On the Blow-Up of Solutions to Liouville-Type Equations

Tonia Ricciardi; Gabriella Zecca

doi:10.1515/ans-2015-5015

Article Publicly Available

On the Blow-Up of Solutions to Liouville-Type Equations

Tonia Ricciardi and Gabriella Zecca

Published/Copyright: December 8, 2015

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 16 Issue 1

Abstract

We estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points.

Keywords: Liouville Equation; Peak Solutions; Mean Field Equation

MSC 2010: 35J20; 35J60

1 Introduction

In the article [6], Nagasaki and Suzuki considered the Liouville-type problem

(1.1) { - Δ ⁢ u = ρ ⁢ f ⁢ ( u ) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

where Ω⊂ℝ2 is a smooth bounded domain, ρ>0, and f:ℝ→ℝ is a smooth function such that

(1.2) f ⁢ ( t ) = e t + φ ⁢ ( t ) with ⁢ φ ⁢ ( t ) = o ⁢ ( e t ) ⁢ as ⁢ t → + ∞ .

Equations of the form (1.1) are of actual interest in several contexts, including turbulent Euler flows, chemotaxis, and the Nirenberg problem in geometry; see, e.g., [5] and the references therein. A recent example is given by the mean field equation

(1.3) { - Δ ⁢ u = λ ⁢ ∫ [ - 1 , 1 ] α ⁢ e α ⁢ u ⁢ 𝒫 ⁢ ( d ⁢ α ) ∬ [ - 1 , 1 ] × Ω e α ⁢ u ⁢ 𝒫 ⁢ ( d ⁢ α ) ⁢ 𝑑 x in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

which was derived in [7] for turbulent flows with variable intensities, where 𝒫∈ℳ⁢([-1,1]) is a probability measure related to the vortex intensity distribution. In this case, setting

f ⁢ ( t ) = ∫ [ - 1 , 1 ] α ⁢ e α ⁢ t ⁢ 𝒫 ⁢ ( d ⁢ α ) , ρ = λ ⁢ ( ∬ [ - 1 , 1 ] × Ω e α ⁢ u ⁢ 𝒫 ⁢ ( d ⁢ α ) ⁢ 𝑑 x ) - 1 ,

it is readily seen that if 𝒫⁢({1})>0, then along a blow-up sequence, (1.3) is of the form (1.1). See [10, 11, 13, 12] for more details, where the existence of solutions by variational arguments and blow-up analysis are also considered. Blow-up solution sequences for (1.3) have also been recently constructed in [9] following the approach introduced in [4].

In [6], Nagasaki and Suzuki derived a concentration-compactness principle for (1.1), mass quantization, and the location of blow-up points, under some additional technical assumptions for f. More precisely, they assumed that

(1.4) | φ ⁢ ( t ) - φ ′ ⁢ ( t ) | ≤ 𝒢 ⁢ ( t ) for some ⁢ 𝒢 ∈ C 1 ⁢ ( ℝ , ℝ ) ⁢ satisfying ⁢ 𝒢 ⁢ ( t ) + | 𝒢 ′ ⁢ ( t ) | ≤ C ⁢ e γ ⁢ t ⁢ with ⁢ γ < 1 4

and

(1.5) f ⁢ ( t ) ≥ 0 for all ⁢ t ≥ 0 .

By a complex analysis approach, they established the following result.

Theorem 1.1

Theorem 1.1 ([6])

Let f satisfy assumptions (1.2), (1.4), and (1.5). Let un be a solution sequence to (1.1) with ρ=ρn→0. Suppose un converges to some nontrivial function u0. Then,

u 0 ⁢ ( x ) = 8 ⁢ π ⁢ ∑ j = 1 m G Ω ⁢ ( x , p j )

for some p1,…,pm∈Ω, m∈ℕ, where GΩ denotes the Green’s function for the Dirichlet problem on Ω. Furthermore, at each blow-up point pj, j=1,…,m, there holds that

∇ ⁡ [ G Ω ⁢ ( x , p j ) + 1 2 ⁢ π ⁢ log ⁡ | x - p j | ] | x = p j + ∇ ⁡ [ ∑ i ≠ j G Ω ⁢ ( p j , p i ) ] = 0 .

The original estimates in [6] are involved and require the technical assumption γ∈(0,14). It should be mentioned that this assumption was later weakened to the natural assumption γ∈(0,1) in [14], by taking a different viewpoint on the line of [1].

Here, we are interested in revisiting the complex analysis framework introduced in [6]. In particular, we study the effect of the lower-order terms which naturally appear when the equation is considered on a compact Riemannian 2-manifold. We observe that, although the very elaborate key L∞-estimate obtained in [6], namely, Proposition 1.2 below, may be extended in a straightforward manner to the case of manifolds (see Appendix A for the details), the lower-order terms are naturally estimated only in L1. Therefore, we are led to consider an L1-framework, which turns out to be significantly simpler and which holds under the weaker assumption γ∈(0,12). As a byproduct, we obtain a quick proof of mass quantization and blow-up point location for the case γ∈(0,12).

In order to state our results, for a function u∈C2⁢(Ω), we define the quantity

(1.6) S ⁢ ( u ) = u z 2 2 - u z ⁢ z ,

where

∂ z = ∂ x - i ⁢ ∂ y 2 , ∂ z ¯ = ∂ x + i ⁢ ∂ y 2 .

Then, if u is a solution to (1.1), we have

∂ z ¯ ⁡ [ S ⁢ ( u ) ] = - ρ 4 ⁢ u z ⁢ [ f ⁢ ( u ) - f ′ ⁢ ( u ) ] = ρ 4 ⁢ u z ⁢ [ φ ⁢ ( u ) - φ ′ ⁢ ( u ) ] .

In particular, in the Liouville case f⁢(u)=eu, the function S⁢(u) is holomorphic. Therefore, the complex derivative ∂z¯⁡[S⁢(u)] may be viewed as an estimate of the “distance” between the equation in (1.1) and the standard Liouville equation.

We recall that the main technical estimate in [6] is given by the following proposition.

Proposition 1.2

Proposition 1.2 ([6])

Let uρ be a blow-up sequence for (1.1). Assume (1.2), (1.4), and (1.5). Then,

∥ ∂ z ¯ ⁡ S ⁢ ( u ) ∥ L ∞ ⁢ ( Ω ) = ρ 4 ⁢ ∥ ∇ ⁡ u ρ ⁢ ( f ′ ⁢ ( u ρ ) - f ⁢ ( u ρ ) ) ∥ L ∞ ⁢ ( Ω ) → 0 .

It is natural to expect that corresponding results should hold on a compact Riemannian 2-manifold (M,g) without boundary. We show that, in fact, the L∞-convergence as stated in Proposition 1.2 still holds true on M (see Proposition A.1 in Appendix A). However, a modified point of view is needed in order to suitably locally define a function S corresponding to (1.6), such that the lower-order terms may be controlled, as well as to prove its convergence to a holomorphic function in some suitable norm, so that the mass quantization and the location of the blow-up points may be derived. As we shall see, our point of view holds under the weaker assumption γ∈(0,12) and is significantly simpler than the original L∞-framework.

More precisely, on a compact Riemannian 2-manifold without boundary (M,g), we consider the problem

(1.7) { - Δ g ⁢ u = ρ ⁢ f ⁢ ( u ) - c ρ in ⁢ M , ∫ M u ⁢ 𝑑 x = 0 ,

where cρ=ρ⁢|M|-1⁢∫Mf⁢(u)⁢𝑑x∈ℝ, dx denotes the volume element on M, and Δg denotes the Laplace–Beltrami operator. We assume that f⁢(t)=et+φ⁢(t) satisfies (1.2) and, moreover, that

(1.8) | φ ⁢ ( t ) - φ ′ ⁢ ( t ) | ≤ 𝒢 ⁢ ( t ) for some ⁢ 𝒢 ∈ C 1 ⁢ ( ℝ , ℝ ) ⁢ satisfying ⁢ 𝒢 ⁢ ( t ) + | 𝒢 ′ ⁢ ( t ) | ≤ C ⁢ e γ ⁢ t ⁢ with ⁢ γ < 1 2

and

(1.9) f ⁢ ( t ) ≥ - C for all ⁢ t ≥ 0 .

In the spirit of [3], we assume that along a blow-up sequence we have

(1.10) ρ ⁢ ∫ M f ⁢ ( u ) ⁢ 𝑑 x ≤ C .

In particular, without loss of generality, we may assume that

(1.11) c ρ → c 0 as ⁢ ρ → 0 + .

We note that (1.9) implies that u≥-C. We now define the modified quantity corresponding to S⁢(u). Let 𝒮={p1,…,pm} denote the blow-up set. Let p∈𝒮 and denote X=(x1,x2). We consider a local isothermal chart (Ψ,𝒰) such that ℬε⁢(p)⊂𝒰, Ψ⁢(p)=0, ℬε⁢(p)∩𝒮=∅, g⁢(X)=eξ⁢(X)⁢(d⁢x12+d⁢x22), and ξ⁢(0)=0. For the sake of simplicity, we identify here functions on M with their pullback functions to B=B⁢(0,r)=Ψ⁢(ℬε⁢(p)). We denote by GB⁢(X,Y) the Green’s function of ΔX=∂x12+∂x22 on B. We set

(1.12) K ⁢ ( X ) = - ∫ B G B ⁢ ( X , Y ) ⁢ e ξ ⁢ ( Y ) ⁢ 𝑑 Y + c 1 ⁢ z

with c1∈ℂ defined by

(1.13) ∂ z ⁡ [ ξ ⁢ ( z , z ¯ ) + c 0 ⁢ K ⁢ ( z , z ¯ ) ] | z = 0 = 0 ,

where c0 is defined in (1.11). Let u denote a solution sequence to (1.7). We define w⁢(z)=u-cρ⁢K, so that -Δ⁢w=eξ⁢ρ⁢f⁢(u) in B. Finally, consider S⁢(w), where S is defined in (1.6). Our main estimate is given in the following theorem.

Theorem 1.3

Assume that f⁢(t)=et+φ⁢(t) satisfies (1.2), (1.8), and (1.9). Let uρ be a blow-up solution sequence for (1.7). Then,

for every 1≤s<(γ+12)-1,

ρ ⁢ ∥ ∇ ⁡ u ρ ⁢ ( f ′ ⁢ ( u ρ ) - f ⁢ ( u ρ ) ) ∥ L s ⁢ ( M ) → 0 as ⁢ ρ → 0 + ;
for every blow-up point p∈𝒮, the function S⁢(w)→S0 in L1⁢(B) as ρ→0+, where S0 is holomorphic in B.

Consequently, we derive the following corollary.

Corollary 1.4

Assume that f⁢(t)=et+φ⁢(t) satisfies (1.2), (1.8), and (1.9). Suppose un converges to some nontrivial function u0. Then,

(1.14) u 0 ⁢ ( x ) = 8 ⁢ π ⁢ ∑ j = 1 m G M ⁢ ( x , p j ) .

Moreover, for all p∈𝒮, we have the relation

(1.15) [ ∇ X ⁡ ( ∑ q ∈ 𝒮 ∖ { p } G M ⁢ ( Ψ - 1 ⁢ ( X ) , q ) + G M ⁢ ( Ψ - 1 ⁢ ( X ) , p ) + 1 2 ⁢ π ⁢ log ⁡ | X | + 1 8 ⁢ π ⁢ ξ ⁢ ( X ) ) ] | X = 0 = 0 .

We provide the proofs of Theorem 1.3 and Corollary 1.4 in Section 2. For the sake of completeness and in order to readily allow a comparison with the L∞-framework employed in [6], in Appendix A we extend Proposition 1.2 to the case of Riemannian 2-manifolds without boundary.

Throughout this note, we denote by C>0 a constant whose actual value may vary from line to line.

2 Proof of Theorem 1.3

We begin by establishing the following result.

Lemma 2.1

Let u be a solution to (1.7). For every r>0, we have

(2.1) r ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ 𝑑 x ≤ C ,

where C=C⁢(r,M,φ,c0).

Proof.

We multiply the equation -Δg⁢u=ρ⁢f⁢(u)-cρ by e-r⁢u. Integrating, we have

r ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ 𝑑 x = ∫ M e - r ⁢ u ⁢ Δ g ⁢ u ⁢ 𝑑 x

= - ρ ⁢ ∫ M e - r ⁢ u ⁢ f ⁢ ( u ) ⁢ 𝑑 x + c ρ ⁢ ∫ M e - r ⁢ u ⁢ 𝑑 x

≤ ρ ⁢ ∫ M e - r ⁢ u ⁢ | φ ⁢ ( u ) | ⁢ 𝑑 x + c ρ ⁢ ∫ M e r ⁢ C ⁢ 𝑑 x

≤ ρ ⁢ ∫ M e - r ⁢ u ⁢ | φ ⁢ ( u ) | ⁢ 𝑑 x + c ρ ⁢ e r ⁢ C ⁢ | M | ,

since u≥-C. Using the assumptions on φ, there exists t0>0 such that |g⁢(u)|<eu for u>t0, so that

r ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ 𝑑 x ≤ C + ρ ⁢ ( ∫ { u > t 0 } e ( 1 - r ) ⁢ u ⁢ 𝑑 x + ∫ { u ≤ t 0 } e - r ⁢ u ⁢ | φ ⁢ ( u ) | ⁢ 𝑑 x ) ≤ C + ρ ⁢ ( ∫ M e u ⁢ 𝑑 x + ∫ { u ≤ t 0 } e - r ⁢ u ⁢ | φ ⁢ ( u ) | ⁢ 𝑑 x ) ,

and the claim follows using again the fact that u≥-C. ∎

The following proposition proves Theorem 1.3 (i).

Proposition 2.2

Let u be a solution to (1.7). Then, for every 1≤s<(γ+12)-1 and for every ε>0, we have

∥ ∇ ⁡ u ⁢ ( f ′ ⁢ ( u ) - f ⁢ ( u ) ) ∥ L s ⁢ ( M ) ≤ C ⁢ ρ - γ - ε

for 0<ρ<1.

Proof.

In view of (1.8), we have

0 ≤ | f ⁢ ( u ) - f ′ ⁢ ( u ) | ≤ C ⁢ e γ ⁢ u .

Hence,

(2.2) ∥ ( f ⁢ ( u ) - f ′ ⁢ ( u ) ) ⁢ ∇ ⁡ u ∥ L s ≤ C ⁢ ∥ e γ ⁢ u ⁢ ∇ ⁡ u ∥ L s .

Moreover, (1.10) implies that

∫ M e u ⁢ 𝑑 x ≤ c ⁢ ρ - 1 .

Then, for every 1≤q<γ-1, using Hölder’s inequality we have

(2.3) ∥ e γ ⁢ u ∥ L q ⁢ ( M ) ≤ C ⁢ | M | 1 q - γ ⁢ ρ - γ .

Let 0<r<1-s⁢(γ+12). By Lemma 2.1, for

q = s + r γ 1 - s 2 < 1 γ ,

using Hölder’s inequality again, we have

(2.4) ∥ e γ ⁢ u ⁢ ∇ ⁡ u ∥ L s ⁢ ( M ) s = ∫ M e ( s ⁢ γ + r ) ⁢ u ⁢ ( e - r ⁢ u ⁢ | ∇ ⁡ u | s ) ⁢ 𝑑 x ≤ ( ∫ M e γ ⁢ u ⁢ q ⁢ 𝑑 x ) 1 - s 2 ⁢ ( ∫ M e - 2 ⁢ r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ 𝑑 x ) s 2 ≤ C ⁢ ∥ e γ ⁢ u ∥ L q ⁢ ( M ) s + r γ .

Then, by (2.3) and (2.4) we have

(2.5) ∥ e γ ⁢ u ⁢ ∇ ⁡ u ∥ L s ⁢ ( M ) ≤ C ⁢ ρ - γ - r s .

Combining (2.2) and (2.5), the claim is proved. ∎

Let p∈𝒮. We denote by (Ψ,𝒰) an isothermal chart satisfying

𝒰 ¯ ∩ 𝒮 = { p } , Ψ ⁢ ( 𝒰 ) = 𝒪 ⊂ ℝ 2 , Ψ ⁢ ( p ) = 0 , g ⁢ ( X ) = e ξ ⁢ ( X ) ⁢ ( d ⁢ x 1 2 + d ⁢ x 2 2 ) , ξ ⁢ ( 0 ) = 0 ,

where X=(x1,x2) denotes a coordinate system on 𝒪. We consider ε>0 sufficiently small so that ℬ⁢(p,ε)⋐𝒰 and let B=B⁢(0,r)=Ψ⁢(ℬ⁢(p,ε)). The Laplace–Beltrami operator Δg is then mapped to the operator e-ξ⁢(X)⁢ΔX on 𝒪, where ΔX=∂x122+∂x222. By GB⁢(X,Y) we denote the Green’s function of ΔX on B, namely,

{ - Δ X ⁢ G B ⁢ ( X , Y ) = δ Y in ⁢ B , G B ⁢ ( X , Y ) = 0 on ⁢ ∂ ⁡ B .

We recall from (1.12) that

K ⁢ ( X ) = - ∫ B G B ⁢ ( X , Y ) ⁢ e ξ ⁢ ( Y ) ⁢ 𝑑 Y + c 1 ⁢ z

with c1 the constant defined by (1.13), namely,

∂ z ⁡ [ ξ ⁢ ( z , z ¯ ) + c 0 ⁢ K ⁢ ( z , z ¯ ) ] | z = 0 = 0 ,

where c0=limρ→0⁡cρ. Then, K∈C∞⁢(B) and

Δ X ⁢ K = e ξ in ⁢ B ¯ .

Let uρ be a blow-up solution sequence for (1.7). As ρ→0, u→u0 in Cloc∞⁢(M∖𝒮), u-u0∈W1,q⁢(M) for 1≤q<2, and f⁢(u)→f⁢(u0) in Cloc∞⁢(M∖𝒮), we have Δg⁢u→Δg⁢u0 in Cloc∞⁢(M∖𝒮), so that

Δ g ⁢ u 0 = c 0 in ⁢ M ∖ 𝒮 .

We consider the following functions defined in B:

(2.6)

u ~ = u ∘ Ψ - 1 , u ~ 0 = u 0 ∘ Ψ - 1 ,

w ⁢ ( z ) = u ~ - c ρ ⁢ K , w 0 ⁢ ( z ) = u ~ 0 - c 0 ⁢ K ,

S ⁢ ( w ) = w z ⁢ z - 1 2 ⁢ w z 2 , S 0 = w 0 ⁢ z ⁢ z - 1 2 ⁢ w 0 ⁢ z 2 .

The following proposition proves Theorem 1.3 (ii).

Proposition 2.3

The complex function S0 defined in (2.6) is holomorphic in B and S→S0 in L1⁢(B).

Proof.

By (2.6) we have

- Δ X ⁢ w = ρ ⁢ f ⁢ ( u ~ ) ⁢ e ξ and w z = u ~ z - c ρ ⁢ K z .

Then, using ΔX=4⁢∂z⁢z¯ we compute

∂ z ¯ ⁡ [ S ⁢ ( w ) ] = 1 4 ⁢ ( ∂ z ⁡ Δ X ⁢ w - w z ⁢ Δ X ⁢ w )

= - ρ 4 ⁢ e ξ ⁢ ( f ⁢ ( u ~ ) ⁢ ξ z + u ~ z ⁢ f ′ ⁢ ( u ~ ) ) + ρ 4 ⁢ e ξ ⁢ f ⁢ ( u ~ ) ⁢ ( u ~ z - c ρ ⁢ K z )

(2.7) = ρ 4 ⁢ e ξ ⁢ ( f ⁢ ( u ~ ) - f ′ ⁢ ( u ~ ) ) ⁢ u ~ z - ρ 4 ⁢ e ξ ⁢ f ⁢ ( u ~ ) ⁢ ( ξ z + c 0 ⁢ K z ) + ( c 0 - c ρ ) ⁢ ρ 4 ⁢ e ξ ⁢ f ⁢ ( u ~ ) ⁢ K z .

Using (2) we derive that

(2.8) ∂ z ¯ ⁡ S → 0 in ⁢ L 1 ⁢ ( B ) .

Indeed, this follows by Proposition 2.2, (1.13), and by the fact that |ρ⁢f⁢(u~)|→*a⁢δ0⁢(d⁢x) for some a>0. On the other hand, by (2.6), since u→u0 in Cloc∞⁢(M∖𝒮), we have

w → w 0 in ⁢ C loc ∞ ⁢ ( B ¯ ∖ { 0 } )

and then

(2.9) S → S 0 in ⁢ C loc ∞ ⁢ ( B ¯ ∖ { 0 } ) .

At this point, we set Ξ=(ξ1,ξ2) and ζ=ξ1+i⁢ξ2 and we observe that by the Cauchy integral formula we may write

(2.10) [ S ⁢ ( w ) ] ⁢ ( ζ ) = 1 π ⁢ ∫ B ∂ z ¯ ⁡ S ⁢ ( z ) ζ - z ⁢ 𝑑 X + i 2 ⁢ π ⁢ ∫ + ∂ ⁡ B [ S ⁢ ( w ) ] ⁢ ( z ) ζ - z ⁢ 𝑑 z = g ⁢ ( ζ ) + h ⁢ ( ζ ) .

We have

(2.11) h ⁢ ( ζ ) → h 0 ⁢ ( ζ ) = i 2 ⁢ π ⁢ ∫ + ∂ ⁡ B S 0 ⁢ ( z ) ζ - z ⁢ 𝑑 z in ⁢ C loc 0 ⁢ ( B )

and h0 is holomorphic in B. On the other hand, we have

(2.12) g → 0 in ⁢ L 1 ⁢ ( B ) .

To prove (2.12), it is sufficient to observe that for every z∈B=B⁢(0,r), we have B⊂B⁢(z,2⁢r) and then

∥ g ∥ L 1 ⁢ ( B ) ≤ ∬ B × B | ∂ z ¯ ⁡ S ⁢ ( z ) | ⁢ 1 | ζ - z | ⁢ 𝑑 X ⁢ 𝑑 Ξ ≤ ∫ B | ∂ z ¯ ⁡ S ⁢ ( z ) | ⁢ ( ∫ B ⁢ ( z , 2 ⁢ r ) 1 | ζ - z | ⁢ 𝑑 Ξ ) ⁢ 𝑑 X = 4 ⁢ π ⁢ r ⁢ ∫ B | ∂ z ¯ ⁡ S ⁢ ( z ) | ⁢ 𝑑 X ,

which tends to zero by (2.8). Combining (2.10), (2.11), and (2.12), we have

S → h 0 in ⁢ L 1 ⁢ ( B ) ⁢ as ⁢ ρ → 0 ,

and hence, up to subsequences,

S → h 0 a.e. in ⁢ B ⁢ as ⁢ ρ → 0 ,

so that by (2.9),

S 0 ⁢ ( ζ ) = h 0 ⁢ ( ζ ) for all ⁢ ζ ∈ B ∖ { 0 } .

This completes our proof. ∎

Finally, we use the following result from [2].

Proposition 2.4

Proposition 2.4 ([2])

For B=B⁢(0,1)⊂ℝn, n≥2, the conditions v∈W1,p⁢(B), 1<p<∞, and Δ⁢v=0 in B∖{0} imply that H=v-ℓ⁢E is harmonic in B, where ℓ is some constant and

E ⁢ ( x ) = { | x | 2 - n if ⁢ n > 2 , log ⁡ | x | if ⁢ n = 2 .

Now, we are ready to prove Corollary 1.4. By GM we denote the Green’s function on the manifold M, defined by

{ - Δ g ⁢ G M ⁢ ( x , y ) = δ y - 1 | M | ∫ M G M ⁢ ( x , y ) ⁢ 𝑑 x = 0 .

Proof of Corollary 1.4.

Assume that p∈𝒮. Let us start by observing that w0 in (2.6) is harmonic in B∖{0} by definition and that w0∈W1,q⁢(B) for all 1<q<2. Hence, also by using Proposition 2.4, we have

w 0 ⁢ ( z ) = ℓ ⁢ log ⁡ 1 | z | + H ⁢ ( z ) ,

where H is harmonic in B and ℓ≠0. Then, using the fact that

∂ z ⁡ log ⁡ | z | = 1 2 ⁢ ∂ z ⁡ log ⁡ ( z ⁢ z ¯ ) = ( 2 ⁢ z ) - 1 ,

we compute

w 0 ⁢ z = - ℓ 2 ⁢ z + H z , w 0 ⁢ z ⁢ z = ℓ 2 ⁢ z 2 + H z ⁢ z .

Therefore,

S 0 = w 0 ⁢ z ⁢ z - 1 2 ⁢ w 0 ⁢ z 2 = ℓ 2 ⁢ z 2 + H z ⁢ z - 1 2 ⁢ ( ℓ 2 ⁢ z - H z ) 2 = ℓ ⁢ ( 4 - ℓ ) 8 ⁢ z 2 + ℓ 2 ⁢ z ⁢ H z + H z ⁢ z - 1 2 ⁢ H z 2 .

By Proposition 2.3, we know that S0 is holomorphic. Hence, we can conclude that ℓ=4 and Hz⁢(0)=0. Since

(2.13) H = w 0 - 4 ⁢ log ⁡ 1 | z |

is harmonic in B, we have

Δ X ⁢ ( u ~ 0 - 4 ⁢ log ⁡ 1 | z | ) = c 0 ⁢ e ξ in ⁢ B ⁢ ( 0 , r )

and, therefore,

Δ g ⁢ ( u 0 ⁢ ( x ) - 8 ⁢ π ⁢ G M ⁢ ( x , p ) ) = c 0 - 8 ⁢ π | M | + h p in ⁢ ℬ ⁢ ( p , ε )

for some harmonic function hp. Arguing similarly for each p∈𝒮={p1,p2,…,pm}, we conclude that

Δ g ⁢ ( u 0 ⁢ ( x ) - 8 ⁢ π ⁢ ∑ j = 1 m G M ⁢ ( x , p j ) ) = c 0 - 8 ⁢ π ⁢ m | M | in ⁢ M .

In particular, we obtain

u 0 ⁢ ( x ) - 8 ⁢ π ⁢ ∑ j = 1 m G M ⁢ ( x , p j ) = constant in ⁢ M .

Observing that ∫Mu0=0, this completes the proof of (1.14). To obtain (1.15) it is sufficient to observe that, in view of (2.13) and (1.13), we have

0 = 1 8 ⁢ π ⁢ ∂ z ⁡ H ⁢ ( X ) | X = 0

= ∂ z ⁡ [ ∑ q ∈ 𝒮 G M ⁢ ( Ψ - 1 ⁢ ( X ) , q ) + 1 2 ⁢ π ⁢ log ⁡ | X | ] | X = 0 - [ m | M | ⁢ ∂ z ⁡ K ⁢ ( X ) ] | X = 0

= ∂ z ⁡ [ ∑ q ∈ 𝒮 G M ⁢ ( Ψ - 1 ⁢ ( X ) , q ) + 1 2 ⁢ π ⁢ log ⁡ | X | - 1 8 ⁢ π ⁢ ξ ⁢ ( X ) ] | X = 0 .

Now, Corollary 1.4 is completely established. ∎

Funding statement: The first author acknowledges the support of FP7-MC-2009-IRSES-247486 “MaNEqui”.

A The L∞-estimate on M

In this appendix, for the sake of completeness and in order to outline the original arguments in [6], so that the simplification of our L1-approach may be seen, we check that Proposition 1.2 may be actually extended to (1.7) on a compact Riemannian 2-manifold (M,g) without boundary with minor modifications. We consider a solution sequence for (1.7). We assume that f satisfies (1.2), (1.4), and (1.5). Moreover, we assume (1.10), so that cρ→c0 as ρ→0+. We show the following proposition.

Proposition A.1

Let u be a solution to (1.7). Then,

ρ ⁢ ∥ ∇ ⁡ u ⁢ ( f ′ ⁢ ( u ) - f ⁢ ( u ) ) ∥ L ∞ ⁢ ( M ) → 0 as ⁢ ρ → 0 .

The proof relies on the following relation, due to Obata.

Lemma A.2

Lemma A.2 ([8])

Let w=w⁢(x)>0 be a solution to

(A.1) Δ ⁢ w = | ∇ ⁡ w | 2 w + F ⁢ ( w ) on ⁢ M ,

where F is a C1-function. Then, there holds the identity

(A.2) div ⁡ V = J + 1 2 ⁢ | ∇ ⁡ w | 2 ⁢ w - 2 ⁢ ( F ⁢ ( w ) + w ⁢ F ′ ⁢ ( w ) ) ,

where, in local coordinates,

V j = w - 1 ⁢ { ∇ ⁡ ( ∂ ⁡ w ∂ ⁡ x i ) ⋅ ∇ ⁡ w - 1 2 ⁢ ∂ ⁡ w ∂ ⁡ x i ⁢ Δ ⁢ w } , j = 1 , 2 ,

and

J = w - 1 ⁢ { ∑ i , j = 1 2 ( ∂ 2 ⁡ w ∂ ⁡ x i ⁢ ∂ ⁡ x j ) 2 - 1 2 ⁢ ( Δ ⁢ w ) 2 } ≥ 0 .

Lemma A.3

Let u be a solution to (1.7). Then, for every r>0, there holds

(A.3) ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ ( 2 ⁢ r ⁢ f ⁢ ( u ) - f ′ ⁢ ( u ) ) ≤ 2 ⁢ r ⁢ c ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 .

Proof.

Let u be a solution to (1.7). Denoting w=e-r⁢u, it is easy to see that Obata’s assumption (A.1) is satisfied by the function w with

F ⁢ ( w ) = r ⁢ e - r ⁢ u ⁢ ( ρ ⁢ f ⁢ ( u ) - c ρ ) .

On the other hand, we have

F ⁢ ( w ) + w ⁢ F ′ ⁢ ( w ) = ρ ⁢ e - r ⁢ u ⁢ ( 2 ⁢ r ⁢ f ⁢ ( u ) - f ′ ⁢ ( u ) ) - 2 ⁢ r ⁢ e - r ⁢ u ⁢ c ρ .

In view of Obata’s identity (A.2), we conclude that

∫ M | ∇ ⁡ w | 2 w 2 ⁢ ( F ⁢ ( w ) + w ⁢ F ⁢ ( w ) ) ≤ 2 ⁢ ∫ M div ⁡ V = 0 .

In particular, since

∇ ⁡ w w = - r ⁢ ∇ ⁡ u ,

by the last inequality we obtain

∫ M r 2 ⁢ | ∇ ⁡ u | 2 ⁢ ( F ⁢ ( w ) + w ⁢ F ′ ⁢ ( w ) ) = r 2 ⁢ ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ ( 2 ⁢ r ⁢ f ⁢ ( u ) - f ′ ⁢ ( u ) ) - 2 ⁢ r 3 ⁢ c ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ≤ 0 . ∎

We note that combining (A.3) and (2.1), for 12<r<1, we obtain

(A.4) ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ f ⁢ ( u ) ≤ C ⁢ ( 1 + ρ ⁢ ∫ M e - ( r - γ ) ⁢ u ⁢ | ∇ ⁡ u | 2 ) .

Since γ<14, combining (2.1) and (A.4) we obtain

ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ f ⁢ ( u ) ⁢ 𝑑 x ≤ C

and then, since u≥-C, using (2.1) again we have

(A.5) ρ ⁢ ∫ M e - r ⁢ u ⁢ | ∇ ⁡ u | 2 ⁢ | f ⁢ ( u ) | ⁢ 𝑑 x ≤ C if ⁢ 1 2 < r < 1 .

For r>0, we define

G r ⁢ ( t ) = ∫ 0 t e - r 2 ⁢ s ⁢ | f ⁢ ( s ) | ⁢ 𝑑 s .

Then, (A.5) may be written in the form

(A.6) ∥ ∇ ⁡ G r ⁢ ( u ) ∥ L 2 ⁢ ( M ) ≤ C ρ .

Lemma A.4

There holds

(A.7) ∥ G r ⁢ ( u ) ∥ L 1 ⁢ ( M ) ≤ C ρ .

Proof.

The proof can be easily obtained as in Lemma 2.1. Let us observe that in our assumption, for every 12<r<1, we have

(A.8) ∫ { x ∈ M : u ⁢ ( x ) ≥ 0 } G r ⁢ ( u ) ⁢ 𝑑 x ≤ 2 r ⁢ ∫ { u ≥ 0 } | f ⁢ ( u ) | ⁢ 𝑑 x ≤ C ⁢ ( ∫ M | f ⁢ ( u ) | ⁢ 𝑑 x ) 1 2 ≤ C ρ .

On the other hand, since -u≤C, we have

(A.9) ∫ { x ∈ M : u ⁢ ( x ) ≤ 0 } | G r ⁢ ( u ) | ⁢ 𝑑 x ≤ C ⁢ ∫ { u ≤ 0 } 𝑑 x ⁢ ∫ u 0 e C ⁢ r 2 ≤ C ⁢ e C ⁢ r 2 ⁢ | M | ≤ C .

Combining (A.8) and (A.9), we conclude the proof of (A.7). ∎

Reducing (A.6) to

∥ ∇ ⁡ G r ⁢ ( u ) ∥ L p ⁢ ( M ) ≤ C ρ for ⁢ 1 < p < 2 ,

and using (A.7) and the Sobolev embedding, we obtain

∥ G r ⁢ ( u ) ∥ L p * ⁢ ( M ) ≤ C ρ , 1 p * = 1 p - 1 2 .

Moreover, we have

| f ⁢ ( t ) | 1 2 ⁢ σ ≤ C ⁢ ( | G r ⁢ ( t ) | + 1 )

for σ=11-r (>2). We choose 12<r<1 such that

(A.10) ( γ + 1 2 ) ⁢ σ < 3 2 .

Arguing as in [6], for every ε>0, we obtain

(A.11) ∥ f ⁢ ( u ) ∥ L p ⁢ ( M ) ≤ C ⁢ ρ - σ + σ - 1 p - ε , 1 < p < ∞ ,

and, for q>2,

(A.12) ∥ ∇ ⁡ u ∥ L q ⁢ ( M ) ≤ C ⁢ ρ ( - 1 2 + 1 q ) ⁢ ( σ - 1 ) - ε .

Now, we conclude the proof of Proposition A.1.

Proof of Proposition A.1.

There holds

(A.13) ∥ ( f ′ ⁢ ( u ) - f ⁢ ( u ) ) ⁢ ∇ ⁡ u ∥ L ∞ ⁢ ( M ) ≤ C ⁢ ∥ e γ ⁢ u ⁢ ∇ ⁡ u ∥ L ∞ ⁢ ( M ) = C γ ⁢ ∥ ∇ ⁡ e γ ⁢ u ∥ L ∞ ⁢ ( M ) .

Moreover, by (1.7) we have

- Δ g ⁢ e γ ⁢ u = - γ 2 ⁢ e γ ⁢ u ⁢ | ∇ ⁡ u | 2 + ρ ⁢ γ ⁢ e γ ⁢ u ⁢ f ⁢ ( u ) - c ρ ⁢ γ ⁢ e γ ⁢ u in ⁢ M .

Hence, for p>2, we have

∥ ∇ ⁡ e γ ⁢ u ∥ L ∞ ⁢ ( M ) ≤ C ⁢ ( ∥ Δ g ⁢ e γ ⁢ u ∥ L p ⁢ ( M ) + ∥ e γ ⁢ u ∥ L 1 ⁢ ( M ) ) ≤ C ⁢ ( ∥ e γ ⁢ u ⁢ | ∇ ⁡ u | 2 ∥ L p ⁢ ( M ) + ρ ⁢ ∥ e γ ⁢ u ⁢ f ⁢ ( u ) ∥ L p ⁢ ( M ) + ∥ c ρ ⁢ e γ ⁢ u ∥ L p ⁢ ( M ) ) .

Now, observing that eu≤C⁢(f⁢(u)+1), by (A.11) we obtain

(A.14) ρ ⁢ ∥ e γ ⁢ u ⁢ f ⁢ ( u ) ∥ L p ⁢ ( M ) ≤ C ⁢ ρ ⁢ ∥ e ( γ + 1 ) ⁢ u ∥ L p ⁢ ( M ) = C ⁢ ρ ⁢ ∥ e u ∥ L p ⁢ ( γ + 1 ) ⁢ ( M ) γ + 1 ≤ C ⁢ ρ τ - ε

for every ε>0 with

(A.15) τ = 1 + ( γ + 1 ) ⁢ ( σ - 1 p ⁢ ( γ + 1 ) - σ ) = 1 + σ - 1 p - σ ⁢ ( γ + 1 ) .

Hence, as p↓2, we have

(A.16) τ ↑ 1 + 1 2 ⁢ ( σ - 1 ) - σ ⁢ ( γ + 1 ) > - 1

by (A.10). On the other hand, by (2.3), for 1≤p<1γ, we have

∥ c ρ ⁢ e γ ⁢ u ∥ L p ⁢ ( M ) ≤ C ⁢ ρ - γ .

Moreover, if q>12⁢γ(>2), then

∥ e γ ⁢ u ⁢ | ∇ ⁡ u | 2 ∥ L p ⁢ ( M ) ≤ ∥ e γ ⁢ u ∥ L p ⁢ q ⁢ ( M ) ⁢ ∥ ∇ ⁡ u | ∥ L 2 ⁢ p ⁢ q ′ ⁢ ( M ) 2 ,

where q⁢q′=q+q′. By (A.12), for every ε>0 and since 2⁢p⁢q′>2, we have

∥ ∇ ⁡ u ∥ L 2 ⁢ p ⁢ q ′ ⁢ ( M ) 2 ≤ C ⁢ ρ ( - 1 + 1 p ⁢ q ′ ) ⁢ ( σ - 1 ) - ε .

Using again (A.11), for every ε>0, we have

∥ e γ ⁢ u ∥ L p ⁢ q ⁢ ( M ) ≤ C ⁢ ∥ e u ∥ L p ⁢ q ⁢ γ ⁢ ( M ) γ ≤ C ⁢ ρ - γ ⁢ σ + σ - 1 p ⁢ q - ε .

Then, for every ε>0, we have

(A.17) ∥ e γ ⁢ u ⁢ | ∇ ⁡ u | 2 ∥ L p ⁢ ( M ) ≤ C ⁢ ρ τ - ε

with τ defined by (A.15). Combining (A.13)–(A.14) and (A.16)–(A.17), we complete the proof. ∎

References

[1] Bethuel F., Brezis H. and Hélein F., Ginzburg–Landau Vortices, Progr. Nonlinear Differential Equations Appl. 13, Birkhäuser, Boston, 1994. 10.1007/978-1-4612-0287-5Search in Google Scholar

[2] Brézis H. and Lions P. L., A note on isolated singularities for linear elliptic equations, Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7A, Academic Press, New York (1981), 263–266. Search in Google Scholar

[3] Brezis H. and Merle F., Uniform estimates and blow-up behavior for solutions of -Δ⁢u=V⁢(x)⁢eu in two dimensions, Comm. Partial Differential Equations 16 (1991), no. 8–9, 1223–1253. 10.1080/03605309108820797Search in Google Scholar

[4] Esposito P., Grossi M. and Pistoia A., On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 2, 227–257. 10.1016/j.anihpc.2004.12.001Search in Google Scholar

[5] Lin C.-S., An expository survey on the recent development of mean field equations, Discrete Contin. Dyn. Syst. 19 (2007), no. 2, 387–410. 10.3934/dcds.2007.19.387Search in Google Scholar

[6] Nagasaki K. and Suzuki T., Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially-dominated nonlinearities, Asymptot. Anal. 3 (1990), no. 2, 173–188. 10.3233/ASY-1990-3205Search in Google Scholar

[7] Neri C., Statistical mechanics of the N-point vortex system with random intensities on a bounded domain, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 3, 381–399. 10.1016/j.anihpc.2003.05.002Search in Google Scholar

[8] Obata M., The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), no. 2, 247–258. 10.4310/jdg/1214430407Search in Google Scholar

[9] Pistoia A. and Ricciardi T., Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence, preprint 2015, http://arxiv.org/abs/1505.05304. 10.1088/0951-7715/29/2/271Search in Google Scholar

[10] Ricciardi T. and Suzuki T., Duality and best constant for a Trudinger–Moser inequality involving probability measures, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 7, 1327–1348. 10.4171/JEMS/462Search in Google Scholar

[11] Ricciardi T. and Zecca G., Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics, Differential Integral Equations 25 (2012), no. 3–4, 201–222. 10.57262/die/1356012734Search in Google Scholar

[12] Ricciardi T. and Zecca G., Mean field equations with probability measure in 2D-turbulence, Ric. Mat. 63 (2014), no. 1, suppl., S255–S264. 10.1007/s11587-014-0208-6Search in Google Scholar

[13] Ricciardi T. and Zecca G., Mass quantization and minimax solutions for Neri’s mean field equation in 2D-turbulence, J. Differential Equations 260 (2016), no. 1, 339–369. 10.1016/j.jde.2015.08.045Search in Google Scholar

[14] Ye D., Une remarque sur le comportement asymptotique des solutions de -Δ⁢u=λ⁢f⁢(u), C. R. Acad. Sci. Paris Sér I Math. 325 (1997), no. 12, 1279–1282. 10.1016/S0764-4442(97)82353-1Search in Google Scholar

Received: 2015-01-28

Accepted: 2015-05-19

Published Online: 2015-12-08

Published in Print: 2016-02-01

Articles in the same Issue

https://doi.org/10.1515/ans-2015-5015

Keywords for this article

Liouville Equation; Peak Solutions; Mean Field Equation