## 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.

Show Summary Details# On the Blow-Up of Solutions to Liouville-Type Equations

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## Abstract

## 1 Introduction

#### ([6])

#### ([6])

## 2 Proof of Theorem 1.3

#### Proof.

#### Proof.

#### Proof.

#### ([2])

#### Proof of Corollary 1.4.

## A The ${L}^{\mathrm{\infty}}$-estimate on *M*

#### ([8])

#### Proof.

#### Proof.

#### Proof of Proposition A.1.

## References

## About the article

## Citing Articles

*Nonlinear Analysis: Real World Applications*, 2017, Volume 38, Page 222*Calculus of Variations and Partial Differential Equations*, 2016, Volume 55, Number 6

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Editor-in-Chief: Ahmad, Shair

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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

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

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

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

$f(t)={e}^{t}+\phi (t)\mathit{\hspace{1em}}\text{with}\phi (t)=o({e}^{t})\text{as}t\to +\mathrm{\infty}.$(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

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

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

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

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

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

and

$f(t)\ge 0\mathit{\hspace{1em}}\text{for all}t\ge 0.$(1.5)

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

*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,*

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

*for some ${p}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{p}_{m}\mathrm{\in}\mathrm{\Omega}$, $m\mathrm{\in}\mathbb{N}$, 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*

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

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

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

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

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

In particular, in the Liouville case $f(u)={e}^{u}$, the function $S(u)$ is holomorphic. Therefore, the complex derivative ${\partial}_{\overline{z}}[S(u)]$ 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.

*Let ${u}_{\rho}$ be a blow-up sequence for (1.1). Assume (1.2), (1.4), and (1.5). Then,*

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

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 ${L}^{\mathrm{\infty}}$-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 $\gamma \in (0,\frac{1}{2})$ and is significantly simpler than the original ${L}^{\mathrm{\infty}}$-framework.

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

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

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

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

and

$f(t)\ge -C\mathit{\hspace{1em}}\text{for all}t\ge 0.$(1.9)

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

$\rho {\int}_{M}f(u)\mathit{d}x\le C.$(1.10)

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

${c}_{\rho}\to {c}_{0}\mathit{\hspace{1em}}\text{as}\rho \to {0}^{+}.$(1.11)

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

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

with ${c}_{1}\in \u2102$ defined by

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

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

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

- (i)
*for every*$1\le s<{(\gamma +\frac{1}{2})}^{-1}$,$\rho {\parallel \nabla {u}_{\rho}({f}^{\prime}({u}_{\rho})-f({u}_{\rho}))\parallel}_{{L}^{s}(M)}\to 0\mathit{\hspace{1em}}\mathit{\text{as}}\rho \to {0}^{+};$

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

Consequently, we derive the following corollary.

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

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

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

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

We begin by establishing the following result.

*Let u be a solution to (1.7). For every $r\mathrm{>}\mathrm{0}$, we have*

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

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

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

$r{\displaystyle {\int}_{M}}{e}^{-ru}{|\nabla u|}^{2}\mathit{d}x={\displaystyle {\int}_{M}}{e}^{-ru}{\mathrm{\Delta}}_{g}u\mathit{d}x$$=-\rho {\displaystyle {\int}_{M}}{e}^{-ru}f(u)\mathit{d}x+{c}_{\rho}{\displaystyle {\int}_{M}}{e}^{-ru}\mathit{d}x$$\le \rho {\displaystyle {\int}_{M}}{e}^{-ru}|\phi (u)|\mathit{d}x+{c}_{\rho}{\displaystyle {\int}_{M}}{e}^{rC}\mathit{d}x$$\le \rho {\displaystyle {\int}_{M}}{e}^{-ru}|\phi (u)|\mathit{d}x+{c}_{\rho}{e}^{rC}|M|,$

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

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

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 $\mathrm{1}\mathrm{\le}s\mathrm{<}{\mathrm{(}\gamma \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{)}}^{\mathrm{-}\mathrm{1}}$ and for every $\epsilon \mathrm{>}\mathrm{0}$, we have*

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

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

In view of (1.8), we have

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

Hence,

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

Moreover, (1.10) implies that

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

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

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

Let $0<r<1-s(\gamma +\frac{1}{2})$. 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

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

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

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

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

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

where $X=({x}_{1},{x}_{2})$ denotes a coordinate system on $\mathcal{\mathcal{O}}$. We consider $\epsilon >0$ sufficiently small so that $\mathcal{\mathcal{B}}(p,\epsilon )\u22d0\mathcal{\mathcal{U}}$ and let $B=B(0,r)=\mathrm{\Psi}(\mathcal{\mathcal{B}}(p,\epsilon ))$. The Laplace–Beltrami operator ${\mathrm{\Delta}}_{g}$ is then mapped to the operator ${e}^{-\xi (X)}{\mathrm{\Delta}}_{X}$ on $\mathcal{\mathcal{O}}$, where ${\mathrm{\Delta}}_{X}={\partial}_{{x}_{1}^{2}}^{2}+{\partial}_{{x}_{2}^{2}}^{2}$. By ${G}_{B}(X,Y)$ we denote the Green’s function of ${\mathrm{\Delta}}_{X}$ on *B*, namely,

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

We recall from (1.12) that

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

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

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

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

${\mathrm{\Delta}}_{X}K={e}^{\xi}\mathit{\hspace{1em}}\text{in}\overline{B}.$

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

${\mathrm{\Delta}}_{g}{u}_{0}={c}_{0}\mathit{\hspace{1em}}\text{in}M\setminus \mathcal{\mathcal{S}}.$

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(z)=\stackrel{~}{u}-{c}_{\rho}K,$${w}_{0}(z)={\stackrel{~}{u}}_{0}-{c}_{0}K,$$S(w)={w}_{zz}-{\displaystyle \frac{1}{2}}{w}_{z}^{2},$${S}_{0}={w}_{0zz}-{\displaystyle \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 ${L}^{\mathrm{1}}\mathit{}\mathrm{(}B\mathrm{)}\mathrm{.}$
*

By (2.6) we have

$-{\mathrm{\Delta}}_{X}w=\rho f(\stackrel{~}{u}){e}^{\xi}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{w}_{z}={\stackrel{~}{u}}_{z}-{c}_{\rho}{K}_{z}.$

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

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

Using (2) we derive that

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

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

$w\to {w}_{0}\mathit{\hspace{1em}}\text{in}{C}_{\mathrm{loc}}^{\mathrm{\infty}}(\overline{B}\setminus \{0\})$

and then

$S\to {S}_{0}\mathit{\hspace{1em}}\text{in}{C}_{\mathrm{loc}}^{\mathrm{\infty}}(\overline{B}\setminus \{0\}).$(2.9)

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

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

We have

$h(\zeta )\to {h}_{0}(\zeta )=\frac{i}{2\pi}{\int}_{+\partial B}\frac{{S}_{0}(z)}{\zeta -z}\mathit{d}z\mathit{\hspace{1em}}\text{in}{C}_{\mathrm{loc}}^{0}(B)$(2.11)

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

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

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

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

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

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

and hence, up to subsequences,

$S\to {h}_{0}\mathit{\hspace{1em}}\text{a.e. in}B\text{as}\rho \to 0,$

so that by (2.9),

${S}_{0}(\zeta )={h}_{0}(\zeta )\mathit{\hspace{1em}}\text{for all}\zeta \in B\setminus \{0\}.$

This completes our proof. ∎

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

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

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

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

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

Assume that $p\in \mathcal{\mathcal{S}}$. Let us start by observing that ${w}_{0}$ in (2.6) is harmonic in $B\setminus \{0\}$ by definition and that ${w}_{0}\in {W}^{1,q}(B)$ for all $1<q<2$. Hence, also by using Proposition 2.4, we have

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

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

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

we compute

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

Therefore,

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

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

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

is harmonic in *B*, we have

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

and, therefore,

${\mathrm{\Delta}}_{g}({u}_{0}(x)-8\pi {G}_{M}(x,p))={c}_{0}-\frac{8\pi}{|M|}+{h}_{p}\mathit{\hspace{1em}}\text{in}\mathcal{\mathcal{B}}(p,\epsilon )$

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

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

In particular, we obtain

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

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

Now, Corollary 1.4 is completely established. ∎

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 $(M,g)$ 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 ${c}_{\rho}\to {c}_{0}$ as $\rho \to {0}^{+}$. We show the following proposition.

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

$\rho {\parallel \nabla u({f}^{\prime}(u)-f(u))\parallel}_{{L}^{\mathrm{\infty}}(M)}\to 0\mathit{\hspace{1em}}\mathit{\text{as}}\rho \to 0.$

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

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

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

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

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

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

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

$\rho {\int}_{M}{e}^{-ru}{|\nabla u|}^{2}(2rf(u)-{f}^{\prime}(u))\le 2r{c}_{\rho}{\int}_{M}{e}^{-ru}{|\nabla u|}^{2}.$(A.3)

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

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

On the other hand, we have

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

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

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

In particular, since

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

by the last inequality we obtain

${\int}_{M}{r}^{2}{|\nabla u|}^{2}(F(w)+w{F}^{\prime}(w))={r}^{2}\rho {\int}_{M}{e}^{-ru}{|\nabla u|}^{2}(2rf(u)-{f}^{\prime}(u))-2{r}^{3}{c}_{\rho}{\int}_{M}{e}^{-ru}{|\nabla u|}^{2}\le 0.\mathit{\u220e}$

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

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

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

$\rho {\int}_{M}{e}^{-ru}{|\nabla u|}^{2}f(u)\mathit{d}x\le C$

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

$\rho {\int}_{M}{e}^{-ru}{|\nabla u|}^{2}|f(u)|\mathit{d}x\le C\mathit{\hspace{1em}}\text{if}\frac{1}{2}<r<1.$(A.5)

For $r>0$, we define

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

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

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

*There holds*

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

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

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

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

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

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

Reducing (A.6) to

${\parallel \nabla {G}_{r}(u)\parallel}_{{L}^{p}(M)}\le \frac{C}{\sqrt{\rho}}\mathit{\hspace{1em}}\text{for}1<p<2,$

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

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

Moreover, we have

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

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

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

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

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

and, for $q>2$,

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

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

There holds

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

Moreover, by (1.7) we have

$-{\mathrm{\Delta}}_{g}{e}^{\gamma u}=-{\gamma}^{2}{e}^{\gamma u}{|\nabla u|}^{2}+\rho \gamma {e}^{\gamma u}f(u)-{c}_{\rho}\gamma {e}^{\gamma u}\mathit{\hspace{1em}}\text{in}M.$

Hence, for $p>2$, we have

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

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

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

for every $\epsilon >0$ with

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

Hence, as $p\downarrow 2$, we have

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

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

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

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

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

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

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

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

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

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

${\parallel {e}^{\gamma u}{|\nabla u|}^{2}\parallel}_{{L}^{p}(M)}\le C{\rho}^{\tau -\epsilon}$(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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**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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[1]

F. De Marchis and T. Ricciardi

[2]

Tonia Ricciardi and Ryo Takahashi

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