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On semilinear elliptic equations with borderline Hardy potentials

In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgren-type monotonicity formula is used to determine the exact asymptotic behavior of solutions.


Introduction
On a domain Ω ⊆ R N , N 3, containing the origin, let us consider the following problem where h is possibly singular at the origin but negligible with respect to the Hardy potential and f is a nonlinearity subcritical with respect to the critical Sobolev exponent. Looking at equation (1), one may observe that the best constant for the classical Hardy inequality appears in front of the inverse square potential; this can be considered as a borderline situation for several points of view, from the variational setting to the existence and qualitative behavior of solutions. Recent papers were devoted to equations and differential inequalities involving elliptic operators with inverse square potentials in the borderline situation, see [5,11,12,13,20,28,29]. In [13] the authors study necessary conditions for the existence of nonnegative distributional solutions of the differential inequality where B R denotes the ball of radius R centered at the origin. The logarithmic term appearing in the above inequality is related to an improved version of the Hardy inequality, see for example [1,3,10,22]. In [11], the author studies existence of positive distributional solutions of the nonlinear elliptic equation −∆u − λ u |x| 2 + b(x)h(u) = 0 in Ω \ {0}, satisfying some prescribed asymptotic behaviors at the origin, where Ω ⊂ R N , N 3 is a domain containing the origin and λ ∈ (−∞, (N −2) 2 /4]. These prescribed asymptotic behaviors are related to the following fundamental solutions respectively in the cases λ ∈ (−∞, (N − 2) 2 /4) and λ = (N − 2) 2 /4. Similar results were obtained in [4] for equations with elliptic operators in divergence form.
In [12] the author studies a singular elliptic Dirichlet problem with a power type nonlinearity and a forcing term: where Ω ⊂ R N , N 3, is a domain containing the origin, p > 1, t > 0, f a smooth, bounded, nonnegative function, and c ∈ (0, (N −2) 2 /4]. In [12], the author provides a classification of different kind of solutions of problem (2), both of distributional and variational type. In the present paper, we are going to introduce an analogous terminology for solutions to (1), a classification of which will be provided as a byproduct of our main result, see section 2 for details.
We also mention that, in [5] the authors study existence and nonexistence of solutions of the equation in (2) with t = 0.
In the spirit of [12], in the present paper we concentrate our attention on local solutions to (1) belonging to a suitable functional space related to the borderline case of the Hardy inequality. To this purpose, in section 2 we introduce the Hilbert space H(ω) defined as completion of C ∞ c (ω \{0}) with respect to a scalar product related to the Hardy potential appearing in (1) (see (4)). Here ω represents a bounded domain with ∂ω ∈ C 1 .
The purpose of this paper is to classify the possible asymptotic behaviors of solutions to (1) near the singularity of the Hardy potential. Some results in this direction were obtained in [14,15,16,17] for different kinds of problems: in [15,17] Schrödinger equations with electromagnetic potentials, in [16] Schrödinger equations with inverse square many-particle potentials; finally in [14] the authors study the asymptotic behavior of solutions of a singular elliptic equation near a corner of the boundary. For other results concerning elliptic equations with singular inverse square potentials see also [18,19,23,24,27].
The results of the present paper are closely related to the ones obtained in [15,17]. If we drop the magnetic part of the electromagnetic potential, the equation studied in [15,17] becomes where a ∈ L ∞ (S N −1 ). In [15,17] the quadratic form associated to the linear operator −∆− a(x/|x|) |x| 2 is assumed to satisfy a coercivity type condition. More precisely, it is required that the first eigenvalue µ 1 (0, a) of the spherical operator −∆ S N −1 − a(θ) satisfies µ 1 (0, a) > − N −2 2 2 .
Due to this coercivity, it was quite natural in that setting looking for H 1 -solutions to (3), i.e. functions u ∈ H 1 (Ω) satisfying (3) in a variational sense, whereas, in the borderline situation considered in the present paper, it is reasonable to replace the classical H 1 Sobolev space with the above mentioned H space.
In the proof of our main result (Theorem 2.1 below) we perform an Almgren-type monotonicity procedure (see [2,21]) and provide a characterization of the leading term in the asymptotic expansion by means of a Cauchy's integral type representation formula.
As an application of the main result, we also prove an a priori estimate and a unique continuation principle for solutions to (1); see [25] for questions related to unique continuation principles for elliptic equations with singular potentials. This paper is organized as follows. In Section 2 we introduce the assumptions of the main result and explain in details what we mean by a H-solution of (1). In Section 3 we describe the main properties of the space H, while in Section 4 we reformulate (1) in cylindrical variables, introducing an auxiliary equation in a cylinder of R N +1 . In Section 5 we study the Almgren-type function associated to the problem, which is combined in Section 6 with a blow-up argument to characterize the leading term in the asymptotic expansion of solutions of (1) near the origin, thus proving the main theorem.

Assumptions and main results
We first introduce the assumptions on the potential h and the nonlinearity f . We assume that h satisfies in Ω \ {0} for some C h > 0 and ε > 0. It is not restrictive to assume that ε ∈ (0, 2). Let f satisfy x ∈ Ω and all s ∈ R, 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). In order to state the main result of this paper, a suitable variational formulation for solutions of (1) has to be introduced (see also [12,20,29]). For any bounded domain ω ⊂ R N containing the origin and satisfying ∂ω ∈ C 1 , let us define H(ω) as the completion of the space C ∞ c (ω \ {0}) with respect to the scalar product The form in (4) is actually a scalar product on C ∞ c (ω \ {0}) as detailed in Section 3. For any domain Ω ⊆ R N satisfying 0 ∈ Ω (with ∂Ω not necessarily in C 1 ), we define the space H loc (Ω) as the space of functions u ∈ H 1 loc (Ω \ {0}) such that u |ω ∈ H(ω) for any domain ω ⋐ Ω with ∂ω ∈ C 1 .
We are ready to provide a rigorous definition for solutions to (1). Let h, f satisfy respectively (H) and (F): by a solution of (1) we mean a function u ∈ H loc (Ω) such that We observe that every term in the above identity is well-defined in view of Proposition 3.2 and Proposition 3.5.
The following result follows immediately from Corollary 2.2.   (1) and h, f are smooth outside 0, then u is a strong solution in the sense of [12].
As another byproduct of Theorem 2.1, we also have the following version of the Strong Unique Continuation Principle for an elliptic equation with a singular coefficient. Notation.
-For all r > 0, B r denotes the open ball {x ∈ R N : |x| < r} in R N with center at 0 and radius r.
Clearly T is an isomorphism between vector spaces. Let us denote by µ the standard volume measure on the cylinder C, by ∇ C the gradient associated with the standard Riemannian metric of C, and by (t, θ) the generic element of C.
We observe that (·, ·) H(ω) as defined in (4) is actually a scalar product on C ∞ c (ω \ {0}) since the following identities hold for any u ∈ C ∞ c (ω \ {0}), see [8,9,30]: We also define, for any ω as above, the weighted Sobolev space H µ (C ω ) as the completion of C ∞ c (C ω ) with respect to the norm By density and continuity it is possible to extend T as a linear and continuous operator from H(ω) to H µ (C ω ). In this way T : H(ω) → H µ (C ω ) becomes an isometric isomorphism. The following proposition relates H(ω) with the classical Sobolev space H 1 (ω).
We omit the proof of Proposition 3.1 which can be easily obtained by classical density arguments. We observe the inclusion H 1 (ω) ⊂ H(ω) is actually strict, since the function |x| − N −2 2 belongs to H(ω) but not to H 1 (ω).

and direct calculations, for any open set
Then {u n } is a Cauchy sequence in H 1 (A) for any A as above and hence u ∈ H(ω) may be seen as a function in H 1 loc (ω \ {0}). Moreover, by (4), (7) and (8) it is clear that and hence {u n } is a Cauchy sequence in L 2 (ω). This completes the proof of the proposition.
By Proposition 3.3 and the definition of H 0 (ω), the following Sobolev type embedding follows.
Proof. Let u ∈ H(ω). Then by Proposition 3.2 we deduce that u ∈ L q (ω \ B δ ) for any δ > 0 such that B δ ⊂ ω. Moreover, arguing as in (9), we infer that there exists a constant C(N, q, δ) depending only on N, q, δ, ω such that Let us prove that for some fixed δ > 0 chosen as above u ∈ L q (B δ ). To this purpose let η ∈ C ∞ c (ω) be a radial function such that 0 η 1 and . Then by (7), (8), (10) we obtain . This shows that {u n η} is a Cauchy sequence in L q (ω). Since u n η → uη pointwise then uη ∈ L q (ω). In particular u ∈ L q (B δ ). Moreover proceeding as in (13) we also have that Combining (12) and (14) we conclude that H(ω) ⊂ L q (ω) with continuous embedding.
for any ϕ ∈ C ∞ c (Ω). Moreover, for any bounded domain ω with ∂ω ∈ C 1 and ω ⊂ Ω, we have that Proof. Let ϕ ∈ C ∞ c (Ω) and let ω be an open domain with smooth boundary satisfying ω ⊂ Ω and supp ϕ ⊂ ω. Since u ∈ H loc (Ω) then u ∈ H(ω) and there exists a sequence By Proposition 3.5 we also have that u n → u in L q (ω) for any 1 q < 2 * . By Hölder inequality and hence passing to the limit in (18) we obtain On the other hand, by the convergence u n → u in H 1 loc (ω \ {0}), see Proposition 3.2, we obtain where the last identity follows from assumptions (H) and (F) and the fact that ϕ m → ϕ in L q (ω) for any 1 q < 2N N −2 . Combining (19) and (20) obtain (15). The proof of (16) follows by the following density argument: . Now it is enough to pass to the limit as m → +∞ in (20) with v m in place of ϕ.
. Therefore we are allowed to integrate by parts to obtain The proof of (17) follows passing to the limit as m → +∞.
4. An equivalent problem on the cylinder C Reformulation of (5) in cylindrical variables yields the following characterizations of solutions to (1). (1) in the sense of (5). If ω is a bounded domain with ∂ω ∈ C 1 and ω ⊂ Ω, then the function v : is a weak solution of the equation in the sense that for every w ∈ H µ (C ω ), where ν ∂Cω denotes the exterior normal vector to ∂C ω on C.
The following corollary is an immediate consequence of (23).  (1) in the sense of (5). For any t ∈ R, let Then for any t such that C t ⊂ C Ω , the function v := T u ∈ H µ (C t ) satisfies for any w ∈ H µ (C t ).
In order to study solutions to (21), the properties of space H µ have to be investigated. The next results go in this direction.
is an equivalent norm in H µ (C t ); more precisely, there exists a constant C > 0 such that, for all t ∈ R and v ∈ H µ (C t ), Proof. The embedding H µ (C t ) ֒→ L 2 (Γ t ) and its compactness are just a consequence of the fact that T : H(ω) → H µ (C ω ) is an isometric isomorphism combined with Proposition 3.2 and compactness of classical Sobolev trace embeddings. To show that the quadratic form in (25) is an equivalent norm in H µ (C t ), we notice that, for all v ∈ C ∞ c (C t ), integration by parts yields On the other hand, (27) also implies The conclusion then follows by density.
The following lemma provides a Hardy type inequality with boundary terms.
The following Hardy-Sobolev type inequality holds.
Proof. From Proposition 3.5, there exists c N,q > 0 such that for all u ∈ H(B 1 ). Performing the change of variable v(s, θ) = T u(s − t, θ) in the above inequality for all t ∈ R and taking into account (26), we obtain the stated inequality with C N,q = c 2 N,q C.

The Almgren frequency function
In this section, our purpose would be to construct an appropriate Almgren-type frequency function for the solution to problem (5). Since for a general function u ∈ H(ω), the norm u H(ω) cannot be expressed in an integral form, we prefer to look for an Algrem-type function associated with the function v := T u.
Lemma 5.1. Let Ω ⊂ R N a domain a let u ∈ H loc (Ω) be a solution of (5), u ≡ 0, with h satisfying (H) and f satisfying (F). Let H = H(t) be the function defined in (29). Then there existst > 0 such that H(t) > 0 for any t >t.
Proof. Let us argue by contradiction and assume that there exists t n → +∞ such that H(t n ) = 0; in particular v = 0 on Γ tn and v ∈ H µ,0 (C tn ). From (24), (H), (F), and Lemmas 4.4 and 4.5 which implies that v ≡ 0 in C tn for n sufficiently large. Hence u ≡ 0 in a neighborhood of the origin and, by classical unique continuation principles for second order elliptic equations with locally bounded coefficients (see e.g. [31]) we conclude that u = 0 a.e. in Ω, a contradiction.
By virtue of Lemma 5.1, the Almgren-type frequency function is well defined in (t, +∞). In order to obtain a suitable representation for the derivative of D we need the following Pohozaev-type identity.
Proposition 5.2. Let h satisfy (H), f satisfy (F), and u ∈ H loc (Ω) be a solution to (5). Let R > 0 be such that B R ⊂ Ω. Then for every t ∈ (− log R, +∞) the function v := T u ∈ H µ (C − log R ) satisfies Let {s k } ⊂ R be an increasing sequence such that s k → +∞ and From Remark 3.6, u ∈ C 1 (Ω \ {0}) and hence v ∈ C 1 (C Ω ). Since v is a weak solution of (21) in C ω for any bounded domain ω satisfying ∂ω ∈ C 1 and ω ⊂ Ω, testing (21) with ∂v ∂s (we recall that ∂v ∂s ∈ H 1 µ (C t \ C s k ) in view of Remark 3.6) and using (23) we obtain By (32) we infer that lim k→+∞ Γs k ∂v ∂s Moreover an integration by parts in the left hand side of (33) yields Passing to the limit as k → +∞ in (33), by (H), (F), (32), (34) and (35) we arrive to the conclusion.
In the next lemma we provide a useful representation for the derivative of D. Ct in a distributional sense and for a.e. t ∈ (− log R, +∞).

Proof.
Since the proof directly follows from (31).
The derivative of H is computed in the next lemma.
Lemma 5.4. Under the same assumptions of Proposition 5.2 let H be as in (29). Then H is differentiable in (− log R, +∞) and for any t ∈ (log R, +∞).
which, together with the identity Let us now compute the derivative of N .
In order to show that the Almgren function N admits a finite limit as t → +∞ we need some preliminary estimates which will be proved in the next lemmas.
for which yields the conclusion if t is chosen sufficiently large.
Next we provide an estimate on the function ν 2 introduced in Lemma 5.5.
Lemma 5.7. Under the same assumptions as in Lemma 5.5 we have where C 1 and C 2 are two positive constant depending only on N, h, f, u but independent of t, α := min{ε, 2, 2N p − N + 2}, and g ∈ L 1 (t, +∞), g 0 a.e., satisfies

Proof. From (H) and (F) it follows that
and hence we also have g ∈ L 1 (t, +∞).
Combining (38), (39) with Lemma 4.4, Lemma 4.5 and (37) we obtain The statements of the lemma follow from this last estimate and the definition of g.
We can now prove that the function N admits a finite limit as t → +∞.
Lemma 5.8. Under the same assumptions as in Lemma 5.5, the limit γ := lim t→+∞ N (t) exists and is finite. Moreover γ 0.
Proof. From Lemma 5.6 we have that On the other hand, by Lemma 5.5, Schwarz inequality, and Lemma 5.7, we have and in turn d dt e C1 +∞ t (e −αs +g(s)) ds (N (t) + 1) Since the right hand side in the above line belongs to L 1 (t, +∞), after integration we deduce that N is bounded from above and hence, by (42) and Lemma 5.7, it follows that N ′ is the sum of the nonpositive function ν 1 and of the integrable function ν 2 . This implies that exists and it is necessarily finite since N is bounded. This limit is necessarily nonnegative in view of (41).
Proof. By Lemma 5.4 and Lemma 5.8, we have By Lemma 5.7 we then obtain where C is a constant depending only on N, h, f, u, ε, N, p and α is as in Lemma 5.7. Estimate (43) follows after integration in the last inequality. On the other hand, for any σ > 0 there exists λ(σ) > 0 such that Estimate (44) follows after integration.

A blow-up argument
Convergence of the frequency function N as t → +∞ is a fundamental tool in the following blow-up argument. Hereafter, we denote as 0 = µ 1 < µ 2 . . . µ n . . . the eigenvalues of −∆ S N −1 with the usual notation of repeating them as many times as their multiplicity. Hence we have that µ 1 = λ 0 = 0 and if k > 1 and ℓ−1 n=0 m n < k ℓ n=0 m n , then µ k = λ ℓ . Lemma 6.1. Under the same assumptions as in Lemma 5.5, let us define the family of functions for any t 0 and θ ∈ S N −1 .
Let γ be the limit introduced in Lemma 5.8. Then (i) there exists k 0 ∈ N \ {0} such that γ = √ µ κ0 ; (ii) for any sequence λ n → +∞ there exists a subsequence λ n k and an eigenfunction ψ of −∆ S N −1 corresponding to the eigenvalue µ k0 such that ψ L 2 (S N −1 ) = 1 and {w λn k } converges to the function w(s, θ) := e − √ µ k 0 s ψ(θ) weakly in H µ (C 0 ), strongly in H µ (C t ) for any t > 0, and strongly in C 1,α loc (C 0 ) for any α ∈ (0, 1). Proof. By (37) and the definition of w λ we see that and by Lemma 4.3 and Lemma 5.8 we deduce that {w λ } λ>t is bounded in H µ (C 0 ). Let {λ n } n∈N be a sequence such that λ n → +∞. Then there exists a subsequence λ n k and a function w ∈ H µ (C 0 ) such that w λn k ⇀ w in H µ (C 0 ). Let us show that w weakly solves the equation ∆ C w = 0 in C 0 . By direct computation one sees that w λ weakly solves the equation H(λ) e −2t f (t+λ, θ, H(λ)w λ (t, θ)), in C 0 , and hence, for any φ ∈ H µ,0 (C 0 ), We estimate the two terms in the right hand side of (46). By (H), Lemma 4.4, and boundedness of {w λ } in H µ (C 0 ), we have as λ → +∞. On the other hand by (F) we have One can show that the first term in the right hand side of (48) tends to zero as λ → +∞ by proceeding as in (47). Let us prove that also the second term in right hand side of (48) tends to zero. Indeed, by Hölder inequality, Lemma 4.5 and boundedness of as λ → +∞ since p < 2 * . Passing to the limit in (46) along the sequence {λ n k } and using (47), (48), (49) we obtain C0 ∇ C w · ∇ C φ dµ = 0 for any φ ∈ H µ,0 (C 0 ) .
This proves that w is harmonic in C 0 . Testing with a function φ ∈ H µ (C 0 ) we also have Moreover, since Γ0 w 2 λ dS = 1, by compactness of the trace map we also have that We claim that w λn k → w strongly in H µ (C t ) for any t > 0. To this aim, we first observe that, by direct computation, the function u λ = T −1 w λ is actually a rescaling of the function u, i.e.
Passing to the limit as k → +∞, proceeding as in (47)-(49) and using (54) and the fact that w λn k → w in L 2 (Γ t ), we obtain ∇ C w λn k → ∇ C w in L 2 (C t ) and in turn, thanks to Lemma 4.3, w λn k → w strongly in H µ (C t ) for all t > 0.
According with (30), it is reasonable to associate to every solution w λ of (45) the Almgren-type frequency function for any t 0 and λ > t, where By direct computation it follows true: for any t > 0 and λ > t.
Indeed if there exists t > 0 such that Ct |∇ C w| 2 dµ + Γt w 2 dS = 0 then by a classical unique continuation property we deduce that w is identically zero in C 0 in contradiction with (51). Moreover a contradiction. Therefore N λn k (t) → N w (t) := Ct |∇ C w| 2 dµ Γt w 2 dS for any t > 0. Combining this with (55) and Lemma 5.8 we deduce that N w (t) = γ for any t > 0.
This means that N w is constant and in particular, for almost every t > 0, by Lemma 5.5 we have The condition Γt w ∂w ∂s dS 2 = Γt ∂w ∂s 2 dS Γt w 2 dS implies that, for almost every t > 0, the functions θ → w(t, θ) and θ → ∂w ∂s (t, θ) are parallel as vectors of L 2 (S N −1 ) and hence there exists a function η depending only on t such that for any t > 0.
Clearly the function η(t) = (w(t, θ)) −1 ∂w ∂t (t, θ) is well defined and continuous for any t > 0 thanks to (59). After integration in (61) we deduce that w admits the representation It is not restrictive assuming that S N −1 ψ 2 dS = 1. Inserting the above representation of w into the equation ∆ C w = 0, it follows that there exist k 0 ∈ N \ {0} and c 1 , c 2 ∈ R such that Since w ∈ H µ (C 0 ) and Γ0 w 2 dS = 1, then necessarily c 1 = 0 and c 2 = 1, so that we may write Finally, inserting (62) into (60), we infer that γ = √ µ k0 , thus completing the proof.
The next lemma provides an upper bound for the function v. Proof. Estimate (64) follows from (63) and (43). In order to prove (63) we proceed by contradiction and assume that there exists a sequence s n → +∞ such that Putting λ n := s n − 1 and dividing both sides of the last inequality by H(λ n ) we infer (65) sup with w λn as in Lemma 6.1. By Lemma 6.1, along a suitable subsequence {λ n k } we have hence contradicting (65).
We now describe the behavior of H(t) as t → +∞. exists and belongs to (0, +∞).
Proof. By Lemma 5.4, Lemma 5.8, and direct computations we obtain is integrable in a neighborhood of infinity. This implies that the right hand side of (67) admits a limit as t → +∞. This proves that the limit in (66) exists; on the other hand by (43) it is necessarily finite. It remains to prove that it is strictly positive.
Let R be such that B R ⊂ Ω and let T := − log R. For any k ∈ N let us denote by ψ k an eigenfunction of −∆ S N −1 corresponding to the eigenvalue µ k and suppose that the set {ψ k } k 1 is an orthonormal basis of L 2 (S N −1 ). For any t T , we define the functions e −2t h(t, θ)v(t, θ) + e −2t f (t, θ, v(t, θ)) ψ k (θ) dS(θ) .
By definition of ϕ k and the Parseval identity we have H(t) = with ψ as in Lemma 6.1. This contradicts the fact that ψ L 2 (S N −1 ) = 1 and that ψ belongs to the space generated by ψ j0 , . . . , ψ j0+m−1 . The proof is thereby complete.