The Cheeger constant as limit of Sobolev-type constants

Abstract

Let $\Omega$ be a bounded, smooth domain of ${\mathbb {R}}^{N},$ $N\ge 2.$ For $1<p<N$ and $0<q(p)<p^{*}:=\frac{Np}{N-p}$, let

$$\begin{aligned} \lambda _{p,q(p)}:=\inf \left\{ \int _{\Omega }\left| \nabla u\right| ^{p}\textrm{d}x:u\in W_{0}^{1,p}(\Omega ) \ \text {and} \ \int _{\Omega }\left| u\right| ^{q(p)}\textrm{d}x=1\right\} . \end{aligned}$$

We prove that if $\lim _{p\rightarrow 1^{+}}q(p)=1,$ then $\lim _{p\rightarrow 1^{+}}\lambda _{p,q(p)}=h(\Omega )$, where $h(\Omega )$ denotes the Cheeger constant of $\Omega .$ Moreover, we study the behavior of the positive solutions $w_{p,q(p)}$ to the Lane–Emden equation $-{\text {div}} (\left| \nabla w\right| ^{p-2}\nabla w)=\left| w\right| ^{q-2}w,$ as $p\rightarrow 1^{+}.$

1 Introduction

Let $\Omega$ be a smooth, bounded domain of ${\mathbb {R}}^{N},$ $N\ge 2.$ For $1<p<N$ and $0<q\le p^{*}:=\frac{Np}{N-p}$, let

$$\begin{aligned} \lambda _{p,q}:=\inf \left\{ \left\| \nabla u\right\| _{p}^{p}:u\in W_{0}^{1,p}(\Omega ) \ \text {and} \ \left\| u\right\| _{q}=1\right\} , \end{aligned}$$

(1.1)

where

$$\begin{aligned} \left\| u\right\| _{r}:=\left( \int _{\Omega }\left| u\right| ^{r}\textrm{d}x\right) ^{\frac{1}{r}},\quad r>0. \end{aligned}$$

We recall that $\left\| \cdot \right\| _{r}$ is the standard norm of the Lebesgue space $L^{r}(\Omega )$ if $r\ge 1,$ but it is not a norm if $0<r<1$.

Note from (1.1) that

$$\begin{aligned} \lambda _{p,q}\le \frac{\left\| \nabla u\right\| _{p}^{p}}{\left\| u\right\| _{q}^{p}} \ \text { for all } \ u\in W_{0}^{1,p}(\Omega )\setminus \left\{ 0\right\} \end{aligned}$$

(1.2)

since the above quotients are homogeneous.

When $0<q<p^{*},$ the existence of a minimizer $u_{p,q}$ for the constrained minimization problem (1.1) follows from standard arguments of the Calculus of Variations. Moreover, $u_{p,q}$ is a weak solution to the Dirichlet problem for the p-Laplacian operator

$$\begin{aligned} \left\{ \begin{array}{ll} -{\text{ div }}\left( \left| \nabla u\right| ^{p-2}\nabla u\right) =\lambda _{p,q}\left| u\right| ^{q-2}u &{}{} \text{ in } \quad \Omega \\ u=0 &{}{} \text{ on } \quad \partial \Omega . \end{array}\right. \end{aligned}$$

(1.3)

In consequence, $u_{p,q}>0$ in $\Omega$ and $u_{p,q}\in C^{1,\alpha } (\overline{\Omega })$ for some $0<\alpha <1$. (We refer to Giacomoni et al. [9, Theorem 1(i)] for the regularity of $u_{p,q}$ when $0<q<1$, in which case (1.3) is singular.) These facts are well known when $1\le q<p^{*},$ since $\lambda _{p,q}$ is the best constant in the Sobolev (compact) embedding $W_{0}^{1,p}(\Omega )\hookrightarrow L^{q}(\Omega ).$

It is worth mentioning that $u_{p,q}$ is the only positive minimizer to (1.1) in the sublinear case: $0<q<p.$ However, this uniqueness property might fail in the superlinear, subcritical case: $p<q<p^{*}.$ For examples and a discussion about this issue, we recommend the recent paper [4] by Brasco and Lindgren, where an important result is established for general smooth bounded domains: the uniqueness of the minimizer $u_{p,q}$ whenever $2<p<q$ and q is sufficiently close to p. We stress that such a uniqueness result for $1<p<2$ is not yet available in the literature.

When $q=p^{*},$ the infimum $\lambda _{p,p^{*}}$ cannot be attained in $W_{0}^{1,p}(\Omega )$ if $\Omega \not ={\mathbb {R}}^{N}.$ Actually, $\lambda _{p,p^{*}}$ does not depend on $\Omega$ as it coincides with the well-known Sobolev constant $S_{N,p},$ that is:

$$\begin{aligned} \lambda _{p,p^{*}}=S_{N,p}:=N\omega _{N}^{\frac{p}{N}}\left( \frac{N-p}{p-1}\right) ^{p-1}\left( \frac{\Gamma (N/p)\Gamma (1+N-N/p)}{\Gamma (N)}\right) ^{\frac{p}{N}}, \end{aligned}$$

(1.4)

where $\Gamma (t)=\int _{0}^{\infty }s^{t-1}e^{-s}\textrm{d}s$ is the Gamma function and $\omega _{N}:=\pi ^{N/2}/\Gamma (1+N/2)$ is the N-dimensional Lebesgue volume of the unit ball of ${\mathbb {R}}^{N}.$

According to [2, Theorem 9] by Anello et al., for each fixed $p\in (1,N)$ the function

$$\begin{aligned} (0,p^{*}]\ni q\mapsto \lambda _{p,q}\left| \Omega \right| ^{\frac{p}{q}} \end{aligned}$$

is decreasing and absolutely continuous on compact sets of $(0,p^{*}].$ The same result, but for $q\in [1,p^{*}],$ had already been obtained by Ercole [8].

As for q varying with p, Kawohl and Fridman proved in [10] that

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\lambda _{p,p}=h(\Omega ) \end{aligned}$$

(1.5)

where $h(\Omega )$ is the Cheeger constant of $\Omega .$

We recall that

$$\begin{aligned} h(\Omega ):=\inf \left\{ \frac{P(E)}{\left| E\right| }:E\subset \overline{\Omega } \ \text { and } \ \left| E\right| >0\right\} , \end{aligned}$$

where P(E) stands for the perimeter of E in ${\mathbb {R}}^{N}$ and $\left| E\right|$ stands for the N-dimensional Lebesgue volume of E.

The Cheeger problem consists of finding a subset $E\subset \overline{\Omega }$ such that

$$\begin{aligned} h(\Omega )=\frac{P(E)}{\left| E\right| }. \end{aligned}$$

Such a subset E is called Cheeger set of $\Omega .$

We notice from (1.4) that

$$\begin{aligned} \lim \limits _{p\rightarrow 1^{+}}\lambda _{p,p^{*}}=N\omega _{N}^{\frac{1}{N} }=\left| \Omega \right| ^{\frac{1}{N}}h(\Omega ^{\star }) \end{aligned}$$

(1.6)

where $\Omega ^{\star }$ denotes the ball of ${\mathbb {R}}^{N}$ centered at the origin such that $\left| \Omega ^{\star }\right| =\left| \Omega \right| .$ The second equality in (1.6) is due to the fact that balls are Cheeger sets of themselves (i.e. they are calibrable). Hence, as $R=(\left| \Omega \right| /\omega _{N})^{\frac{1}{N}}$ is the radius of $\Omega ^{\star },$ one has $h(\Omega ^{\star })=N/R=N(\omega _{N}/\left| \Omega \right| )^{\frac{1}{N}}.$ It is well known that $h(\Omega ^{\star })\le h(\Omega ),$ the equality occurs if and only if $\Omega$ is a ball.

Owing to (1.5), when $p\rightarrow 1^{+}$, the minimizer $u_{p,p}$ converges in $L^{1}(\Omega )$ (after passing to a subsequence) to a function $u_{1}$ whose the t-superlevel sets $E_{t}:=\left\{ x\in \Omega :u_{1}(x)>t\right\}$ are Cheeger sets, for almost every $t>0.$ As shown in [10], these properties are obtained from a variational version of the Cheeger problem in the BV setting, which we briefly present in Sect. 2.

The approach of solving the Cheeger problem by a p-Laplacian approximation, as $p\rightarrow 1^{+},$ has been extended by Butazzo, Carlier and Comte in [6] to a slightly more general Cheeger problem where the volume and the perimeter are weighted by two positive weight functions. In that paper, after showing that such an approach does not provide a criterion for determining the maximal Cheeger set, they introduced an alternative approximation method in the BV setting, based in concave penalizations, to select maximal Cheeger sets.

In this paper we suppose that q varies with p along a more general path, $q=q(p)$ for $p\in (1,p^{*}),$ and study the behavior of $\lambda _{p,q(p)}$ when $p\rightarrow 1^{+}$ and $q(p)\rightarrow 1.$ Adapting an estimate from Ercole [8] (see Lemma 3.1 below) and making use of (1.5), we extend the results of Kawohl and Fridman [10]. Our main result, which will be proved in Sect. 3, is stated as follows.

Theorem 1.1

If $0<q(p)<p^{*}$ and $\lim \nolimits _{p\rightarrow 1^{+}}q(p)=1,$ then

$$\begin{aligned}{} & {} \lim _{p\rightarrow 1^{+}}\lambda _{p,q(p)}=h(\Omega ), \end{aligned}$$

(1.7)

$$\begin{aligned}{} & {} \lim _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{1}=1 \end{aligned}$$

(1.8)

and

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{\infty }^{q(p)-p}=1. \end{aligned}$$

(1.9)

Moreover, any sequence $\left( u_{p_{n},q(p_{n})}\right) ,$ with $p_{n}\rightarrow 1^{+},$ admits a subsequence that converges in $L^{1} (\Omega )$ to a nonnegative function $u\in L^{1}(\Omega )\cap L^{\infty } (\Omega )$ such that:

• (a) $\left\| u\right\| _{1}=1,$

• (b) $\dfrac{1}{\left| \Omega \right| }\le \left\| u\right\| _{\infty }\le \dfrac{h(\Omega )^{N}}{\left| \Omega \right| h(\Omega ^{\star })^{N}},$ and

• (c) for almost every $t\ge 0,$ the t-superlevel set

  $$\begin{aligned} E_{t}:=\left\{ x\in \Omega :u(x)>t\right\} \end{aligned}$$

  is a Cheeger set.

Besides allowing $q(p)\rightarrow 1^{-}$, which embraces (1.3) in its singular form, our approach holds for every family of extremals $u_{p,q(p)}$ in the superlinear, subcritical case: $p<q(p)<p^{*}.$

It is simple to verify that the function

$$\begin{aligned} v_{p,q}:=\lambda _{p,q}^{\frac{1}{q-p}}u_{p,q}, \end{aligned}$$

(1.10)

is a positive weak solution to the Lane–Emden-type problem

$$\begin{aligned} \left\{ \begin{array}{ll} -{\text{ div }}\left( \left| \nabla w\right| ^{p-2}\nabla w\right) =\left| w\right| ^{q-2}w &{}{} \text{ in } \quad \Omega \\ w=0 &{}{} \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

(1.11)

The next corollary is stated to solutions to (1.11) in the form (1.10). It is an immediate consequence of (1.7) and (1.9) since

$$\begin{aligned} \left\| v_{p,q}\right\| _{q}^{q-p}=\lambda _{p,q}\left\| u_{p,q}\right\| _{q}^{q-p}=\lambda _{p,q}\quad \text {and}\quad \left\| v_{p,q}\right\| _{\infty }^{q-p}=\lambda _{p,q}\left\| u_{p,q}\right\| _{\infty }^{q-p}. \end{aligned}$$

Corollary 1.2

If $0<q(p)<p^{*}$ and $\lim \nolimits _{p\rightarrow 1^{+}}q(p)=1,$ then

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left\| v_{p,q(p)}\right\| _{q(p)} ^{q(p)-p}=h(\Omega )=\lim _{p\rightarrow 1^{+}}\left\| v_{p,q(p)}\right\| _{\infty }^{q(p)-p}. \end{aligned}$$

In the particular case where $q(p)\equiv 1$, this result had already been obtained in [5] by Bueno and Ercole, without using (1.5).

As it is well known, (1.11) has a unique positive weak solution when $0<q<p,$ which is, of course, that given by (1.10). However, there are examples of smooth domains for which (1.11) has multiple positive weak solutions when $p<q<p^{*},$ which may be of the form (1.10) or not (see [4] and references therein). By the way, it is plain to check that

$$\begin{aligned} \lambda _{p,q}=\left\| v_{p,q}\right\| _{q}^{q-p}=\min \left\{ \left\| w\right\| _{q}^{q-p}:w\text { is a weak solution to }(1.11)\right\} . \end{aligned}$$

(1.12)

Corollary 1.2 deals with the behavior of positive weak solutions to (1.11) that attains the minimum in (1.12). Aiming to cover a wider class of positive weak solutions $w_{p,q}$ to (1.11), including those satisfying $\left\| w_{p,q}\right\| _{q}^{q-p}>\lambda _{p,q}$, we provide the following stronger result, which will be proved in Sect. 4 by using Picone’s inequality (see [1, 3]).

Theorem 1.3

Let $w_{p,q(p)}\in W_{0}^{1,p}(\Omega )$ be a positive weak solution to (1.11), with $p<q(p)<p^{*}.$ Then, either

$$\begin{aligned} \limsup _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{\infty } ^{q(p)-p}=+\infty \end{aligned}$$

(1.13)

or

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)} ^{q(p)-p}=h(\Omega )=\lim _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p}. \end{aligned}$$

The alternative (1.13) can be replaced with (see Remark 4.1)

$$\begin{aligned} \limsup _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)} ^{q(p)-p}=+\infty . \end{aligned}$$

We believe that determining whether this alternative (or its equivalent version (1.13)) is actually possible is a very interesting open question that we plan to study in the near future.

2 The Cheeger problem in the BV setting

In this section, we assume that $\Omega$ is a Lipschitz bounded domain and collect some definitions, properties and basic results related to the variational version of the Cheeger problem in the BV setting. For details, we refer to Carlier and Comte [7] and Parini [11].

The total variation of $u\in L^{1}(\Omega )$ is defined as

$$\begin{aligned} \left| Du\right| (\Omega ):=\sup \left\{ \int _{\Omega } u{\text {div}}\varphi \textrm{d}x:\varphi \in C_{c}^{1}(\Omega ;{\mathbb {R}}^{N})\quad \text {and}\quad \left\| \varphi \right\| _{L^{\infty } }\le 1\right\} . \end{aligned}$$

The space $BV(\Omega )$ of the functions $u\in L^{1}(\Omega )$ of bounded variation in $\Omega$ (i.e.$\,\left| Du\right| (\Omega )<\infty$), endowed with the norm

$$\begin{aligned} \left\| u\right\| _{BV}:=\left\| u\right\| _{1}+\left| Du\right| (\Omega ), \end{aligned}$$

is a Banach space compactly embedded into $L^{1}(\Omega )$. Moreover, the functional $BV(\Omega )\ni u\mapsto \left| Du\right| (\Omega )$ is lower semicontinuous in $L^{1}(\Omega ).$

The Cheeger constant is also characterized as (see [11, Proposition 3.1])

$$\begin{aligned} h(\Omega )=\inf _{BV_{0}(\Omega )}\frac{\left| Du\right| ({\mathbb {R}} ^{N})}{\left\| u\right\| _{1}} \end{aligned}$$

(2.1)

where

$$\begin{aligned} BV_{0}(\Omega ):=\left\{ u\in BV({\mathbb {R}}^{N}):\left\| u\right\| _{1}>0 \ \text {and} \ u\equiv 0\text { in }{\mathbb {R}}^{N}\setminus \overline{\Omega }\right\} \end{aligned}$$

and

$$\begin{aligned} \left| Du\right| ({\mathbb {R}}^{N})=\left| Du\right| (\Omega )+\int _{\partial \Omega }\left| u\right| \textrm{d} {\mathcal {H}}^{N-1} \end{aligned}$$

(${\mathcal {H}}^{N-1}$ stands for the $(N-1)$-Hausdorff measure in ${\mathbb {R}}^{N}$).

Proposition 2.1

([7, Corollary 1(2)]) Let $\left( u_{n}\right) \subset BV_{0}(\Omega )$ be such that $u_{n}\rightarrow u$ in $L^{1}({\mathbb {R}}^{N}).$ Then,

$$\begin{aligned} \left| Du\right| ({\mathbb {R}}^{N})\le \liminf _{n\rightarrow \infty }\left| Du_{n}\right| ({\mathbb {R}}^{N}). \end{aligned}$$

Proposition 2.2

Suppose that

$$\begin{aligned} h(\Omega )=\frac{\left| Du\right| ({\mathbb {R}}^{N})}{\left\| u\right\| _{1}} \end{aligned}$$

for some $u\in BV_{0}(\Omega ).$ Then,

$$\begin{aligned} E_{t}:=\left\{ x\in \Omega :u(x)>t\right\} \end{aligned}$$

is a Cheeger set for almost every $t\ge 0.$

Inversely, if $E\subset \overline{\Omega }$ is a Cheeger set of $\Omega ,$ then

$$\begin{aligned} h(\Omega )=\frac{\left| D\chi _{E}\right| ({\mathbb {R}}^{N})}{\left\| \chi _{E}\right\| _{1}} \end{aligned}$$

where $\chi _{E}$ stands for the characteristic function of E in ${\mathbb {R}}^{N}.$

Proof

Combining Coarea formula and Cavalieri’s principle, we find

$$\begin{aligned} 0=\left| Du\right| ({\mathbb {R}}^{N})-h(\Omega )\left\| u\right\| _{1}=\int _{0}^{\infty }(P(E_{t})-h(\Omega )\left| E_{t}\right| )\textrm{d}t. \end{aligned}$$

(2.2)

As $\left| E_{t}\right| >0$ a.e.$\,t\ge 0$, we have that $P(E_{t} )-h(\Omega )\left| E_{t}\right| \ge 0$ a.e.$\,t\ge 0.$ Therefore, it follows from (2.2) that

$$\begin{aligned} h(\Omega )=\frac{P(E_{t})}{\left| E_{t}\right| }\quad \text {a.e.}\quad t\ge 0. \end{aligned}$$

Now, if $E\subset \overline{\Omega }$ is a Cheeger set of $\Omega ,$ then $\chi _{E}\in BV_{0}(\Omega ).$ As $P(E)=\left| D\chi _{E}\right| ({\mathbb {R}}^{N})$ and $\left\| \chi _{E}\right\| _{1}=\left| E\right|$, we have

$$\begin{aligned} h(\Omega )=\frac{P(E)}{\left| E\right| }=\frac{\left| D\chi _{E}\right| ({\mathbb {R}}^{N})}{\left\| \chi _{E}\right\| _{1}}. \end{aligned}$$

$\square$

3 Proof of Theorem 1.1

We recall from the Introduction that $u_{p,q}$ (for $1<p<N$ and $0<q<p^{*}$) denotes the positive minimizer of the constrained minimization problem (1.1), so that $u_{p,q}\in W_{0}^{1,p}(\Omega ),$

$$\begin{aligned} u_{p,q}>0\text { in }\Omega , \ \left\| u_{p,q}\right\| _{q}=1, \ \lambda _{p,q}=\left\| \nabla u_{p,q}\right\| _{p}^{p}, \end{aligned}$$

and $u_{p,q}$ is a weak solution to (1.3).

If $q=p$, the Dirichlet problem (1.3) is homogeneous and thus it can be recognized as an eigenvalue problem. In this setting, $\lambda _{p,p}$ is known as the first eigenvalue of the Dirichlet p-Laplacian. Actually, $\lambda _{p,p}$ is simple in the sense that the set of its corresponding eigenfunctions is generated by $u_{p,p},$ that is, $w\in W_{0}^{1,p}(\Omega )$ is a nontrivial weak solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -{\text {div}}\left( \left| \nabla u\right| ^{p-2}\nabla u\right) =\lambda _{p,p}\left| u\right| ^{p-2}u &{} \textrm{in} \quad \Omega \\ u=0 &{} \textrm{on} \quad \partial \Omega \end{array} \right. \end{aligned}$$

(3.1)

if and only if $w=ku_{p,p}$ for some $k\in {\mathbb {R}}{\setminus }\left\{ 0\right\} .$

In this section, we prove Theorem 1.1 by assuming that $\partial \Omega$ is smooth enough to ensure that $u_{p,q}\in C^{1}(\overline{\Omega }).$ In consequence, $u_{p,q}\in BV_{0}(\Omega )$ (after extended as zero on ${\mathbb {R}}^{N}\setminus \overline{\Omega }$) and

$$\begin{aligned} \left| Du_{p,q}\right| ({\mathbb {R}}^{N})=\left\| \nabla u_{p,q}\right\| _{1}, \end{aligned}$$

since

$$\begin{aligned} \left| Du_{p,q}\right| (\Omega )=\left\| \nabla u_{p,q}\right\| _{1}\quad \textrm{and}\quad \int _{\partial \Omega }\left| u_{p,q}\right| \textrm{d}{\mathcal {H}}^{N-1}=0. \end{aligned}$$

The next result is adapted from Lemma 5 of Ercole [8] established there for $1\le q<p^{*}$.

Lemma 3.1

Let $u\in W_{0}^{1,p}(\Omega )\cap C^{1}(\overline{\Omega })$ be a positive weak solution to the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -{\text{ div }}\left( \left| \nabla u\right| ^{p-2}\nabla u\right) =\lambda \left| u\right| ^{q-2}u &{}{} \text{ in } \quad \Omega \\ u=0 &{}{} \text{ on } \quad \partial \Omega \end{array} \right. \end{aligned}$$

(3.2)

with $0\le q<p^{*}$ and $\lambda >0.$ If $\sigma \ge 1,$ then

$$\begin{aligned} C_{\lambda ,\sigma ,q}\left\| u\right\| _{\infty }^{\frac{N(p-q)+p\sigma }{p}}\le \left\| u\right\| _{\sigma }^{\sigma }, \end{aligned}$$

(3.3)

where

$$\begin{aligned} C_{\lambda ,\sigma ,q}:=\left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1}I_{\sigma ,q} \end{aligned}$$

and

$$\begin{aligned} I_{\sigma ,q}:=\left\{ \begin{array}{ll} \sigma \int _{0}^{1}\left( 1-\tau \right) ^{\frac{N(p-1)}{p}}\tau ^{\frac{N(1-q)}{p}+\sigma -1}\textrm{d}\tau &{} \text {if} \quad 0\le q<1\\ \sigma \int _{0}^{1} (1-\tau )^{\frac{N(p-1)}{p}}\tau ^{\sigma -1}\textrm{d}\tau &{} \text {if}\quad 1\le q<p^{*}. \end{array} \right. \end{aligned}$$

Proof

For each $0<t<\left\| u\right\| _{\infty }$, let us define

$$\begin{aligned} (u-t)_{+}:=\max \left\{ u-t,0\right\} \quad \text {and}\quad A_{t}:=\left\{ x\in \Omega :u(x)>t\right\} . \end{aligned}$$

As $(u-t)_{+}\in W_{0}^{1,p}(\Omega )$ and u is a positive weak solution to (3.2), we have

$$\begin{aligned} \int _{A_{t}}\left| \nabla u\right| ^{p}\textrm{d}x= & {} \lambda \int _{\Omega }\left| \nabla u\right| ^{p-2}\nabla u\cdot \nabla (u-t)_{+}\textrm{d}x \nonumber \\= & {} \lambda \int _{\Omega }u^{q-1}(u-t)_{+}\textrm{d} x=\lambda \int _{A_{t}}u^{q-1}(u-t)\textrm{d}x. \end{aligned}$$

(3.4)

We also have

$$\begin{aligned} \left( \int _{A_{t}}(u-t)\textrm{d}x\right) ^{p}\le \left| A_{t} \right| ^{p-\frac{p}{p^{*}}}\left( \int _{A_{t}}(u-t)^{p^{*} }\textrm{d}x\right) ^{\frac{p}{p^{*}}}\le \frac{\left| A_{t} \right| ^{p-\frac{p}{p^{*}}}}{\lambda _{p,p^{*}}}\int _{A_{t} }\left| \nabla u\right| ^{p}\textrm{d}x, \end{aligned}$$

(3.5)

where we have used Hölder’s inequality and (1.2). Note that

$$\begin{aligned} \lambda _{p,p^{*}}\le \frac{\left\| \nabla (u-t)_{+}\right\| _{p}^{p} }{\left\| (u-t)_{+}\right\| _{p^{*}}^{p}}=\frac{ {\displaystyle \int _{A_{t}}} \left| \nabla u\right| ^{p}\textrm{d}x}{\left( {\displaystyle \int _{A_{t}}} (u-t)^{p^{*}}\textrm{d}x\right) ^{\frac{p}{p^{*}}}}. \end{aligned}$$

We divide the remaining of the proof in two cases.

Case 1. $0\le q<1.$ As

$$\begin{aligned} \int _{A_{t}}u^{q-1}(u-t)\textrm{d}x\le t^{q-1}\int _{A_{t}}(\left\| u\right\| _{\infty }-t)\textrm{d}x, \end{aligned}$$

(3.6)

we obtain from (3.4) the estimate

$$\begin{aligned} \int _{A_{t}}\left| \nabla u\right| ^{p}\textrm{d}x\le \lambda t^{q-1}(\left\| u\right\| _{\infty }-t)\left| A_{t}\right| . \end{aligned}$$

(3.7)

Combining (3.7) and (3.5), we obtain the inequalities

$$\begin{aligned} \lambda _{p,p^{*}}\left| A_{t}\right| ^{-p+\frac{p}{p^{*}} }\left( \int _{A_{t}}(u-t)\textrm{d}x\right) ^{p}\le \int _{A_{t}}\left| \nabla u\right| ^{p}\textrm{d}x\le \lambda t^{q-1}(\left\| u\right\| _{\infty }-t)\left| A_{t}\right| \end{aligned}$$

which lead to

$$\begin{aligned} \frac{\lambda _{p,p^{*}}t^{1-q}}{\lambda (\left\| u\right\| _{\infty }-t)}\left( \int _{A_{t}}(u-t)\textrm{d}x\right) ^{p}\le \left| A_{t}\right| ^{p(1-\frac{1}{p^{*}}+\frac{1}{p})}=\left| A_{t}\right| ^{p(\frac{N+1}{N})}. \end{aligned}$$

(3.8)

Now, let us define the function

$$\begin{aligned} g(t):=\int _{A_{t}}(u-t)\textrm{d}x. \end{aligned}$$

It is simple to verify that

$$\begin{aligned} g(t)=\int _{t}^{\left\| u\right\| _{\infty }}\left| A_{s}\right| \textrm{d}s, \end{aligned}$$

so that

$$\begin{aligned} g^{\prime }(t)=-\left| A_{t}\right| . \end{aligned}$$

Then, (3.8) can be rewritten as

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p(N+1)} }\left( \frac{t^{1-q}}{\left\| u\right\| _{\infty }-t}\right) ^{\frac{N}{p(N+1)}}\le -g^{\prime }(t)g(t)^{-\frac{N}{N+1}}. \end{aligned}$$

(3.9)

Integration of the right-hand side of (3.9) over $[t,\left\| u\right\| _{\infty }]$ yields

$$\begin{aligned} -\int _{t}^{\left\| u\right\| _{\infty }}g^{\prime }(s)g(s)^{-\frac{N}{N+1}}\textrm{d}s= & {} (N+1)g(t)^{\frac{1}{N+1}}-(N+1)g(\left\| u\right\| _{\infty })^{\frac{1}{N+1}}\nonumber \\\le & {} (N+1)g(t)^{\frac{1}{N+1}} \end{aligned}$$

(3.10)

whereas integration of the function at the left-hand side of (3.9) over $[t,\left\| u\right\| _{\infty }]$ yields

$$\begin{aligned} \int _{t}^{\left\| u\right\| _{\infty }}\left( \frac{s^{1-q}}{\left\| u\right\| _{\infty }-s}\right) ^{\frac{N}{p(N+1)}}\textrm{d}s&\ge t^{\frac{N(1-q)}{p(N+1)}}\int _{t}^{\left\| u\right\| _{\infty }}\left( \left\| u\right\| _{\infty }-s\right) ^{-\frac{N}{p(N+1)}} \textrm{d}s\nonumber \\&=\frac{p(N+1)}{p+N(p-1)}t^{\frac{N(1-q)}{p(N+1)}}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{p+N(p-1)}{p(N+1)}}. \end{aligned}$$

(3.11)

Thus, after integrating (3.9) we obtain from (3.10) and (3.11) the inequality

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1}t^{\frac{N(1-q)}{p}}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{p+N(p-1)}{p}}\le g(t). \end{aligned}$$

(3.12)

As $g(t)\le (\left\| u\right\| _{\infty }-t)\left| A_{t}\right|$, it follows from (3.12) that

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1}t^{\frac{N(1-q)}{p}}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{N(p-1)}{p}}\le \left| A_{t}\right| . \end{aligned}$$

Now, for a given $\sigma \ge 1,$ we multiply the latter inequality by $\sigma t^{\sigma -1}$ and integrate over $[0,\left\| u\right\| _{\infty }]$ to get (3.3) after noticing that

$$\begin{aligned} \sigma \int _{0}^{\left\| u\right\| _{\infty }}\left| A_{t}\right| t^{\sigma -1}\textrm{d}t=\int _{\Omega }u^{\sigma }\textrm{d}x, \end{aligned}$$

and that the change of variable $t=\left\| u\right\| _{\infty }\tau$ yields

$$\begin{aligned} \int _{0}^{\left\| u\right\| _{\infty }}t^{\frac{N(1-q)}{p}+\sigma -1}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{N(p-1)}{p} }\textrm{d}t=\left\| u\right\| _{\infty }^{\frac{N(p-q)}{p}+\sigma } \int _{0}^{1}\left( 1-\tau \right) ^{\frac{N(p-1)}{p}}\tau ^{\frac{N(1-q)}{p}+\sigma -1}\textrm{d}\tau . \end{aligned}$$

Case 2. $1\le q<p^{*}.$ The factor $t^{q-1}$ in (3.6) can be replaced with $\left\| u\right\| _{\infty }^{q-1},$ so that (3.9) and (3.11) become

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p(N+1)} }\left( \frac{\left\| u\right\| _{\infty }^{1-q}}{\left\| u\right\| _{\infty }-t}\right) ^{\frac{N}{p(N+1)}}\le -g^{\prime }(t)g(t)^{-\frac{N}{N+1}} \end{aligned}$$

and

$$\begin{aligned} \int _{t}^{\left\| u\right\| _{\infty }}\left( \frac{\left\| u\right\| _{\infty }^{1-q}}{\left\| u\right\| _{\infty }-s}\right) ^{\frac{N}{p(N+1)}}\textrm{d}s=\frac{p(N+1)}{p+N(p-1)}\left\| u\right\| _{\infty }^{\frac{N(1-q)}{p(N+1)}}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{p+N(p-1)}{p(N+1)}}, \end{aligned}$$

respectively. Hence, we obtain from (3.10) that

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p(N+1)}} \frac{p(N+1)}{p+N(p-1)}\left\| u\right\| _{\infty }^{\frac{N(1-q)}{p(N+1)}}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{p+N(p-1)}{p(N+1)}}\le (N+1)g(t)^{\frac{1}{N+1}}. \end{aligned}$$

Then, using that $g(t)\le (\left\| u\right\| _{\infty }-t)\left| A_{t}\right|$, the latter inequality leads to

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda }\right) ^{\frac{N}{p}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1}\left\| u\right\| _{\infty } ^{\frac{N(1-q)}{p}}\left( \left\| u\right\| _{\infty }-t\right) ^{\frac{N(p-1)}{p}}\le \left| A_{t}\right| . \end{aligned}$$

(3.13)

Multiplying (3.13) by $\sigma t^{\sigma -1}$ and integrating over $[0,\left\| u\right\| _{\infty }]$, we arrive at (3.3) with

$$\begin{aligned} I_{\sigma ,q}=\sigma {\displaystyle \int _{0}^{1}} (1-\tau )^{\frac{N(p-1)}{p}}\tau ^{\sigma -1}\textrm{d}\tau . \end{aligned}$$

$\square$

Remark 3.2

The estimate (3.3) can be rewritten as

$$\begin{aligned} C_{\lambda ,\sigma ,q}\left\| u\right\| _{\infty }^{\frac{N}{p^{*} }(p^{*}-q)+(\sigma -q)}\le \left\| u\right\| _{\sigma }^{\sigma }. \end{aligned}$$

In the sequel, $e_{p}$ denotes the $L^{\infty }$-normalized minimizer corresponding to $\lambda _{p,p}$, that is:

$$\begin{aligned} e_{p}:=\dfrac{u_{p,p}}{\left\| u_{p,p}\right\| _{\infty }}. \end{aligned}$$

(3.14)

As $e_{p}$ is also a positive weak solution to the homogeneous Dirichlet problem (3.1), Lemma 3.1 applied to $e_{p,}$ with $q=p,$ $\sigma =1$ and $\lambda =\lambda _{p,p},$ yields

$$\begin{aligned} \left( \frac{\lambda _{p,p^{*}}}{\lambda _{p,p}}\right) ^{\frac{N}{p} }\left( \frac{p}{p+N(p-1)}\right) ^{N+1} {\displaystyle \int _{0}^{1}} (1-\tau )^{\frac{N(p-1)}{p}}\textrm{d}\tau \le \left\| e_{p}\right\| _{1}. \end{aligned}$$

Hence, we have

$$\begin{aligned} 0<\left| \Omega \right| \left( \frac{h(\Omega ^{\star })}{h(\Omega )}\right) ^{N}\le \liminf _{p\rightarrow 1^{+}}\left\| e_{p}\right\| _{1}, \end{aligned}$$

(3.15)

since

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left( \frac{\lambda _{p,p^{*}}}{\lambda _{p,p} }\right) ^{\frac{N}{p}}=\left| \Omega \right| \left( \frac{h(\Omega ^{\star })}{h(\Omega )}\right) ^{N} \end{aligned}$$

and

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1} =\lim _{p\rightarrow 1^{+}} {\displaystyle \int _{0}^{1}} (1-\tau )^{\frac{N(p-1)}{p}}\textrm{d}\tau =1. \end{aligned}$$

(3.16)

Lemma 3.3

If $q_{n}\rightarrow 1$ and $p_{n}\rightarrow 1^{+},$ then (up to a subsequence) $e_{p_{n}}$ converges in $L^{1}(\Omega )$ to a function e. Moreover,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\Omega }e_{p_{n}}^{q_{n}}\textrm{d} x=\lim _{n\rightarrow \infty }\int _{\Omega }e_{p_{n}}^{p_{n}}\textrm{d} x=\left\| e\right\| _{1}>0. \end{aligned}$$

(3.17)

Proof

We have $\left\| e_{p}\right\| _{1}\le \left\| e_{p}\right\| _{\infty }\left| \Omega \right| =\left| \Omega \right|$ and, by Hölder inequality,

$$\begin{aligned} \left\| \nabla e_{p}\right\| _{1}\le \left\| \nabla e_{p}\right\| _{p}\left| \Omega \right| ^{1-\frac{1}{p}}=\lambda _{p,p}^{\frac{1}{p} }\left| \Omega \right| ^{1-\frac{1}{p}}. \end{aligned}$$

Hence, it follows from (1.5) that the family $\left( e_{p}\right)$ is uniformly bounded in $BV(\Omega ).$ Therefore, owing to the compactness of the embedding $BV(\Omega )\hookrightarrow L^{1}(\Omega )$, we can assume that (up to a subsequence) $e_{p_{n}}$ converges to a function e in $L^{1}(\Omega )$ and also pointwise almost everywhere in $\Omega .$ In view of (3.15), the convergence in $L^{1}(\Omega )$ shows that $\left\| e\right\| _{1}>0.$ As the nonnegative functions $e_{p_{n}}^{q_{n}}$ and $e_{p_{n}}^{p_{n}}$ are dominated by 1,  the convergence a.e. in $\Omega$ leads to the equalities in (3.17). $\square$

Lemma 3.4

If $0<q(p)<p^{*}$ and $\lim \nolimits _{p\rightarrow 1^{+}}q(p)=1,$ then

$$\begin{aligned} \limsup _{p\rightarrow 1^{+}}\lambda _{p,q(p)}\le h(\Omega ) \end{aligned}$$

(3.18)

and

$$\begin{aligned} \limsup _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{1}\le 1. \end{aligned}$$

(3.19)

Proof

Let us take $p_{n}\rightarrow 1^{+}$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\lambda _{p_{n},q(p_{n})}=L:=\limsup _{p\rightarrow 1^{+}}\lambda _{p,q(p)}. \end{aligned}$$

Using (1.2) for $\lambda _{p_{n},q(p_{n})}$ and the definition of $e_{p_{n}}$, we have that

$$\begin{aligned} \lambda _{p_{n},q(p_{n})}\le \frac{\left\| \nabla e_{p_{n}}\right\| _{p_{n}}^{p_{n}}}{\left\| e_{p_{n}}\right\| _{q(p_{n})}^{p_{n}}} =\lambda _{p_{n},p_{n}}\left( \frac{\left\| e_{p_{n}}\right\| _{p_{n}} }{\left\| e_{p_{n}}\right\| _{q(p_{n})}}\right) ^{p_{n}}. \end{aligned}$$

Hence, we can apply Lemma 3.3 to get (3.18) from (1.5), since

$$\begin{aligned} L=\lim _{n\rightarrow \infty }\lambda _{p_{n},q(p_{n})}\le \lim _{n\rightarrow \infty }\lambda _{p_{n},p_{n}}\frac{\lim _{n\rightarrow \infty }\left\| e_{p_{n}}\right\| _{p_{n}}}{\lim _{n\rightarrow \infty }\left\| e_{p_{n} }\right\| _{q(p_{n})}}=\lim _{n\rightarrow \infty }\lambda _{p_{n},p_{n} }=h(\Omega ). \end{aligned}$$

Using Hölder’s inequality and exploiting (1.2) with respect to $\lambda _{p,p},$ we obtain

$$\begin{aligned} \left\| u_{p,q(p)}\right\| _{1}\le \left\| u_{p,q(p)}\right\| _{p}\left| \Omega \right| ^{1-\frac{1}{p}}\le \lambda _{p,p}^{-\frac{1}{p}}\left\| \nabla u_{p,q(p)}\right\| _{p}\left| \Omega \right| ^{1-\frac{1}{p}}=\lambda _{p,p}^{-\frac{1}{p}}\lambda _{p,q(p)}^{\frac{1}{p}}\left| \Omega \right| ^{1-\frac{1}{p}}. \nonumber \\ \end{aligned}$$

(3.20)

Hence, (3.19) follows from (1.5) and (3.18). $\square$

Lemma 3.5

If $0<q(p)<p^{*}$ and $\lim \nolimits _{p\rightarrow 1^{+}}q(p)=1,$ then

$$\begin{aligned} \frac{1}{\left| \Omega \right| }\le \liminf _{p\rightarrow 1^{+} }\left\| u_{p,q(p)}\right\| _{\infty }\quad \textrm{and}\quad \limsup _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{\infty }\le \frac{h(\Omega )^{N}}{\left| \Omega \right| h(\Omega ^{\star })^{N}}. \end{aligned}$$

(3.21)

Proof

The first estimate in (3.21) is immediate since

$$\begin{aligned} 1=\left\| u_{p,q(p)}\right\| _{q(p)}^{q(p)}\le \left\| u_{p,q(p)} \right\| _{\infty }^{q(p)}\left| \Omega \right| . \end{aligned}$$

According to Remark 3.2, we have that

$$\begin{aligned} C_{p}\left\| u_{p,q(p)}\right\| _{\infty }^{\frac{N(p^{*} -q(p))}{p^{*}}}\le \left\| u_{p,q(p)}\right\| _{q(p)}^{q(p)}=1 \end{aligned}$$

(3.22)

where

$$\begin{aligned} C_{p}:=\left( \frac{\lambda _{p,p^{*}}}{\lambda _{p,q(p)}}\right) ^{\frac{N}{p}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1}I_{p} \end{aligned}$$

and

$$\begin{aligned} I_{p}:=\left\{ \begin{array}{ll} q(p)\int _{0}^{1}\left( 1-\tau \right) ^{\frac{N(p-1)}{p}}\tau ^{\frac{N(1-q(p))}{p}+q(p)-1}\textrm{d}\tau &{} \text {if} \quad 0\le q(p)<1\\ q(p) {\displaystyle \int _{0}^{1}} (1-\tau )^{\frac{N(p-1)}{p}}\tau ^{q(p)-1}\textrm{d}\tau &{} \text {if} \quad 1\le q(p)<p^{*}. \end{array} \right. \end{aligned}$$

It follows from (3.22) that

$$\begin{aligned} \left\| u_{p,q(p)}\right\| _{\infty }\le C_{p}^{-\frac{p^{*} }{N(p^{*}-q(p))}}. \end{aligned}$$

As

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\frac{p^{*}}{N(p^{*}-q(p))}=1=\lim _{p\rightarrow 1^{+}}\left( \frac{p+N(p-1)}{p}\right) ^{N+1}=\lim _{p\rightarrow 1^{+}}I_{p} \end{aligned}$$

and

$$\begin{aligned} C_{p}^{-1}=\left( \frac{\lambda _{p,q(p)}}{\lambda _{p,p^{*}}}\right) ^{\frac{N}{p}}\left( \frac{p+N(p-1)}{p}\right) ^{N+1}I_{p}^{-1}, \end{aligned}$$

we obtain the second estimate in (3.21) from (1.6) and (3.18). $\square$

Proof of Theorem 1.1

Of course, (1.9) follows directly from (3.21).

Let us prove (1.8). If $0<q(p)<1,$ then Hölder’s inequality yields

$$\begin{aligned} 1=\left\| u_{p,q(p)}\right\| _{q(p)}^{q(p)}\le \left\| u_{p,q(p)} \right\| _{1}\left| \Omega \right| ^{1-q(p)} \end{aligned}$$

so that

$$\begin{aligned} 1=\lim _{p\rightarrow 1^{+}}\frac{1}{\left| \Omega \right| ^{1-q(p)}} \le \liminf _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{1}. \end{aligned}$$

As for $1\le q(p)<p^{*},$ we first note from (3.21) that

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{\infty } ^{q(p)-1}=1. \end{aligned}$$

Then, taking into account that

$$\begin{aligned} 1=\left\| u_{p,q(p)}\right\| _{q(p)}^{q(p)}\le \left\| u_{p,q(p)} \right\| _{\infty }^{q(p)-1}\left\| u_{p,q(p)}\right\| _{1}, \end{aligned}$$

we get

$$\begin{aligned} 1=\lim _{p\rightarrow 1^{+}}\frac{1}{\left\| u_{p,q(p)}\right\| _{\infty }^{q(p)-1}}\le \liminf _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{1}. \end{aligned}$$

We have thus proved the estimate

$$\begin{aligned} 1\le \liminf _{p\rightarrow 1^{+}}\left\| u_{p,q(p)}\right\| _{1} \end{aligned}$$

which, in view of (3.19), leads us to (1.8).

Exploiting (1.2) with respect to $\lambda _{p,p}$ again (see (3.20)), we obtain from (1.5) and (1.8) that

$$\begin{aligned} h(\Omega )=\lim _{p\rightarrow 1^{+}}\left( \lambda _{p,p}^{-1}\left| \Omega \right| ^{1-p}\left\| u_{p,q(p)}\right\| _{1}^{p}\right) \le \liminf _{p\rightarrow 1^{+}}\lambda _{p,q(p)}. \end{aligned}$$

Bearing in mind (3.18), this proves (1.7).

In order to complete the proof, let us take $p_{n}\rightarrow 1^{+}$ and set

$$\begin{aligned} q_{n}:=q(p_{n}) \ \text {and} \ u_{n}:=u_{p_{n},q_{n}}. \end{aligned}$$

Then, $\lambda _{p_{n},q_{n}}=\left\| \nabla u_{n}\right\| _{p_{n} }^{p_{n}},$ $\left\| u_{n}\right\| _{q_{n}}=1,$ and $\lim _{n\rightarrow \infty }q_{n}=1.$ Moreover, it follows from (1.8) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| u_{n}\right\| _{1}=1. \end{aligned}$$

(3.23)

We note that

$$\begin{aligned} \left| Du_{n}\right| ({\mathbb {R}}^{N})=\left| Du_{n}\right| (\Omega )=\left\| \nabla u_{n}\right\| _{1}\le \left\| \nabla u_{n}\right\| _{p_{n}}\left| \Omega \right| ^{1-\frac{1}{p_{n}} }=\lambda _{p_{n},q_{n}}^{\frac{1}{p_{n}}}\left| \Omega \right| ^{1-\frac{1}{p_{n}}}. \end{aligned}$$

Hence, (1.7) implies that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left| Du_{n}\right| ({\mathbb {R}}^{N} )\le \lim _{n\rightarrow \infty }\lambda _{p_{n},q_{n}}^{\frac{1}{p_{n}} }\left| \Omega \right| ^{1-\frac{1}{p_{n}}}=h(\Omega ). \end{aligned}$$

(3.24)

We conclude from (3.23) and (3.24) that the sequence $\left( u_{n}\right)$ is bounded in $BV(\Omega ).$ Thus, by the compactness of the embedding $BV(\Omega )\hookrightarrow L^{1}(\Omega )$ we can assume (up to passing to a subsequence) that $u_{n}\rightarrow u,$ in $L^{1}(\Omega )$ and also pointwise almost everywhere in $\Omega .$ Extending $u_{n}$ as zero on ${\mathbb {R}}^{N}\setminus \overline{\Omega }$, we have that $u_{n}$ converges in $L^{1}({\mathbb {R}}^{N})$ to u extended as zero on ${\mathbb {R}} ^{N}\setminus \overline{\Omega }.$

Owing to (3.23), we have $\left\| u\right\| _{1}=1,$ which confirms item (a) and also implies that $u\in BV_{0}(\Omega ).$ Hence, it follows from (2.1) that

$$\begin{aligned} h(\Omega )\le \frac{\left| Du\right| ({\mathbb {R}}^{N})}{\left\| u\right\| _{1}}=\left| Du\right| ({\mathbb {R}}^{N}). \end{aligned}$$

Moreover, Proposition 2.1 and (3.24) yield

$$\begin{aligned} \left| Du\right| ({\mathbb {R}}^{N})\le \liminf _{n\rightarrow \infty }\left| Du_{n}\right| ({\mathbb {R}}^{N})\le h(\Omega ), \end{aligned}$$

showing that $\left| Du\right| ({\mathbb {R}}^{N})=h(\Omega ).$ Then, item (c) is consequence of Proposition 2.2.

Now, let us prove item (b). Let us fix $r>1$ and $\epsilon >0.$ As $q_{n}\rightarrow 1,$ we have that $q_{n}<r$ for all $n\ge n_{0}$ and some $n_{0}\in {\mathbb {N}}.$ Moreover, owing to the second estimate in (3.21) we can also assume that

$$\begin{aligned} \left\| u_{n}\right\| _{\infty }\le \frac{h(\Omega )^{N}}{\left| \Omega \right| h(\Omega ^{\star })^{N}}+\epsilon , \ \text {for all} \ n\ge n_{0}. \end{aligned}$$

(3.25)

By Hölder’s inequality, we have

$$\begin{aligned} 1=\left\| u_{n}\right\| _{q_{n}}\le \left\| u_{n}\right\| _{r}\left| \Omega \right| ^{\frac{1}{q_{n}}-\frac{1}{r}}, \end{aligned}$$

so that

$$\begin{aligned} \left| \Omega \right| ^{\frac{1}{r}-\frac{1}{q_{n}}}\le \left\| u_{n}\right\| _{r}, \ \text {for all} \ n\ge n_{0}. \end{aligned}$$

(3.26)

We also have

$$\begin{aligned} \left\| u_{n}\right\| _{r}^{r}\le \left\| u_{n}\right\| _{\infty }^{r-1}\left\| u_{n}\right\| _{1}\le \left( \frac{h(\Omega )^{N} }{\left| \Omega \right| h(\Omega ^{\star })^{N}}+\epsilon \right) ^{r-1}\left\| u_{n}\right\| _{1},\quad \text {for all }n\ge n_{0}. \end{aligned}$$

(3.27)

Convergence dominated theorem and (3.25) imply that $u_{n}\rightarrow u$ in $L^{r}(\Omega ).$ Hence, (3.26) and (3.27) imply that

$$\begin{aligned} \left| \Omega \right| ^{\frac{1}{r}-1}\le \left\| u\right\| _{r}\le \left( \frac{h(\Omega )^{N}}{\left| \Omega \right| h(\Omega ^{\star })^{N}}+\epsilon \right) ^{\frac{r-1}{r}}\left\| u\right\| _{1}^{\frac{1}{r}}=\left( \frac{h(\Omega )^{N}}{\left| \Omega \right| h(\Omega ^{\star })^{N}}+\epsilon \right) ^{\frac{r-1}{r}}. \end{aligned}$$

(3.28)

As r and $\epsilon$ are arbitrarily fixed, (3.28) implies that $u\in L^{\infty }(\Omega )$ and

$$\begin{aligned} \left| \Omega \right| ^{-1}\le \left\| u\right\| _{\infty } \le \frac{h(\Omega )^{N}}{\left| \Omega \right| h(\Omega ^{\star })^{N}}. \end{aligned}$$

$\square$

As mentioned in the Introduction, right after Corollary 1.2, Bueno and Ercole proved in [5] that

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\left\| v_{p,1}\right\| _{1}^{1-p} =h(\Omega )=\left\| v_{p,1}\right\| _{\infty }^{1-p}. \end{aligned}$$

As $\lambda _{p,1}=\left\| v_{p,1}\right\| _{1}^{1-p},$ a fact that was not noticed in [5], the first equality above leads directly to

$$\begin{aligned} \lim _{p\rightarrow 1^{+}}\lambda _{p,1}=h(\Omega ), \end{aligned}$$

(3.29)

which is (1.7) in the case where $q(p)\equiv 1.$ Thus, (3.29) combined with (1.5) and the monotonicity of the function $q\mapsto \lambda _{p,q}\left| \Omega \right| ^{\frac{p}{q}}$ also produces (1.7) for $q(p)\in (1,p).$ However, this combination does not lead to the same result for $q(p)\in (0,1)\cup (p,p^{*})$ as, for example, $q(p)=p^{\beta }$ with $\beta <0$ or $\beta >1$ (and p close to $1^{+}$). Our approach combining (1.5) with Lemma 3.1 provides a unified proof to (1.7) as well as allows us to estimate the limit function u.

4 Proof of Theorem 1.3

In this section, we prove Theorem 1.3, by applying Picone’s inequality to $w_{p,q(p)}$ and $e_{p},$ where $e_{p}$ is the first eigenfunction defined in (3.14).

Proof of Theorem 1.3

As

$$\begin{aligned} \lambda _{p,q(p)}\le \frac{\left\| \nabla w_{p,q(p)}\right\| _{p}^{p} }{\left\| w_{p,q(p)}\right\| _{q(p)}^{p}}=\left\| w_{p,q(p)} \right\| _{q(p)}^{q(p)-p}\le \left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p}\left| \Omega \right| ^{\frac{q(p)-p}{q(p)}}, \end{aligned}$$

we obtain from (1.7) that

$$\begin{aligned} h(\Omega )\le \liminf _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)}^{q(p)-p}\le \liminf _{p\rightarrow 1^{+}}\left\| w_{p,q(p)} \right\| _{\infty }^{q(p)-p}. \end{aligned}$$

(4.1)

Applying Picone’s inequality and using that $w_{p,q(p)}$ is a weak solution to (1.11), we find

$$\begin{aligned} \lambda _{p,p}\int _{\Omega }e_{p}^{p}\textrm{d}x&=\int _{\Omega }\left| \nabla e_{p}\right| ^{p}\textrm{d}x\\&\ge \int _{\Omega }\left| \nabla w_{p,q(p)}\right| ^{p-2}\nabla w_{p,q(p)}\cdot \nabla (\frac{e_{p}^{p}}{w_{p,q(p)}^{p-1}})\textrm{d}x\\&=\int _{\Omega }w_{p,q(p)}^{q(p)-1}\frac{e_{p}^{p}}{w_{p,q(p)}^{p-1} }\textrm{d}x=\int _{\Omega }w_{p,q(p)}^{q(p)-p}e_{p}^{p}\textrm{d}x. \end{aligned}$$

Hence,

$$\begin{aligned} \left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p}\int _{\Omega }W_{p} ^{q(p)-p}e_{p}^{p}\text {d}x\le \lambda _{p,p}\int _{\Omega }e_{p}^{p}\text {d}x \end{aligned}$$

(4.2)

where

$$\begin{aligned} W_{p}:=\frac{w_{p,q(p)}}{\left\| w_{p,q(p)}\right\| _{\infty }}. \end{aligned}$$

Now, let us assume that

$$\begin{aligned} L:=\limsup _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p}<\infty . \end{aligned}$$

Using again that $w_{p,q(p)}$ is a weak solution to (1.11), we have from Lemma 3.1, with $\lambda =\sigma =1,$ that

$$\begin{aligned} C_{p}\left\| w_{p,q(p)}\right\| _{\infty }^{\frac{N(p-q(p))}{p} }\left\| w_{p,q(p)}\right\| _{\infty }\le \left\| w_{p,q(p)} \right\| _{1} \end{aligned}$$

(4.3)

where

$$\begin{aligned} C_{p}:=\lambda _{p,p^{*}}^{\frac{N}{p}}\left( \frac{p}{p+N(p-1)}\right) ^{N+1}\sigma {\displaystyle \int _{0}^{1}} (1-\tau )^{\frac{N(p-1)}{p}}\textrm{d}\tau . \end{aligned}$$

It follows from (4.3) that

$$\begin{aligned} \frac{C_{p}}{\left\| w_{p,q(p)}\right\| _{\infty }^{\frac{N(q(p)-p)}{p}} }\le \left\| W_{p}\right\| _{1}, \end{aligned}$$

so that (1.6) and (3.16) yield

$$\begin{aligned} 0<\frac{\left| \Omega \right| h(\Omega ^{\star })^{N}}{L^{N}}\le \liminf _{p\rightarrow 1^{+}}\left\| W_{p}\right\| _{1}. \end{aligned}$$

(4.4)

We also have that

$$\begin{aligned} \left\| \nabla w_{p,q(p)}\right\| _{1}\le \left| \Omega \right| ^{1-\frac{1}{p}}\left\| \nabla w_{p,q(p)}\right\| _{p}=\left| \Omega \right| ^{1-\frac{1}{p}}\left\| w_{p,q(p)}\right\| _{q(p)}^{\frac{q(p)}{p}} \end{aligned}$$

so that

$$\begin{aligned} \left| DW_{p}\right| (\Omega )=\left\| \nabla W_{p}\right\| _{1}\le \left| \Omega \right| ^{1-\frac{1}{p}}\left\| w_{p,q(p)} \right\| _{\infty }^{\frac{q(p)-p}{p}}\left\| W_{p}\right\| _{q(p)}^{\frac{q(p)}{p}}. \end{aligned}$$

Hence, as $\left\| W_{p}\right\| _{q(p)}^{\frac{q(p)}{p}}\le \left| \Omega \right| ^{\frac{1}{p}}$ and $\left\| W_{p}\right\| _{1} \le \left| \Omega \right| ,$ we conclude that the family $\left( W_{p}\right)$ is uniformly bounded in $BV(\Omega ).$

Now, let $p_{n}\rightarrow 1^{+}$ be such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| w_{p_{n},q(p_{n})}\right\| _{\infty }^{q(p_{n})-p_{n}}=L. \end{aligned}$$

Owing to the compactness of $BV(\Omega )\hookrightarrow L^{1}(\Omega )$, we can assume (passing to subsequences, if necessary) that $W_{p_{n}}\rightarrow W$ in $L^{1}(\Omega )$ and also pointwise almost everywhere in $\Omega .$ It follows from (4.4) that $W>0$ a.e. in $\Omega$ and this implies that $W_{p_{n}}^{q(p_{n})-p_{n}}\rightarrow 1$ pointwise almost everywhere in $\Omega .$ As $\left\| W_{p_{n}}^{q(p_{n})-p_{n}}e_{p_{n}}^{p_{n} }\right\| _{\infty }\le 1,$ dominated convergence theorem and Lemma 3.3 guarantee that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\Omega }W_{p_{n}}^{q(p_{n})-p_{n}}e_{p_{n} }^{p_{n}}\textrm{d}x=\left\| e\right\| _{1}>0. \end{aligned}$$

Hence, (4.2) and (1.5) yield

$$\begin{aligned} L\left\| e\right\| _{1}\le h(\Omega )\left\| e\right\| _{1}, \end{aligned}$$

so that $L\le h(\Omega ).$ Combining this inequality with (4.1), we conclude that

$$\begin{aligned} h(\Omega )=\lim _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)}^{q(p)-p}=\lim _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p}. \end{aligned}$$

$\square$

Remark 4.1

One can derive from Remark 3.2 that if $0<q(p)<p^{*},$ then

$$\begin{aligned} \liminf _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)} ^{q(p)-p}=\liminf _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p} \end{aligned}$$

and

$$\begin{aligned} \limsup _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)} ^{q(p)-p}=\limsup _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{\infty }^{q(p)-p}. \end{aligned}$$

Thus, the alternative (1.13) in the statement of Theorem 1.3 can be replaced with

$$\begin{aligned} \limsup _{p\rightarrow 1^{+}}\left\| w_{p,q(p)}\right\| _{q(p)} ^{q(p)-p}=+\infty . \end{aligned}$$