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Advanced Nonlinear Studies

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Volume 16, Issue 1

Issues

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

Tonia Ricciardi
  • Corresponding author
  • Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy
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/ Gabriella Zecca
  • Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy
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Published Online: 2015-12-08 | DOI: https://doi.org/10.1515/ans-2015-5015

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

{-Δu=ρf(u) in Ω,u=0 on Ω,(1.1)

where Ω2 is a smooth bounded domain, ρ>0, and f: is a smooth function such that

f(t)=et+φ(t)with φ(t)=o(et) as t+.(1.2)

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

{-Δu=λ[-1,1]αeαu𝒫(dα)[-1,1]×Ωeαu𝒫(dα)𝑑x in Ω,u=0 on Ω,(1.3)

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

f(t)=[-1,1]αeαt𝒫(dα),ρ=λ([-1,1]×Ωeαu𝒫(dα)𝑑x)-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

|φ(t)-φ(t)|𝒢(t)for some 𝒢C1(,) satisfying 𝒢(t)+|𝒢(t)|Ceγt with γ<14(1.4)

and

f(t)0for all t0.(1.5)

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

([6])

Let f satisfy assumptions (1.2), (1.4), and (1.5). Let un be a solution sequence to (1.1) with ρ=ρn0. Suppose un converges to some nontrivial function u0. Then,

u0(x)=8πj=1mGΩ(x,pj)

for some p1,,pmΩ, m, 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

[GΩ(x,pj)+12πlog|x-pj|]|x=pj+[ijGΩ(pj,pi)]=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 uC2(Ω), we define the quantity

S(u)=uz22-uzz,(1.6)

where

z=x-iy2,z¯=x+iy2.

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

z¯[S(u)]=-ρ4uz[f(u)-f(u)]=ρ4uz[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.

([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)=ρ4uρ(f(uρ)-f(uρ))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

{-Δgu=ρf(u)-cρin M,Mu𝑑x=0,(1.7)

where cρ=ρ|M|-1Mf(u)𝑑x, 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

|φ(t)-φ(t)|𝒢(t)for some 𝒢C1(,) satisfying 𝒢(t)+|𝒢(t)|Ceγt with γ<12(1.8)

and

f(t)-Cfor all t0.(1.9)

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

ρMf(u)𝑑xC.(1.10)

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

cρc0as ρ0+.(1.11)

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 p𝒮 and denote X=(x1,x2). We consider a local isothermal chart (Ψ,𝒰) such that ε(p)𝒰, Ψ(p)=0, ε(p)𝒮=, g(X)=eξ(X)(dx12+dx22), 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(ε(p)). We denote by GB(X,Y) the Green’s function of ΔX=x12+x22 on B. We set

K(X)=-BGB(X,Y)eξ(Y)𝑑Y+c1z(1.12)

with c1 defined by

z[ξ(z,z¯)+c0K(z,z¯)]|z=0=0,(1.13)

where c0 is defined in (1.11). Let u denote a solution sequence to (1.7). We define w(z)=u-cρK, so that -Δw=eξρf(u) in B. Finally, consider S(w), where S is defined in (1.6). Our main estimate is given in the following theorem.

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,

  • (i)

    for every 1s<(γ+12)-1,

    ρuρ(f(uρ)-f(uρ))Ls(M)0 as ρ0+;

  • (ii)

    for every blow-up point p𝒮 , the function S(w)S0 in L1(B) as ρ0+ , where S0 is holomorphic in B.

Consequently, we derive the following corollary.

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

u0(x)=8πj=1mGM(x,pj).(1.14)

Moreover, for all p𝒮, we have the relation

[X(q𝒮{p}GM(Ψ-1(X),q)+GM(Ψ-1(X),p)+12πlog|X|+18πξ(X))]|X=0=0.(1.15)

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.

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

rMe-ru|u|2𝑑xC,(2.1)

where C=C(r,M,φ,c0).

Proof.

We multiply the equation -Δgu=ρf(u)-cρ by e-ru. Integrating, we have

rMe-ru|u|2𝑑x=Me-ruΔgu𝑑x=-ρMe-ruf(u)𝑑x+cρMe-ru𝑑xρMe-ru|φ(u)|𝑑x+cρMerC𝑑xρMe-ru|φ(u)|𝑑x+cρerC|M|,

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

rMe-ru|u|2𝑑xC+ρ({u>t0}e(1-r)u𝑑x+{ut0}e-ru|φ(u)|𝑑x)C+ρ(Meu𝑑x+{ut0}e-ru|φ(u)|𝑑x),

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

The following proposition proves Theorem 1.3(i).

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

u(f(u)-f(u))Ls(M)Cρ-γ-ε

for 0<ρ<1.

Proof.

In view of (1.8), we have

0|f(u)-f(u)|Ceγu.

Hence,

(f(u)-f(u))uLsCeγuuLs.(2.2)

Moreover, (1.10) implies that

Meu𝑑xcρ-1.

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

eγuLq(M)C|M|1q-γρ-γ.(2.3)

Let 0<r<1-s(γ+12). By Lemma 2.1, for

q=s+rγ1-s2<1γ,

using Hölder’s inequality again, we have

eγuuLs(M)s=Me(sγ+r)u(e-ru|u|s)𝑑x(Meγuq𝑑x)1-s2(Me-2ru|u|2𝑑x)s2CeγuLq(M)s+rγ.(2.4)

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

eγuuLs(M)Cρ-γ-rs.(2.5)

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

Let p𝒮. We denote by (Ψ,𝒰) an isothermal chart satisfying

𝒰¯𝒮={p},Ψ(𝒰)=𝒪2,Ψ(p)=0,g(X)=eξ(X)(dx12+dx22),ξ(0)=0,

where X=(x1,x2) denotes a coordinate system on 𝒪. We consider ε>0 sufficiently small so that (p,ε)𝒰 and let B=B(0,r)=Ψ((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,

{-ΔXGB(X,Y)=δYin B,GB(X,Y)=0on B.

We recall from (1.12) that

K(X)=-BGB(X,Y)eξ(Y)𝑑Y+c1z

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

z[ξ(z,z¯)+c0K(z,z¯)]|z=0=0,

where c0=limρ0cρ. Then, KC(B) and

ΔXK=eξin B¯.

Let uρ be a blow-up solution sequence for (1.7). As ρ0, uu0 in Cloc(M𝒮), u-u0W1,q(M) for 1q<2, and f(u)f(u0) in Cloc(M𝒮), we have ΔguΔgu0 in Cloc(M𝒮), so that

Δgu0=c0in M𝒮.

We consider the following functions defined in B:

u~=uΨ-1,u~0=u0Ψ-1,w(z)=u~-cρK,w0(z)=u~0-c0K,S(w)=wzz-12wz2,S0=w0zz-12w0z2.

The following proposition proves Theorem 1.3(ii).

The complex function S0 defined in (2.6) is holomorphic in B and SS0 in L1(B).

Proof.

By (2.6) we have

-ΔXw=ρf(u~)eξandwz=u~z-cρKz.

Then, using ΔX=4zz¯ we compute

z¯[S(w)]=14(zΔXw-wzΔXw)=-ρ4eξ(f(u~)ξz+u~zf(u~))+ρ4eξf(u~)(u~z-cρKz)=ρ4eξ(f(u~)-f(u~))u~z-ρ4eξf(u~)(ξz+c0Kz)+(c0-cρ)ρ4eξf(u~)Kz.(2.7)

Using (2) we derive that

z¯S0in L1(B).(2.8)

Indeed, this follows by Proposition 2.2, (1.13), and by the fact that |ρf(u~)|*aδ0(dx) for some a>0. On the other hand, by (2.6), since uu0 in Cloc(M𝒮), we have

ww0in Cloc(B¯{0})

and then

SS0in Cloc(B¯{0}).(2.9)

At this point, we set Ξ=(ξ1,ξ2) and ζ=ξ1+iξ2 and we observe that by the Cauchy integral formula we may write

[S(w)](ζ)=1πBz¯S(z)ζ-z𝑑X+i2π+B[S(w)](z)ζ-z𝑑z=g(ζ)+h(ζ).(2.10)

We have

h(ζ)h0(ζ)=i2π+BS0(z)ζ-z𝑑zin Cloc0(B)(2.11)

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

g0in L1(B).(2.12)

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

gL1(B)B×B|z¯S(z)|1|ζ-z|𝑑X𝑑ΞB|z¯S(z)|(B(z,2r)1|ζ-z|𝑑Ξ)𝑑X=4πrB|z¯S(z)|𝑑X,

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

Sh0in L1(B) as ρ0,

and hence, up to subsequences,

Sh0 a.e. in B as ρ0,

so that by (2.9),

S0(ζ)=h0(ζ)for all ζB{0}.

This completes our proof. ∎

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

([2])

For B=B(0,1)n, n2, the conditions vW1,p(B), 1<p<, and Δv=0 in B{0} imply that H=v-E is harmonic in B, where is some constant and

E(x)={|x|2-nif 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

{-ΔgGM(x,y)=δy-1|M|MGM(x,y)𝑑x=0.

Proof of Corollary 1.4.

Assume that p𝒮. Let us start by observing that w0 in (2.6) is harmonic in B{0} by definition and that w0W1,q(B) for all 1<q<2. Hence, also by using Proposition 2.4, we have

w0(z)=log1|z|+H(z),

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

zlog|z|=12zlog(zz¯)=(2z)-1,

we compute

w0z=-2z+Hz,w0zz=2z2+Hzz.

Therefore,

S0=w0zz-12w0z2=2z2+Hzz-12(2z-Hz)2=(4-)8z2+2zHz+Hzz-12Hz2.

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

H=w0-4log1|z|(2.13)

is harmonic in B, we have

ΔX(u~0-4log1|z|)=c0eξ in B(0,r)

and, therefore,

Δg(u0(x)-8πGM(x,p))=c0-8π|M|+hpin (p,ε)

for some harmonic function hp. Arguing similarly for each p𝒮={p1,p2,,pm}, we conclude that

Δg(u0(x)-8πj=1mGM(x,pj))=c0-8πm|M|in M.

In particular, we obtain

u0(x)-8πj=1mGM(x,pj)=constantin 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=18πzH(X)|X=0=z[q𝒮GM(Ψ-1(X),q)+12πlog|X|]|X=0-[m|M|zK(X)]|X=0=z[q𝒮GM(Ψ-1(X),q)+12πlog|X|-18πξ(X)]|X=0.

Now, Corollary 1.4 is completely established. ∎

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 cρc0 as ρ0+. We show the following proposition.

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

ρu(f(u)-f(u))L(M)0as ρ0.

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

([8])

Let w=w(x)>0 be a solution to

Δw=|w|2w+F(w)on M,(A.1)

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

divV=J+12|w|2w-2(F(w)+wF(w)),(A.2)

where, in local coordinates,

Vj=w-1{(wxi)w-12wxiΔw},j=1,2,

and

J=w-1{i,j=12(2wxixj)2-12(Δw)2}0.

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

ρMe-ru|u|2(2rf(u)-f(u))2rcρMe-ru|u|2.(A.3)

Proof.

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

F(w)=re-ru(ρf(u)-cρ).

On the other hand, we have

F(w)+wF(w)=ρe-ru(2rf(u)-f(u))-2re-rucρ.

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

M|w|2w2(F(w)+wF(w))2MdivV=0.

In particular, since

ww=-ru,

by the last inequality we obtain

Mr2|u|2(F(w)+wF(w))=r2ρMe-ru|u|2(2rf(u)-f(u))-2r3cρMe-ru|u|20.

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

ρMe-ru|u|2f(u)C(1+ρMe-(r-γ)u|u|2).(A.4)

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

ρMe-ru|u|2f(u)𝑑xC

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

ρMe-ru|u|2|f(u)|𝑑xCif 12<r<1.(A.5)

For r>0, we define

Gr(t)=0te-r2s|f(s)|𝑑s.

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

Gr(u)L2(M)Cρ.(A.6)

There holds

Gr(u)L1(M)Cρ.(A.7)

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

{xM:u(x)0}Gr(u)𝑑x2r{u0}|f(u)|𝑑xC(M|f(u)|𝑑x)12Cρ.(A.8)

On the other hand, since -uC, we have

{xM:u(x)0}|Gr(u)|𝑑xC{u0}𝑑xu0eCr2CeCr2|M|C.(A.9)

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

Reducing (A.6) to

Gr(u)Lp(M)Cρfor 1<p<2,

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

Gr(u)Lp*(M)Cρ,1p*=1p-12.

Moreover, we have

|f(t)|12σC(|Gr(t)|+1)

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

(γ+12)σ<32.(A.10)

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

f(u)Lp(M)Cρ-σ+σ-1p-ε,1<p<,(A.11)

and, for q>2,

uLq(M)Cρ(-12+1q)(σ-1)-ε.(A.12)

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

Proof of Proposition A.1.

There holds

(f(u)-f(u))uL(M)CeγuuL(M)=CγeγuL(M).(A.13)

Moreover, by (1.7) we have

-Δgeγu=-γ2eγu|u|2+ργeγuf(u)-cργeγuin M.

Hence, for p>2, we have

eγuL(M)C(ΔgeγuLp(M)+eγuL1(M))C(eγu|u|2Lp(M)+ρeγuf(u)Lp(M)+cρeγuLp(M)).

Now, observing that euC(f(u)+1), by (A.11) we obtain

ρeγuf(u)Lp(M)Cρe(γ+1)uLp(M)=CρeuLp(γ+1)(M)γ+1Cρτ-ε(A.14)

for every ε>0 with

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

Hence, as p2, we have

τ1+12(σ-1)-σ(γ+1)>-1(A.16)

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

cρeγuLp(M)Cρ-γ.

Moreover, if q>12γ (>2), then

eγu|u|2Lp(M)eγuLpq(M)u|L2pq(M)2,

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

uL2pq(M)2Cρ(-1+1pq)(σ-1)-ε.

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

eγuLpq(M)CeuLpqγ(M)γCρ-γσ+σ-1pq-ε.

Then, for every ε>0, we have

eγu|u|2Lp(M)Cρτ-ε(A.17)

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

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About the article

Received: 2015-01-28

Accepted: 2015-05-19

Published Online: 2015-12-08

Published in Print: 2016-02-01


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


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 75–85, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5015.

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