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# Advanced Nonlinear Studies

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Volume 16, Issue 1

# 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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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ 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.

MSC 2010: 35J20; 35J60

## 1 Introduction

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

$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =\rho f\left(u\right)\hfill & & \hfill \text{in}\mathrm{\Omega },\\ \hfill u& =0\hfill & & \hfill \text{on}\partial \mathrm{\Omega },\end{array}$(1.1)

where $\mathrm{\Omega }\subset {ℝ}^{2}$ is a smooth bounded domain, $\rho >0$, and $f:ℝ\to ℝ$ is a smooth function such that

$f\left(t\right)={e}^{t}+\phi \left(t\right)\mathit{ }\text{with}\phi \left(t\right)=o\left({e}^{t}\right)\text{as}t\to +\mathrm{\infty }.$(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

$\left\{\begin{array}{cccc}\hfill -\mathrm{\Delta }u& =\lambda {\int }_{\left[-1,1\right]}\frac{\alpha {e}^{\alpha u}\mathcal{𝒫}\left(d\alpha \right)}{{\iint }_{\left[-1,1\right]×\mathrm{\Omega }}{e}^{\alpha u}\mathcal{𝒫}\left(d\alpha \right)𝑑x}\hfill & & \hfill \text{in}\mathrm{\Omega },\\ \hfill u& =0\hfill & & \hfill \text{on}\partial \mathrm{\Omega },\end{array}$(1.3)

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

$f\left(t\right)={\int }_{\left[-1,1\right]}\alpha {e}^{\alpha t}\mathcal{𝒫}\left(d\alpha \right),\rho =\lambda {\left({\iint }_{\left[-1,1\right]×\mathrm{\Omega }}{e}^{\alpha u}\mathcal{𝒫}\left(d\alpha \right)𝑑x\right)}^{-1},$

it is readily seen that if $\mathcal{𝒫}\left(\left\{1\right\}\right)>0$, 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

$|\phi \left(t\right)-{\phi }^{\prime }\left(t\right)|\le \mathcal{𝒢}\left(t\right)\mathit{ }\text{for some}\mathcal{𝒢}\in {C}^{1}\left(ℝ,ℝ\right)\text{satisfying}\mathcal{𝒢}\left(t\right)+|{\mathcal{𝒢}}^{\prime }\left(t\right)|\le C{e}^{\gamma t}\text{with}\gamma <\frac{1}{4}$(1.4)

and

$f\left(t\right)\ge 0\mathit{ }\text{for all}t\ge 0.$(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 ${u}_{n}$ be a solution sequence to (1.1) with $\rho \mathrm{=}{\rho }_{n}\mathrm{\to }\mathrm{0}$. Suppose ${u}_{n}$ converges to some nontrivial function ${u}_{\mathrm{0}}$. Then,

${u}_{0}\left(x\right)=8\pi \sum _{j=1}^{m}{G}_{\mathrm{\Omega }}\left(x,{p}_{j}\right)$

for some ${p}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{p}_{m}\mathrm{\in }\mathrm{\Omega }$, $m\mathrm{\in }ℕ$, where ${G}_{\mathrm{\Omega }}$ denotes the Green’s function for the Dirichlet problem on Ω. Furthermore, at each blow-up point ${p}_{j}$, $j\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}m$, there holds that

${\nabla \left[{G}_{\mathrm{\Omega }}\left(x,{p}_{j}\right)+\frac{1}{2\pi }\mathrm{log}|x-{p}_{j}|\right]|}_{x={p}_{j}}+\nabla \left[\sum _{i\ne j}{G}_{\mathrm{\Omega }}\left({p}_{j},{p}_{i}\right)\right]=0.$

The original estimates in [6] are involved and require the technical assumption $\gamma \in \left(0,\frac{1}{4}\right)$. It should be mentioned that this assumption was later weakened to the natural assumption $\gamma \in \left(0,1\right)$ 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 ${L}^{\mathrm{\infty }}$-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 ${L}^{1}$. Therefore, we are led to consider an ${L}^{1}$-framework, which turns out to be significantly simpler and which holds under the weaker assumption $\gamma \in \left(0,\frac{1}{2}\right)$. As a byproduct, we obtain a quick proof of mass quantization and blow-up point location for the case $\gamma \in \left(0,\frac{1}{2}\right)$.

In order to state our results, for a function $u\in {C}^{2}\left(\mathrm{\Omega }\right)$, we define the quantity

$S\left(u\right)=\frac{{u}_{z}^{2}}{2}-{u}_{zz},$(1.6)

where

${\partial }_{z}=\frac{{\partial }_{x}-i{\partial }_{y}}{2},{\partial }_{\overline{z}}=\frac{{\partial }_{x}+i{\partial }_{y}}{2}.$

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

${\partial }_{\overline{z}}\left[S\left(u\right)\right]=-\frac{\rho }{4}{u}_{z}\left[f\left(u\right)-{f}^{\prime }\left(u\right)\right]=\frac{\rho }{4}{u}_{z}\left[\phi \left(u\right)-{\phi }^{\prime }\left(u\right)\right].$

In particular, in the Liouville case $f\left(u\right)={e}^{u}$, the function $S\left(u\right)$ is holomorphic. Therefore, the complex derivative ${\partial }_{\overline{z}}\left[S\left(u\right)\right]$ 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 ${u}_{\rho }$ be a blow-up sequence for (1.1). Assume (1.2), (1.4), and (1.5). Then,

${\parallel {\partial }_{\overline{z}}S\left(u\right)\parallel }_{{L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)}=\frac{\rho }{4}{\parallel \nabla {u}_{\rho }\left({f}^{\prime }\left({u}_{\rho }\right)-f\left({u}_{\rho }\right)\right)\parallel }_{{L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)}\to 0.$

It is natural to expect that corresponding results should hold on a compact Riemannian 2-manifold $\left(M,g\right)$ without boundary. We show that, in fact, the ${L}^{\mathrm{\infty }}$-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 $\gamma \in \left(0,\frac{1}{2}\right)$ and is significantly simpler than the original ${L}^{\mathrm{\infty }}$-framework.

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

$\left\{\begin{array}{cccc}\hfill -{\mathrm{\Delta }}_{g}u& =\rho f\left(u\right)-{c}_{\rho }\hfill & & \hfill \text{in}M,\\ \hfill {\int }_{M}u𝑑x& =0,\hfill & & \end{array}$(1.7)

where ${c}_{\rho }=\rho {|M|}^{-1}{\int }_{M}f\left(u\right)𝑑x\in ℝ$, dx denotes the volume element on M, and ${\mathrm{\Delta }}_{g}$ denotes the Laplace–Beltrami operator. We assume that $f\left(t\right)={e}^{t}+\phi \left(t\right)$ satisfies (1.2) and, moreover, that

$|\phi \left(t\right)-{\phi }^{\prime }\left(t\right)|\le \mathcal{𝒢}\left(t\right)\mathit{ }\text{for some}\mathcal{𝒢}\in {C}^{1}\left(ℝ,ℝ\right)\text{satisfying}\mathcal{𝒢}\left(t\right)+|{\mathcal{𝒢}}^{\prime }\left(t\right)|\le C{e}^{\gamma t}\text{with}\gamma <\frac{1}{2}$(1.8)

and

$f\left(t\right)\ge -C\mathit{ }\text{for all}t\ge 0.$(1.9)

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

$\rho {\int }_{M}f\left(u\right)𝑑x\le C.$(1.10)

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

${c}_{\rho }\to {c}_{0}\mathit{ }\text{as}\rho \to {0}^{+}.$(1.11)

We note that (1.9) implies that $u\ge -C$. We now define the modified quantity corresponding to $S\left(u\right)$. Let $\mathcal{𝒮}=\left\{{p}_{1},\mathrm{\dots },{p}_{m}\right\}$ denote the blow-up set. Let $p\in \mathcal{𝒮}$ and denote $X=\left({x}_{1},{x}_{2}\right)$. We consider a local isothermal chart $\left(\mathrm{\Psi },\mathcal{𝒰}\right)$ such that ${\mathcal{ℬ}}_{\epsilon }\left(p\right)\subset \mathcal{𝒰}$, $\mathrm{\Psi }\left(p\right)=0$, ${\mathcal{ℬ}}_{\epsilon }\left(p\right)\cap \mathcal{𝒮}=\mathrm{\varnothing }$, $g\left(X\right)={e}^{\xi \left(X\right)}\left(d{x}_{1}^{2}+d{x}_{2}^{2}\right)$, and $\xi \left(0\right)=0$. For the sake of simplicity, we identify here functions on M with their pullback functions to $B=B\left(0,r\right)=\mathrm{\Psi }\left({\mathcal{ℬ}}_{\epsilon }\left(p\right)\right)$. We denote by ${G}_{B}\left(X,Y\right)$ the Green’s function of ${\mathrm{\Delta }}_{X}={\partial }_{{x}_{1}}^{2}+{\partial }_{{x}_{2}}^{2}$ on B. We set

$K\left(X\right)=-{\int }_{B}{G}_{B}\left(X,Y\right){e}^{\xi \left(Y\right)}𝑑Y+{c}_{1}z$(1.12)

with ${c}_{1}\in ℂ$ defined by

${{\partial }_{z}\left[\xi \left(z,\overline{z}\right)+{c}_{0}K\left(z,\overline{z}\right)\right]|}_{z=0}=0,$(1.13)

where ${c}_{0}$ is defined in (1.11). Let u denote a solution sequence to (1.7). We define $w\left(z\right)=u-{c}_{\rho }K$, so that $-\mathrm{\Delta }w={e}^{\xi }\rho f\left(u\right)$ in B. Finally, consider $S\left(w\right)$, where S is defined in (1.6). Our main estimate is given in the following theorem.

Assume that $f\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{e}^{t}\mathrm{+}\phi \mathit{}\mathrm{\left(}t\mathrm{\right)}$ satisfies (1.2), (1.8), and (1.9). Let ${u}_{\rho }$ be a blow-up solution sequence for (1.7). Then,

• (i)

for every $1\le s<{\left(\gamma +\frac{1}{2}\right)}^{-1}$,

$\rho {\parallel \nabla {u}_{\rho }\left({f}^{\prime }\left({u}_{\rho }\right)-f\left({u}_{\rho }\right)\right)\parallel }_{{L}^{s}\left(M\right)}\to 0\mathit{ }\mathit{\text{as}}\rho \to {0}^{+};$

• (ii)

for every blow-up point $p\in \mathcal{𝒮}$ , the function $S\left(w\right)\to {S}_{0}$ in ${L}^{1}\left(B\right)$ as $\rho \to {0}^{+}$ , where ${S}_{0}$ is holomorphic in B.

Consequently, we derive the following corollary.

Assume that $f\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}{e}^{t}\mathrm{+}\phi \mathit{}\mathrm{\left(}t\mathrm{\right)}$ satisfies (1.2), (1.8), and (1.9). Suppose ${u}_{n}$ converges to some nontrivial function ${u}_{\mathrm{0}}$. Then,

${u}_{0}\left(x\right)=8\pi \sum _{j=1}^{m}{G}_{M}\left(x,{p}_{j}\right).$(1.14)

Moreover, for all $p\mathrm{\in }\mathcal{𝒮}$, we have the relation

${\left[{\nabla }_{X}\left(\sum _{q\in \mathcal{𝒮}\setminus \left\{p\right\}}{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),q\right)+{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),p\right)+\frac{1}{2\pi }\mathrm{log}|X|+\frac{1}{8\pi }\xi \left(X\right)\right)\right]|}_{X=0}=0.$(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 ${L}^{\mathrm{\infty }}$-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 $r\mathrm{>}\mathrm{0}$, we have

$r{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}𝑑x\le C,$(2.1)

where $C\mathrm{=}C\mathit{}\mathrm{\left(}r\mathrm{,}M\mathrm{,}\phi \mathrm{,}{c}_{\mathrm{0}}\mathrm{\right)}$.

#### Proof.

We multiply the equation $-{\mathrm{\Delta }}_{g}u=\rho f\left(u\right)-{c}_{\rho }$ by ${e}^{-ru}$. Integrating, we have

$r{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}𝑑x={\int }_{M}{e}^{-ru}{\mathrm{\Delta }}_{g}u𝑑x$$=-\rho {\int }_{M}{e}^{-ru}f\left(u\right)𝑑x+{c}_{\rho }{\int }_{M}{e}^{-ru}𝑑x$$\le \rho {\int }_{M}{e}^{-ru}|\phi \left(u\right)|𝑑x+{c}_{\rho }{\int }_{M}{e}^{rC}𝑑x$$\le \rho {\int }_{M}{e}^{-ru}|\phi \left(u\right)|𝑑x+{c}_{\rho }{e}^{rC}|M|,$

since $u\ge -C$. Using the assumptions on φ, there exists ${t}_{0}>0$ such that $|g\left(u\right)|<{e}^{u}$ for $u>{t}_{0}$, so that

$r{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}𝑑x\le C+\rho \left({\int }_{\left\{u>{t}_{0}\right\}}{e}^{\left(1-r\right)u}𝑑x+{\int }_{\left\{u\le {t}_{0}\right\}}{e}^{-ru}|\phi \left(u\right)|𝑑x\right)\le C+\rho \left({\int }_{M}{e}^{u}𝑑x+{\int }_{\left\{u\le {t}_{0}\right\}}{e}^{-ru}|\phi \left(u\right)|𝑑x\right),$

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

The following proposition proves Theorem 1.3(i).

Let u be a solution to (1.7). Then, for every $\mathrm{1}\mathrm{\le }s\mathrm{<}{\mathrm{\left(}\gamma \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\right)}}^{\mathrm{-}\mathrm{1}}$ and for every $\epsilon \mathrm{>}\mathrm{0}$, we have

${\parallel \nabla u\left({f}^{\prime }\left(u\right)-f\left(u\right)\right)\parallel }_{{L}^{s}\left(M\right)}\le C{\rho }^{-\gamma -\epsilon }$

for $\mathrm{0}\mathrm{<}\rho \mathrm{<}\mathrm{1}$.

#### Proof.

In view of (1.8), we have

$0\le |f\left(u\right)-{f}^{\prime }\left(u\right)|\le C{e}^{\gamma u}.$

Hence,

${\parallel \left(f\left(u\right)-{f}^{\prime }\left(u\right)\right)\nabla u\parallel }_{{L}^{s}}\le C{\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{s}}.$(2.2)

Moreover, (1.10) implies that

${\int }_{M}{e}^{u}𝑑x\le c{\rho }^{-1}.$

Then, for every $1\le q<{\gamma }^{-1}$, using Hölder’s inequality we have

${\parallel {e}^{\gamma u}\parallel }_{{L}^{q}\left(M\right)}\le C{|M|}^{\frac{1}{q}-\gamma }{\rho }^{-\gamma }.$(2.3)

Let $0. By Lemma 2.1, for

$q=\frac{s+\frac{r}{\gamma }}{1-\frac{s}{2}}<\frac{1}{\gamma },$

using Hölder’s inequality again, we have

${\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{s}\left(M\right)}^{s}={\int }_{M}{e}^{\left(s\gamma +r\right)u}\left({e}^{-ru}{|\nabla u|}^{s}\right)𝑑x\le {\left({\int }_{M}{e}^{\gamma uq}𝑑x\right)}^{1-\frac{s}{2}}{\left({\int }_{M}{e}^{-2ru}{|\nabla u|}^{2}𝑑x\right)}^{\frac{s}{2}}\le C{\parallel {e}^{\gamma u}\parallel }_{{L}^{q}\left(M\right)}^{s+\frac{r}{\gamma }}.$(2.4)

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

${\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{s}\left(M\right)}\le C{\rho }^{-\gamma -\frac{r}{s}}.$(2.5)

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

Let $p\in \mathcal{𝒮}$. We denote by $\left(\mathrm{\Psi },\mathcal{𝒰}\right)$ an isothermal chart satisfying

$\overline{\mathcal{𝒰}}\cap \mathcal{𝒮}=\left\{p\right\},\mathrm{\Psi }\left(\mathcal{𝒰}\right)=\mathcal{𝒪}\subset {ℝ}^{2},\mathrm{\Psi }\left(p\right)=0,g\left(X\right)={e}^{\xi \left(X\right)}\left(d{x}_{1}^{2}+d{x}_{2}^{2}\right),\xi \left(0\right)=0,$

where $X=\left({x}_{1},{x}_{2}\right)$ denotes a coordinate system on $\mathcal{𝒪}$. We consider $\epsilon >0$ sufficiently small so that $\mathcal{ℬ}\left(p,\epsilon \right)⋐\mathcal{𝒰}$ and let $B=B\left(0,r\right)=\mathrm{\Psi }\left(\mathcal{ℬ}\left(p,\epsilon \right)\right)$. The Laplace–Beltrami operator ${\mathrm{\Delta }}_{g}$ is then mapped to the operator ${e}^{-\xi \left(X\right)}{\mathrm{\Delta }}_{X}$ on $\mathcal{𝒪}$, where ${\mathrm{\Delta }}_{X}={\partial }_{{x}_{1}^{2}}^{2}+{\partial }_{{x}_{2}^{2}}^{2}$. By ${G}_{B}\left(X,Y\right)$ we denote the Green’s function of ${\mathrm{\Delta }}_{X}$ on B, namely,

$\left\{\begin{array}{cccc}\hfill -{\mathrm{\Delta }}_{X}{G}_{B}\left(X,Y\right)& ={\delta }_{Y}\hfill & & \hfill \text{in}B,\\ \hfill {G}_{B}\left(X,Y\right)& =0\hfill & & \hfill \text{on}\partial B.\end{array}$

We recall from (1.12) that

$K\left(X\right)=-{\int }_{B}{G}_{B}\left(X,Y\right){e}^{\xi \left(Y\right)}𝑑Y+{c}_{1}z$

with ${c}_{1}$ the constant defined by (1.13), namely,

${{\partial }_{z}\left[\xi \left(z,\overline{z}\right)+{c}_{0}K\left(z,\overline{z}\right)\right]|}_{z=0}=0,$

where ${c}_{0}={lim}_{\rho \to 0}{c}_{\rho }$. Then, $K\in {C}^{\mathrm{\infty }}\left(B\right)$ and

${\mathrm{\Delta }}_{X}K={e}^{\xi }\mathit{ }\text{in}\overline{B}.$

Let ${u}_{\rho }$ be a blow-up solution sequence for (1.7). As $\rho \to 0$, $u\to {u}_{0}$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, $u-{u}_{0}\in {W}^{1,q}\left(M\right)$ for $1\le q<2$, and $f\left(u\right)\to f\left({u}_{0}\right)$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, we have ${\mathrm{\Delta }}_{g}u\to {\mathrm{\Delta }}_{g}{u}_{0}$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, so that

${\mathrm{\Delta }}_{g}{u}_{0}={c}_{0}\mathit{ }\text{in}M\setminus \mathcal{𝒮}.$

We consider the following functions defined in B:

$\stackrel{~}{u}=u\circ {\mathrm{\Psi }}^{-1},$${\stackrel{~}{u}}_{0}={u}_{0}\circ {\mathrm{\Psi }}^{-1},$$w\left(z\right)=\stackrel{~}{u}-{c}_{\rho }K,$${w}_{0}\left(z\right)={\stackrel{~}{u}}_{0}-{c}_{0}K,$$S\left(w\right)={w}_{zz}-\frac{1}{2}{w}_{z}^{2},$${S}_{0}={w}_{0zz}-\frac{1}{2}{w}_{0z}^{2}.$

The following proposition proves Theorem 1.3(ii).

The complex function ${S}_{\mathrm{0}}$ defined in (2.6) is holomorphic in B and $S\mathrm{\to }{S}_{\mathrm{0}}$ in ${L}^{\mathrm{1}}\mathit{}\mathrm{\left(}B\mathrm{\right)}\mathrm{.}$

#### Proof.

By (2.6) we have

$-{\mathrm{\Delta }}_{X}w=\rho f\left(\stackrel{~}{u}\right){e}^{\xi }\mathit{ }\text{and}\mathit{ }{w}_{z}={\stackrel{~}{u}}_{z}-{c}_{\rho }{K}_{z}.$

Then, using ${\mathrm{\Delta }}_{X}=4{\partial }_{z\overline{z}}$ we compute

${\partial }_{\overline{z}}\left[S\left(w\right)\right]=\frac{1}{4}\left({\partial }_{z}{\mathrm{\Delta }}_{X}w-{w}_{z}{\mathrm{\Delta }}_{X}w\right)$$=-\frac{\rho }{4}{e}^{\xi }\left(f\left(\stackrel{~}{u}\right){\xi }_{z}+{\stackrel{~}{u}}_{z}{f}^{\prime }\left(\stackrel{~}{u}\right)\right)+\frac{\rho }{4}{e}^{\xi }f\left(\stackrel{~}{u}\right)\left({\stackrel{~}{u}}_{z}-{c}_{\rho }{K}_{z}\right)$$=\frac{\rho }{4}{e}^{\xi }\left(f\left(\stackrel{~}{u}\right)-{f}^{\prime }\left(\stackrel{~}{u}\right)\right){\stackrel{~}{u}}_{z}-\frac{\rho }{4}{e}^{\xi }f\left(\stackrel{~}{u}\right)\left({\xi }_{z}+{c}_{0}{K}_{z}\right)+\left({c}_{0}-{c}_{\rho }\right)\frac{\rho }{4}{e}^{\xi }f\left(\stackrel{~}{u}\right){K}_{z}.$(2.7)

Using (2) we derive that

${\partial }_{\overline{z}}S\to 0\mathit{ }\text{in}{L}^{1}\left(B\right).$(2.8)

Indeed, this follows by Proposition 2.2, (1.13), and by the fact that $|\rho f\left(\stackrel{~}{u}\right)|\stackrel{*}{\to }a{\delta }_{0}\left(dx\right)$ for some $a>0$. On the other hand, by (2.6), since $u\to {u}_{0}$ in ${C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(M\setminus \mathcal{𝒮}\right)$, we have

$w\to {w}_{0}\mathit{ }\text{in}{C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(\overline{B}\setminus \left\{0\right\}\right)$

and then

$S\to {S}_{0}\mathit{ }\text{in}{C}_{\mathrm{loc}}^{\mathrm{\infty }}\left(\overline{B}\setminus \left\{0\right\}\right).$(2.9)

At this point, we set $\mathrm{\Xi }=\left({\xi }_{1},{\xi }_{2}\right)$ and $\zeta ={\xi }_{1}+i{\xi }_{2}$ and we observe that by the Cauchy integral formula we may write

$\left[S\left(w\right)\right]\left(\zeta \right)=\frac{1}{\pi }{\int }_{B}\frac{{\partial }_{\overline{z}}S\left(z\right)}{\zeta -z}𝑑X+\frac{i}{2\pi }{\int }_{+\partial B}\frac{\left[S\left(w\right)\right]\left(z\right)}{\zeta -z}𝑑z=g\left(\zeta \right)+h\left(\zeta \right).$(2.10)

We have

$h\left(\zeta \right)\to {h}_{0}\left(\zeta \right)=\frac{i}{2\pi }{\int }_{+\partial B}\frac{{S}_{0}\left(z\right)}{\zeta -z}𝑑z\mathit{ }\text{in}{C}_{\mathrm{loc}}^{0}\left(B\right)$(2.11)

and ${h}_{0}$ is holomorphic in B. On the other hand, we have

$g\to 0\mathit{ }\text{in}{L}^{1}\left(B\right).$(2.12)

To prove (2.12), it is sufficient to observe that for every $z\in B=B\left(0,r\right)$, we have $B\subset B\left(z,2r\right)$ and then

${\parallel g\parallel }_{{L}^{1}\left(B\right)}\le {\iint }_{B×B}|{\partial }_{\overline{z}}S\left(z\right)|\frac{1}{|\zeta -z|}𝑑X𝑑\mathrm{\Xi }\le {\int }_{B}|{\partial }_{\overline{z}}S\left(z\right)|\left({\int }_{B\left(z,2r\right)}\frac{1}{|\zeta -z|}𝑑\mathrm{\Xi }\right)𝑑X=4\pi r{\int }_{B}|{\partial }_{\overline{z}}S\left(z\right)|𝑑X,$

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

$S\to {h}_{0}\mathit{ }\text{in}{L}^{1}\left(B\right)\text{as}\rho \to 0,$

and hence, up to subsequences,

$S\to {h}_{0}\mathit{ }\text{a.e. in}B\text{as}\rho \to 0,$

so that by (2.9),

${S}_{0}\left(\zeta \right)={h}_{0}\left(\zeta \right)\mathit{ }\text{for all}\zeta \in B\setminus \left\{0\right\}.$

This completes our proof. ∎

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

#### ([2])

For $B\mathrm{=}B\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{\subset }{ℝ}^{n}$, $n\mathrm{\ge }\mathrm{2}$, the conditions $v\mathrm{\in }{W}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{\left(}B\mathrm{\right)}$, $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty }$, and $\mathrm{\Delta }\mathit{}v\mathrm{=}\mathrm{0}$ in $B\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$ imply that $H\mathrm{=}v\mathrm{-}\mathrm{\ell }\mathit{}E$ is harmonic in B, where $\mathrm{\ell }$ is some constant and

$E\left(x\right)=\left\{\begin{array}{cc}{|x|}^{2-n}\hfill & \mathit{\text{if}}n>2,\hfill \\ \mathrm{log}|x|\hfill & \mathit{\text{if}}n=2.\hfill \end{array}$

Now, we are ready to prove Corollary 1.4. By ${G}_{M}$ we denote the Green’s function on the manifold M, defined by

$\left\{\begin{array}{cc}\hfill -{\mathrm{\Delta }}_{g}{G}_{M}\left(x,y\right)& ={\delta }_{y}-\frac{1}{|M|}\hfill \\ \hfill {\int }_{M}{G}_{M}\left(x,y\right)𝑑x& =0.\hfill \end{array}$

#### Proof of Corollary 1.4.

Assume that $p\in \mathcal{𝒮}$. Let us start by observing that ${w}_{0}$ in (2.6) is harmonic in $B\setminus \left\{0\right\}$ by definition and that ${w}_{0}\in {W}^{1,q}\left(B\right)$ for all $1. Hence, also by using Proposition 2.4, we have

${w}_{0}\left(z\right)=\mathrm{\ell }\mathrm{log}\frac{1}{|z|}+H\left(z\right),$

where H is harmonic in B and $\mathrm{\ell }\ne 0$. Then, using the fact that

${\partial }_{z}\mathrm{log}|z|=\frac{1}{2}{\partial }_{z}\mathrm{log}\left(z\overline{z}\right)={\left(2z\right)}^{-1},$

we compute

${w}_{0z}=-\frac{\mathrm{\ell }}{2z}+{H}_{z},{w}_{0zz}=\frac{\mathrm{\ell }}{2{z}^{2}}+{H}_{zz}.$

Therefore,

${S}_{0}={w}_{0zz}-\frac{1}{2}{w}_{0z}^{2}=\frac{\mathrm{\ell }}{2{z}^{2}}+{H}_{zz}-\frac{1}{2}{\left(\frac{\mathrm{\ell }}{2z}-{H}_{z}\right)}^{2}=\frac{\mathrm{\ell }\left(4-\mathrm{\ell }\right)}{8{z}^{2}}+\frac{\mathrm{\ell }}{2z}{H}_{z}+{H}_{zz}-\frac{1}{2}{H}_{z}^{2}.$

By Proposition 2.3, we know that ${S}_{0}$ is holomorphic. Hence, we can conclude that $\mathrm{\ell }=4$ and ${H}_{z}\left(0\right)=0$. Since

$H={w}_{0}-4\mathrm{log}\frac{1}{|z|}$(2.13)

is harmonic in B, we have

${\mathrm{\Delta }}_{X}\left({\stackrel{~}{u}}_{0}-4\mathrm{log}\frac{1}{|z|}\right)={c}_{0}{e}^{\xi }\mathit{ }\text{in}B\left(0,r\right)$

and, therefore,

${\mathrm{\Delta }}_{g}\left({u}_{0}\left(x\right)-8\pi {G}_{M}\left(x,p\right)\right)={c}_{0}-\frac{8\pi }{|M|}+{h}_{p}\mathit{ }\text{in}\mathcal{ℬ}\left(p,\epsilon \right)$

for some harmonic function ${h}_{p}$. Arguing similarly for each $p\in \mathcal{𝒮}=\left\{{p}_{1},{p}_{2},\mathrm{\dots },{p}_{m}\right\}$, we conclude that

${\mathrm{\Delta }}_{g}\left({u}_{0}\left(x\right)-8\pi \sum _{j=1}^{m}{G}_{M}\left(x,{p}_{j}\right)\right)={c}_{0}-\frac{8\pi m}{|M|}\mathit{ }\text{in}M.$

In particular, we obtain

${u}_{0}\left(x\right)-8\pi \sum _{j=1}^{m}{G}_{M}\left(x,{p}_{j}\right)=\text{constant}\mathit{ }\text{in}M.$

Observing that ${\int }_{M}{u}_{0}=0$, 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={\frac{1}{8\pi }{\partial }_{z}H\left(X\right)|}_{X=0}$$={{\partial }_{z}\left[\sum _{q\in \mathcal{𝒮}}{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),q\right)+\frac{1}{2\pi }\mathrm{log}|X|\right]|}_{X=0}-{\left[\frac{m}{|M|}{\partial }_{z}K\left(X\right)\right]|}_{X=0}$$={{\partial }_{z}\left[\sum _{q\in \mathcal{𝒮}}{G}_{M}\left({\mathrm{\Psi }}^{-1}\left(X\right),q\right)+\frac{1}{2\pi }\mathrm{log}|X|-\frac{1}{8\pi }\xi \left(X\right)\right]|}_{X=0}.$

Now, Corollary 1.4 is completely established. ∎

## A The ${L}^{\mathrm{\infty }}$-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 ${L}^{1}$-approach may be seen, we check that Proposition 1.2 may be actually extended to (1.7) on a compact Riemannian 2-manifold $\left(M,g\right)$ 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 ${c}_{\rho }\to {c}_{0}$ as $\rho \to {0}^{+}$. We show the following proposition.

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

$\rho {\parallel \nabla u\left({f}^{\prime }\left(u\right)-f\left(u\right)\right)\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}\to 0\mathit{ }\mathit{\text{as}}\rho \to 0.$

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

#### ([8])

Let $w\mathrm{=}w\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{>}\mathrm{0}$ be a solution to

$\mathrm{\Delta }w=\frac{{|\nabla w|}^{2}}{w}+F\left(w\right)\mathit{ }\mathit{\text{on}}M,$(A.1)

where F is a ${C}^{\mathrm{1}}$-function. Then, there holds the identity

$divV=J+\frac{1}{2}{|\nabla w|}^{2}{w}^{-2}\left(F\left(w\right)+w{F}^{\prime }\left(w\right)\right),$(A.2)

where, in local coordinates,

${V}_{j}={w}^{-1}\left\{\nabla \left(\frac{\partial w}{\partial {x}_{i}}\right)\cdot \nabla w-\frac{1}{2}\frac{\partial w}{\partial {x}_{i}}\mathrm{\Delta }w\right\},j=1,2,$

and

$J={w}^{-1}\left\{\sum _{i,j=1}^{2}{\left(\frac{{\partial }^{2}w}{\partial {x}_{i}\partial {x}_{j}}\right)}^{2}-\frac{1}{2}{\left(\mathrm{\Delta }w\right)}^{2}\right\}\ge 0.$

Let u be a solution to (1.7). Then, for every $r\mathrm{>}\mathrm{0}$, there holds

$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}\left(2rf\left(u\right)-{f}^{\prime }\left(u\right)\right)\le 2r{c}_{\rho }{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}.$(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

$F\left(w\right)=r{e}^{-ru}\left(\rho f\left(u\right)-{c}_{\rho }\right).$

On the other hand, we have

$F\left(w\right)+w{F}^{\prime }\left(w\right)=\rho {e}^{-ru}\left(2rf\left(u\right)-{f}^{\prime }\left(u\right)\right)-2r{e}^{-ru}{c}_{\rho }.$

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

${\int }_{M}\frac{{|\nabla w|}^{2}}{{w}^{2}}\left(F\left(w\right)+wF\left(w\right)\right)\le 2{\int }_{M}divV=0.$

In particular, since

$\frac{\nabla w}{w}=-r\nabla u,$

by the last inequality we obtain

${\int }_{M}{r}^{2}{|\nabla u|}^{2}\left(F\left(w\right)+w{F}^{\prime }\left(w\right)\right)={r}^{2}\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}\left(2rf\left(u\right)-{f}^{\prime }\left(u\right)\right)-2{r}^{3}{c}_{\rho }{\int }_{M}{e}^{-ru}{|\nabla u|}^{2}\le 0.\mathit{∎}$

We note that combining (A.3) and (2.1), for $\frac{1}{2}, we obtain

$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}f\left(u\right)\le C\left(1+\rho {\int }_{M}{e}^{-\left(r-\gamma \right)u}{|\nabla u|}^{2}\right).$(A.4)

Since $\gamma <\frac{1}{4}$, combining (2.1) and (A.4) we obtain

$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}f\left(u\right)𝑑x\le C$

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

$\rho {\int }_{M}{e}^{-ru}{|\nabla u|}^{2}|f\left(u\right)|𝑑x\le C\mathit{ }\text{if}\frac{1}{2}(A.5)

For $r>0$, we define

${G}_{r}\left(t\right)={\int }_{0}^{t}{e}^{-\frac{r}{2}s}\sqrt{|f\left(s\right)|}𝑑s.$

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

${\parallel \nabla {G}_{r}\left(u\right)\parallel }_{{L}^{2}\left(M\right)}\le \frac{C}{\sqrt{\rho }}.$(A.6)

There holds

${\parallel {G}_{r}\left(u\right)\parallel }_{{L}^{1}\left(M\right)}\le \frac{C}{\sqrt{\rho }}.$(A.7)

#### Proof.

The proof can be easily obtained as in Lemma 2.1. Let us observe that in our assumption, for every $\frac{1}{2}, we have

${\int }_{\left\{x\in M:u\left(x\right)\ge 0\right\}}{G}_{r}\left(u\right)𝑑x\le \frac{2}{r}{\int }_{\left\{u\ge 0\right\}}\sqrt{|f\left(u\right)|}𝑑x\le C{\left({\int }_{M}|f\left(u\right)|𝑑x\right)}^{\frac{1}{2}}\le \frac{C}{\sqrt{\rho }}.$(A.8)

On the other hand, since $-u\le C$, we have

${\int }_{\left\{x\in M:u\left(x\right)\le 0\right\}}|{G}_{r}\left(u\right)|𝑑x\le C{\int }_{\left\{u\le 0\right\}}𝑑x{\int }_{u}^{0}{e}^{\frac{Cr}{2}}\le C{e}^{\frac{Cr}{2}}|M|\le C.$(A.9)

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

Reducing (A.6) to

${\parallel \nabla {G}_{r}\left(u\right)\parallel }_{{L}^{p}\left(M\right)}\le \frac{C}{\sqrt{\rho }}\mathit{ }\text{for}1

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

${\parallel {G}_{r}\left(u\right)\parallel }_{{L}^{{p}^{*}}\left(M\right)}\le \frac{C}{\sqrt{\rho }},\frac{1}{{p}^{*}}=\frac{1}{p}-\frac{1}{2}.$

Moreover, we have

${|f\left(t\right)|}^{\frac{1}{2\sigma }}\le C\left(|{G}_{r}\left(t\right)|+1\right)$

for $\sigma =\frac{1}{1-r}$ ($>2$). We choose $\frac{1}{2} such that

$\left(\gamma +\frac{1}{2}\right)\sigma <\frac{3}{2}.$(A.10)

Arguing as in [6], for every $\epsilon >0$, we obtain

${\parallel f\left(u\right)\parallel }_{{L}^{p}\left(M\right)}\le C{\rho }^{-\sigma +\frac{\sigma -1}{p}-\epsilon },1(A.11)

and, for $q>2$,

${\parallel \nabla u\parallel }_{{L}^{q}\left(M\right)}\le C{\rho }^{\left(-\frac{1}{2}+\frac{1}{q}\right)\left(\sigma -1\right)-\epsilon }.$(A.12)

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

#### Proof of Proposition A.1.

There holds

${\parallel \left({f}^{\prime }\left(u\right)-f\left(u\right)\right)\nabla u\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}\le C{\parallel {e}^{\gamma u}\nabla u\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}=\frac{C}{\gamma }{\parallel \nabla {e}^{\gamma u}\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}.$(A.13)

Moreover, by (1.7) we have

$-{\mathrm{\Delta }}_{g}{e}^{\gamma u}=-{\gamma }^{2}{e}^{\gamma u}{|\nabla u|}^{2}+\rho \gamma {e}^{\gamma u}f\left(u\right)-{c}_{\rho }\gamma {e}^{\gamma u}\mathit{ }\text{in}M.$

Hence, for $p>2$, we have

${\parallel \nabla {e}^{\gamma u}\parallel }_{{L}^{\mathrm{\infty }}\left(M\right)}\le C\left({\parallel {\mathrm{\Delta }}_{g}{e}^{\gamma u}\parallel }_{{L}^{p}\left(M\right)}+{\parallel {e}^{\gamma u}\parallel }_{{L}^{1}\left(M\right)}\right)\le C\left({\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel }_{{L}^{p}\left(M\right)}+\rho {\parallel {e}^{\gamma u}f\left(u\right)\parallel }_{{L}^{p}\left(M\right)}+{\parallel {c}_{\rho }{e}^{\gamma u}\parallel }_{{L}^{p}\left(M\right)}\right).$

Now, observing that ${e}^{u}\le C\left(f\left(u\right)+1\right)$, by (A.11) we obtain

$\rho {\parallel {e}^{\gamma u}f\left(u\right)\parallel }_{{L}^{p}\left(M\right)}\le C\rho {\parallel {e}^{\left(\gamma +1\right)u}\parallel }_{{L}^{p}\left(M\right)}=C\rho {\parallel {e}^{u}\parallel }_{{L}^{p\left(\gamma +1\right)}\left(M\right)}^{\gamma +1}\le C{\rho }^{\tau -\epsilon }$(A.14)

for every $\epsilon >0$ with

$\tau =1+\left(\gamma +1\right)\left(\frac{\sigma -1}{p\left(\gamma +1\right)}-\sigma \right)=1+\frac{\sigma -1}{p}-\sigma \left(\gamma +1\right).$(A.15)

Hence, as $p↓2$, we have

$\tau ↑1+\frac{1}{2}\left(\sigma -1\right)-\sigma \left(\gamma +1\right)>-1$(A.16)

by (A.10). On the other hand, by (2.3), for $1\le p<\frac{1}{\gamma }$, we have

${\parallel {c}_{\rho }{e}^{\gamma u}\parallel }_{{L}^{p}\left(M\right)}\le C{\rho }^{-\gamma }.$

Moreover, if $q>\frac{1}{2\gamma }$ $\left(>2\right)$, then

${\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel }_{{L}^{p}\left(M\right)}\le {{\parallel {e}^{\gamma u}\parallel }_{{L}^{pq}\left(M\right)}\parallel \nabla u|\parallel }_{{L}^{2p{q}^{\prime }}\left(M\right)}^{2},$

where $q{q}^{\prime }=q+{q}^{\prime }$. By (A.12), for every $\epsilon >0$ and since $2p{q}^{\prime }>2$, we have

${\parallel \nabla u\parallel }_{{L}^{2p{q}^{\prime }}\left(M\right)}^{2}\le C{\rho }^{\left(-1+\frac{1}{p{q}^{\prime }}\right)\left(\sigma -1\right)-\epsilon }.$

Using again (A.11), for every $\epsilon >0$, we have

${\parallel {e}^{\gamma u}\parallel }_{{L}^{pq}\left(M\right)}\le C{\parallel {e}^{u}\parallel }_{{L}^{pq\gamma }\left(M\right)}^{\gamma }\le C{\rho }^{-\gamma \sigma +\frac{\sigma -1}{pq}-\epsilon }.$

Then, for every $\epsilon >0$, we have

${\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel }_{{L}^{p}\left(M\right)}\le C{\rho }^{\tau -\epsilon }$(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

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,

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