---
abstract: |
    For each $q\in(0,1)$ let $$\lambda_{q}(\Omega):=\inf\left\{  \left\Vert \nabla v\right\Vert
    _{L^{p}(\Omega)}^{p}:v\in W_{0}^{1,p}(\Omega)\;\mathrm{and}\;\int_{\Omega
    }\left\vert v\right\vert ^{q}\mathrm{d}x=1\right\}  ,$$ where $p>1$ and $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ We first show that$$0<\mu(\Omega):=\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)\left\vert
    \Omega\right\vert ^{\frac{p}{q}}<\infty,$$ where $\left\vert \Omega\right\vert =\int_{\Omega}\mathrm{d}x.$ Then, we prove that $$\mu(\Omega)=\min\left\{  \left\Vert \nabla v\right\Vert _{L^{p}(\Omega)}^{p}:v\in W_{0}^{1,p}(\Omega)\;\mathrm{and}\;\lim_{q\rightarrow0^{+}}\left(
    \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert
    ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=1\right\}$$ and that $\mu(\Omega)$ is reached by a function $u\in W_{0}^{1,p}(\Omega),$ which is positive in $\Omega,$ belongs to $C^{0,\alpha}(\overline{\Omega}),$ for some $\alpha\in(0,1),$ and satisfies$$\mathrm{\ }-\operatorname{div}(\left\vert \nabla u\right\vert ^{p-2}\nabla
    u)=\mu(\Omega)\left\vert \Omega\right\vert ^{-1}u^{-1}\quad\mathrm{in\quad
    \Omega,\quad and\quad}\int_{\Omega}\log u\mathrm{d}x=0.$$ We also show that $\mu(\Omega)^{-1}$ is the best constant $C$ in the following log-Sobolev type inequality$$\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
    v\right\vert ^{p}\mathrm{d}x\right)  \leq C\left\Vert \nabla v\right\Vert
    _{L^{p}(\Omega)}^{p},\quad v\in W_{0}^{1,p}(\Omega)$$ and that this inequality becomes an equality if, and only if, $v$ is a scalar multiple of $u$ and $C=\mu(\Omega)^{-1}.\medskip$

    **2010 Mathematics Subject Classification.**  35B40; 35J25; 35J92. 

    **Keywords:**
author:
- |
    [**Grey Ercole[^1]  and Gilberto de Assis Pereira**]{}\
    [*Departamento de Matemática - Universidade Federal de Minas Gerais*]{}\
    [*Belo Horizonte, MG, 30.123-970, Brazil.*]{}\
    [grey@mat.ufmg.br, gilbertoapereira@yahoo.com.br]{}
title: '*On a singular minimizing problem*'
---

Introduction
============

Let $p>1$ be fixed and let $\Omega\subset\mathbb{R}^{N},$ $N\geq2,$ be a bounded and smooth domain.  For each $q\in(0,1)$ let us define $$\lambda_{q}(\Omega):=\inf\left\{  \left\Vert \nabla v\right\Vert _{p}^{p}:v\in
W_{0}^{1,p}(\Omega)\;\mathrm{and}\;\int_{\Omega}\left\vert v\right\vert
^{q}\mathrm{d}x=1\right\}  , \label{lambq}$$ where $\left\Vert \cdot\right\Vert _{s}$ denotes the standard norm of the Lebesgue space $L^{s}(\Omega),$ $1\leq s\leq\infty.$

As proved in [@Anello], $\lambda_{q}(\Omega)$ is achieved by a positive function $u_{q}\in W_{0}^{1,p}(\Omega)\cap C^{1}(\Omega)$ satisfying the singular Dirichlet problem $$\left\{
\begin{array}
[c]{ll}-\Delta_{p}v=\lambda_{q}(\Omega)\left\vert v\right\vert ^{q-2}v &
\mathrm{in\ }\Omega\\
v=0 & \mathrm{on\ }\partial\Omega,
\end{array}
\right.   \label{qbvp}$$ in the weak sense, where $\Delta_{p}v=\operatorname{div}\left(  \left\vert
\nabla v\right\vert ^{p-2}\nabla v\right)  $ is the $p$-Laplacian operator.  Moreover, it follows from [@GST1 Theorem 1.1 (i)] that $u_{q}\in
C^{1,\alpha}(\overline{\Omega}),$ for some $\alpha\in(0,1).$

In this paper we first show that $$0<\mu(\Omega):=\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{q}}<\infty, \label{mu1}$$ where $\left\vert D\right\vert $ stands for the $N$-dimensional Lebesgue volume of $D\subset\mathbb{R}^{N},$ $\left\vert
D\right\vert =\int_{D}\mathrm{d}x.$

Then, we prove that $$\mu(\Omega)=\min\left\{  \left\Vert \nabla v\right\Vert _{p}^{p}:v\in
W_{0}^{1,p}(\Omega)\;\mathrm{and}\;\lim_{q\rightarrow0^{+}}\left(  \frac
{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert
^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=1\right\}  \label{defmu}$$ and that the minimum is reached by a function $u\in W_{0}^{1,p}(\Omega),$ which is positive in $\Omega,$ belongs to $C^{0,\alpha}(\overline{\Omega}),$ for some $\alpha\in(0,1),$ and satisfies:

1.   $u=\lim_{q\rightarrow0^{+}}\left\vert \Omega\right\vert ^{\frac
    {1}{q}}u_{q}$ in $W_{0}^{1,p}(\Omega);$

2.   $-\Delta_{p}u=\mu(\Omega)\left\vert \Omega\right\vert ^{-1}u^{-1}$ in $\Omega;$ and

3.   $\int_{\Omega}\log u\mathrm{d}x=0.$

Exploring (\[defmu\]) we also prove that $\mu(\Omega)^{-1}$ is the best constant $C$ in the following log-Sobolev type inequality$$\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
v\right\vert ^{p}\mathrm{d}x\right)  \leq C\left\Vert \nabla v\right\Vert
_{p}^{p},\quad v\in W_{0}^{1,p}(\Omega),$$ and that $\mu(\Omega)^{-1}$ is reached if, and only if, $v$ is a scalar multiple of $u$, which is the unique case where the inequality becomes an equality.  Up to our knowledge, these facts are entirely new. 

It is easy to check that for each fixed $\lambda>0$ the function $u_{\lambda
}:=\left(  \frac{\lambda\left\vert \Omega\right\vert }{\mu(\Omega)}\right)
^{\frac{1}{p}}u\ $is a positive weak solution of the singular problem $$\left\{
\begin{array}
[c]{ll}-\Delta_{p}v=\lambda v^{-1} & \mathrm{in\ }\Omega,\\
v=0 & \mathrm{on\ }\partial\Omega. 
\end{array}
\right.   \label{lsing}$$

The function $u_{\lambda}$ is, in fact, the unique positive solution of (\[lsing\]).  This uniqueness result follows from a simple and well-known inequality involving vectors of $\mathbb{R}^{N}$. Existence and regularity of weak solutions for (\[lsing\]) were first studied in the particular case $p=2$ (see [@CRT; @LM; @Stuart]), whereas the case $p>1$ has received more attention in the last decade (see [@Chu; @GST; @GST1; @Mohammed] and references therein). 

We remark that the differentiability of the functional $v\in W_{0}^{1,p}(\Omega)\mapsto\lambda{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x\in\lbrack-\infty,\infty)$ is a delicate question, which makes it difficult to apply variational methods to obtain the positive solution of (\[lsing\]).  Thus, $u_{\lambda}$ has generally been obtained by nonvariational methods, mainly the sub-super solution method.  As for regularity, it is proved in [@GST1 Theorem 2.2 (ii)] that $u_{\lambda}\in C^{0,\alpha}(\overline{\Omega}),$ for some $\alpha\in(0,1).$

We emphasize that besides providing a new existence proof of $u_{\lambda},$ we show that $$\int_{\Omega}\log u_{\lambda}\mathrm{d}x=\frac{\left\vert \Omega\right\vert
}{p}\log\left(  \frac{\lambda\left\vert \Omega\right\vert }{\mu(\Omega
)}\right)  \in(-\infty,\infty).$$ This property of $u_{\lambda}$ was not known up to now.  It comes from the connection between (\[lsing\]) and the minimizing problem (\[defmu\]). 

Also in this paper, we show that the formal energy functional associated with (\[lsing\]), $$J_{\lambda}(v):=\left\{
\begin{array}
[c]{ll}\dfrac{1}{p}{\displaystyle\int_{\Omega}}
\left\vert \nabla v\right\vert ^{p}\mathrm{d}x-\lambda{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x, & \mathrm{if}\;{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x\in(-\infty,\infty)\\
\infty, & \mathrm{if}\;{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x=-\infty,
\end{array}
\right.$$ attains its minimum value $\dfrac{\lambda\left\vert \Omega\right\vert }{p}\left(  1-\log\left(  \frac{\lambda\left\vert \Omega\right\vert }{\mu(\Omega)}\right)  \right)  $ only at the functions $u_{\lambda}$ and $-u_{\lambda}.$

We end the paper by describing the asymptotic behavior of the pair $(\lambda_{q}(\Omega),\left\Vert u_{q}\right\Vert _{\infty}),$ as $q\rightarrow0^{+}.$ That is, we determine when these quantities either go to $0$ or to $\infty$ or remain bounded from above and from below, when $q\rightarrow0^{+}.$ More precisely, we obtain directly from (\[mu1\]) that $$\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=\left\{
\begin{array}
[c]{ll}\infty & \mathrm{if}\;\left\vert \Omega\right\vert <1\\
\mu(\Omega) & \mathrm{if}\;\left\vert \Omega\right\vert =1\\
0 & \mathrm{if}\;\left\vert \Omega\right\vert >1,
\end{array}
\right.   \label{assy1}$$ and apply lower and upper estimates (derived in Section \[S1\]) to show that$$\lim_{q\rightarrow0^{+}}\left\Vert u_{q}\right\Vert _{\infty}=\left\{
\begin{array}
[c]{ll}\infty & \mathrm{if}\quad\left\vert \Omega\right\vert <1\\
0 & \mathrm{if}\quad\left\vert \Omega\right\vert >1
\end{array}
\right.   \label{assy2}$$ and that $$0<A\mu(\Omega)^{\frac{1}{p}}\leq\lim_{q\rightarrow0^{+}}\left\Vert
u_{q}\right\Vert _{\infty}\leq B\mu(\Omega)^{\frac{1}{p}},\quad\mathrm{if}\;\left\vert \Omega\right\vert =1, \label{assy3}$$ where $A$ and $B$ are positive constants that depend only on $N$ and $p.$

The result in (\[assy1\]) for the case $\left\vert \Omega\right\vert <1$ has recently been obtained in [@Anello].  The cases $\left\vert \Omega
\right\vert \geq1$ in (\[assy1\]) as well as (\[assy2\]) and (\[assy3\]) are new observations. 

Thus, (\[assy1\]), (\[assy2\]) and (\[assy3\]) provide complementary information on how the function $q\in(0,p^{\star})\mapsto(\lambda_{q}(\Omega),\left\Vert u_{q}\right\Vert _{\infty})\in\mathbb{R}^{2}$ behaves at the endpoints of its domain.  In fact, the behavior of this function as $q\rightarrow p^{\star}$ is well known: $$\lim_{q\rightarrow p^{\star}}\lambda_{q}(\Omega)=\left\{
\begin{array}
[c]{ll}S_{N,p}, & \;\mathrm{if}\quad1<p<N\\
0, & \;\mathrm{if}\quad p=N>1\\
\Lambda_{p}(\Omega), & \;\mathrm{if}\quad p>N,
\end{array}
\right.   \label{asyl}$$ and $$\lim_{q\rightarrow p^{\star}}\left\Vert u_{q}\right\Vert _{\infty}=\left\{
\begin{array}
[c]{ll}\infty, & \;\mathrm{if}\quad1<p<N\\
C_{N}, & \;\mathrm{if}\quad p=N>1\\
1, & \;\mathrm{if}\quad p>N,
\end{array}
\right.   \label{asyu}$$ where $$\Lambda_{p}(\Omega):=\min\left\{  \left\Vert \nabla u\right\Vert _{p}^{p}:u\in
W_{0}^{1,p}(\Omega)\text{ \ and \ }\left\Vert u\right\Vert _{\infty
}=1\right\}  ,$$ $S_{N,p}$ is the well-known Sobolev constant, defined by$$S_{N,p}:=\pi^{\frac{p}{2}}N\left(  \frac{N-p}{p-1}\right)  ^{p-1}\left(
\frac{\Gamma(N/p)\Gamma(1+N-N/p)}{\Gamma(1+N/2)\Gamma(N)}\right)  ^{\frac
{p}{N}}, \label{SNp}$$ $\Gamma$ denoting the Gamma Function, and $C_{N}$ is a positive constant that does not depend on $\Omega.$

For (\[asyl\]) and (\[asyu\]) we refer to [@Anello; @GEJMAA], [@RenWei-p] and [@GGilb], respectively to the cases $1<p<N,$ $p=N>1$ and $p>N>1.$

Preliminaries\[S1\]
===================

In this section, we present some properties of the weak solutions of the singular Dirichlet problem $$\left\{
\begin{array}
[c]{ll}-\Delta_{p}v=\lambda v^{q-1} & \mathrm{in\ }\Omega,\quad0\leq q<1,\quad
\lambda>0\\
v>0 & \mathrm{in\ }\Omega,\\
v=0 & \mathrm{on\ }\partial\Omega
\end{array}
\right.   \label{singdiric}$$ which will be used in the paper.  A weak solution of (\[singdiric\]) is a function $v\in W_{0}^{1,p}(\Omega)$ such that $\operatorname{essinf}_{K}v>0$ in each compact $K\subset\Omega$ and $$\int_{\Omega}\left\vert \nabla v\right\vert ^{p-2}\nabla v\cdot\nabla
\varphi\mathrm{d}x=\lambda\int_{\Omega}v^{q-1}\varphi\mathrm{d}x,\;\mathrm{for\ all}\;\varphi\in W_{0}^{1,p}(\Omega).\label{weak+}$$

Next, we present a simple uniqueness proof for (\[singdiric\]), which makes use of the following well-known inequality: $$(\left\vert x\right\vert ^{p-2}x-\left\vert y\right\vert ^{p-2}y)\cdot
(x-y)\geq0\;\mathrm{for\ all}\;x,y\in\mathbb{R}^{N}.  \label{vectors}$$

\[propuniq\]Let $u_{1},u_{2}\in W_{0}^{1,p}(\Omega)$ be weak solutions of (\[singdiric\]).  Then, $u_{2}=u_{1}$ in $\Omega.$

Taking $\varphi=u_{2}-u_{1}$ in (\[weak+\]) we obtain $$\int_{\Omega}(\left\vert \nabla u_{2}\right\vert ^{p-2}\nabla u_{2}-\left\vert
\nabla u_{1}\right\vert ^{p-2}\nabla u_{1})\cdot\nabla(u_{2}-u_{1})\mathrm{d}x=\lambda\int_{\Omega}\left(  u_{2}^{q-1}-u_{1}^{q-1}\right)
(u_{2}-u_{1})\mathrm{d}x.  \label{weak++}$$

It follows from (\[vectors\]) that the integrand in the left-hand side of (\[weak++\]) is nonnegative.  It is easy to see that the integrand of the right-hand side of (\[weak++\]) cannot be positive.  Thus, both of them must be null almost everywhere in $\Omega,$ which implies that $u_{2}=u_{1}$ in $\Omega.$

In the sequel we derive estimates for the weak solutions of (\[singdiric\]) depending explicitly on $q\in\lbrack0,1).$

Let us recall that $$\lambda_{q}(D^{\ast})\leq\lambda_{q}(D),\quad0<q<p^{\star} \label{aux1}$$ where $D$ is a general bounded and smooth domain of $\mathbb{R}^{N}$ and $D^{\ast}$ is the ball centered at the origin and with the same volume as $D,$ that is, $\left\vert D^{\ast}\right\vert =\left\vert D\right\vert .$

Inequality (\[aux1\]) comes from well known properties of Schwarz symmetrization (see [@Kw]) and, among other important utilities, it provides a lower bound for $\lambda_{q}(D)$ in terms of $\left\vert
D\right\vert $ and $\lambda_{q}(B_{1}),$ where $B_{1}$ denotes the unit ball of $\mathbb{R}^{N}.$ In fact, one can show that $$\lambda_{q}(D^{\ast})=\lambda_{q}(B_{1})\left(  \frac{\left\vert D^{\ast
}\right\vert }{\omega_{N}}\right)  ^{1-\frac{p}{N}-\frac{p}{q}},\quad
0<q<p^{\star} \label{aux2}$$ where $\omega_{N}=\left\vert B_{1}\right\vert .$

Hence, by combining (\[aux1\]) and (\[aux2\]) one obtains the following version of the well-known Poincaré-Sobolev inequality$$\left(  \int_{D}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}\leq\frac{\left\vert D\right\vert ^{\frac{p}{q}+\frac{p}{N}-1}}{\lambda
_{q}(B_{1})\left\vert B_{1}\right\vert ^{\frac{p}{q}+\frac{p}{N}-1}}\left\Vert
\nabla v\right\Vert _{L^{p}(D)}^{p},\;\mathrm{for\ all}\;v\in W_{0}^{1,p}(D)\;\mathrm{and}\;0<q<p^{\star}.$$ When $q=p$ we have $$\int_{D}|v|^{p}\mathrm{d}x\leq\frac{|D|^{\frac{p}{N}}}{C_{N,p}}\int_{D}|\nabla
v|^{p}\,\mathrm{d}x,\;\mathrm{for\ all}\;v\in W_{0}^{1,p}(D), \label{D}$$ where $C_{N,p}=\lambda_{p}(B_{1})\left\vert B_{1}\right\vert ^{\frac{p}{N}},$ a positive constant that depends only on $p$ and $N.$

The following lemma is an adaptation of [@GECCM Theorem 4.1] which, in its turn, is based on classical set level techniques (see [@Bandle; @Ladyz]). 

\[linfty\]If $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (\[singdiric\]), then $u\in L^{\infty}(\Omega)$ and$$\left\Vert u\right\Vert _{\infty}\leq K_{N,p}\left\vert \Omega\right\vert
^{\frac{p}{N(p-q)}}\lambda^{\frac{1}{p-q}}, \label{x4b}$$ where $K_{N,p}$ is a positive constant depending only on $N$ and $p.$

For each $t>0,$ let $$E_{t}:=\left\{  x\in\Omega:u>t\right\}  \;\mathrm{and}\;\left(  u-t\right)
_{+}:=\max\left\{  u-t,0\right\}  \in W_{0}^{1,p}\left(  \Omega\right)  .$$

Let us suppose $\left\vert E_{t}\right\vert >0.$ Since $$\int_{\Omega}\left\vert \nabla u\right\vert ^{p-2}\nabla u\cdot\nabla\left(
u-t\right)  _{+}\mathrm{d}x=\int_{E_{t}}\left\vert \nabla u\right\vert
^{p}\mathrm{d}x$$ and$$\int_{\Omega}u^{q-1}\left(  u-t\right)  _{+}\mathrm{d}x=\int_{E_{t}}u^{q-1}\left(  u-t\right)  \mathrm{d}x\leq t^{q-1}\int_{E_{t}}\left(
u-t\right)  \mathrm{d}x,$$ (note that $q-1<0$) we obtain from (\[weak+\]) that$$\int_{E_{t}}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x\leq\lambda
t^{q-1}\int_{E_{t}}\left(  u-t\right)  \mathrm{d}x.  \label{x1}$$

Now, we estimate $\int_{E_{t}}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x$ from below.  For this, we apply Hölder inequality and the estimate (\[D\]) with $D=E_{t}$ to obtain$$\left(  \int_{E_{t}}\left(  u-t\right)  \mathrm{d}x\right)  ^{p}\leq
|E_{t}|^{p-1}\int_{E_{t}}\left(  u-t\right)  ^{p}\mathrm{d}x\leq\frac
{|E_{t}|^{p-1}|E_{t}|^{\frac{p}{N}}}{C_{N,p}}\int_{E_{t}}|\nabla
u|^{p}\,\mathrm{d}x.$$ Hence, $$C_{N,p}|E_{t}|^{-\frac{p}{N}+1-p}\left(  \int_{E_{t}}\left(  u-t\right)
\mathrm{d}x\right)  ^{p}\leq\int_{E_{t}}\left\vert \nabla u\right\vert
^{p}\mathrm{d}x$$ and, by taking into account (\[x1\]), we get$$C_{N,p}|E_{t}|^{-\frac{p}{N}+1-p}\left(  \int_{E_{t}}\left(  u-t\right)
\mathrm{d}x\right)  ^{p}\leq\lambda t^{q-1}\int_{E_{t}}\left(  u-t\right)
\mathrm{d}x,$$ which is equivalent to$$\left(  \int_{E_{t}}\left(  u-t\right)  \mathrm{d}x\right)  ^{p-1}\leq
\frac{\lambda}{C_{N,p}}t^{q-1}\left\vert E_{t}\right\vert ^{\frac{p+N(p-1)}{N}}.$$

This latter inequality can be rewritten as$$\left(  \int_{E_{t}}\left(  u-t\right)  \mathrm{d}x\right)  ^{\frac
{N(p-1)}{p+N(p-1)}}\leq\left(  \frac{\lambda}{C_{N,p}}t^{q-1}\right)
^{\frac{N}{p+N(p-1)}}\left\vert E_{t}\right\vert .  \label{x2}$$

Let us define $$f(t):=\int_{E_{t}}\left(  u-t\right)  \mathrm{d}x=\int_{t}^{\infty}\left\vert
E_{s}\right\vert \mathrm{d}s,$$ where the second equality follows from Cavalieri’s principle. 

Since $f^{\prime}\left(  t\right)  =-\left\vert E_{t}\right\vert $ the inequality in (\[x2\]) can be rewritten as $$t^{\frac{(1-q)N}{p+N(p-1)}}\leq-\left(  \frac{\lambda}{C_{N,p}}\right)
^{\frac{N}{p+N(p-1)}}f\left(  t\right)  ^{-\frac{N(p-1)}{p+N(p-1)}}f^{\prime
}\left(  t\right)  .  \label{x3}$$

Integration of (\[x3\]) yields$$\begin{aligned}
\frac{p+N(p-1)}{p+N(p-q)}t^{\frac{p+N(p-q)}{p+N(p-1)}}  &  \leq\left(
\frac{\lambda}{C_{N,p}}\right)  ^{\frac{N}{p+N(p-1)}}\frac{p+N(p-1)}{p}\left[
f(0)^{\frac{p}{p+N(p-1)}}-f(t)^{\frac{p}{p+N(p-1)}}\right] \\
&  \leq\left(  \frac{\lambda}{C_{N,p}}\right)  ^{\frac{N}{p+N(p-1)}}\frac{p+N(p-1)}{p}\left(  \left\Vert u\right\Vert _{1}\right)  ^{\frac
{p}{p+N(p-1)}}<\infty.\end{aligned}$$ We have concluded that if $\left\vert E_{t}\right\vert >0$ then $t\leq K,$ where $K$ is a positive constant that does not depend on $t.$ Of course, this implies that $\left\Vert u\right\Vert _{\infty}<\infty.$

Hence, we have$$\begin{aligned}
\frac{p+N(p-1)}{p+N(p-q)}t^{\frac{p+N(p-q)}{p+N(p-1)}}  &  \leq\left(
\frac{\lambda}{C_{N,p}}\right)  ^{\frac{N}{p+N(p-1)}}\frac{p+N(p-1)}{p}\left(
\left\Vert u\right\Vert _{1}\right)  ^{\frac{p}{p+N(p-1)}}\\
&  \leq\left(  \frac{\lambda}{C_{N,p}}\right)  ^{\frac{N}{p+N(p-1)}}\frac{p+N(p-1)}{p}\left(  \left\Vert u\right\Vert _{\infty}\left\vert
\Omega\right\vert \right)  ^{\frac{p}{p+N(p-1)}}$$ and then, after making $t\rightarrow\left\Vert u\right\Vert _{\infty},$ we obtain $$\left\Vert u\right\Vert _{\infty}\leq\left(  \frac{\lambda}{C_{N,p}}\right)
^{\frac{1}{p-q}}\left(  \frac{p+N(p-q)}{p}\right)  ^{\frac{p+N(p-1)}{N(p-q)}}\left\vert \Omega\right\vert ^{\frac{p}{N(p-q)}}$$ which leads to (\[x4b\]) with$$K_{N,p}:=\sup_{0\leq q\leq1}C_{N,p}^{-\frac{1}{p-q}}\left(  \frac{p+N(p-q)}{p}\right)  ^{\frac{p+N(p-1)}{N(p-q)}}.$$

In the next lemma, $\phi_{p}\in W_{0}^{1,p}(\Omega)$ denotes the $p$-torsion function of $\Omega,$ that is, the weak solution of the $p$-torsional creep problem$$\left\{
\begin{array}
[c]{ll}-\Delta_{p}v=1 & \mathrm{in\ }\Omega,\\
v=0 & \mathrm{on\ }\partial\Omega. 
\end{array}
\right.$$ It is well known that the function $\phi_{p}$ is positive in $\Omega$ and belongs to $C^{1,\alpha}(\overline{\Omega}),$ for some $\alpha\in(0,1).$

\[lowb\]If $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (\[singdiric\]), then $$0<\left(  K_{N,p}\left\vert \Omega\right\vert ^{\frac{p}{N(p-q)}}\right)
^{\frac{q-1}{p-1}}\lambda^{\frac{1}{p-q}}\phi_{p}(x)\leq u(x),\quad
\mathrm{for\ almost\ every}\;x\in\Omega.  \label{a1}$$

Let $0\leq\varphi\in W_{0}^{1,p}(\Omega)$ and $c:=\left(  \lambda\left\Vert
u\right\Vert _{\infty}^{q-1}\right)  ^{\frac{1}{p-1}}.$ Since $$\begin{aligned}
\int_{\Omega}\left\vert \nabla u\right\vert ^{p-2}\nabla u\cdot\nabla
\varphi\mathrm{d}x  &  =\lambda\int_{\Omega}u^{q-1}\varphi\mathrm{d}x\\
&  \geq\lambda\left\Vert u\right\Vert _{\infty}^{q-1}\int_{\Omega}\varphi\mathrm{d}x\\
&  =\int_{\Omega}c^{p-1}\varphi\mathrm{d}x=\int_{\Omega}\left\vert
\nabla(c\phi_{p})\right\vert ^{p-2}\nabla(c\phi_{p})\cdot\nabla\varphi
\mathrm{d}x,\end{aligned}$$ the weak comparison principle guarantees that $$c\phi_{p}(x)\leq u(x),\quad\mathrm{for\ almost\ every}\;x\in\Omega.$$ This leads to (\[a1\]) since (\[x4b\]) implies that $$c\geq\lambda^{\frac{1}{p-1}}(K_{N,p}\left\vert \Omega\right\vert ^{\frac
{p}{N(p-q)}}\lambda^{\frac{1}{p-q}})^{\frac{q-1}{p-1}}=\left(  K_{N,p}\left\vert \Omega\right\vert ^{\frac{p}{N(p-q)}}\right)  ^{\frac{q-1}{p-1}}\lambda^{\frac{1}{p-q}},\quad\mathrm{for\ almost\ every}\;x\in\Omega.$$

\[regu\]It follows from Lemma \[linfty\] and Lemma \[lowb\] that if $u\in W_{0}^{1,p}(\Omega)$ is a weak solution of (\[singdiric\]) then $$0<A\lambda^{\frac{p-1}{p-q}}\leq\lambda u(x)^{q-1}\leq B\lambda^{\frac
{p-1}{p-q}}\phi_{p}(x)^{q-1},\quad\mathrm{for\ almost\ every}\;x\in\Omega,$$ where $A$ and $B$ are positive constants that depend only on $N,p$ and $\left\vert \Omega\right\vert .$ This fact implies that if $\Omega^{\prime}$ is a subdomain of $\Omega$ such that $\overline{\Omega^{\prime}}\subset
\Omega,$ then $\lambda u^{q-1}$ is bounded in $\overline{\Omega^{\prime}}.$

The main results\[S2\]
======================

One can check, as a simple application of the Hölder inequality, that for each $\ v\in L^{1}(\Omega)$ the function $q\in(0,1]\mapsto\left(  \frac
{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert
^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}$ is increasing.  This fact has two immediate consequences: it implies that$$0\leq\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert
}\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=\inf_{0<s\leq1}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega
}\left\vert v\right\vert ^{s}\mathrm{d}x\right)  ^{\frac{1}{s}}\leq
\frac{\left\Vert v\right\Vert _{1}}{\left\vert \Omega\right\vert },\quad v\in
L^{1}(\Omega)$$ and also that the function $q\in(0,1]\mapsto\lambda_{q}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{q}}$ is decreasing, so that we can define$$\mu(\Omega):=\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{q}}=\sup_{0<s\leq1}\lambda_{s}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{s}}.$$ Of course, $$0<\lambda_{1}(\Omega)\left\vert \Omega\right\vert ^{p}\leq\mu(\Omega
)\leq\infty.  \label{lowbmu}$$

Our first goal in this section is to show that $\mu(\Omega)<\infty.$

One has$$\lim_{q\rightarrow0^{+}}\left(  \int_{0}^{1}(1-t^{\frac{1}{q}})^{N}\mathrm{d}t\right)  ^{\frac{1}{q}}=e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots
-\frac{1}{N}},\quad N\geq2.$$

We have$$\lim_{q\rightarrow0^{+}}\left(  \int_{0}^{1}(1-t^{\frac{1}{q}})^{N}\mathrm{d}t\right)  ^{\frac{1}{q}}=\lim_{s\rightarrow\infty}\left(  \int
_{0}^{1}(1-t^{s})^{N}\mathrm{d}t\right)  ^{s}=e^{L}.$$ where$$\begin{aligned}
L  &  :=\lim_{s\rightarrow\infty}\frac{\ln\left(  \int_{0}^{1}(1-t^{s})^{N}\mathrm{d}t\right)  }{s^{-1}}\\
&  =\lim_{s\rightarrow\infty}\frac{s^{2}N\int_{0}^{1}(1-t^{s})^{N-1}t^{s}\ln
t\mathrm{d}t}{\int_{0}^{1}(1-t^{s})^{N}\mathrm{d}t}=\lim_{s\rightarrow\infty
}s^{2}N\int_{0}^{1}(1-t^{s})^{N-1}t^{s}\ln t\mathrm{d}t.\end{aligned}$$

After making the change of variable $\tau=t^{s}$ in the latter integral, we obtain$$L=N\lim_{s\rightarrow\infty}\int_{0}^{1}(1-\tau)^{N-1}\tau^{\frac{1}{s}}\ln\tau\mathrm{d}\tau=N\int_{0}^{1}(1-\tau)^{N-1}\ln\tau\mathrm{d}\tau.$$

In order to finish the proof, it is enough to verify that$$N\int_{0}^{1}(1-\tau)^{N-1}\ln\tau\mathrm{d}\tau=-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N},\quad N\geq2.  \label{IN}$$ For this, let $I(N):=N\int_{0}^{1}(1-\tau)^{N-1}\ln\tau\mathrm{d}\tau.$ After some simple calculations one can show that $$I(N+1)=I(N)-\frac{1}{N+1},\quad N\geq2.  \label{INa}$$

It is easy to check that $I(2)=-1-\dfrac{1}{2}.$ Hence, by using the recursive formula (\[INa\]), we arrive at (\[IN\]). 

\[P3\]Suppose that $\Omega$ is a bounded domain, star-shaped with respect to $x_{0}\in\mathbb{R}^{N}.$ There exists $\rho\in C(\overline{\Omega})$ such that: $0<\rho\leq1$ in $\Omega,$ $\rho(x_{0})=1,$ $\rho=0$ on $\partial\Omega$ and$$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert \rho\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}.$$

In particular, any function $v\in W_{0}^{1,p}(\Omega)$ such that $v\geq
\rho\quad\mathrm{in\ }\Omega$ satisfies $$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}\geq e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}>0.$$

We will assume in this proof, without loss of generality, that $x_{0}=0.$

For each $0\not =x\in\overline{\Omega},$ let $r(x)$ be the unique positive number such that $$r(x)x\in\partial\Omega.$$ Of course, $r(x)\geq1$ and $r(x)\rightarrow\infty$ as $x\rightarrow0.$ Moreover, if $x\in\Omega$ and $\alpha>0$ is such that $\alpha x\in\Omega,$ then $r(\alpha x)\alpha x=r(x)x,$ so that$$r(\alpha x)=\frac{r(x)}{\alpha}.$$

Let us define $\rho:\overline{\Omega}\mapsto\lbrack0,1]$ by$$\rho(x):=\left\{
\begin{array}
[c]{ll}1-\dfrac{1}{r(x)} & \mathrm{if}\quad x\in\overline{\Omega}\quad,\quad
x\not =0.\\
1 & \mathrm{if}\quad x=0. 
\end{array}
\right.$$ The graph of $\rho$ in $\mathbb{R}^{N}\times\mathbb{R}$ is the cone of base $\Omega,$ height $1$ and vertex at the point $(0,1)\in\mathbb{R}^{N}\times\mathbb{R}.$

For each $t\in\lbrack0,1)$ the change of variable $x=(1-t)y$ yields $$\left\vert \left\{  \rho(x)>t\right\}  \right\vert =\int_{\left\{
\rho(x)>t\right\}  }\mathrm{d}x=\int_{\left\{  \rho(y)>0\right\}  }(1-t)^{N}\mathrm{d}y=(1-t)^{N}\left\vert \Omega\right\vert .  \label{cone1}$$ Indeed, by taking $\alpha=(1-t)$ one has$$\rho(\alpha y)=1-\frac{1}{r(\alpha y)}=1-\frac{\alpha}{r(y)}=1-\alpha
+\alpha\left(  1-\frac{1}{r(y)}\right)  =1-\alpha+\alpha\rho(y).$$ It follows that $$t<\rho((1-t)y)=t+(1-t)\rho(y)\Longleftrightarrow0<\rho(y).$$

Thus, (\[cone1\]) and Cavalieri’s principle yield $$\int_{\Omega}\rho^{q}\mathrm{d}x=\int_{0}^{1}\left\vert \left\{  \rho
(x)^{q}>t\right\}  \right\vert \mathrm{d}t=\int_{0}^{1}\left\vert \left\{
\rho(x)>t^{\frac{1}{q}}\right\}  \right\vert \mathrm{d}t=\int_{0}^{1}(1-t^{\frac{1}{q}})^{N}\left\vert \Omega\right\vert \mathrm{d}t,$$ so that$$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\rho^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=\lim_{q\rightarrow
0^{+}}\left(  \int_{0}^{1}(1-t^{\frac{1}{q}})^{N}\mathrm{d}t\right)
^{\frac{1}{q}}=e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}.$$

If $\Omega=B_{R}$ is the ball centered at the origin with radius $R,$ then $\rho(x)=1-\dfrac{\left\vert x\right\vert }{R}$ and$$\frac{1}{\left\vert B_{R}\right\vert }\int_{B_{R}}\rho^{q}\mathrm{d}x=\int
_{0}^{1}(1-t^{\frac{1}{q}})^{N}\mathrm{d}t.$$

In the proof of the following theorem we will write $\Omega$ as a finite union of star-shaped subdomains.  This decomposition is quite general in the sense that it is valid for bounded domains with low regularity as, for instance, those with Lipschitz boundary (see [@Galdi Lemma II.1.3]). 

\[main1\]There exists $v\in W_{0}^{1,p}(\Omega)$ such that $v>0$ in $\Omega$ and $$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=1.$$

Moreover,$$\mu(\Omega)\leq\left\Vert \nabla v\right\Vert _{p}^{p}.  \label{mufin}$$

Let $\Omega_{1},\Omega_{2},\ldots,\Omega_{m}$ be star-shaped subdomains of $\Omega$ such that $\Omega=\bigcup\limits_{j=1}^{m}\Omega_{j}$ (not necessarily disjoint). 

According Lemma \[P3\], for each $j\in\Lambda:=\left\{  1,2,\ldots
,m\right\}  $ we can take $v_{j}\in W_{0}^{1,p}(\Omega_{j})$ such that $v_{j}>0$ in $\Omega_{j}$ and $$e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}\leq\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega_{j}\right\vert }\int_{\Omega_{j}}\left\vert v_{j}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=\inf_{0<s\leq1}\left(  \frac{1}{\left\vert \Omega_{j}\right\vert }\int_{\Omega_{j}}\left\vert v_{j}\right\vert ^{s}\mathrm{d}x\right)
^{\frac{1}{s}}.$$ Thus,$$e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}\leq\left(  \frac
{1}{\left\vert \Omega_{j}\right\vert }\int_{\Omega_{j}}\left\vert
v_{j}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}},\quad j\in
\Lambda,\quad0<q\leq1.$$

By extending $v_{j}$ to zero outside $\Omega_{j}$ we can consider that $v_{j}$ belongs to $W_{0}^{1,p}(\Omega).$ Thus, $$V:=\sum_{j=1}^{m}v_{j}\in W_{0}^{1,p}(\Omega).$$

Now, let $q_{n}\rightarrow0^{+}$ and, for each $n\in\mathbb{N},$ let $j_{n}\in\Lambda$ be such that $$\frac{1}{\left\vert \Omega_{j_{n}}\right\vert }\int_{\Omega_{j_{n}}}\left\vert
v_{j_{n}}\right\vert ^{q_{n}}\mathrm{d}x=\min\left\{  \frac{1}{\left\vert
\Omega_{j}\right\vert }\int_{\Omega_{j}}\left\vert v_{j}\right\vert ^{q_{n}}\mathrm{d}x:j\in\Lambda\right\}  .$$ Then, for each fixed $n\in\mathbb{N}$ we have$$\begin{aligned}
\int_{\Omega}\left\vert V\right\vert ^{q_{n}}\mathrm{d}x  &  =\sum_{j=1}^{m}\left(  \frac{1}{\left\vert \Omega_{j}\right\vert }\int_{\Omega_{j}}\left\vert v_{j}\right\vert ^{q_{n}}\mathrm{d}x\right)  \left\vert \Omega
_{j}\right\vert \\
&  \geq\frac{1}{\left\vert \Omega_{j_{n}}\right\vert }\int_{\Omega_{j_{n}}}\left\vert v_{j_{n}}\right\vert ^{q_{n}}\mathrm{d}x\sum_{j=1}^{m}\left\vert
\Omega_{j}\right\vert \\
&  \geq\frac{\left\vert \Omega\right\vert }{\left\vert \Omega_{j_{n}}\right\vert }\int_{\Omega_{j_{n}}}\left\vert v_{j_{n}}\right\vert ^{q_{n}}\mathrm{d}x\geq\left\vert \Omega\right\vert (e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}})^{q_{n}},\end{aligned}$$ from which we conclude that $$\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert
V\right\vert ^{q_{n}}\mathrm{d}x\right)  ^{\frac{1}{q_{n}}}\geq e^{-1-\frac
{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}>0.$$ It follows that$$\theta:=\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert
}\int_{\Omega}\left\vert V\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}\geq e^{-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{N}}>0.$$

Therefore, the function $v:=\theta^{-1}V$ belongs to $W_{0}^{1,p}(\Omega),$ is positive in $\Omega$ and satisfies$$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=1.$$

Now, (\[mufin\]) follows immediately since$$\mu(\Omega)=\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{q}}\leq\lim_{q\rightarrow0^{+}}\frac{\left\Vert
\nabla v\right\Vert _{p}^{p}}{\left(  \int_{\Omega}\left\vert v\right\vert
^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}}\left\vert \Omega\right\vert ^{\frac
{p}{q}}=\left\Vert \nabla v\right\Vert _{p}^{p}.$$

It might be interesting to know an explicit lower bound for an abstract minimum such as $\mu(\Omega).$ Thus, by combining (\[lowbmu\]) with (\[aux1\]) we have $$\lambda_{1}(\Omega^{\ast})\left\vert \Omega\right\vert ^{p}\leq\lambda
_{1}(\Omega)\left\vert \Omega\right\vert ^{p}\leq\mu(\Omega), \label{lowbmu1}$$ where $\Omega^{\ast}$ denotes the ball centered at the origin with radius $R=(\left\vert \Omega\right\vert /\omega_{N})^{\frac{1}{N}},$ so that $\left\vert \Omega^{\ast}\right\vert =\left\vert \Omega\right\vert .$ It is a known fact (see [@GEJMAA]) that $\lambda_{1}(D)=\left\Vert \phi
_{p,D}\right\Vert _{1}^{1-p},$ where $D$ is a bounded domain and $\phi_{p,D}$ denotes its $p$-torsion function.  Since the $p$-torsion function of a ball $B_{R}$ of radius $R$ is explicitly given by $$\phi_{p,B_{R}}(x)=\frac{p-1}{p}N^{-\frac{1}{p-1}}(R^{\frac{p}{p-1}}-\left\vert
x\right\vert ^{\frac{p}{p-1}}),\quad0\leq\left\vert x\right\vert \leq R,$$ we can compute $\lambda_{1}(\Omega^{\ast})$ explicitly and so obtain, from (\[lowbmu1\]), the following estimate$$N\left(  N+\frac{p}{p-1}\right)  ^{p-1}(\omega_{N})^{\frac{p}{N}}\left\vert
\Omega\right\vert ^{1-\frac{p}{N}}\leq\mu(\Omega).  \label{lowbmu2}$$

For the sake of clarity, we will make use of the following scaling property in the next proof: $$\lambda_{q}(\Omega)\left\vert \Omega\right\vert ^{\frac{p}{q}}=\left\vert
\Omega\right\vert ^{1-\frac{N}{p}}\lambda_{q}(\Omega_{1}) $$ where $\Omega_{1}:=\left\{  \left\vert \Omega\right\vert ^{-\frac{1}{N}}x:x\in\Omega\right\}  $ is such that $\left\vert \Omega_{1}\right\vert =1.$ Thus, $$\mu(\Omega)=\left\vert \Omega\right\vert ^{1-\frac{N}{p}} \mu(\Omega_{1}). 
\label{scaling}$$ Let us define, $$\mathcal{M}(\Omega):=\left\{  v\in W_{0}^{1,p}(\Omega):\lim_{q\rightarrow
0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert
v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=1\right\}  .$$ It is easy to check that $\mathcal{M}(\Omega)$ has infinitely many elements by combining Lemma \[P3\] with the construction in the proof of Theorem \[main1\]. 

As pointed out in the Introduction, for each $q\in(0,1)$ there exist $\alpha_{q}\in(0,1)$ and $u_{q}\in W_{0}^{1,p}(\Omega)\cap C^{1,\alpha_{q}}(\overline{\Omega})$ such that $$u_{q}>0\quad\mathrm{in}\quad\Omega,\quad\lambda_{q}(\Omega)=\left\Vert \nabla
u_{q}\right\Vert _{p}^{p},\quad\int_{\Omega}\left\vert u_{q}\right\vert
^{q}\mathrm{d}x=1\label{ann1}$$ and$$\int_{\Omega}\left\vert \nabla u_{q}\right\vert ^{p-2}\nabla u_{q}\cdot
\nabla\varphi\mathrm{d}x=\lambda_{q}(\Omega)\int_{\Omega}u_{q}^{q-1}\varphi\mathrm{d}x\quad\mathrm{for}\,\mathrm{all}\;\varphi\in W_{0}^{1,p}(\Omega).\label{ann2}$$ The existence of $u_{q}$ satisfying (\[ann1\]) and (\[ann2\]) is proved in [@Anello], whereas the Hölder regularity of $u_{q}$ follows directly from [@GST1 Theorem 2.2 (i)].  Let us observe that the proof of (\[ann2\]) made in [@Anello] is restricted to the functions $\varphi\in
W_{0}^{1,p}(\Omega)$ such that $\operatorname*{supp}\varphi\subset\Omega.$ However, this restriction can be dropped by using arguments of [@GST; @Mohammed] based on Fatou’s lemma combined with the density of $C_{c}^{\infty
}(\Omega)$ in $W_{0}^{1,p}(\Omega).$ We will make use of these arguments in the next proof. 

\[theoexist\]For each $q\in(0,1),$ let $u_{q}\in W_{0}^{1,p}(\Omega)\cap
C^{1,\alpha_{q}}(\overline{\Omega})$ satisfying (\[ann1\]) and (\[ann2\]).  There exists $u\in\mathcal{M}(\Omega)\cap C^{0,\alpha}(\overline{\Omega}),$ for some $\alpha\in(0,1),$ such that:

1.   $u=\lim\limits_{q\rightarrow0^{+}}(\left\vert \Omega\right\vert
    ^{\frac{1}{q}}u_{q})$ $\mathrm{in}$ $W_{0}^{1,p}(\Omega);$

2.   $\mu(\Omega)=\left\Vert \nabla u\right\Vert _{p}^{p}=\min\left\{
    \left\Vert \nabla v\right\Vert _{p}^{p}:v\in\mathcal{M}(\Omega)\right\}  ;$

3.   $-\Delta_{p}u=\frac{\mu(\Omega)}{\left\vert \Omega\right\vert
    }u^{-1},\quad\mathrm{in}\quad\Omega;$

4.   $0<A\mu(\Omega)^{\frac{1}{p}}\phi_{p}(x)\leq u(x)\leq B\mu
    (\Omega)^{\frac{1}{p}},$ for almost every $x\in\Omega,$ where $A$ and $B$ are positive constants depending only on $N,$ $p$ and $\left\vert \Omega
    \right\vert .$

Taking (\[scaling\]) into account, we assume in this proof, without loss of generality, that $\left\vert \Omega\right\vert =1.$ Thus, $$\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=\lim_{q\rightarrow0^{+}}\left\Vert
\nabla u_{q}\right\Vert _{p}^{p}=\mu(\Omega)\in(0,\infty).  \label{l01}$$

Since $$\lambda_{q}(\Omega)\left(  \int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}\leq\left\Vert \nabla v\right\Vert _{p}^{p}\quad\mathrm{for}\,\mathrm{all}\;v\in W_{0}^{1,p}(\Omega), \label{c}$$ we have $$\mu(\Omega)\leq\left\Vert \nabla v\right\Vert _{p}^{p}\quad\mathrm{for}\,\mathrm{all}\;v\in\mathcal{M}(\Omega).  \label{g}$$

It follows from (\[l01\]) that there exist $q_{n}\rightarrow0^{+}$ and $u\in
W_{0}^{1,p}(\Omega)$ such that $u\geq0$ in $\Omega,$ $u_{q_{n}}\rightharpoonup
u$ (weakly) in $W_{0}^{1,p}(\Omega)$ and $u_{q_{n}}\rightarrow u$ pointwise almost everywhere in $\Omega.$ Hence,$$\left\Vert \nabla u\right\Vert _{p}\leq\liminf_{n\rightarrow\infty}\left\Vert
\nabla u_{q_{n}}\right\Vert _{p}=\lim_{n\rightarrow\infty}\left\Vert \nabla
u_{q_{n}}\right\Vert _{p}=\lim_{n\rightarrow\infty}\lambda_{q_{n}}(\Omega)^{\frac{1}{p}}=\mu(\Omega)^{\frac{1}{p}}.  \label{d}$$

We note from (\[c\]), with $v=u,$ that $$\mu(\Omega)\lim_{q\rightarrow0^{+}}\left(  \int_{\Omega}\left\vert
u\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}\leq\left\Vert \nabla
u\right\Vert _{p}^{p}.$$ Combining this estimate with (\[d\]) we obtain $$\lim_{q\rightarrow0^{+}}\left(  \int_{\Omega}\left\vert u\right\vert
^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}\leq1.  \label{a}$$

On the other hand, for each $s\in(0,1)$ and every $n$ large enough (such that $q_{n}<s$), we have$$1=\left(  \int_{\Omega}\left\vert u_{q_{n}}\right\vert ^{q_{n}}\mathrm{d}x\right)  ^{\frac{1}{q_{n}}}\leq\left(  \int_{\Omega}\left\vert u_{q_{n}}\right\vert ^{s}\mathrm{d}x\right)  ^{\frac{1}{s}}.$$ Hence,$$1\leq\lim_{n\rightarrow\infty}\left(  \int_{\Omega}\left\vert u_{q_{n}}\right\vert ^{s}\mathrm{d}x\right)  ^{\frac{1}{s}}=\left(  \int_{\Omega
}\left\vert u\right\vert ^{s}\mathrm{d}x\right)  ^{\frac{1}{s}},$$ where we have used Dominated Convergence Theorem, since$$0\leq u_{q_{n}}\leq K_{N,p}\lambda_{q_{n}}(\Omega)^{\frac{1}{p-q_{n}}}\leq
K_{N,p}\mu(\Omega)^{\frac{1}{p-q_{n}}}$$ according Lemma \[linfty\].  Thus, we conclude that $$1\leq\lim_{s\rightarrow0^{+}}\left(  \int_{\Omega}\left\vert u\right\vert
^{s}\mathrm{d}x\right)  ^{\frac{1}{s}}.  \label{b}$$

Gathering (\[a\]) and (\[b\]) we obtain$$\lim_{q\rightarrow0^{+}}\left(  \int_{\Omega}\left\vert u\right\vert
^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=1.$$ It follows that $u\in\mathcal{M}(\Omega)$ and thus, by combining (\[g\]) and (\[d\]) we conclude that$$\mu(\Omega)^{\frac{1}{p}}=\left\Vert \nabla u\right\Vert _{p}=\lim
_{n\rightarrow\infty}\left\Vert \nabla u_{q_{n}}\right\Vert _{p}, \label{f}$$ which ends the proof of the claim . 

Taking into account the weak convergence $u_{q_{n}}\rightharpoonup u,$ the second equality in (\[f\]) implies that $u_{q_{n}}\rightarrow u$ (strongly) in $W_{0}^{1,p}(\Omega).$ In view of (\[ann2\]) we have$$\int_{\Omega}\left\vert \nabla u_{q_{n}}\right\vert ^{p-2}\nabla u_{q_{n}}\cdot\nabla\varphi\mathrm{d}x=\lambda_{q_{n}}(\Omega)\int_{\Omega}u_{q_{n}}^{q_{n}-1}\varphi\mathrm{d}x,\quad\mathrm{for}\,\mathrm{all}\;\varphi\in
W_{0}^{1,p}(\Omega).$$

Strong convergence $u_{q_{n}}\rightarrow u$ in $W_{0}^{1,p}(\Omega)$ guarantees that$$\lim_{n\rightarrow\infty}\int_{\Omega}\left\vert \nabla u_{q_{n}}\right\vert
^{p-2}\nabla u_{q_{n}}\cdot\nabla\varphi\mathrm{d}x=\int_{\Omega}\left\vert
\nabla u\right\vert ^{p-2}\nabla u\cdot\nabla\varphi\mathrm{d}x,\quad
\mathrm{for}\,\mathrm{all}\;\varphi\in W_{0}^{1,p}(\Omega) \label{h3}$$ and (\[l01\]) guarantees that$$\lim_{n\rightarrow\infty}\lambda_{q_{n}}(\Omega)\int_{\Omega}u_{q_{n}}^{q_{n}-1}\varphi\mathrm{d}x=\mu(\Omega)\lim_{n\rightarrow\infty}\int_{\Omega
}u_{q_{n}}^{q_{n}-1}\varphi\mathrm{d}x,\quad\mathrm{for}\,\mathrm{all}\;\varphi\in W_{0}^{1,p}(\Omega).  \label{h2}$$

Let us first assume that $\varphi\subset\Omega.$ Then, Dominated Convergence Theorem yields $$\lim_{n\rightarrow\infty}\int_{\Omega}u_{q_{n}}^{q_{n}-1}\varphi
\mathrm{d}x=\int_{\Omega}u^{-1}\varphi\mathrm{d}x, \label{h1}$$ since Lemma \[linfty\] and Lemma \[lowb\] imply that $0<c_{1}\leq
u_{q_{n}}^{q_{n}-1}\varphi\leq c_{2}$ in $\varphi,$ where the constants $c_{1}$ and $c_{2}$ are uniform with respect to $n.$ Hence, by gathering (\[h3\]), (\[h2\]) and (\[h1\]) we have$$\int_{\Omega}\left\vert \nabla u\right\vert ^{p-2}\nabla u\cdot\nabla
\varphi\mathrm{d}x=\mu(\Omega)\int_{\Omega}\varphi u^{-1}\mathrm{d}x,\quad\mathrm{for}\,\mathrm{all}\;\varphi\in C_{c}^{\infty}(\Omega). 
\label{h4}$$

Thus, in order to prove we need to show that (\[h4\]) holds, in fact, for any $\varphi\in W_{0}^{1,p}(\Omega),$ which reduces to prove that (\[h1\]) holds for any $\varphi\in W_{0}^{1,p}(\Omega).$ We prove this by following arguments of [@GST; @Mohammed].  So, let $w\in W_{0}^{1,p}(\Omega)$ be arbitrary and take a sequence $\left\{  \xi_{n}\right\}  \subset
C_{c}^{\infty}(\Omega)$ of nonnegative functions such that $\xi_{n}\rightarrow\left\vert w\right\vert ,$ strongly in $W_{0}^{1,p}(\Omega)$ and pointwise almost everywhere in $\Omega.$ Hence, by applying: Fatou’s lemma, (\[h4\]) and Hölder’s inequality, we obtain$$\begin{aligned}
\left\vert \int_{\Omega}wu^{-1}\mathrm{d}x\right\vert  &  \leq\int_{\Omega
}\left\vert w\right\vert u^{-1}\mathrm{d}x\\
&  \leq\liminf_{n\rightarrow\infty}\int_{\Omega}\xi_{n}u^{-1}\mathrm{d}x\\
&  =\lim_{n\rightarrow\infty}\int_{\Omega}\left\vert \nabla u\right\vert
^{p-2}\nabla u\cdot\nabla\xi_{n}\mathrm{d}x\leq\left\Vert \nabla u\right\Vert
_{p}^{p-1}\lim_{n\rightarrow\infty}\left\Vert \nabla\xi_{n}\right\Vert
_{p}=\left\Vert \nabla u\right\Vert _{p}^{p-1}\left\Vert \nabla w\right\Vert
_{p}.\end{aligned}$$

Now, let $\varphi$ be an arbitrary function in $W_{0}^{1,p}(\Omega)$ and take $\left\{  \varphi_{n}\right\}  \subset C_{c}^{\infty}(\Omega)$ such that $\varphi_{n}\rightarrow\varphi$ strongly in $W_{0}^{1,p}(\Omega).$ Then, by using $\varphi_{n}-\varphi$ in the place of $w,$ we obtain$$\lim_{n\rightarrow\infty}\left\vert \int_{\Omega}(\varphi_{n}-\varphi
)u^{-1}\mathrm{d}x\right\vert \leq\left\Vert \nabla u\right\Vert _{p}^{p-1}\lim_{n\rightarrow\infty}\left\Vert \nabla(\varphi_{n}-\varphi
)\right\Vert _{p}=0,$$ which yields $$\lim_{n\rightarrow\infty}\int_{\Omega}\varphi_{n}u^{-1}\mathrm{d}x=\int_{\Omega}\varphi u^{-1}\mathrm{d}x.  \label{h5}$$

Since $\varphi_{n}\in C_{c}^{\infty}(\Omega)$ we obtain from (\[h4\]) that $$\lim_{n\rightarrow\infty}\int_{\Omega}\varphi_{n}u^{-1}\mathrm{d}x=\mu
(\Omega)^{-1}\lim_{n\rightarrow\infty}\int_{\Omega}\left\vert \nabla
u\right\vert ^{p-2}\nabla u\cdot\nabla\varphi_{n}\mathrm{d}x=\mu(\Omega
)^{-1}\int_{\Omega}\left\vert \nabla u\right\vert ^{p-2}\nabla u\cdot
\nabla\varphi\mathrm{d}x.  \label{h6}$$

Therefore, by combining (\[h5\]) with (\[h6\]) we conclude that (\[h1\]) holds true for any $\varphi\in W_{0}^{1,p}(\Omega),$ which proves the claim . 

Claim now follows after combining Lemma \[linfty\] with Lemma \[lowb\].  Theorem 2.2 (ii) of [@GST1] implies that $u\in C^{0,\alpha
}(\overline{\Omega})$ for some $\alpha\in(0,1).$

Claim follows from the uniqueness of the weak solutions of $$\left\{
\begin{array}
[c]{ll}-\Delta_{p}w=\frac{\mu(\Omega)}{\left\vert \Omega\right\vert }w^{-1} &
\mathrm{in\ }\Omega,\\
w>0 & \mathrm{in\ }\Omega,\\
w=0 & \mathrm{on\ }\partial\Omega,
\end{array}
\right.   \label{weakl0}$$ combined with the fact that $u_{q_{n}}\rightarrow u$ strongly in $W_{0}^{1,p}(\Omega).$ Indeed, these facts together imply that $u$ is the unique limit function of the family $\left\{  u_{q}\right\}  ,$ as $q\rightarrow
0^{+}.$

Our next goal is to prove that the solution $u$ of (\[weakl0\]) satisfies $$\int_{\Omega}\log u\mathrm{d}x=0.$$

\[into\]Let $v\in L^{1}(\Omega).$ Then $\log\left\vert v\right\vert $ is Lebesgue measurable in $\Omega$ and$$-\infty\leq\lim_{q\rightarrow0^{+}}\int_{\Omega}\left\vert v\right\vert
^{q}\log\left\vert v\right\vert \mathrm{d}x=\int_{\Omega}\log\left\vert
v\right\vert \mathrm{d}x\leq2e^{-1}\left\Vert v\right\Vert _{1}.$$

For every $x\in\Omega$ such that $\left\vert v(x)\right\vert \leq1$ the function $q\in(0,1]\mapsto-\left\vert v(x)\right\vert ^{q}\log\left\vert
v(x)\right\vert \in\lbrack0,\infty]$ is decreasing and $$\lim_{q\rightarrow0^{+}}\left[  -\left\vert v(x)\right\vert ^{q}\log\left\vert
v(x)\right\vert \right]  =-\log\left\vert v(x)\right\vert .$$ Therefore, it follows directly from Lebesgue’s Monotone Convergence Theorem that $\log\left\vert v\right\vert $ is Lebesgue measurable in the set $\left\{  x\in\Omega:\left\vert v(x)\right\vert \leq1\right\}  $ and$$-\infty\leq\lim_{q\rightarrow0^{+}}\int_{\left\{  \left\vert v\right\vert
\leq1\right\}  }\left\vert v\right\vert ^{q}\log\left\vert v\right\vert
\mathrm{d}x=\int_{\left\{  \left\vert v\right\vert \leq1\right\}  }\log\left\vert v\right\vert \mathrm{d}x\leq0.$$

Now, for every $x\in\Omega$ such that $\left\vert v(x)\right\vert \geq1$ the function $q\in(0,1]\mapsto\left\vert v(x)\right\vert ^{q}\log\left\vert
v(x)\right\vert \in\lbrack0,\infty]$ is increasing and $$\lim_{q\rightarrow0^{+}}\left\vert v(x)\right\vert ^{q}\log\left\vert
v(x)\right\vert =\log\left\vert v(x)\right\vert .$$ Moreover, for every $q\in(0,\frac{1}{2})$ one has $$0\leq\left\vert v(x)\right\vert ^{q}\log\left\vert v(x)\right\vert
\leq\left\vert v(x)\right\vert ^{\frac{1}{2}}\log\left\vert v(x)\right\vert
\leq2e^{-1}\left\vert v(x)\right\vert ^{\frac{1}{2}}\left\vert v(x)\right\vert
^{\frac{1}{2}}=2e^{-1}\left\vert v(x)\right\vert \in L^{1}(\Omega),$$ since $\max\limits_{t\geq1}t^{-\frac{1}{2}}\log t=2e^{-1}.$ Therefore, Lebesgue’s Dominated Convergence Theorem implies that $\log\left\vert
v\right\vert $ is integrable in the set $\left\{  x\in\Omega:\left\vert
v(x)\right\vert \geq1\right\}  $ and that$$0\leq\lim_{q\rightarrow0^{+}}\int_{\left\{  \left\vert v\right\vert
\geq1\right\}  }\left\vert v\right\vert ^{q}\log\left\vert v\right\vert
\mathrm{d}x=\int_{\left\{  \left\vert v\right\vert \geq1\right\}  }\log\left\vert v\right\vert \mathrm{d}x\leq2e^{-1}\left\Vert v\right\Vert
_{1}.$$

Thus, we have that $\log\left\vert v\right\vert $ is Lebesgue measurable in $\Omega$ and $$\begin{aligned}
\lim_{q\rightarrow0^{+}}\int_{\Omega}\left\vert v\right\vert ^{q}\log\left\vert v\right\vert \mathrm{d}x  &  =\lim_{q\rightarrow0^{+}}\int_{\left\{  \left\vert v\right\vert \leq1\right\}  }\left\vert v\right\vert
^{q}\log\left\vert v\right\vert \mathrm{d}x+\lim_{q\rightarrow0^{+}}\int_{\left\{  \left\vert v\right\vert >1\right\}  }\left\vert v\right\vert
^{q}\log\left\vert v\right\vert \mathrm{d}x\\
&  =\int_{\left\{  \left\vert v\right\vert \leq1\right\}  }\log\left\vert
v\right\vert \mathrm{d}x+\int_{\left\{  \left\vert v\right\vert >1\right\}
}\log\left\vert v\right\vert \mathrm{d}x\\
&  =\int_{\Omega}\log\left\vert v\right\vert \mathrm{d}x\leq2e^{-1}\left\Vert
v\right\Vert _{1}.\end{aligned}$$

\[intlog1\]For $v\in L^{1}(\Omega)$ define $$\theta_{v}:=\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega
\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)
^{\frac{1}{q}}\in\lbrack0,\infty)\;\mathrm{and}\;\beta_{v}:=\frac
{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert v\right\vert
\mathrm{d}x\in\lbrack-\infty,\infty).$$ Then $$\beta_{v}=\log\theta_{v}.  \label{logl}$$

It follows from L’Hôpital’s rule and Proposition \[into\] that$$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\lim_{q\rightarrow0^{+}}\int_{\Omega}\left\vert v\right\vert ^{q}\log\left\vert v\right\vert
\mathrm{d}x\right)  =\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert v\right\vert \mathrm{d}x\right)  .$$ Hence, (\[logl\]) follows. 

The following corollary is an immediate consequence of the previous proposition. 

\[log0\]Let $v\in W_{0}^{1,p}(\Omega).$ Then $v\in\mathcal{M}(\Omega)$ if, and only if, $$\int_{\Omega}\log\left\vert v\right\vert \mathrm{d}x=0.$$ In particular, $\int_{\Omega}\log\left\vert u\right\vert \mathrm{d}x=0,$ where $u$ is the solution of (\[weakl0\]). 

Minimizing the energy functional
--------------------------------

In this subsection, $u$ denotes the solution of (\[weakl0\]).  As we have shown, $u$ minimizes the functional $v\mapsto\left\Vert \nabla v\right\Vert
_{p}^{p}$ on $\mathcal{M}(\Omega).$ Let us show that $u$ is the unique, up to sign, with this property. 

\[unicM0\]Let $v\in\mathcal{M}(\Omega)$ such that $\mu(\Omega)=\left\Vert
\nabla v\right\Vert _{p}^{p}.$ Then, $v$ does not change sign in $\Omega.$

Let $\Omega_{+}:=\left\{  x\in\Omega:v(x)>0\right\}  ,$ $\Omega_{-}=\left\{
x\in\Omega:v(x)<0\right\}  ,$ $a_{+}:=\frac{\left\vert \Omega_{+}\right\vert
}{\left\vert \Omega\right\vert }$ and $a_{-}:=\frac{\left\vert \Omega
_{-}\right\vert }{\left\vert \Omega\right\vert }.$ For $0<q<1,$ we have$$\begin{aligned}
\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left\vert
v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}  &  =\left(  \frac{a_{+}}{\left\vert \Omega_{+}\right\vert }\int_{\Omega_{+}}\left\vert v_{+}\right\vert ^{q}\mathrm{d}x+\frac{a_{-}}{\left\vert \Omega_{-}\right\vert
}\int_{\Omega_{-}}\left\vert v_{-}\right\vert ^{q}\mathrm{d}x\right)
^{\frac{1}{q}}\\
&  \leq a_{+}\left(  \frac{1}{\left\vert \Omega_{+}\right\vert }\int
_{\Omega_{+}}\left\vert v_{+}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}+a_{-}\left(  \frac{1}{\left\vert \Omega_{-}\right\vert }\int_{\Omega_{-}}\left\vert v_{-}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}.\end{aligned}$$

It follows that$$\begin{aligned}
1 &  =\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert
}\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}\\
&  \leq(a_{+})\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega
_{+}\right\vert }\int_{\Omega_{+}}\left\vert v_{+}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}+(a_{-})\lim_{q\rightarrow0^{+}}\left(
\frac{1}{\left\vert \Omega_{-}\right\vert }\int_{\Omega_{-}}\left\vert
v_{-}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}.\end{aligned}$$

Let us suppose, by contradiction, that both $\left\vert \Omega_{+}\right\vert
$ and $\left\vert \Omega_{-}\right\vert $ are positive.  Without loss of generality, we assume that $$\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega_{-}\right\vert
}\int_{\Omega_{-}}\left\vert v_{-}\right\vert ^{q}\mathrm{d}x\right)
^{\frac{1}{q}}\leq\theta_{+}:=\lim_{q\rightarrow0^{+}}\left(  \frac
{1}{\left\vert \Omega_{+}\right\vert }\int_{\Omega_{+}}\left\vert
v_{+}\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}.$$ Then$$1\leq(a_{+}+a_{-})\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert
\Omega_{+}\right\vert }\int_{\Omega_{+}}\left\vert v_{+}\right\vert
^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=\theta_{+}.$$

It follows that $\theta_{+}^{-1}v_{+}\in\mathcal{M}(\Omega)$ and$$\mu(\Omega)\leq\left\Vert \nabla(\theta_{+}^{-1}v_{+})\right\Vert _{p}^{p}\leq\left\Vert \nabla(v_{+})\right\Vert _{p}^{p}\leq\left\Vert \nabla
v\right\Vert _{p}^{p}=\mu(\Omega).$$ Thus, $$\mu(\Omega)=\left\Vert \nabla v_{+}\right\Vert _{p}^{p}=\left\Vert \nabla
v\right\Vert _{p}^{p}=\left\Vert \nabla v_{+}\right\Vert _{p}^{p}+\left\Vert
\nabla v_{-}\right\Vert _{p}^{p},$$ implying that $\left\Vert \nabla v_{-}\right\Vert _{p}^{p}=0$ and, therefore, that $v_{-}=0$ $\mathrm{a.e}$ in $\Omega.$ This implies that $\left\vert \Omega_{-}\right\vert =0,$ which is a contradiction. 

\[unicM\]Let $v\in\mathcal{M}(\Omega)$ such that $\mu(\Omega)=\left\Vert
\nabla v\right\Vert _{p}^{p}.$ Then, $v=\pm u$ $\mathrm{a.e}$ in $\Omega.$

According Lemma \[unicM0\], we can assume, without loss of generality, that $v\geq0$ in $\Omega.$ Since $0<q<1,$ it is simple to check that $$\left(  \int_{\Omega}\left(  \frac{u+v}{2}\right)  ^{q}\mathrm{d}x\right)
^{\frac{1}{q}}\geq\left(  \int_{\Omega}\left(  \frac{u}{2}\right)
^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}+\left(  \int_{\Omega}\left(  \frac
{v}{2}\right)  ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}.$$ Thus,$$h:=\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\left(  \frac{u+v}{2}\right)  ^{q}\mathrm{d}x\right)  ^{\frac
{1}{q}}\geq\frac{1}{2}+\frac{1}{2}=1.  \label{h+}$$ Of course, $\dfrac{u+v}{2h}\in\mathcal{M}(\Omega).$ Thus, $$\mu(\Omega)\leq\left\Vert \nabla\left(  \dfrac{u+v}{2h}\right)  \right\Vert
_{p}^{p}\leq\frac{1}{(2h)^{p}}\left(  \left\Vert \nabla u\right\Vert
_{p}+\left\Vert \nabla v\right\Vert _{p}\right)  ^{p}=\frac{\left(
2\mu(\Omega)^{\frac{1}{p}}\right)  ^{p}}{(2h)^{p}}=\frac{\mu(\Omega)}{h^{p}},
\label{h-}$$ which implies that $h\leq1.$ Therefore, (\[h+\]) and (\[h-\]) imply that $h=1,$ and this fact in (\[h-\]) yields$$\left\Vert \nabla(u+v)\right\Vert _{p}=\left\Vert \nabla u\right\Vert
_{p}+\left\Vert \nabla v\right\Vert _{p}.$$ We conclude from this equality that $u=v$ in $\Omega.$

The next corollary shows that $\mu(\Omega)^{-1}$ is the best constant $C$ in the following log-Sobolev type inequality$$\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
v\right\vert ^{p}\mathrm{d}x\right)  \leq C\left\Vert \nabla v\right\Vert
_{p}^{p},\quad v\in W_{0}^{1,p}(\Omega)\label{logsob}$$ and that when $C=\mu(\Omega)^{-1}$ this inequality becomes an equality if, and only if, $v$ is a scalar multiple of $u.$

One has$$\mu(\Omega)=\min\left\{  \frac{\left\Vert \nabla v\right\Vert _{p}^{p}}{\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert v\right\vert ^{p}\mathrm{d}x\right)  }:v\in W_{0}^{1,p}(\Omega)\;\mathrm{and}\;\int_{\Omega}\log\left\vert v\right\vert
^{p}\mathrm{d}x>-\infty\right\}  .$$ Moreover, the minimum is reached only by scalar multiples of $u.$

Let $v\in W_{0}^{1,p}(\Omega)$ be such that $\int_{\Omega}\log\left\vert
v\right\vert ^{p}\mathrm{d}x>-\infty.$ Since $$\lambda_{q}(\Omega)\left\vert \Omega\right\vert ^{\frac{p}{q}}\leq
\frac{\left\Vert \nabla v\right\Vert _{p}^{p}}{\left(  \frac{1}{\left\vert
\Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}}$$ we obtain$$\mu(\Omega)=\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{q}}\leq\frac{\left\Vert \nabla v\right\Vert
_{p}^{p}}{\lim\limits_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert
\Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}}=\frac{\left\Vert \nabla v\right\Vert _{p}^{p}}{\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert v\right\vert ^{p}\mathrm{d}x\right)  }.$$

Since $\int_{\Omega}\log\left\vert u\right\vert \mathrm{d}x=0,$ we have$$\frac{\left\Vert \nabla u\right\Vert _{p}^{p}}{\exp\left(  \frac{1}{\left\vert
\Omega\right\vert }\int_{\Omega}\log\left\vert u\right\vert ^{p}\mathrm{d}x\right)  }=\left\Vert \nabla u\right\Vert _{p}^{p}=\mu(\Omega).$$

Of course, if $v\in W_{0}^{1,p}(\Omega)$ is such that $$\frac{\left\Vert \nabla v\right\Vert _{p}^{p}}{\exp\left(  \frac{1}{\left\vert
\Omega\right\vert }\int_{\Omega}\log\left\vert v\right\vert ^{p}\mathrm{d}x\right)  }=\mu(\Omega)$$ then $\theta_{v}^{-1}v\in\mathcal{M}(\Omega),$ where $$\theta_{v}=\lim\limits_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert
\Omega\right\vert }\int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{1}{q}}=\exp\left(  \frac{1}{\left\vert \Omega\right\vert
}\int_{\Omega}\log\left\vert v\right\vert ^{p}\mathrm{d}x\right)  >0,$$ and $\left\Vert \nabla(\theta_{v}^{-1}v)\right\Vert _{p}^{p}=\mu(\Omega).$ Hence, $\theta_{v}^{-1}v=\pm u,$ implying that $v$ is a scalar multiple of $u.$

Gathering (\[lowbmu2\]) and (\[logsob\]), with $C=\mu(\Omega)^{-1},$ we obtain$$\exp\left(  \frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
v\right\vert ^{p}\mathrm{d}x\right)  \leq C_{N,p,\left\vert \Omega\right\vert
}\left\Vert \nabla v\right\Vert _{p}^{p},\quad v\in W_{0}^{1,p}(\Omega),$$ where $C_{N,p,\left\vert \Omega\right\vert }:=N^{-1}\left(  N+\frac{p}{p-1}\right)  ^{1-p}(\omega_{N})^{-\frac{p}{N}}\left\vert \Omega\right\vert
^{\frac{p}{N}-1}.$

Now, let us define $J:W_{0}^{1,p}(\Omega)\mapsto(-\infty,\infty],$ the formal energy functional, by$$J(v):=\left\{
\begin{array}
[c]{ll}\dfrac{1}{p}{\displaystyle\int_{\Omega}}
\left\vert \nabla v\right\vert ^{p}\mathrm{d}x-\dfrac{\mu(\Omega)}{\left\vert
\Omega\right\vert }{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x, & \mathrm{if}\;{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x\in(-\infty,\infty)\\
\infty, & \mathrm{if}\;{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x=-\infty. 
\end{array}
\right.$$

We are going to show that $u$ is the unique minimizer of $J$ in $W_{0}^{1,p}(\Omega).$

\[intlog2\]One has$$J(v)\geq\frac{\mu(\Omega)}{p}-\frac{\mu(\Omega)}{p}\log\left(  \frac
{\mu(\Omega)}{\left\Vert \nabla v\right\Vert _{p}^{p}}\right)  -\frac
{\mu(\Omega)}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
v\right\vert \mathrm{d}x,\quad\mathrm{for}\,\mathrm{all}\;v\in W_{0}^{1,p}(\Omega).  \label{Jmin0}$$

Let $v\in W_{0}^{1,p}(\Omega)$ and consider the function$$g(t):=\frac{t^{p}}{p}\left\Vert \nabla v\right\Vert _{p}^{p}-\mu(\Omega)\log
t-\frac{\mu(\Omega)}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
v\right\vert \mathrm{d}x,\quad t>0.$$ It is easy to check that $J(tv)=g(\left\vert t\right\vert )$ and that $\min_{t>0}g(t)$ is the right-hand side of (\[Jmin0\]).  The result then follows, since $J(v)=g(1)\geq\min_{t>0}g(t).$

\[intlog3\]One has $$J(u)=\frac{\mu(\Omega)}{p}=\min\limits_{v\in W_{0}^{1,p}(\Omega)}J(v).$$

Since $$\mu(\Omega)={\displaystyle\int_{\Omega}}
\left\vert \nabla u\right\vert ^{p}\mathrm{d}x\;\mathrm{and}\;{\displaystyle\int_{\Omega}}
\log\left\vert u\right\vert \mathrm{d}x=0$$ we have $J(u)=\dfrac{\mu(\Omega)}{p}.$ Thus, we need only to prove that$$J(v)\geq\frac{\mu(\Omega)}{p},\quad\mathrm{for}\,\mathrm{all}\;v\in
W_{0}^{1,p}(\Omega).\label{Jmin1}$$

Let $v\in W_{0}^{1,p}(\Omega)$ and $\theta_{v}$ as in Proposition \[intlog1\].  If $\theta_{v}=0,$ then $J(v)=\infty$ and (\[Jmin1\]) holds trivially.  If $\theta_{v}>0$ then $\theta_{v}^{-1}v\in\mathcal{M}(\Omega)$ and $$\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert \theta
_{v}^{-1}v\right\vert \mathrm{d}x=0\;\mathrm{and}\;\mu(\Omega)\leq\left\Vert
\nabla(\theta_{v}^{-1}v)\right\Vert _{p}^{p}=\theta_{v}^{-p}\left\Vert \nabla
v\right\Vert _{p}^{p},$$ implying, respectively, that $$\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert
v\right\vert \mathrm{d}x=\log\theta_{v}$$ and $$\begin{aligned}
J(v)  &  \geq\frac{\mu(\Omega)}{p}-\frac{\mu(\Omega)}{p}\log\left(  \frac
{\mu(\Omega)}{\left\Vert \nabla v\right\Vert _{p}^{p}}\right)  -\mu
(\Omega)\log(\theta_{v})\\
&  \geq\frac{\mu(\Omega)}{p}-\frac{\mu(\Omega)}{p}\log\left(  \theta_{v}^{-p}\right)  -\mu(\Omega)\log(\theta_{v})=\frac{\mu(\Omega)}{p}.\end{aligned}$$

If $w\in W_{0}^{1,p}(\Omega)$ minimizes $J,$ then $w=\pm\theta_{w}u$ $\mathrm{a.e.}$ in $\Omega,$ where $$\theta_{w}:=\lim_{q\rightarrow0^{+}}\left(  \frac{1}{\left\vert \Omega
\right\vert }\int_{\Omega}\left\vert w\right\vert ^{q}\mathrm{d}x\right)
^{\frac{1}{q}}.$$

It follows from Lemma \[intlog2\] that $$\dfrac{\mu(\Omega)}{p}=J(w)\geq\frac{\mu(\Omega)}{p}-\frac{\mu(\Omega)}{p}\log\left(  \frac{\mu(\Omega)}{\left\Vert \nabla w\right\Vert _{p}^{p}}\right)  -\frac{\mu(\Omega)}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert w\right\vert \mathrm{d}x,$$ which implies that$$\frac{1}{p}\log\left(  \frac{\mu(\Omega)}{\left\Vert \nabla w\right\Vert
_{p}^{p}}\right)  +\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert w\right\vert \mathrm{d}x\geq0.  \label{Jmin2}$$

Of course, $\theta_{w}\in(0,\infty),$ so that $\theta_{w}^{-1}w\in
\mathcal{M}(\Omega).$ Thus, according Corollary \[log0\], $$\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert \theta
_{w}^{-1}w\right\vert \mathrm{d}x=0,$$ so that$$\log\theta_{w}=\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\log\left\vert w\right\vert \mathrm{d}x.$$ Hence, (\[Jmin2\]) yields $$\frac{1}{p}\log\left(  \frac{\mu(\Omega)}{\left\Vert \nabla w\right\Vert
_{p}^{p}}\right)  +\log\theta_{w}=\log\left(  \frac{\mu(\Omega)}{\left\Vert
\nabla(\theta_{w}^{-1}w)\right\Vert _{p}^{p}}\right)  ^{\frac{1}{p}}\geq0,$$ which is equivalent to $\mu(\Omega)\geq\left\Vert \nabla\theta_{w}^{-1}w\right\Vert _{p}^{p}.$

Since $\mu(\Omega)\leq\left\Vert \nabla\theta_{w}^{-1}w\right\Vert _{p}^{p}$ (recall that $\theta_{w}^{-1}w\in\mathcal{M}(\Omega)$), we conclude that $\mu(\Omega)=\left\Vert \nabla(\theta_{w}^{-1}w)\right\Vert _{p}^{p}.$ So, Theorem \[unicM\] yields $\theta_{w}^{-1}w=\pm u.$

We end this section by remarking that, for each $\lambda>0,$ a simple scaling argument shows that the function $$u_{\lambda}:=\left(  \frac{\lambda\left\vert \Omega\right\vert }{\mu(\Omega
)}\right)  ^{\frac{1}{p}}u$$ is the unique solution of the singular problem$$\left\{
\begin{array}
[c]{ll}-\Delta_{p}v=\lambda v^{-1} & \mathrm{in\ }\Omega,\\
v>0 & \mathrm{in\ }\Omega,\\
v=0 & \mathrm{on\ }\partial\Omega. 
\end{array}
\right.$$ Moreover, since $\int_{\Omega}\log u\mathrm{d}x=0$ and $\mu(\Omega)=\left\Vert
\nabla u\right\Vert _{p}^{p},$ the equality in (\[logsob\]) (with $C=\mu(\Omega)^{-1}$) yields$$\int_{\Omega}\log u_{\lambda}\mathrm{d}x=\frac{\left\vert \Omega\right\vert
}{p}\log\left(  \mu(\Omega)^{-1}\left(  \frac{\lambda\left\vert \Omega
\right\vert }{\mu(\Omega)}\right)  \left\Vert \nabla u\right\Vert _{p}^{p}\right)  =\frac{\left\vert \Omega\right\vert }{p}\log\left(  \frac
{\lambda\left\vert \Omega\right\vert }{\mu(\Omega)}\right)  .$$

We also note that $u_{\lambda}$ and $-u_{\lambda}$ are the unique minimizers of the functional $$J_{\lambda}(v):=\left\{
\begin{array}
[c]{ll}\dfrac{1}{p}{\displaystyle\int_{\Omega}}
\left\vert \nabla v\right\vert ^{p}\mathrm{d}x-\lambda{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x, & \mathrm{if}\;{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x\in(-\infty,\infty)\\
\infty, & \mathrm{if}\;{\displaystyle\int_{\Omega}}
\log\left\vert v\right\vert \mathrm{d}x=-\infty,
\end{array}
\right.$$ being $$J_{\lambda}(\pm u_{\lambda})=\frac{\mu(\Omega)}{p}-\frac{\lambda\left\vert
\Omega\right\vert }{p}\log\left(  \frac{\lambda\left\vert \Omega\right\vert
}{\mu(\Omega)}\right)  .$$

Asymptotics for the pair $(\lambda_{q}(\Omega),\left\Vert
u_{q}\right\Vert _{\infty})$
---------------------------------------------------------

In this subsection we describe the asymptotic behavior of $\lambda_{q}(\Omega),$ as $q\rightarrow0^{+}.$ Of course, if $\left\vert \Omega\right\vert
=1$, then $\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=\mu(\Omega)$ and, according items and of Theorem \[theoexist\]$$0<A\mu(\Omega)^{\frac{1}{p}}\left\Vert \phi_{p}\right\Vert _{\infty}\leq
\lim_{q\rightarrow0^{+}}\left\Vert u_{q}\right\Vert _{\infty}\leq B\mu
(\Omega)^{\frac{1}{p}},$$ where $A$ and $B$ are positive constants depending only on $N$ and $p.$ If $\left\vert \Omega\right\vert \not =1$ we have$$\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=\lim_{q\rightarrow0^{+}}(\lambda_{q}(\Omega)\left\vert \Omega\right\vert ^{\frac{p}{q}})\left(
\lim_{q\rightarrow0^{+}}\left\vert \Omega\right\vert ^{-\frac{p}{q}}\right)
=\mu(\Omega)\left(  \lim_{q\rightarrow0^{+}}\left\vert \Omega\right\vert
^{-\frac{p}{q}}\right)  .$$ Therefore, in this case, we readily obtain$$\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=\left\{
\begin{array}
[c]{ll}\infty & \mathrm{if}\quad\left\vert \Omega\right\vert <1\\
0 & \mathrm{if}\quad\left\vert \Omega\right\vert >1. 
\end{array}
\right.   \label{limlamb}$$

Hence, by combining (\[limlamb\]) with Lemma \[linfty\] and Lemma \[lowb\] we conclude that $$\lim_{q\rightarrow0^{+}}\left\Vert u_{q}\right\Vert _{\infty}=\left\{
\begin{array}
[c]{ll}\infty & \mathrm{if}\quad\left\vert \Omega\right\vert <1\\
0 & \mathrm{if}\quad\left\vert \Omega\right\vert >1. 
\end{array}
\right.$$

We remark that (\[limlamb\]) is also simple to prove without using Theorem \[theoexist\].  In fact, in the case $\left\vert \Omega\right\vert <1$ the monotonicity of the function $q\mapsto\lambda_{q}(\Omega)\left\vert
\Omega\right\vert ^{\frac{p}{q}}$ implies that $$\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)\left\vert \Omega\right\vert ^{\frac{p}{q}}\left\vert
\Omega\right\vert ^{-\frac{p}{q}}\geq\lambda_{1}(\Omega)\left\vert
\Omega\right\vert ^{p}\lim_{q\rightarrow0^{+}}\left\vert \Omega\right\vert
^{-\frac{p}{q}}=\infty.$$ This is the proof given in [@Anello].  As for the case $\left\vert
\Omega\right\vert >1,$ take $v\in W_{0}^{1,p}(\Omega)\cap C(\overline{\Omega
})$ such that $v>0$ in $\Omega$ and then define $\Omega_{\epsilon}:=\left\{
x\in\Omega:v(x)\geq\epsilon\right\}  ,$ where $\epsilon>0$ is such that $\left\vert \Omega_{\epsilon}\right\vert >1.$Then, $$0<\lambda_{q}(\Omega)\leq\frac{\int_{\Omega}\left\vert \nabla v\right\vert
^{p}\mathrm{d}x}{\left(  \int_{\Omega}\left\vert v\right\vert ^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}}\leq\frac{\int_{\Omega}\left\vert \nabla
v\right\vert ^{p}\mathrm{d}x}{\left(  \int_{\Omega_{\epsilon}}\epsilon
^{q}\mathrm{d}x\right)  ^{\frac{p}{q}}}=\frac{\int_{\Omega}\left\vert \nabla
v\right\vert ^{p}\mathrm{d}x}{\epsilon^{p}\left\vert \Omega_{\epsilon
}\right\vert ^{\frac{p}{q}}}.$$ Thus, by making $q\rightarrow0^{+},$ we obtain $\lim_{q\rightarrow0^{+}}\lambda_{q}(\Omega)=0$, since $\lim_{q\rightarrow0^{+}}\left\vert
\Omega_{\epsilon}\right\vert ^{-\frac{p}{q}}=0.$

Acknowledgments
===============

The first author thanks the support of Fundação de Amparo à Pesquisa do Estado de Minas Gerais (Fapemig)/Brazil (CEX-PPM-00165) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)/Brazil (483970/2013-1 and 306590/2014-0). 

[99]{}

G. Anello, F. Faraci and A. Iannizzotto: On a problem of Huang concerning best constants in Sobolev embeddings, *Ann.  Mat.  Pura Appl.* ** 194** (2015) 767–779. 

C. Bandle: Rayleigh-Faber-Krahn inequalities and quasilinear elliptic boundary value problems, *Nonlinear analysis and applications*: to V. Lakshmikantham on his 80th birthday.  Vol.  1,2, Kluwer Acad.  Publ.,  Dordrecht, 2003, p.p 227–240. 

Y. Chu and W. Gao: Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms, *Boundary Value Problems* **2013**:229 (2013). 

M. G. Crandall, P. H. Rabinowitz and L. Tartar: On a Dirichlet problem with singular nonlinearity, *Comm.  Partial Differential Equations* **2** (1977) 193–222. 

G. Ercole: Absolute continuity of the best Sobolev constant, *J. Math.  Anal.  Appl.* ** 404** (2013) 420–428. 

G. Ercole: On the resonant Lane-Emden problem for the p-Laplacian, *Comm.  Contemp.  Math. ** 16*** (2014) 1350023 (22 pages).*  *

G. Ercole and G. A. Pereira: Asymptotics for the best Sobolev constants and their extremal functions, Math.  Nachr. ( to appear) DOI 10.1002/mana.201500263. 

G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, New York, 2011. 

J. Giacomoni, I. Schindler and P. Takáč: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, *Ann.  Scuola Norm.  Sup.  Pisa Cl.  Sci.* ** 6** (2007) 117–158**.** 

J. Giacomoni, I. Schindler and P. Takáč: Singular quasilinear elliptic equations and Hölder regularity, *C. R. Acad.  Sci.  Paris, Ser.  I* **350** (2012) 383–388. 

B. Kawohl: Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math.,  vol.  1150, Springer-Verlag, Berlin, 1985. 

O. Ladyzhenskaya and N. Ural’tseva: Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 

A. C. Lazer and P. J. Mckenna: On a singular nonlinear elliptic boundary value problem. * Proc.  Am.  Math.  Soc.* ** 111** (1991) 721–730. 

A. Mohammed: Positive solutions of the p-Laplace equation with singular nonlinearity, J. Math.  Anal.  Appl. ** 352** (2009) 234–245. 

X. Ren and J. Wei: *Counting peaks of solutions to some quasilinear elliptic equations with large exponents,* J. Differential Equations ** **117** (1995) 28–55. 

C. A. Stuart: Existence and approximation of solutions of nonlinear elliptic equations, *Math.  Z.* **147** (1976) 53–63. 

[^1]: Corresponding author
