ). The problem of establishing the asymptotic behavior of solutions to elliptic equations near an isolated singular point has been studied by several authors in a variety of contexts, see e.g. [13] for Fuchsian type elliptic operators and [5] for Fuchsian type weighted operators. The asymptotics we derive in this work is not contained in the aforementioned papers, which prove the existence of the limit at the singularity of any quotient of two positive solutions in some linear and semilinear cases which, however, do not not include the perturbed linear case and the critical nonlinear case treated here. Moreover, besides proving the existence of such a limit, we also obtain a Cauchy type representation formula for it, see (4) and (8).
We also quote [12], where asymptotics at infinity is established for perturbed inverse square potentials and in some particular nonradial case. Hölder continuity results for degenerate elliptic equations with singular weights and asymptotic analysis of the behavior of solutions near the pole are contained in [8].
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 In order to discuss the positivity properties of the Schrödinger operator L a in D 1,2 (R N ), we consider the best constant in the associated Hardy-type inequality (1) Λ N (a) := sup By the classical Hardy inequality, Λ N (a) 4 (N −2) 2 ess sup S N −1 a, where ess sup S N −1 a denotes the essential supremum of a in S N −1 . We also notice that, in the dipole case, by rotation invariance, Λ N λ x |x| · d does not depend on d.
It is easy to verify that the quadratic form associated to L a is positive definite in D 1,2 (R N ) if and only if Λ N (a) < 1. The relation between the value Λ N (a) and the first eigenvalue of the angular component of the operator on the sphere S N −1 is discussed in section 2, see Lemmas 2.4 and 2.5. More precisely, Lemma 2.5 ensures that the quadratic form associated to L a is positive definite if and only if where µ 1 = µ 1 (a, N ) is the first eigenvalue of the operator −∆ S N −1 − a(θ) on S N −1 (see Lemma 2.1). We denote by ψ 1 the associated positive L 2 -normalized eigenfunction and set In the spirit of the well-known Riemann removable singularity theorem, we describe the behavior of solutions to linear Schrödinger equations with a dipole-type singularity localized in a neighborhood of 0.
Then the function We notice that (4) is actually a Cauchy's integral type formula for u. Moreover the term at the right hand side is independent of R. In the case in which the perturbation q is radial then an analogous formula holds also for changing sign solutions to (3), see Remark 4.2.
If the perturbing potential q satisfies some proper summability condition, instead of the stronger control on the blow-up rate at the singularity required in Theorem 1.1, a Brezis-Kato type argument, see [2], allows us to derive an upper estimate on the behavior of solutions. For any q 1, we denote as L q (ϕ 2 * , Ω) the weighted L q -space endowed with the norm N −2 is the critical Sobolev exponent and and ϕ denotes the weight function (5) ϕ(x) := |x| σ ψ 1 (x/|x|).
The following Brezis-Kato type result holds.
, Ω) for some s > N/2. Then, for any Ω ′ ⋐ Ω, there exists a positive constant depending only on N , a, V L s (ϕ 2 * ,Ω) , dist(Ω ′ , ∂Ω), and diam Ω, such that for any weak H 1 (Ω)solution u of The Brezis-Kato procedure can be applied also to semilinear problems with at most critical growth, thus providing an upper bound for solutions and then reducing the semilinear problem to a linear one with enough control on the potential at the singularity to apply Theorem 1.1 and to recover the exact asymptotic behavior.
Notation. We list below some notation used throughout the paper.
-B(a, r) denotes the ball {x ∈ R N : |x − a| < r} in R N with center at a and radius r.
-dV denotes the volume element on the sphere S N −1 .
-the symbol ess sup stands for essential supremum.

Spectrum of the angular component
Due to the structure of the dipole-type potential of equation (3), a natural approach to describe the solutions seems to be the separation of variables. To employ such a technique, we need, as a starting point, the description of the spectrum of the angular part of dipole Schrödinger operators.
From (9), [18,Lemma 1.1], and the optimality of the best constant in Hardy's inequality, it follows that thus proving the left part of the inequality stated in (v).
The asymptotic behavior of eigenvalues µ k as k → +∞ is described by Weyl's law, which is recalled in the theorem below. We refer to [14,15] for a proof.
for some positive constant C(N, a) depending only on N and a.
The following lemma provides an estimate of the L ∞ -norm of eigenfunctions of the operator −∆ S N −1 − a(θ) in terms of the corresponding eigenvalues. For classical results about L ∞ estimates of eigenfunctions of Schrödinger operators we refer to [16, §4] and references therein.
For a ∈ L ∞ (S N −1 ) and k ∈ N \ {0}, let ψ k be a L 2 -normalized eigenfunction of the Schrödinger operator −∆ S N −1 − a(θ) on the sphere associated to the k-th eigenvalue µ k , i.e.
Then, there exists a constant C 1 depending only on N and a such that where ⌊·⌋ denotes the floor function, i.e. ⌊x⌋ := min{j ∈ Z : j x}.
Proof. Using classical elliptic regularity theory and bootstrap methods, we can easily prove that for any j ∈ N there exists a constant C(N, j), depending only on j and N , such that Choosing j = N −1 4 + 1, by Sobolev's inclusions we deduce that , thus implying the required estimate.
We notice that the supremum in (12) is achieved due to the compactness of the embedding H 1 (S N −1 ) ֒→ L 2 (S N −1 ). As a direct consequence of the above lemma, it is possible to compare Λ N (a) with the best constant in Hardy's inequality Indeed, if a is not constant, there holds 4 (N − 2) 2 whereas, if a(θ) = κ for a.e. θ ∈ S N −1 and for some κ ∈ R, then Λ N (a) = 4κ/(N − 2) 2 .
Let us consider the quadratic form associated to the Schrödinger operator L a , i.e.
The problem of positivity of Q a is solved in the following lemma.
Lemma 2.5. Let a ∈ L ∞ (S N −1 ). The following conditions are equivalent: Proof. The equivalence between i) and ii) follows from the definition of Λ N (a), see (1). On the other hand, [18, Proposition 1.3 and Lemma 1.1] ensure that i) is equivalent to iii).
Remark 2.6. We notice that in the case of dipole potentials, namely if a(θ) = λ x·d |x| , then, rewriting (12) in spherical coordinates and exploiting the symmetry with respect to the dipole axis, Λ N (a) can be characterized as In dimension N = 3, a Taylor's expansion of Λ N λ x·d |x| near λ = 0 can be found in [11].
There is no explicit formula for the values Λ N λ x·d |x| . When N = 4, it can be expressed in terms of Mathieu's special functions. The results of a numerical approximation of Λ N λ x·d |x| performed with a finite difference method are listed in table 1, which highlights how the dipole Hardy-type constant detaches the classical Hardy constant more and more as the dimension grows.

A Brezis-Kato type lemma
In this section, we follow the procedure developed by Brezis and Kato in [2] to control from above the behavior of solutions to Schrödinger equations with dipole type potentials, in order to prove Theorem 1.2.
Let us consider the weight ϕ introduced in (5) and define the weighted By the Caffarelli-Kohn-Nirenberg inequality (see [3] and [4]) and the definition of ϕ, it follows that, for any w ∈ D 1,2 ϕ (Ω), for some positive constant C N,a depending only on N and a.
Proof of Theorem 1.2. Let u be a weak H 1 (Ω)-solution to (6). It is easy to verify that ϕ(x) := |x| σ ψ 1 (x/|x|) ∈ H 1 (Ω) satisfies (in a weak H 1 (Ω)-sense and in a classical sense in Ω \ {0}) Then v := u ϕ ∈ H 1 ϕ (Ω) turns out to be a weak solution to (14). Let R > 0 be such that Setting, for any n ∈ N, n 1, r k , and r n = 1 n 2 , and using iteratively Lemma 3.1, we obtain that, for any n ∈ N, n 1, and, for some constant Hence ∞ n=1 b n converges to some positive sum depending only on V L s (ϕ 2 * ,Ω) , dist(Ω ′ , ∂Ω), N , and a, hence is finite and depends only on N , a, V L s (ϕ 2 * ,Ω) , and dist(Ω ′ , ∂Ω). Hence, from (23), we deduce that there exists a positive constant C depending only on N , a, V L s (ϕ 2 * ,Ω) , dist(Ω ′ , ∂Ω), and diam Ω, such that Letting n → +∞ we deduce that v is essentially bounded in Ω ′ with respect to the measure where v L ∞ (ϕ 2 * ,Ω ′ ) denotes the essential supremum of v with respect to the measure ϕ 2 * dx. Since ϕ 2 * dx is absolutely continuous with respect to the Lebesgue measure and viceversa, there thus completing the proof.
If the potential V in equation (6) belongs to L N/2 (ϕ 2 * , Ω) (but to L s (ϕ 2 * , Ω) for no s > N/2), although we can no more derive an L ∞ -bound for u/ϕ, we can obtain for u/ϕ as high summability as we like.
, Ω). Then, for any Ω ′ ⋐ Ω and for any weak Proof. The proof follows closely the proofs of Theorem 1.2 and Lemma 3.1. However, since we only require V ∈ L N/2 (ϕ 2 * , Ω), we have that for any q there exists ℓ q such that but we can no more estimate ℓ q in terms of q, as we did in (17) thanks to the summability assumption V ∈ L s (ϕ 2 * , Ω) for some s > N/2. Hence we still arrive at an estimate of type (23) but we have no control on the product in (24) as n → +∞.

Behavior of solutions at singularities
The procedure followed in this section to prove Theorem 1.1 relies in comparison methods and separation of variables. Indeed we will evaluate the asymptotics of solutions to problem (3) by trapping them between functions which solve analogous problems with radial perturbing potentials. To this aim, the first step consists in deriving the asymptotic behavior of solutions to Schrödinger equations with a potential which is given by a radial perturbation of the dipole-type singular term. In this case, it is possible to expand the solution in Fourier series, thus separating the radial and angular variables, and to estimate the behavior of the Fourier coefficients in order to establish which of them is dominant near the singularity.
where h ∈ L ∞ loc (0, R) ∩ L p (0, R) for some p > N/2. Then, for any r ∈ (0, R), there exists a positive constant C (depending on h, R, r, a, ε, and u) such that where σ is defined in (2). Moreover, there exists a positive constant C (depending on u, h, N , and a) such that, for any θ ∈ S N −1 , and, for any r ∈ (0, R), Furthermore, for any r ∈ (0, R), there exists a positive constantC (depending on h, R, r, a, ε, but not on u) such that Proof. Let r ∈ (0, R). We can assume, without loss of generality, that R > 1 and r = 1. Indeed, setting w(x) := u(rx), we notice that w ∈ H 1 (B(0, R/r)) and weakly solves Hence, it is enough to prove the statement for R > 1 and r = 1, being the general case easily obtainable from scaling.
Let R > 1, r = 1 and u ∈ H 1 (B(0, R)), u 0 a.e. in B(0, R), u ≡ 0, be a weak solution of (25). By standard regularity theory, u ∈ C 0 B(0, 1) \ B(0, s) for any s ∈ (0, 1). For any k ∈ N \ {0}, let ψ k be a L 2 -normalized eigenfunction of the operator −∆ S N −1 − a(θ) on the sphere associated to the k-th eigenvalue µ k , i.e. satisfying (11). We can choose the functions ψ k in such a way that they form an orthonormal basis of L 2 (S N −1 ), hence u can be expanded as where ρ = |x| ∈ (0, 1], θ = x/|x| ∈ S N −1 , and The Parseval identity yields and hence Equations (25) and (11) imply that, for every k, A direct calculation shows that, for some c k 1 , c k 2 ∈ R, For the sake of notation, we set Without loss of generality, we can assume that , for every k. From Hölder's inequality and Lemma 2.3, it follows that where ω N = S N −1 dV (θ). In particular is finite. Since u ∈ L 2 * (B(0, 1)), from (37), (39), and the fact that ρ σ − k ψ k (θ) ∈ L 2 * (B(0, 1)), we conclude that there must be Since u ∈ C 0 B(0, 1) \ B(0, s) for any s ∈ (0, 1), it makes sense to evaluate ϕ k at ρ = 1 and, from (32) and (40), we have that From (32), (41), and (42), we deduce that From above, (38), and standard elliptic estimates (which allow to estimate u outside the singularity in terms of its H 1 -norm) we obtain that, for some positive constantc depending only on N , R, a, and h, Arguing as in (38), we find that (45) From (35), (36), and (45), we can estimate ϕ k as i.e. j k is the unique integer number such that Notice that ε 1 3 σ + k + N −2 2 implies j k 2. To prove the claim, we observe that, from (34) and (46) it follows that In a similar way, from (41) and (46) we deduce that Summing up (48) and (49), we obtain , and hence, from (35), Using (50), we can improve our estimates of A k (ρ) and B k (ρ) thus obtaining Summing up we find that An iteration of the above argument (j k − 1) times easily leads to estimate (47). Claim 1 is thereby proved.
Claim 2: the function s → s −σ + k +1 h(s)ϕ k (s) belongs to L 1 (0, 1) and Indeed, from (47), (35), (44), and the choice of j k , it follows that , for some positive constant d k depending on k (and on a, R, h N ). We distinguish now two cases.
If j k ε < N −2 2 2 + µ k − ε, then from (34), (41), and (52) we derive that , for some other positive constant d ′ k depending on k (and on a, R, h, N ), and hence, by (44) and the choice of j k , Estimate (53) and the choice of j k imply that the function s → s −σ + k +1 h(s)ϕ k (s) belongs to L 1 (0, 1). Moreover, from (41) and (53) it follows that and the claim is proved. If j k ε = N −2 2 2 + µ k − ε, then from (34), (41), and (52) we derive that for some other positive constant γ k depending on k (and on a, R, h, N ), and hence, Estimate (55) implies that the function s → s −σ + k +1 h(s)ϕ k (s) belongs to L 1 (0, 1). Moreover, from (41) and (55) it follows that and claim 2 is proved also in this case.
and, by standard regularity theory, u ∈ C 0 B(0, 1) \ B(0, s) for any s ∈ (0, 1), the proof of Proposition 4.1 will be complete if we show that In order to obtain (70), we need to prove the following From claim 2 and (51), we know that there exists a constant ℓ k depending on k (and on a, R, h, u, N ) such that |ϕ k (ρ)| ℓ k ρ σ + k . Using the above estimate in (71), we can improve such an estimate as Using the above estimate in (71), we can obtain the following further improvement Arguing by induction, we can easily prove that, for all j ∈ N, , and letting j → +∞, we deduce that ϕ k (ρ) = 0 for all ρ ∈ (0, 1). Claim 3 is thereby proved.
We are now in position to prove (70). Arguing by contradiction, let us assume that (72) and let k 0 > 1 be the smallest index for which Such a k 0 exists in view of claim 3; indeed if c k 1 + 1 0 for all k and u would be identically zero, thus giving rise to a contradiction. Moreover, from (72), we have that k 0 > 1, and, by claim 3, ϕ k ≡ 0 in (0, 1) for all 1 k k 0 − 1. Repeating the same arguments we used above to prove (66), it is now possible to show that where m k0 is the geometric multiplicity of the eigenvalue µ k0 . We notice that the sum at the right hand side is a nontrivial function in L 2 (S N −1 ) which, being k 0 > 1, is orthogonal to the first positive eigenfunction ψ 1 . Hence the right hand side of (73) changes sign in S N −1 . Therefore the limit in (73) implies that u changes sign in a neighborhood of 0, which is in contradiction with the positivity assumption on u. Condition (70) follows and the proof of Proposition 4.1 is now complete.
Remark 4.2. If we let the assumption of positivity of u drop, following the proof of Proposition 4.1, we can still prove a Cauchy's integral type formula for u. More precisely, if u ∈ H 1 (B(0, R)) is a weak solution to (25) in B(0, R) which changes sign in any neighborhood of 0, with a radial potential h ∈ L ∞ loc (0, R) ∩ L p (0, R) for some p > N/2, then, following the notation introduced in (33) and letting k 0 > 1 be the smallest index for which for any θ ∈ S N −1 and r ∈ (0, R) there holds Without the assumption of radial symmetry of the potential, it is still possible to evaluate the exact behavior near the singularity of the first Fourier coefficient ϕ 1 (see (29) and (30)).
In order to extend the result of Proposition 4.1 to the case in which the potential is a non radial perturbation of the dipole-type singular term, we will construct a subsolution and a supersolution which solve equations of type (25) and the behavior of which is consequently known in view of Proposition 4.1.
Estimate (90) and the continuity of u outside the origin imply that there exists a positive constant C (depending on q, R, Ω, a, ε, and u) such that 1 C |x| σ u(x) C|x| σ for all x ∈ B(0, R) \ {0}.
From Proposition 4.1, for any 0 < r r, the functions x → u r (x) |x| σ ψ 1 (x/|x|) and x →ū r (x) |x| σ ψ 1 (x/|x|) have limits as |x| → 0, which, accordingly with (26-27) and taking into account the continuity of functions u r andū r up to the boundary |x| = r, can be computed as In view of (89), there holds that, for any 0 < r r, Letting r → 0, we complete the proof.

Behavior of solutions to the semilinear problem
The L q and L ∞ bounds of solutions to dipole-type linear Schrödinger equations with properly summable potentials, derived in Theorems 1.2 and 3.2, allow us to obtain in the semilinear case analogous estimates.