Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $\gamma\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma\neq1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena.


Introduction and statement of the main result
We are interested in the existence of bubble-tower type solutions for the problem: where Ω ⊂ R 2 is a smooth bounded domain, ρ > 0 is a small constant, γ, τ ∈ (0, 1]. Equation (1.1) arises in the statistical mechanics description of two-dimensional equilibrium turbulence, as initiated by Onsager [19]. More precisely, in an unpublished manuscript reproduced in the review article [5], Onsager derived the following equation (see also [25] for a rigorous derivation): in Ω u =0 on ∂Ω, (1.2) where u denotes the stream function of the two-dimensional flow, λ > 0 is a constant related to the inverse temperature, the positively rotating vortices have unit intensity, γ ∈ (0, 1] denotes the intensity of the negatively rotating vortices and τ 1 ∈ [0, 1] determines a priori the ratio of the number of positively rotating vortices to the total number of vortices. In more recent years, a similar equation was derived by Neri [16], under the assumption that the vortex intensities are independent identically distributed random variables with probability measure P, defined on the (normalized) vortex intensity range  There is a vast literature concerning (1.4)-(1.5), see, e.g., [1,10,13,14] and the references therein.
On the other hand, few results are available for (1.1). The existence of concentrating signchanging solutions was recently established in [20] and mountain pass solutions were obtained in [24]. The special case γ = 1/2 was studied in [9] in relation to the Tzitzéica equation in affine geometry.
Our aim in this article is to construct a family of solutions u ρ to problem (1.1) which concentrate as ρ → 0 + with an arbitrarily prescribed number k ∈ N of sign-changing singular bubbles, on the line of [7].
Correspondingly, we define the Sobolev space (1.10) We establish the following result.
Since the appropriate choice of α i 's leads to α i = α j for i = j, we find that the bubble tower approximate solution W ρ is actually the sum of solutions to different singular Liouville problems. Such new blow-up profiles were observed in [6]. Towers of concentrated solutions to different singular Liouville equations were initially introduced in the article [7], where the case γ = 1 is considered, and which is the main motivation to this work. Moreover, in view of (1.9), 0 ∈ Ω is a critical point for the Robin's function. In fact, identity (1.15) is a general property for concentrating solution sequences for (1.1), and if the concentration occurs at a single point, such a point is necessarily a critical point for Robin's function, see, e.g., Remark 8.1 for a proof. For γ = 1 identity (1.15) was derived in [18]. It is natural to conjecture that the blow-up mass values (1.11)-(1.12) are the only admissible values for m + (0), m − (0), in view of the mass quantization results for the case γ = 1 in [11]. In this respect, a mass quantization property for (1.1) was announced in [26]; a partial result in this direction concerning the minimum values for blow-up masses was obtained in [24].
From the physics interpretation point of view, the solutions u ρ as obtained in Theorem 1.1 yield solutions to (1.2) with total mass and vortex distribution parameter They also yield solutions to (1.3) with total mass given by (1.16) and with no restriction on τ 1 . It may be interesting to note that the "total mass" λ is the quantity on the left hand side in the identity (1.15). A proof of these statements is provided in the Appendix. As already mentioned, our approach to prove Theorem 1.1 is strongly inspired by the singular bubble-tower construction in [7], where the case γ = 1 is considered, in the L p -framework introduced in [4], see also [3]. Nevertheless, the case γ = 1 turns out to be significantly more delicate to handle, and it emphasizes specific analytic and geometric properties of the asymmetry parameter. In fact, the dependence of the singularity coefficients α i and of the concentration parameters δ α i i on γ is rather subtle; in particular, unlike the case γ = 1, the α i 's are never monotonically increasing with respect to i and the concentration parameters δ α i i do not depend linearly with respect to i. Consequently, new ingredients are required in several estimates. Finally, it is interesting to observe that the geometrical symmetry condition (1.9) required for Ω, which ensures invertibility of the linearized operator, depends in a relevant way on γ, if γ ∈ (0, 1] ∩ Q. Notation. For any measurable set A ⊂ Ω we denote by χ A the characteristic function of A. We denote by C > 0 a general constant whose value may vary from line to line. When the integration variable is clear from the context, we omit it. For all φ ∈ H 1 0 (Ω) we set φ := ∇φ L 2 (Ω) .
See Section 3 for the precise values of d i , i = 1, 2, . . . , k and for the precise power decay rate of δ i /δ i+1 . Henceforth, we denote w i := w α i δ i , i = 1, 2, . . . k. The most delicate part of the construction will be to show that if α i , δ i are chosen according to the above definitions, then W ρ approximates a genuine solution to (1.1) up to an error which vanishes as a power of ρ, as ρ → 0 + . This fact, combined with the | ln ρ|-estimate for the norm of the linearized operator (see (2.7) below for the precise statement) will enable us to obtain the desired solution as the fixed point of a contraction mapping.
Then, the error term to be estimated is given by: It is convenient to set: One of the main technical issues will be to show that, provided α i , δ i are chosen as above, there exist p > 1, β = β(k, γ, p) > 0 such that The appropriate choice of the parameters α i , δ i is carried out in Section 3, where some properties necessary for the subsequent estimates are also derived. Then, estimate (2.5) is established in Section 4 and Section 5.
We decompose E + : In short, the choice of α i , δ i will ensure the smallness of the error terms E 1 + , E 1 − , which measure the interaction in the j-th annulus A j between the j-th bubble w α j δ j and all other bubbles. Indeed, as in [7], the errors E 1 + , E 1 − are small inside the j-th annulus A j "because" the choice of α j will cancel the interaction of the j-th bubble and all previous (faster concentrating) bubbles, whereas the choice of δ j will cancel the interaction of the j-th bubble w α j δ j and all subsequent (slower concentrating) bubbles. On the other hand, the error terms E 2 + , E 2 − are estimated by some delicate recursive relations for α i , δ i . Estimation of E 3 + , E 3 − follows from the fact that, outside the j-th annulus A j , the j-th bubble w α j δ j is negligible, up to an error which vanishes as a power of ρ.
Once (2.5) is established, we define (2.6) We note that S ρ = E + + γE − , so that estimate (2.5) provides an estimate for S ρ as well. In Section 6 we show that for any p > 1 there exists c > 0 such that At this point, we can show that there exists ρ 0 > 0 such that for all ρ ∈ (0, ρ 0 ) the equation admits a fixed point φ ρ satisfying φ ρ ≤ Rρ β p | ln ρ| for some β p = β p (τ, γ, k) > 0, p > 1 and R > 0. The function u ρ = W ρ + φ ρ is the desired solution to (1.1). The details of the fixed point argument are contained in Section 7.

Definition and properties of the parameters
In this section we define the parameters α j , δ j , j = 1, 2, . . . , k and we establish some properties which will be used in order to estimate the error terms. The justification of the choice of α j , δ j will be provided in Section 4.
We denote by H(x, y) the regular part of G(x, y): Definition of α j , δ j . The appropriate values for α j , δ j are deduced form the following defining conditions.

(3.4)
Moreover, if k = 2, we obtain the following decay rates for the δ j 's: It will be convenient to set Properties of the singularity coefficients α j . In this subsection we determine α j explicitly in terms of j, for j = 1, 2, . . . , k, and we establish the main properties of the α j 's which will be needed in the sequel.
where A k is defined in (3.5).
The following consequence of Proposition 3.1 will be essential in the proof of the invertibility of the linearized operator L ρ defined in (2.6). Indeed, the kernel of L ρ , is determined by the divisibility properties of α j /2, j = 1, 2, . . . , k. We recall that two integers m, n ∈ N are said to be coprime if they do not admit common divisors.
Corollary 3.1. Suppose γ = m/n with m, n ∈ N, m, n coprime, and suppose that α j /2 ∈ N for some j = 1, 2, . . . , k. Then, there exists k j ∈ N ∪ {0} such that Proof. Suppose j is odd. Then, Since m, n are coprime, it follows that j − 1 = k j m for some k j ∈ N ∪ {0}. Consequently, α j 2 = k j m + 1 + nk j = (m + n)k j + 1, and (3.10) is established for odd j. Similarly, suppose j is even. Then, Since m, n are coprime, it follows that j = k j n for some k j ∈ N. Consequently, Formula (3.10) is completely established.
In order to prove Proposition 3.1 we establish some lemmas.
Proof. Suppose j is odd. Then, in view of (3.2) and the fact that j − 1 is even, we have In view of (3.3), we have We conclude that

and (3.11) is established for odd indices j.
Similarly, suppose j is even. Then, in view of (3.3) and the fact that j − 1 is odd, we have Since j − 1 is odd, we have from (3.2) that We deduce that and the recursive formula (3.11) is also established for all even indices j.
We also use the following results, whose proof is elementary.
Now we can provide the proof of Proposition 3.1.
Proof of Proposition 3.1. Proof of (3.6). We argue by induction. We already know from (3.4) that α 1 = 2 and α 2 = 2(1 + 2γ). Suppose (3.6) holds true for all i < j, with j an odd index. Then, in view of (3.11) and the induction assumption we have: and (3.6) is established in this case. Suppose (3.6) holds true for all i < j, with j an even index. Then, in view of (3.11) and the induction assumption we have: In view of Lemma 3.2, for k odd we deduce that: Similarly, for k even we deduce that 1≤j≤k j odd Proof of (3.8). Recall from (3.6) that if j is even, then α j = 2[(1 + γ)j − 1]. In view of Lemma 3.2, for k odd we deduce that: In view of Lemma 3.2, for k odd we deduce that: Similarly, for k even we deduce that 1 γ 1≤j≤k j even Proof of (3.9). The proof of (3.9) follows from (3.7)-(3.8). However, a proof may also be derived independently from (3.2)-(3.3) and (3.6) with j = k. Indeed, suppose k is odd. In view of (3.2) we have Hence, we may write In view of (3.6) with j = k we have Similarly, suppose k is even. In view of (3.3) we have Hence, we may write Now, in view of (3.6) with j = k we conclude that The asserted formula (3.9) follows.
Indeed, the following holds true.
Properties of the concentration parameters δ j . In this subsection we compute the power decay rates of the concentration parameters δ j as ρ → 0 + , j = 1, 2, . . . , k. Let κ j = κ j (γ, τ, h(0), k) > 0, j = 1, 2, . . . , k be defined by if j is even, With the above definitions, we have: The following power decay rates hold true for all j = 1, 2, . . . , k: where r j = r j (γ, k) > 0 is defined by (3.14) In order to prove Proposition 3.2, we first establish a recursive formula.
if j is even, j = 1, 2, . . . , k − 1. (3.17) Proof. Suppose k is odd. Then, formula (3.2) takes the form Recalling the explicit value of A k as in (3.9) and of α k as in (3.2), we conclude the proof.
Similarly, suppose k is even. Then, formula (3.3) takes the form Recalling the explicit value of A k as in (3.9) and of α k as in (3.3), we conclude the proof.
Suppose j is odd, j ≤ k − 1. Using (3.2) and observing that j + 1 is even we have We deduce that Hence, the asserted recursive formula follows for j odd. Similarly, suppose that j is even, j ≤ k − 1. In view of (3.3) and observing that j + 1 is odd, we have We deduce that and finally The asserted recursive formula is now completely established.
Proof of Proposition 3.2. Proof of the first decay rate in (3.13). We equivalently show that We argue by induction. Suppose k is odd. For j = 0 we have δ α k k = κ k ρ = c k ρ and the formula holds true in this case. For j = 1 we have, since k − 1 is even, , and the formula is verified for j = 1 as well.
Hence, assume that the formula is true for all i ≤ j < k with j even. Then, k − j − 1 is even, , and the formula holds true in this case.
Suppose the formula holds true for j odd. The, k − j − 1 is odd and we have: By induction, the formula is established for k odd. Now, assume that k is even. For j = 0 we have δ α k k = κ k ρ = c k ρ, and the statement is verified. For j = 1 we have Hence, suppose the statement holds true for all i ≤ j, j even. The, j + 1 is odd, k − j − 1 is odd. We have: , and the asserted formula follows.
Finally, suppose the statement holds true for all i ≤ j, j odd. Then, k − j − 1 is even. We compute: . Proof of the second decay rate in (3.13). Using (3.13), if k is odd and j is even, we have Recalling the explicit value of α j and the definition od d j , we derive If k is odd and j is odd, we have Consequently, , and the statement follows for k odd.
If k is even and j is odd, we have . Finally, if k is even and j is even, we have . The proof of the third decay rate in (3.13) is an elementary computation; for the reader's convenience we outline it in the Appendix. 4. The "error function" Θ j (estimation of E 1 ± ) In this section we justify the choice of (3.2)-(3.3) for the parameters α i , δ i . We recall that the shrinking annuli A j are defined by where we set δ 0 = 0 and δ k+1 = +∞. With this definition, for every i, j = 1, 2, . . . , k we have and it is readily checked that: runs off to infinity, if i < j invades whole space, if i = j shrinks to the origin, if i > j.
We define the "error functions" Θ j in A j /δ j by setting Then, we may write and consequently The key point is that Θ j is well estimated in the expanding annulus A j /δ j .
Corollary 4.1. We have, for any p ≥ 1: We devote the remaining part of this section to the proof of Proposition 4.1 and of Corollary 4.1. We note that we may write: , if j is even. (4.5) We recall the expansion of P w α δ . Lemma 4.1. For every α ≥ 2, δ > 0 there holds: Proof. The proof is a direct consequence of the maximum principle, see, e.g., [7].
The following estimates are a key ingredient.
Using these facts, together with the definition of α i , δ i , we show the following essential estimate.
Proof. Suppose j is odd. Then, using the projection expansion (4.6) and Lemma 4.3, we have Now, suppose that j is even. We have, from (4.5): The asserted expansion is completely established. Now we can prove Proposition 4.1.
Proof of Proposition 4.1. We recall from Lemma 4.4 that We observe that in view of (3.18) we have The remaining terms are estimated using Lemma 4.2. Hence, (4.2) and (4.4) are established.
Proof of Corollary 4.1. We begin by showing that Claim 1. If j is odd: (4.8) If j is even, (4.9) Suppose j is odd. Then, using (4.1) Similarly, if j is even, we compute, using (4.1): Claim 2. The following decay estimates hold true.
Indeed, in view of (4.2) we have Since the integrals appearing above are uniformly bounded, the asserted decay rates follow. 5. The error terms R ρ and S ρ (estimation of E 2 ± , E 3 ± ) We recall from Section 2 that f (s) := e s − τ e −γs and Our aim in this section is to obtain power decay estimates for E + L p (Ω) and E − L p (Ω) , for p ≥ 1, p − 1 ≪ 1. More precisely, we establish the following Proposition 5.1. There exists p 0 > 1 such that for every p ∈ [1, p 0 ) there exists β p = β p (τ, γ, k) > 0 such that: and consequently R ρ L p (Ω) + S ρ L p (Ω) = O(ρ β p ) By taking p = 1 in (5.1) and using (2.2) we derive from the above: Corollary 5.1. For any r > 0 there holds, as ρ → 0 + :

(5.2)
Moreover, for any q > 1 we have In order to prove Proposition 5.1 we recall from Section 2 that E + = E 1 The errors E 1 + , E 1 − are already estimated in Corollary 4.1. We estimate E 2 + , E 2 − . To this end, we first establish the following auxiliary estimates.
where for the sake of simplicity it is understood that if j = 1 only the first term on the right hand side exists and if j = k only the second term on the right hand side exists.
Proof. We begin by showing the following. Claim 1. If j is odd: If j is even, Proof of Claim 1. Suppose j is odd. We compute: Similarly, suppose j is even. We compute: Claim 1 is thus established. Claim 2. For any η > 0 we have: Proof of Claim 2. We compute: We estimate, for j ≥ 2: Similarly, for j ≤ k − 1, we have . Lemma 5.2. The following power decay rates hold true. If j is odd: if j is even: Proof. Proof of the first decay rate for j odd. Since j + 1 is even, in view of the recursive formula (3.11) we have γα j = α j+1 − 2(1 + γ) and therefore We deduce that , and the asserted estimate follows.
Proof of the second decay rate for j odd. In view of the recursive formula (3.11), we have α j = α j−1 /γ + 2(1 + 1/γ) and therefore It follows that Since j − 1 is even, in view of the recursive formula (3.17) that δ and the asserted estimate follows. Proof of the first estimate for j is even. Since j + 1 is odd, in view of the recursive formula (3.11), we have: α j /γ = α j+1 − 2(1 + 1/γ). Hence, we may write Since j is even, in view of the recursive formula (3.17) we have that δ , as desired.
Proof of the second estimate for j even. Since j is even, in view of the recursive formula (3.11) that α j /γ = α j−1 + 2(1 + 1/γ) and therefore Since j − 1 is odd, in view of (3.17) we have:

24
A. PISTOIA AND T. RICCIARDI Consequently, as desired. The asserted decay estimates are completely established. Lemma 5.3. There holds: , if i < j.
The asserted decay estimates are thus established.
Now we can provide the proof of Proposition 5.1.
Proof of Proposition 5.1. The proof is a direct consequence of Lemma 5.1, Lemma 5.2 and Lemma 5.3.
Proof of Corollary 5.1. We decompose: Using (5.1) with p = 1, we obtain for some β 1 > 0. On the other hand, by a standard rescaling and (2.2), Hence, (5.2) follows. Proof of (5.3). We have: By rescaling we find On the other hand, in view of (5.1) we have E 1 + L q (Ω) = o(1) and E 2 + L q (Ω) = o(1) as ρ → 0 + . Hence, the first estimate in (5.3) is established. The proof of the second estimate in (5.3) is similar.
Our aim in this section is to establish the following result.
Let Ω satisfy the symmetry assumption (1.9). For any p > 1 there exist ρ 0 > 0 and c > 0 such that for any ρ ∈ (0, ρ 0 ) and for any ψ ∈ L p (Ω) there exists a unique φ ∈ W 2,p (Ω) ∩ H γ solution to We observe that L ρ is formally the same operator appearing in [7]. However, it actually depends significantly on the asymmetry parameter γ ∈ (0, 1] via the parameters α i , δ i . Consequently, we can follow the approach in [7] to prove Proposition 6.1, although some intermediate estimates require a modified proof, due to the different dependence of α i , δ i on i, ρ. In particular, since α i does not depend monotonically on i (see Remark 3.2), the proof of Lemma 6.4-(iv) below differs from the proof of the corresponding estimate (4.18) in [7].
For the sale of completeness, in this section we first outline the scheme of the proof of Proposition 6.1, which is analogous to [7]. We then devote the remaining part of this section to prove in detail Lemma 6.4-(iv). 6.1. Outline of the proof of Proposition 6.1. It is convenient to extend the symmetry assumption (1.9) to a possibly unbounded domain D ⊂ R 2 . Let D ⊂ R 2 be a smooth (possibly unbounded) domain. Namely, we define the following geometrical symmetry property for D: Correspondingly, we define a symmetry property for functions φ : D → R: 3) The following lemma clarifies the role of the symmetry assumption (6.3).
Suppose φ is a solution to Proof. It is shown in [7] that φ is necessarily a bounded solution. In turn, it is shown in [2] that any bounded solution to (6.7) is a linear combination of the functions: In view of Corollary 3.1, α 2 is of the form (3.10). In particular, the functions φ 1 , φ 2 do not satisfy (6.3). The claim follows.
For any α ≥ 2 we define the Banach spaces With these definitions, it is shown in [7] that the embedding i α : H α (R 2 ) ֒→ L α (R 2 ) is compact.
The proof of Proposition 6.2 will be outlined below. Once Proposition 6.2 is established, it is simple to prove Proposition 6.1.
In order to prove Proposition 6.2, for i = 1, 2, . . . , k we define the quantities We first show how Lemma 6.3 implies Proposition 6.2. Then, we devote the remaining part of this section to the proof of Lemma 6.3.
Proof of Proposition 6.2. We use the following identities. For i = 1, 2, . . . , k there holds: (6.9) Using the equations for φ n and P w in , we find where w in = w α i δ in . The first term above vanishes as ρ n → 0 + , in view of the form (6.7) of φ i 0 and of the first integral in (6.9). In order to evaluate the second term, we note that similarly as in [7] we find Therefore, Lemma 6.3 and (6.9) yield In turn, we obtain η k = 0 and η i + 2 k j=i+1 η j = 0 for any i = 1, 2, . . . , k − 1.
Proposition 6.2 is thus established.
We are left to prove the asymptotic behavior of the quantities σ i (ρ n ), i = 1, 2, . . . , k, as stated in Lemma 6.3.

6.2.
Proof of Lemma 6.3. Throughout this subsection, for the sake of simplicity, we omit the index n. In order to establish Lemma 6.3 we set and we denote by P Z i its projection onto H 1 0 (Ω). Then, using the equations for φ and P Z i , we find that the sequence φ satisfies the identity Equivalently, we may write The asserted identities (6.8) will then follow from the following facts.
Lemma 6.4. The following expansions hold.
The proof of Lemma 6.4-(i)-(ii)-(iii) is completely analogous to [7]. On the other hand, the proof of Lemma 6.4-(iv) is different, due to the particular dependence on γ of α i , δ i . Therefore, we provide the proof of Lemma 6.4-(iv). The underlying idea is that in order to control the integrals on the expanding domain Ω j n it is convenient to decompose Ω j n = B R j ∪ (Ω j n \ B R j ), with R j suitably defined as follows.
Lemma 6.5. The following properties hold.
Lemma 6.6. The following expansions hold for the function Z i (x) defined in (6.10).
(i) For any x ∈ Ω there holds where R j is defined in (6.12).
Proof. The proof readily follows from the definition of Z i in (6.10) and Lemma 6.5-(ii).
Estimate (6.11) in Lemma 6.4-(iv) is thus completely established. In turn, the proof of (6.8) follows. Hence, the proof of Proposition 6.1 is complete. In this section we conclude the proof of Theorem 1.1 by obtaining a solution u ρ to problem (1.1) in the form u ρ = W ρ +φ ρ , with φ ρ the fixed point of a contraction mapping. Indeed, we establish the following existence result.
Proposition 7.1. For p > 1 sufficiently close to 1 there exist ρ 0 > 0 and R > 0 such that for any ρ ∈ (0, ρ 0 ) there exists a unique solution φ ρ ∈ H γ to the problem Here H γ is the space of γ-symmetric Sobolev functions defined in (1.10) andβ p > 0 is the exponent obtained in Proposition 5.1.
We equivalently seek a fixed point φ ∈ H γ for the operator T ρ : H γ → H γ defined by where R ρ , S ρ , N ρ are the operators defined in (2.4)-(2.6).
In the sequel we shall use the Moser-Trudinger inequality [15,27] in the following form.
We readily check that Consequently, Using the Mean Value Theorem, we have |e a − e b − a + b| ≤ e |a|+|b| |a − b|(|a| + |b|), for all a, b ∈ R. Taking a = φ 1 , b = φ 2 we derive Setting we estimate: By Hölder's inequality with r −1 + s −1 + t −1 = 1 we obtain In view of (5.3) we have Ω (ρ pr e prWρ ) dx where s k > 0 is defined in (3.15). Now, the Moser-Trudinger inequality as in Lemma 7.1 yields We conclude that Let I 2 be defined by Hence, by analogous estimates as above, we conclude the proof of the desired estimates. Now we can prove the main result of this section.
Claim 2. T ρ is a contraction in B ρ,R .
Finally, we are able to provide the proof of our main result.

Appendix
We collect in this Appendix the proof of some complementary results stated in Section 1, as well as some proofs.
For the sake of completeness, we check the following fact which was stated in Section 1.
On the other hand, Hence, (8.1) is verified for k odd. Suppose k is even.
On the other hand, and (8.1) is verified for k even, as well.