# Local boundedness of minimizers under unbalanced Orlicz growth conditions

Andrea Cianchi  Andrea Cianchi, Dipartimento di Matematica e Informatica “U.
Dini”
Università di Firenze
Viale Morgagni 67/a
50134 Firenze
Italy<EMAIL_ADDRESS>and  Mathias Schäffner  Mathias Schäffner,
Institut für Mathematik, MLU Halle-Wittenberg, Theodor-Lieser-Straße 5, 06120
Halle (Saale), Germany<EMAIL_ADDRESS>
###### Abstract.

Local minimizers of integral functionals of the calculus of variations are
analyzed under growth conditions dictated by different lower and upper bounds
for the integrand. Growths of non-necessarily power type are allowed. The
local boundedness of the relevant minimizers is established under a suitable
balance between the lower and the upper bounds. Classical minimizers, as well
as quasi-minimizers are included in our discussion. Functionals subject to so-
called $p,q$-growth conditions are embraced as special cases and the
corresponding sharp results available in the literature are recovered.

###### Key words and phrases:

Local minimizers, local boundedness, unbalanced Orlicz growth, Orlicz-Sobolev
inequalities

###### 2000 Mathematics Subject Classification:

49N60

##  1\. Introduction

We are concerned with the local boundedness of local minimizers, or quasi-
minimizers, of integral functionals of the form

(1.1) $\mathcal{F}(u,\Omega)=\int_{\Omega}f(x,u,\nabla u)\,dx,$

where $\Omega$ is an open set in $\mathbb{R}^{n}$, with $n\geq 2$, and
$f:\Omega\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ is a Carathéodory
function subject to proper structure and growth conditions. Besides its own
interest, local boundedness is needed to ensure certain higher regularity
properties of minimizers. Interestingly, some regularity results for
minimizers admit variants that require weaker hypotheses under the a priori
assumption of their local boundedness.

Local boundedness of local minimizers of the functional $\mathcal{F}$ is
classically guaranteed if $f(x,t,\xi)$ is subject to lower and upper bounds in
terms of positive multiples of $|\xi|^{p}$, for some $p\geq 1$. This result
can be traced back to the work of Ladyzhenskaya and Ural’ceva [LaUr], which,
in turn, hinges upon methods introduced by De Giorgi in his regularity theory
for linear elliptic equations with merely measurable coefficients.

The study of functionals built on integrands $f(x,t,\xi)$ bounded from below
and above by different powers $|\xi|^{p}$ and $|\xi|^{q}$, called with
$p,q$-growth in the literature, was initiated some fifty years ago. A
regularity theory for minimizers under assumptions of this kind calls for
additional structure conditions on $f$, including convexity in the gradient
variable. As shown in various papers starting from the nineties of the last
century, local minimizers of functional with $p,q$-growth are locally bounded
under diverse structure conditions, provided that the difference between $q$
and $p$ is not too large, depending on the dimension $n$. This issue was
addressed in [MaPa, MoNa] and, more recently, in [CMM1, CMM2]. Related
questions are considered in [BMS, FuSb, Str] in connection with anisotropic
growth conditions. By contrast, counterexamples show that unbounded minimizers
may exist if the exponents $p$ and $q$ are too far apart [Gia, Ho, Ma1, Ma2].
The gap between the assumptions on $p$ and $q$ in these examples and in the
regularity result has recently been filled in the paper [HiSch], where the
local boundedness of minimizers is established for the full range of exponents
$p$ and $q$ excluded from the relevant counterexamples. An extension of the
techniques from [HiSch] has recently been applied in [DeGr] to extend the
boundedness result to obstacle problems.

In the present paper, the conventional realm of polynomial growths is
abandoned and the question of local boundedness of local minimizers, and
quasi-minimizers, is addressed under bounds on $f$ of Orlicz type. More
specifically, the growth of $f$ is assumed to be governed by Young functions,
namely nonnegative convex functions vanishing at $0$. The local boundedness of
minimizers in the case when lower and upper bounds on $f$ are imposed in terms
of the same Young function follows via a result from [Ci3], which also deals
with anisotropic Orlicz growths. The same problem for solutions to elliptic
equations is treated in [Ko].

Our focus here is instead on the situation when different Young functions
$A(|\xi|)$ and $B(|\xi|)$ bound $f(x,t,\xi)$ from below and above. Functionals
with $p,q$-growth are included as a special instance. A sharp balance
condition between the Young functions $A$ and $B$ is exhibited for any local
minimizer of the functional $\mathcal{F}$ to be locally bounded. Bounds on
$f(x,t,\xi)$ depending on a function $E(|t|)$ are also included in our
discussion. Let us mention that results in the same spirit can be found in the
paper [DMP], where, however, more restrictive non-sharp assumptions are
imposed.

The global boundedness of global minimizers of functionals and of solutions to
boundary value problems for elliptic equations subject to Orlicz growth
conditions has also been examined in the literature and is the subject e.g. of
[Al, BCM, Ci2, Ta1, Ta2]. Note that, unlike those concerning the local
boundedness of local minimizers and local solutions to elliptic equations,
global boundedness results in the presence of prescribed boundary conditions
just require lower bounds in the gradient variable for integrands of
functionals or equation coefficients. Therefore, the question of imposing
different lower and upper bounds does not arise with this regard.

Beyond boundedness, several further aspects of the regularity theory of
solutions to variational problems and associated Euler equations, under
unbalanced lower and upper bounds, have been investigated. The early papers
[Ma2, Ma3] have been followed by various contributions on this topic, a very
partial list of which includes [BCM, BeMi, BeSch2, BeSch3, BoBr, BCSV, BGS,
ByOh, CKP1, CKP2, CoMi, DeMi1, DeMi3, ELP, ELM, HäOk1, Ma4]. A survey of
investigations around this area can be found in [MiRa]. In particular, results
from [BCM, BoBr, CKP1, DeMi2, HäOk2] demonstrate the critical role of local
boundedness for higher regularity of local minimizers, which we alluded to
above.

##  2\. Main result

We begin by enucleating a basic case of our result for integrands in (1.1)
which do not depend on $u$. Namely, we consider functionals of the form

(2.1) $\mathcal{F}(u,\Omega)=\int_{\Omega}f(x,\nabla u)\,dx,$

where

$f:\Omega\times\mathbb{R}^{n}\to\mathbb{R}.$

A standard structure assumption to be fulfilled by $f$ is that

(2.2) $\text{ the function}\quad\mathbb{R}^{n}\ni\xi\mapsto
f(x,\xi)\quad\text{ is convex for a.e. $x\in\Omega$.}$

Next, an $A,B$-growth condition on $f$ is imposed, in the sense that

(2.3) $\displaystyle A(|\xi|)-L\leq f(x,\xi)\leq B(|\xi|)+L\quad\text{for a.e.
$x\in\Omega$ and every $\xi\in\mathbb{R}^{n}$,}$

where $A$ is a Young function and $B$ is a Young function satisfying the
$\Delta_{2}$-condition near infinity. By contrast, the latter condition is not
required on the lower bound $A$.

The function $A$ dictates the natural functional framework for the trial
functions $u$ in the minimization problem for $\mathcal{F}$. It is provided by
the Orlicz-Sobolev class $V^{1}_{\rm loc}K^{A}(\Omega)$ of those weakly
differentiable functions on $\Omega$ such that

$\int_{\Omega^{\prime}}A(|\nabla u|)\,dx<\infty$

for every open set $\Omega^{\prime}\Subset\Omega$.
Besides standard local minimizers, we can as well deal with so-called quasi-
minimizers, via the very same approach. A function $u\in V^{1}_{\rm
loc}K^{A}(\Omega)$ is said to be a local quasi-minimizer of $\mathcal{F}$ if

$\mathcal{F}(u,\Omega^{\prime})<\infty$

for every open set $\Omega^{\prime}\Subset\Omega$, and there exists a constant
$Q\geq 1$ such that

(2.4) $\mathcal{F}(u,{\rm supp}\,\varphi)\leq Q\mathcal{F}(u+\varphi,{\rm
supp}\,\varphi)$

for every $\varphi\in V^{1}_{\rm loc}K^{A}(\Omega)$ such that ${\rm
supp}\,\varphi\Subset\Omega$. Plainly, $u$ is a standard local minimizer of
$\mathcal{F}$ provided that inequality (2.4) holds with $Q=1$.

Throughout the paper, we shall assume that

(2.5)
$\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-1}}\,dt=\infty.$

Indeed, if $A$ grows so fast near infinity that

(2.6)
$\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-1}}\,dt<\infty,$

then every function $u\in V^{1}_{\rm loc}K^{A}(\Omega)$ is automatically
bounded, irrespective of whether it minimizes $\mathcal{F}$ or not. This is
due to the inclusion

(2.7) $V^{1}_{\rm loc}K^{A}(\Omega)\subset L^{\infty}_{\rm loc}(\Omega),$

which holds as a consequence of a Sobolev-Poincaré inequality in Orlicz
spaces.

Heuristically speaking, our result ensures that any local quasi-minimizer of
$\mathcal{F}$ as in (2.1) is locally bounded, provided that the function $B$
does not grow too quickly near infinity compared to $A$. The maximal
admissible growth of $B$ is described through the sharp Sobolev conjugate
$A_{n-1}$ of $A$ in dimension $n-1$, whose definition is recalled in the next
section. More precisely, if

(2.8) $n\geq
3\quad\text{and}\quad\displaystyle\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-2}}\,dt=\infty,$

then $B$ has to be dominated by $A_{n-1}$ near infinity, in the sense that

(2.9) $\displaystyle B(t)\leq A_{n-1}(Lt)\qquad\mbox{for $t\geq t_{0}$,}$

for some positive constants $L$ and $t_{0}$.
On the other hand, in the regime complementary to (2.8), namely in either of
the following cases

(2.10) $\begin{cases}n=2\\\ n\geq
3\quad\text{and}\quad\displaystyle\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-2}}\,dt<\infty,\end{cases}$

no additional hypothesis besides the $\Delta_{2}$-condition near infinity is
needed on $B$. Notice that, by an Orlicz-Poincaré-Sobolev inequality on
$\mathbb{S}^{n-1}$, both options in (2.10) entail that $V^{1}_{\rm
loc}K^{A}(\mathbb{S}^{n-1})\subset L^{\infty}_{\rm loc}(\mathbb{S}^{n-1})$.

Altogether, our boundedness result for functionals of the form (2.1) reads as
follows.

######  Theorem 2.1.

Let $f:\Omega\times\mathbb{R}^{n}\to\mathbb{R}$ be a Carathéodory function
satisfying the structure assumption (2.2). Suppose that the growth condition
(2.3) holds for some Young functions $A$ and $B$, such that $B\in\Delta_{2}$
near infinity. Assume that either condition (2.10) is in force, or condition
(2.8) is in force and $B$ fulfills estimate (2.9). Then any local quasi-
minimizer of the functional $\mathcal{F}$ in (2.1) is locally bounded in
$\Omega$.

Assume now that $\mathcal{F}$ has the general form (1.1), and hence

$f:\Omega\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}.$

Plain convexity in the gradient variable is no longer sufficient, as a
structure assumption, for a local boundedness result to hold. One admissible
strengthening consists of coupling it with a kind of almost monotonicity
condition in the $u$ variable. Precisely, one can suppose that

(2.11) $\begin{cases}\text{ the function}\quad\mathbb{R}^{n}\ni\xi\mapsto
f(x,t,\xi)\quad\text{ is convex for a.e. $x\in\Omega$ and every
$t\in\mathbb{R}$,}\\\ f(x,t,\xi)\leq Lf(x,s,\xi)+E(|s|)+L\quad\text{if
$|t|\leq|s|$, for a.e. $x\in\Omega$ and every
$\xi\in\mathbb{R}^{n}$,}\end{cases}$

where $L$ is a positive constant and $E:[0,\infty)\to[0,\infty)$ is a non-
decreasing function fulfilling the $\Delta_{2}$-condition near infinity.
An alternate condition which still works is the joint convexity of $f$ in the
couple $(t,\xi)$, in the sense that

(2.12) $\text{ the
function}\quad\mathbb{R}\times\mathbb{R}^{n}\ni(t,\xi)\mapsto
f(x,t,\xi)\quad\text{ is convex for a.e. $x\in\Omega$.}$

The growth of $f$ is governed by the following bounds:

(2.13) $\displaystyle A(|\xi|)-E(|t|)-L\leq f(x,t,\xi)\leq
B(|\xi|)+E(|t|)+L\quad\text{for a.e. $x\in\Omega$ and every $t\in\mathbb{R}$
and $\xi\in\mathbb{R}^{n}$,}$

where $A$ is a Young function, $B$ is a Young function satisfying the
$\Delta_{2}$-condition near infinity, and $E$ is the same function as in
(2.11), if this assumption is in force.

The appropriate function space for trial functions in the definition of quasi-
minimizer of the functional $\mathcal{F}$ is still $V^{1}_{\rm
loc}K^{A}(\Omega)$, and the definition given in the special case (2.1) carries
over to the present general framework.

The bound to be imposed on the function $B$ is the same as in the $u$-free
case described above. On the other hand, the admissible growth of the function
$E$ is dictated by the Sobolev conjugate $A_{n}$ of $A$ in dimension $n$.
Specifically, we require that

(2.14) $\displaystyle E(t)\leq A_{n}(Lt)\qquad\mbox{for $t\geq t_{0}$,}$

for some positive constants $L$ and $t_{0}$.

Our comprehensive result then takes the following form.

######  Theorem 2.2.

Let $f:\Omega\times\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ be a
Carathéodory function satisfying either the structure assumption (2.11) or
(2.12). Suppose that the growth condition (2.13) holds for some Young
functions $A$ and $B$ and a non-decreasing function $E$, such that
$B,E\in\Delta_{2}$ near infinity. Assume that either condition (2.10) is in
force, or condition (2.8) is in force and $B$ fulfills estimate (2.9).
Moreover, assume that $E$ fulfills estimate (2.14). Then any local quasi-
minimizer of the functional $\mathcal{F}$ in (1.1) is locally bounded in
$\Omega$.

Our approach to Theorems 2.1 and 2.2 follows along the lines of De Giorgi’s
regularity result for linear equations with merely measurable coefficients, on
which, together with Moser’s iteration technique, all available proofs of the
local boundedness of local solutions to variational problems or elliptic
equations are virtually patterned. The main novelties in the present framework
amount to the use of sharp Poincaré and Sobolev inequalities in Orlicz spaces
and to an optimized form of the Caccioppoli-type inequality. The lack of
homogeneity of non-power type Young functions results in Orlicz-Sobolev
inequalities whose integral form necessarily involves a gradient term on both
sides. This creates new difficulties, that also appear, again because of the
non-homogeneity of Young functions, in deriving the optimized Caccioppoli
inequality. The latter requires an ad hoc process in the choice of trial
functions in the definition of quasi-minimizers. The advantage of the use of
the relevant Caccioppoli inequality is that its proof only calls into play
Sobolev-type inequalities on $(n-1)$-dimensional spheres, instead of
$n$-dimensional balls. This allows for growths of the function $B$ dictated by
the $(n-1)$-dimensional Sobolev conjugate of $A$. By contrast, a more standard
choice of trial functions would only permit slower growths of $B$, not
exceeding the $n$-dimensional Sobolev conjugate of $A$. Orlicz-Sobolev and
Poincaré inequalities in dimension $n$ just come into play in the proof of
Theorem 2.2, when estimating terms depending on the variable $u$. The trial
function optimization strategy is reminiscent of that used in diverse settings
in recent years. The version exploited in [HiSch] – a variant of [BeSch1] – to
deal with functionals subject to $p,q$-growth conditions is sensitive to the
particular growth of the integrand. The conditions imposed in the situation
under consideration here are so general to force us to resort to a more robust
optimization argument, implemented in Lemma 5.1, Section 5. The latter is
inspired to constructions employed in [BrCa] in the context of div-curl
lemmas, and in [KRS] in the proof of absence of Lavrientiev-phenomena in
vector-valued convex minimization problems.

We conclude this section by illustrating Theorems 2.1 and 2.2 with
applications to a couple of special instances. The former corresponds to
functionals with $p,q$-growth. It not only recovers the available results but
also augments and extends them in some respects. The latter concerns
functionals with “power-times-logarithmic” growths, and provides us with an
example associated with genuinely non-homogenous Young functions.

######  Example 2.1.

In the standard case when

$A(t)=t^{p},$

with $1\leq p\leq n$, Theorem 2.1 recovers a result of [HiSch]. Indeed, if
$n\geq 3$ and $1\leq p<n-1$, we have that $A_{n-1}(t)\approx
t^{\frac{(n-1)p}{(n-1)-p}}$, and assumption (2.9) is equivalent to

(2.15) $B(t)\lesssim t^{\frac{(n-1)p}{(n-1)-p}}\quad\text{near infinity.}$

Here, the relations $\lesssim$ and $\approx$ mean domination and equivalence,
respectively, in the sense of Young functions.
If $p=n-1$, then $A_{n-1}(t)\approx e^{t^{\frac{n-1}{n-2}}}$ near infinity,
whereas if $p>n-1$, then the second alternative condition (2.10) is satisfied.
Hence, if either $n=2$ or $n\geq 3$ and $p\geq n-1$, then any Young function
$B\in\Delta_{2}$ near infinity is admissible.
Condition (2.15) is sharp, since the functionals with $p,q$-growth exhibited
in [Gia, Ho, Ma1, Ma2] admit unbounded local minimizers if assumption (2.15)
is dropped.
Let us point out that the result deduced from Theorem 2.1 also enhances that
of [HiSch], where the function $\xi\mapsto f(x,\xi)$ is assumed to fulfil a
variant of the $\Delta_{2}$-condition, which is not imposed here.
On the other hand, Theorem 2.2 extends the result of [HiSch], where integrands
only depending on $x$ and $\nabla u$ are considered. The conclusion of Theorem
2.2 hold under the same bound (2.15) on the function $B$. Moreover,
$A_{n}(t)\approx t^{\frac{np}{n-p}}$ if $1\leq p<n$ and $A_{n}(t)\approx
e^{t^{\frac{n}{n-1}}}$ near infinity if $p=n$. Hence, if $1\leq p<n$, then
assumption (2.14) reads:

$E(t)\lesssim t^{\frac{np}{n-p}}\quad\text{near infinity.}$

If $p=n$, then any non-decreasing function $E$ satisfying the
$\Delta_{2}$-condition near infinity satisfies assumption (2.14), and it is
therefore admissible.

######  Example 2.2.

Assume that

$A(t)\approx t^{p}(\log t)^{\alpha}\quad\text{near infinity,}$

where $1<p<n$ and $\alpha\in\mathbb{R}$, or $p=1$ and $\alpha\geq 0$, or $p=n$
and $\alpha\leq n-1$. Observe that these restrictions on the exponents $p$ and
$\alpha$ are required for $A$ to be a Young function fulfilling condition
(2.5). From an application of Theorem 2.2 one can deduce that any local
minimizer of $\mathcal{F}$ is locally bounded under the following assumptions.
,
If $n\geq 3$ and $p<n-1$, then we have to require that

$B(t)\lesssim t^{\frac{(n-1)p}{(n-1)-p}}(\log
t)^{\frac{(n-1)\alpha}{(n-1)-p}}\quad\text{near infinity.}$

If either $n=2$, or $n\geq 3$ and $n-1\leq p<n$, then any Young function
$B\in\Delta_{2}$ near infinity is admissible.
Moreover, if $p<n$, then our assumption on $E$ takes the form:

$E(t)\lesssim t^{\frac{np}{n-p}}(\log t)^{\frac{n\alpha}{n-p}}\quad\text{near
infinity.}$

If $p=n$, then any non-decreasing function $E\in\Delta_{2}$ near infinity is
admissible.

##  3\. Orlicz-Sobolev spaces

This section is devoted to some basic definitions and properties from the
theory of Young functions and Orlicz spaces. We refer the reader to the
monograph [RaRe] for a comprehensive presentation of this theory. The Sobolev
and Poincaré inequalities in Orlicz-Sobolev spaces that play a role in our
proofs are also recalled.

Orlicz spaces are defined in terms of Young functions. A function
$A:[0,\infty)\to[0,\infty]$ is called a Young function if it is convex (non
trivial), left-continuous and $A(0)=0$.
The convexity of $A$ and its vanishing at $0$ imply that

(3.1) $\lambda A(t)\leq A(\lambda t)\quad\hbox{for $\lambda\geq 1$ and $t\geq
0$,}$

and that the function

(3.2) $\frac{A(t)}{t}\quad\text{is non-decreasing in $(0,\infty)$.}$

The Young conjugate $\widetilde{A}$ of $A$ is defined by

$\widetilde{A}(t)=\sup\\{\tau t-A(\tau):\,\tau\geq 0\\}\qquad{\rm for}\qquad
t\geq 0\,.$

The following inequalities hold:

(3.3) $s\leq A^{-1}(s)\widetilde{A}^{-1}(s)\leq 2s\qquad\hbox{for $s\geq 0$,}$

where $A^{-1}$ and $\widetilde{A}^{-1}$ denote the generalized right-
continuous inverses of $A$ and $\widetilde{A}$, respectively.
A Young function $A$ is said to satisfy the $\Delta_{2}$-condition globally –
briefly $A\in\Delta_{2}$ globally – if there exists a constant $c$ such that

(3.4) $A(2t)\leq cA(t)\quad\text{for $t\geq 0$.}$

If inequality (3.4) just holds for $t\geq t_{0}$ for some $t_{0}>0$, then we
say that $A$ satisfies the $\Delta_{2}$-condition near infinity, and write
$A\in\Delta_{2}$ near infinity. One has that

(3.5) $A\in\Delta_{2}$ globally [near infinity] if and only if there exists
$q\geq 1$ such that $\frac{tA^{\prime}(t)}{A(t)}\leq q$ for a.e. $t>0$ [$t\geq
t_{0}$].

A Young function $A$ is said to dominate another Young function $B$ globally
if there exists a positive constant $c$ such that

(3.6) $B(t)\leq A(ct)$

for $t\geq 0$. The function $A$ is said to dominate $B$ near infinity if there
exists $t_{0}\geq 0$ such that (3.6) holds for $t\geq t_{0}$. If $A$ and $B$
dominate each other globally [near infinity], then they are called equivalent
globally [near infinity]. We use the notation $B\lesssim A$ to denote that $A$
dominates $B$, and $B\approx A$ to denote that $A$ and $B$ are equivalent.
This terminology and notation will also be adopted for merely nonnegative
functions, which are not necessarily Young functions.

Let $\Omega$ be a measurable set in $\mathbb{R}^{n}$. The Orlicz class
$K^{A}(\Omega)$ built upon a Young function $A$ is defined as

(3.7) $K^{A}(\Omega)=\bigg{\\{}u:\text{$u$ is measurable in $\Omega$ and
$\int_{\Omega}A(|u|)\,dx<\infty$}\bigg{\\}}.$

The set $K^{A}(\Omega)$ is convex for every Young function $A$.
The Orlicz space $L^{A}(\Omega)$ is the linear hull of $K^{A}(\Omega)$. It is
a Banach function space, equipped with the Luxemburg norm defined as

(3.8)
$\|u\|_{L^{A}(\Omega)}=\inf\left\\{\lambda>0:\int_{\Omega}A\left(\frac{|u|}{\lambda}\right)dx\leq
1\right\\}$

for a measurable function $u$. These notions are modified as usual to define
the local Orlicz class $K^{A}_{\rm loc}(\Omega)$ and the local Orlicz space
$L^{A}_{\rm loc}(\Omega)$.
If either $A\in\Delta_{2}$ globally, or $|\Omega|<\infty$ and $A\in\Delta_{2}$
near infinity, then $K^{A}(\Omega)$ is, in fact, a linear space, and
$K^{A}(\Omega)=L^{A}(\Omega)$. Here, $|\Omega|$ denotes the Lebesgue measure
of $\Omega$.
Notice that, in particular, $L^{A}(\Omega)=L^{p}(\Omega)$ if $A(t)=t^{p}$ for
some $p\in[1,\infty)$, and $L^{A}(\Omega)=L^{\infty}(\Omega)$ if $A(t)=0$ for
$t\in[0,1]$ and $A(t)=\infty$ for $t\in(1,\infty)$.
The identity

(3.9) $\|\chi_{E}\|_{L^{A}(\Omega)}=\frac{1}{A^{-1}(1/|E|)}$

holds for every Young function $A$ and any measurable set $E\subset\Omega$.
Here, $\chi_{E}$ stands for the characteristic function of $E$.
The Hölder inequality in Orlicz spaces tells us that

(3.10) $\int_{\Omega}|uv|\,dx\leq
2\|u\|_{L^{A}(\Omega)}\|v\|_{L^{\widetilde{A}}(\Omega)}$

for $u\in L^{A}(\Omega)$ and $v\in L^{\widetilde{A}}(\Omega)$.

Assume now that $\Omega$ is an open set. The homogeneous Orlicz-Sobolev class
$V^{1}K^{A}(\Omega)$ is defined as the convex set

(3.11) $\displaystyle V^{1}K^{A}(\Omega)=\big{\\{}u\in W^{1.1}_{\rm
loc}(\Omega):\,|\nabla u|\in K^{A}(\Omega)\big{\\}}$

and the inhomogeneous Orlicz-Sobolev class $W^{1}K^{A}(\Omega)$ is the convex
set

(3.12) $\displaystyle W^{1}K^{A}(\Omega)=K^{A}(\Omega)\cap
V^{1}K^{A}(\Omega).$

The homogenous Orlicz-Space $V^{1}L^{A}(\Omega)$ and its inhomogenous
counterpart $W^{1}L^{A}(\Omega)$ are accordingly given by

(3.13) $\displaystyle V^{1}L^{A}(\Omega)=\big{\\{}u\in W^{1.1}_{\rm
loc}(\Omega):\,|\nabla u|\in L^{A}(\Omega)\big{\\}}$

and

(3.14) $\displaystyle W^{1}L^{A}(\Omega)=L^{A}(\Omega)\cap
V^{1}L^{A}(\Omega).$

The latter is a Banach space endowed with the norm

(3.15) $\|u\|_{W^{1,A}(\Omega)}=\|u\|_{L^{A}(\Omega)}+\|\nabla
u\|_{L^{A}(\Omega)}.$

Here, and in what follows, we use the notation $\|\nabla u\|_{L^{A}(\Omega)}$
as a shorthand for $\|\,|\nabla u|\,\|_{L^{A}(\Omega)}$.
The local versions $V^{1}_{\rm loc}K^{A}(\Omega)$, $W^{1}_{\rm
loc}K^{A}(\Omega)$, $V^{1}_{\rm loc}L^{A}(\Omega)$, and $W^{1}_{\rm
loc}L^{A}(\Omega)$ of these sets/spaces is obtained by modifying the above
definitions as usual. In the case when $L^{A}(\Omega)=L^{p}(\Omega)$ for some
$p\in[1,\infty]$, the standard Sobolev space $W^{1,p}(\Omega)$ and its
homogeneous version $V^{1,p}(\Omega)$ are recovered.
Orlicz and Orlicz-Sobolev classes of weakly differentiable functions $u$
defined on the $(n-1)$-dimensional unit sphere $\mathbb{S}^{n-1}$ in
$\mathbb{R}^{n}$ also enter our approach. These spaces are defined as in
(3.7), (3.8), (3.11), (3.13), and (3.14), with the Lebesgue measure replaced
with the $(n-1)$-dimensional Hausdorff measure $\mathcal{H}^{n-1}$, and
$\nabla u$ replaced with $\nabla_{\mathbb{S}}u$, the vector field on
$\mathbb{S}^{n-1}$ whose components are the covariant derivatives of $u$

As highlighted in the previous section, sharp embedding theorems and
corresponding inequalities in Orlicz-Sobolev spaces play a critical role in
the formulation of our result and in its proof. As shown in [Ci2] (see also
[Ci1] for an equivalent version), the optimal $n$-dimensional Sobolev
conjugate of a Young function $A$ fulfilling

(3.16) $\int_{0}\biggl{(}\frac{t}{A(t)}\biggr{)}^{\frac{1}{n-1}}\,dt<\infty$

is the Young function $A_{n}$ defined as

(3.17) $A_{n}(t)=A(H_{n}^{-1}(t))\qquad\text{for $t\geq 0$,}$

where the function $H_{n}:[0,\infty)\to[0,\infty)$ is given by

(3.18)
$H_{n}(s)=\biggl{(}\int_{0}^{s}\biggl{(}\frac{t}{A(t)}\biggr{)}^{\frac{1}{n-1}}\,dt\biggr{)}^{\frac{n-1}{n}}\qquad\mbox{for
$s\geq 0$.}$

The function $A_{n-1}$ is defined analogously, by replacing $n$ with $n-1$ in
equations (3.17) and (3.18).
In the statements of Theorems 2.1 and 2.2, the functions $A_{n}$,and $A_{n-1}$
are defined after modifying $A$ near $0$, if necessary, in such a way that
condition (3.16) be satisfied. Assumptions (2.3) and (2.13) are not affected
by the choice of the modified function $A$, thanks to the presence of the
additive constant $L$. Membership of a function in an Orlicz-Sobolev local
class or space associated with $A$ is also not influenced by this choice,
inasmuch as the behavior of $A$ near $0$ is irrelevant (up to additive and/or
multiplicative constants) whenever integrals or norms over sets with finite
measure are concerned.
An optimal Sobolev-Poincaré inequality on balls
$\mathbb{B}_{r}\subset\mathbb{R}^{n}$, centered at $0$ and with radius $r$
reads as follows. In its statement, we adopt the notation

$u_{\mathbb{B}_{r}}=\fint_{\mathbb{B}_{r}}u(x)\,dx,$

where $\fint$ stands for integral average.

######  Theorem A.

Let $n\geq 2$, let $r>0$, and let $A$ be a Young function fulfilling condition
(3.16). Then, there exists a constant $\kappa=\kappa(n)$ such that

(3.19)
$\int_{\mathbb{B}_{r}}A_{n}\Bigg{(}\frac{|u-u_{\mathbb{B}_{r}}|}{\kappa\big{(}\int_{\mathbb{B}_{r}}A(|\nabla
u|)dy\big{)}^{\frac{1}{n}}}\Bigg{)}\,dx\leq\int_{\mathbb{B}_{r}}A(|\nabla
u|)\,dx$

for every $u\in V^{1}K^{A}(\mathbb{B}_{r})$.

As a consequence of inequality (3.19) and of Lemma 4.1, Section 4, the
following inclusion holds:

(3.20) $V^{1}_{\rm loc}K^{A}(\Omega)\subset K^{A}_{\rm loc}(\Omega)$

for any open set $\Omega\subset\mathbb{R}^{n}$ and any Young function $A$.
Thereby,

$V^{1}_{\rm loc}K^{A}(\Omega)=W^{1}_{\rm loc}K^{A}(\Omega).$

Hence, in what follows, the spaces $V^{1}_{\rm loc}K^{A}(\Omega)$ and
$W^{1}_{\rm loc}K^{A}(\Omega)$ will be equally used.

Besides the Sobolev-Poincaré inequality of Theorem A, a Sobolev type
inequality is of use in our applications and is the subject of the following
theorem. Only Part (i) of the statement will be needed. Part (ii)
substantiates inclusion (2.7).

######  Theorem B.

Let $n\geq 2$, let $r>0$, and let $A$ be a Young function fulfilling condition
(3.16).

  * (i)

Assume that condition (2.5) holds. Then, there exists a constant
$\kappa=\kappa(n,r)$ such that

(3.21)
$\int_{\mathbb{B}_{r}}A_{n}\Bigg{(}\frac{|u|}{\kappa\big{(}\int_{\mathbb{B}_{r}}A(|u|)+A(|\nabla
u|)dy\big{)}^{\frac{1}{n}}}\Bigg{)}\,dx\leq\int_{\mathbb{B}_{r}}A(|u|)+A(|\nabla
u|)\,dx$

for every $u\in W^{1}K^{A}(\mathbb{B}_{r})$.

  * (ii)

Assume that condition (2.6) holds. Then, there exists a constant
$\kappa=\kappa(n,r,A)$ such that

(3.22)
$\|u\|_{L^{\infty}(\mathbb{B}_{r})}\leq\kappa\bigg{(}\int_{\mathbb{B}_{r}}A(|u|)+A(|\nabla
u|)\,dx\bigg{)}^{\frac{1}{n}}$

for every $u\in W^{1}K^{A}(\mathbb{B}_{r})$.

In particular, if $r\in[r_{1},r_{2}]$ for some $r_{2}>r_{1}>0$, then the
constant $\kappa$ in inequalities (3.21) and (3.22) depends on $r$ only via
$r_{1}$ and $r_{2}$.

A counterpart of Theorem B for Orlicz-Sobolev functions on the sphere
$\mathbb{S}^{n-1}$ takes the following form.

######  Theorem C.

Let $n\geq 2$ and let $A$ be a Young function such that

(3.23) $\int_{0}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-2}}\,dt<\infty$

if $n\geq 3$.

  * (i)

Assume that $n\geq 3$ and

(3.24)
$\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-2}}\,dt=\infty.$

Then, there exists a constant $\kappa=\kappa(n)$ such that

(3.25)
$\int_{\mathbb{S}^{n-1}}A_{n-1}\biggl{(}\frac{|u|}{\kappa\big{(}\int_{\mathbb{S}^{n-1}}A(|u|)+A(|\nabla_{\mathbb{S}}u|)d\mathcal{H}^{n-1}(y)\big{)}^{\frac{1}{n-1}}}\biggr{)}\,d\mathcal{H}^{n-1}(x)\leq\int_{\mathbb{S}^{n-1}}A(|u|)+A(|\nabla_{\mathbb{S}}u|)\,d\mathcal{H}^{n-1}(x)$

for $u\in W^{1}K^{A}(\mathbb{S}^{n-1})$.

  * (ii)

Assume that one of the following situations occurs:

(3.26) $\begin{cases}n=2&\quad\text{and}\quad\lim_{t\to
0^{+}}\frac{A(t)}{t}>0\\\ \\\ n\geq 3&\quad\text{
and}\quad\displaystyle\int^{\infty}\bigg{(}\frac{t}{A(t)}\bigg{)}^{\frac{1}{n-2}}\,dt<\infty.\end{cases}$

Then, there exists a constant $\kappa=\kappa(n,A)$ such that

(3.27)
$\|u\|_{L^{\infty}(\mathbb{S}^{n-1})}\leq\kappa\biggl{(}\int_{\mathbb{S}^{n-1}}A(|u|)+A(|\nabla_{\mathbb{S}}u|)\,d\mathcal{H}^{n-1}(x)\biggr{)}^{\frac{1}{n-1}}$

for $u\in W^{1}K^{A}(\mathbb{S}^{n-1})$.

Theorems A and B are special cases of [Ci5, Theorems 4.4 and 3.1],
respectively, which hold in any Lipschitz domain in $\mathbb{R}^{n}$ (and for
Orlicz-Sobolev spaces of arbitrary order). The assertions about the dependence
of the constants can be verified via a standard scaling argument. Theorem C
can be derived via arguments analogous to those in the proof of [Ci5, Theorem
3.1]. For completeness, we offer the main steps of the proof.

###### Proof of Theorem C.

_Part (i)._ Let us set

$u_{\mathbb{S}^{n-1}}=\fint_{\mathbb{S}^{n-1}}u(x)\,d\mathcal{H}^{n-1}(x).$

A key step is a Sobolev-Poincaré type inequality, a norm version of (3.19) on
$\mathbb{S}^{n-1}$, which tells us that

(3.28)
$\displaystyle\|u-u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}\leq
c\|\nabla_{\mathbb{S}}u\|_{L^{A}(\mathbb{S}^{n-1})}$

for some constant $c=c(n)$ and for $u\in V^{1}L^{A}(\mathbb{S}^{n-1})$. A
proof of inequality (3.28) rests upon the following symmetrization argument
combined with a one-dimensional Hardy-type inequality in Orlicz spaces.
Set

(3.29) $c_{n}=\mathcal{H}^{n-1}(\mathbb{S}^{n-1})$

and denote by $u^{\circ}:[0,c_{n}]\to[-\infty,\infty]$ the signed decreasing
rearrangement of $u$, defined by

$u^{\circ}(s)=\inf\\{t\in\mathbb{R}:\mathcal{H}^{n-1}(\\{u>t\\})\leq
s\\}\quad\hbox{for $s\in[0,c_{n}]$.}$

Moreover, define the signed symmetral
$u^{\sharp}:\mathbb{S}^{n-1}\to[-\infty,\infty]$ of $u$ as

$u^{\sharp}(x)=u^{\circ}(V(x))\quad\hbox{for $x\in\mathbb{S}^{n-1}$,}$

where $V(x)$ denotes the $\mathcal{H}^{n-1}$-measure of the spherical cap on
$\mathbb{S}^{n-1}$, centered at the north pole on $\mathbb{S}^{n-1}$, whose
boundary contains $x$. Thus, $u^{\sharp}$ is a function, which is
equimeasurable with $u$, and whose level sets are spherical caps centered at
the north pole.
The equimeasurability of the functions $u$, $u^{\circ}$ and $u^{\sharp}$
ensures that

(3.30)
$\|u-u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}=\|u^{\sharp}-u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}=\|u^{\circ}-u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(0,c_{n})}.$

Moreover, since $u^{\circ}(c_{n}/2)$ is a median of $u^{\circ}$ on $(0,c_{n})$
and $u_{\mathbb{S}^{n-1}}$ agrees with the mean value of $u^{\circ}$ over
$(0,c_{n})$, one has that

(3.31)
$\|u^{\circ}-u^{\circ}(c_{n}/2)\|_{L^{A_{n-1}}(0,c_{n})}\geq\tfrac{1}{2}\|u^{\circ}-u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(0,c_{n})}=\tfrac{1}{2}\|u-u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})},$

see e.g. [CMP, Lemma 2.2].
On the other hand, a version of the Pólya-Szegö principle on
$\mathbb{S}^{n-1}$ tells us that $u^{\circ}$ is locally absolutely continuous,
$u^{\sharp}\in V^{1}L^{A}(\mathbb{S}^{n-1})$, and

(3.32)
$\bigg{\|}I_{\mathbb{S}^{n-1}}(s)\,\Big{(}-\frac{du^{\circ}}{ds}\Big{)}\bigg{\|}_{L^{A}(0,c_{n})}=\|\nabla_{\mathbb{S}}u^{\sharp}\,\|_{L^{A}(\mathbb{S}^{n-1})}\leq\|\nabla_{\mathbb{S}}u\|_{L^{A}(\mathbb{S}^{n-1})},$

where $I_{\mathbb{S}^{n-1}}:[0,c_{n}]\to[0,\infty)$ denotes the isoperimetric
function of $\mathbb{S}^{n-1}$ (see [BrZi]). It is well-known that there
exists a positive constant $c=c(n)$ such that

(3.33) $I_{\mathbb{S}^{n-1}}(s)\geq
c\min\\{s,c_{n}-s\\}^{\frac{n-2}{n-1}}\quad\hbox{for $s\in(0,c_{n})$}.$

Hence,

(3.34)
$c\bigg{\|}\min\\{s,c_{n}-s\\}^{\frac{n-2}{n-1}}\,\Big{(}-\frac{du^{\circ}}{ds}\Big{)}\bigg{\|}_{L^{A}(0,c_{n})}\leq\|\nabla_{\mathbb{S}}u\|_{L^{A}(\mathbb{S}^{n-1})},$

The absolute continuity of $u^{\circ}$ ensures that

(3.35)
$u^{\circ}(s)-u^{\circ}(c_{n})=\int_{s}^{c_{n}/2}\bigg{(}-\frac{du^{\circ}}{dr}\bigg{)}\,dr\qquad\text{for
$s\in(0,c_{n})$.}$

Thanks to equations (3.30), (3.31), (3.34), (3.35), and to the symmetry of the
function $\min\\{s,c_{n}-s\\}^{\frac{n-2}{n-1}}$ about $c_{n}/2$, inequality
(3.28) is reduced to the inequality

(3.36)
$\bigg{\|}\int_{s}^{c_{n}/2}r^{-\frac{n-2}{n-1}}\phi(r)\,dr\bigg{\|}_{L^{A_{n-1}}(0,c_{n}/2)}\leq
c\|\phi\|_{L^{A}(0,c_{n}/2)}$

for a suitable constant $c=c(n)$ and for every $\phi\in L^{A}(0,c_{n}/2)$.
Inequality (3.36) is in turn a consequence of [Ci2, inequality (2.7)].

Next, by Lemma 4.2, Section 4, applied with $n$ replaced with $n-1$,

$\frac{1}{\widetilde{A}^{-1}(1/(t))}\frac{1}{A_{n-1}^{-1}(t))}\leq\frac{1}{t^{\frac{n-2}{n-1}}}\quad\text{for
$t>0$.}$

Hence, by inequality (3.10), with $\Omega$ replaced with $\mathbb{S}^{n-1}$,
one has that

(3.37) $\displaystyle\|u_{\mathbb{S}^{n-1}}\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}$
$\displaystyle=|u_{\mathbb{S}^{n-1}}|\|1\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}\leq\frac{2}{c_{n}}\|u\|_{L^{A}(\mathbb{S}^{n-1})}\|1\|_{L^{\widetilde{A}}(\mathbb{S}^{n-1})}\|1\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}$
$\displaystyle=\frac{2}{c_{n}}\frac{1}{\widetilde{A}^{-1}(1/c_{n})}\frac{1}{A_{n-1}^{-1}(1/c_{n})}\|u\|_{L^{A}(\mathbb{S}^{n-1})}\leq\frac{2}{c_{n}^{\frac{1}{n-1}}}\|u\|_{L^{A}(\mathbb{S}^{n-1})}.$

Coupling inequality (3.28) with (3.37) and making use of the triangle
inequality entail that

(3.38) $\displaystyle\|u\|_{L^{A_{n-1}}(\mathbb{S}^{n-1})}\leq
c\big{(}\|\nabla_{\mathbb{S}}u\|_{L^{A}(\mathbb{S}^{n-1})}+\|u\|_{L^{A}(\mathbb{S}^{n-1})}\big{)}$

for some constant $c=c(n)$ and for $u\in W^{1}L^{A}(\mathbb{S}^{n-1})$.
Now set

$M=\int_{\mathbb{S}^{n-1}}A(|\nabla_{\mathbb{S}}u|)+A(|u|)\,d\mathcal{H}^{n-1}(x),$

and apply inequality (3.38) with the function $A$ replaced with the Young
function $A_{M}$ given by

$A_{M}(t)=\frac{A(t)}{M}\qquad\text{for $t\geq 0$.}$

Hence,

(3.39) $\displaystyle\|u\|_{L^{(A_{M})_{n-1}}(\mathbb{S}^{n-1})}\leq
c\big{(}\|\nabla_{\mathbb{S}}u\|_{L^{A_{M}}(\mathbb{S}^{n-1})}+\|u\|_{L^{A_{M}}(\mathbb{S}^{n-1})}\big{)},$

where $(A_{M})_{n-1}$ denotes the function obtained on replacing $A$ with
$A_{M}$ in the definition of $A_{n-1}$. The fact that the constant $c$ in
(3.38) is independent of $A$ is of course crucial in deriving inequality
(3.39). Observe that

(3.40)
$\displaystyle(A_{M})_{n-1}(t)=\frac{1}{M}A_{n-1}\Big{(}\frac{t}{M^{\frac{1}{n-1}}}\Big{)}\qquad\text{for
$t\geq 0$.}$

On the other hand, by the definition of Luxemburg norm and the choice of $M$,

(3.41) $\displaystyle\|u\|_{L^{A_{M}}(\mathbb{S}^{n-1})}\leq
1\quad\text{and}\quad\|\nabla_{\mathbb{S}}u\|_{L^{A}(\mathbb{S}^{n-1})}\leq
1.$

Therefore, by the definition of Luxemburg norm again, inequality (3.39) tells
us that

$\frac{1}{M}\int_{\mathbb{S}^{n-1}}A_{n-1}\bigg{(}\frac{|u(x)|}{2cM^{\frac{1}{n-1}}}\bigg{)}\,d\mathcal{H}^{n-1}(x)\leq
1.$

Hence, inequality (3.25) follows.
_Part (ii)._ First, assume that $n\geq 3$ and the integral condition in (3.26)
holds. Let $\overline{A}$ be the Young function defined as

(3.42)
$\overline{A}(t)=\bigg{(}t^{\frac{n-1}{n-2}}\,\int_{t}^{\infty}\frac{\widetilde{A}(r)}{r^{1+\frac{n-1}{n-2}}}\;dr\bigg{)}^{\widetilde{}}\qquad\text{for
$t\geq 0$,}$

where $(\cdots)^{\widetilde{}}$ stands for the Young conjugate of the function
in parenthesis. Notice that the convergence of integral on the right-hand side
of equation (3.42) is equivalent to the convergence of the integral in (3.26),
see [Ci4, Lemma 2.3]. Since we are assuming that $A$ fulfills condition
(3.23), the same lemma also ensures that

(3.43) $\int_{0}\frac{\widetilde{A}(r)}{r^{1+\frac{n-1}{n-2}}}\;dr<\infty.$

From [CaCi, Theorem 4.1] one has that

(3.44)
$\displaystyle\overline{A}\big{(}c\|u-u_{\mathbb{S}^{n-1}}\|_{L^{\infty}(\mathbb{S}^{n-1})}\big{)}\leq\fint_{\mathbb{S}^{n-1}}A(|\nabla_{\mathbb{S}}u|)\,d\mathcal{H}^{n-1}$

for some positive constant $c=c(n)$ and for $u\in
V^{1}K^{A}(\mathbb{S}^{n-1})$.
Furthermore, by Jensen’s inequality,

(3.45) $\displaystyle
A\big{(}\|u_{\mathbb{S}^{n-1}}\|_{L^{\infty}(\mathbb{S}^{n-1})}\big{)}\leq
A\bigg{(}\fint_{\mathbb{S}^{n-1}}|u|\,d\mathcal{H}^{n-1}\bigg{)}\leq\fint_{\mathbb{S}^{n-1}}A(|u|)\,d\mathcal{H}^{n-1}.$

Thanks to [CaCi, Inequality (4.6)],

(3.46) $\overline{A}(t)\leq A(t)\qquad\text{for $t\geq 0$.}$

Moreover, inequality (3.43) ensures that

(3.47)
$t^{\frac{n-1}{n-2}}\,\int_{t}^{\infty}\frac{\widetilde{A}(r)}{r^{1+\frac{n-1}{n-2}}}\;dr\leq
c\,t^{\frac{n-1}{n-2}}\qquad\text{for $t\geq 0$,}$

where we have set

$c=\int_{0}^{\infty}\frac{\widetilde{A}(r)}{r^{1+\frac{n-1}{n-2}}}\;dr.$

Taking the Young conjugates of both sides of inequality (3.47) results in

(3.48) $\overline{A}(t)\geq ct^{n-1}\qquad\text{for $t\geq 0$,}$

for some constant $c=c(n,A)$. Inequality (3.27) follows, via the triangle
inequality, from inequalities (3.44), (3.45), (3.46) and (3.48).
Assume next that $n=2$ and the limit condition in (3.26) holds. If we denote
by $a$ this limit, then

(3.49) $A(t)\geq at\qquad\text{for $t\geq 0$.}$

A simple one-dimensional argument, coupled with Jensen’s inequality and the
increasing monotonicity of the function $tA^{-1}(1/t)$ shows that

(3.50) $\displaystyle
A\big{(}\tfrac{1}{2\pi}\|u-u_{\mathbb{S}^{1}}\|_{L^{\infty}(\mathbb{S}^{1})}\big{)}\leq\fint_{\mathbb{S}^{1}}A(|\nabla_{\mathbb{S}}u|)\,d\mathcal{H}^{1}$

for $u\in V^{1}K^{A}(\mathbb{S}^{1})$ (see [CaCi, Inequality (4.8) and
below]). Inequality (3.27) now follows from (3.45) (which holds also when
$n=2$), (3.49) and (3.50). ∎

##  4\. Analitic lemmas

Here, we collect a few technical lemmas about one-variable functions. We begin
with two inequalities involving a Young function and its Sobolev conjugate.

######  Lemma 4.1.

Let $n\geq 2$ and let $A$ be a Young function fulfilling condition (3.16).
Then, for every $k>0$ there exists a positive constant $c=c(k,A,n)$ such that

(4.1) $A(t)\leq A_{n}(kt)+c\qquad\text{for $t\geq 0$.}$

###### Proof.

Fix $k>0$. Since $A_{n}(t)=A(H_{n}^{-1}(t))$ and
$\lim_{t\to\infty}\frac{H_{n}^{-1}(t)}{t}=\infty$, there exists $t\geq t_{0}$
such that $A(t)\leq A_{n}(kt)$ for $t\geq t_{0}$. Inequality (4.1) hence
follows, with $c=A(t_{0})$. ∎

######  Lemma 4.2.

Let $n\geq 2$ and let $A$ be a Young function fulfilling condition (3.16).
Then,

(4.2)
$\frac{1}{\widetilde{A}^{-1}(t)}\frac{1}{A_{n}^{-1}(t)}\leq\frac{1}{t^{\frac{1}{n^{\prime}}}}\qquad\text{for
$t>0$.}$

###### Proof.

Hölder’s inequality and property (3.2) imply that

(4.3) $\displaystyle t$
$\displaystyle=\int_{0}^{t}\bigg{(}\frac{A(r)}{r}\bigg{)}^{\frac{1}{n}}\bigg{(}\frac{r}{A(r)}\bigg{)}^{\frac{1}{n}}\,dr\leq\bigg{(}\int_{0}^{t}\frac{A(r)}{r}\,dr\bigg{)}^{\frac{1}{n}}\bigg{(}\int_{0}^{t}\bigg{(}\frac{r}{A(r)}\bigg{)}^{\frac{1}{n-1}}\,dr\bigg{)}^{\frac{1}{n^{\prime}}}$
$\displaystyle\leq\bigg{(}\frac{A(t)}{t}\bigg{)}^{\frac{1}{n}}t^{\frac{1}{n}}H_{n}(t)=A(t)^{\frac{1}{n}}H_{n}(t)\quad\text{for
$t>0$.}$

Hence,

(4.4) $\displaystyle A^{-1}(t)\leq
t^{\frac{1}{n}}H_{n}(A^{-1}(t))\quad\text{for $t\geq 0$.}$

The first inequality in (3.3) and inequality (4.4) imply that

(4.5) $\displaystyle t\leq A^{-1}(t)\widetilde{A}^{-1}(t)\leq
t^{\frac{1}{n}}H_{n}(A^{-1}(t))\widetilde{A}^{-1}(t)\quad\text{for $t\geq
0$.}$

Hence, inequality (4.2) follows. ∎

The next result ensures that the functions $A$, $B$ and $E$ appearing in
assumption (2.13) can be modified near $0$ in such a way that such an
assumption is still fulfilled, possibly with a different constant $L$, and the
conditions imposed on $A$, $B$ and $E$ in Theorem 2.2 are satisfied globally,
instead of just near infinity. Of course, the same applies to the simpler
conditions of Theorem 2.1, where the function $E$ is missing.

######  Lemma 4.3.

Assume that the functions $f$, $A$, $B$ and $E$ are as in Theorem 2.2. Then,
there exist two Young functions
$\widehat{A},\widehat{B}:[0,\infty)\to[0,\infty)$, an increasing function
$\widehat{E}:[0,\infty)\to[0,\infty)$, and constants $\widehat{L}\geq 1$ and
$q>n$ such that:

(4.6) $\displaystyle\widehat{A}(|\xi|)-\widehat{E}(|t|)-\widehat{L}\leq
f(x,t,\xi)\leq\widehat{B}(|\xi|)+\widehat{E}(|t|)+\widehat{L}\qquad\mbox{for
a.e.\ $x\in\Omega$, for every $t\in\mathbb{R}$, and every
$\xi\in\mathbb{R}^{n}$,}$ (4.7) $\displaystyle
t^{\frac{n}{n-1}}\leq\widehat{L}\,\widehat{A}_{n}(t)\quad\text{for $t\geq
0$,}$ (4.8) $\displaystyle\lim_{t\to 0^{+}}\frac{\widehat{A}(t)}{t}>0,$ (4.9)
$\displaystyle\widehat{E}(2t)\leq\widehat{L}\widehat{E}(t)\quad\text{for
$t\geq 0$,}$ (4.10)
$\displaystyle\widehat{E}(t)\leq\widehat{A}_{n}(\widehat{L}t)\quad\text{for
$t\geq 0$,}$ (4.11) $\displaystyle\widehat{B}(\lambda
t)\leq\lambda^{q}\widehat{B}(t)\quad\text{for $t\geq 0$ and $\lambda\geq 1$.}$

Moreover, if assumption (2.8) is in force, then the function $B$ satisfies
assumption (2.9) and

(4.12)
$\displaystyle\widehat{B}(t)\leq\widehat{A}_{n-1}(\widehat{L}t)\qquad\mbox{for
$t\geq 0$;}$

if assumption (2.10) is in force, then

(4.13) $\displaystyle\widehat{B}(t)\leq\widehat{L}t^{q}\quad\text{for $t\geq
0$.}$

Here, $\widehat{A}_{n-1}$ and $\widehat{A}_{n}$ denote the functions defined
as $A_{n-1}$ and $A_{n}$, with $A$ replaced with $\widehat{A}$.

###### Proof.

_Step 1._ _Construction of $\widehat{A}$_. Denote by $t_{1}$ the maximum among
$1$, the constant $t_{0}$ appearing in inequalities (2.14) and (2.9), and the
lower bound for $t$ in the definition of the $\Delta_{2}$-condtion near
infinity for the functions $B$ and $E$. Let us set $a=\frac{A(t_{1})}{t_{1}}$,
and define the Young function $\widehat{A}$ as

(4.14) $\widehat{A}(t)=\begin{cases}at&\mbox{if $0\leq t<t_{1}$}\\\
A(t)&\mbox{if $t\geq t_{1}$.}\end{cases}$

Clearly, $\widehat{A}$ satisfies property (4.8) and

(4.15) $A(t)\leq\widehat{A}(t)\qquad\text{for $t\geq 0$.}$

Also, the convexity of $A$ ensures that

(4.16) $\displaystyle\widehat{A}(t)\geq at\quad\text{for $t\geq 0$.}$

Since

$\widehat{H}_{n}(s)=\biggl{(}\int_{0}^{s}\biggl{(}\frac{t}{\widehat{A}(t)}\biggr{)}^{\frac{1}{n-1}}\,dt\biggr{)}^{\frac{n-1}{n}}\qquad\mbox{for
$s\geq 0$,}$

we deduce that

$\widehat{H}_{n}(s)\leq a^{-\frac{1}{n}}s^{\frac{n-1}{n}}\quad\text{for $s\geq
0$,}$

whence

$a^{\frac{1}{n-1}}t^{\frac{n}{n-1}}\leq{\widehat{H}}_{n}^{-1}(t)\quad\text{for
$t\geq 0$.}$

Inasmuch as $\widehat{A}_{n}=\widehat{A}\circ{\widehat{H}}_{n}^{-1}$, the
latter inequality and inequality (4.16) yield:

$\widehat{A}_{n}(t)\geq\widehat{A}((a^{\frac{1}{n-1}}t^{\frac{n}{n-1}})\geq(at)^{\frac{n}{n-1}}\quad\text{for
$t\geq 0$.}$

This shows that inequality (4.7) holds for sufficiently large $\widehat{L}$.
For later reference, also note that

(4.17) $\widehat{A}_{n}(t)=(at)^{\frac{n}{n-1}}\quad\text{for
$t\in[0,t_{1}]$.}$

Next, we have that

(4.18) $A_{n}(t)\leq\widehat{A}_{n}(t)\qquad\text{for $t\geq 0$.}$

Indeed, inequality (4.15) implies that

$\widehat{H}_{n}(s)\leq H_{n}(s)\qquad\text{for $s\geq 0$.}$

Thus, $H_{n}^{-1}(t)\leq\widehat{H}_{n}^{-1}(t)$ for $t\geq 0$, whence
inequality (4.18) follows, on making use of (4.15) again.
Moreover, there exists $t_{2}\geq t_{1}$, depending on $n$ and $A$, such that

(4.19) $\widehat{A}_{n}(t)\leq A_{n}(2t)\quad\text{for $t\geq t_{2}$.}$

Actually, if $s\geq t_{1}$ and is sufficiently large, then

$\displaystyle\widehat{H}_{n}(s)=\biggl{(}\int_{0}^{s}\biggl{(}\frac{t}{\widehat{A}(t)}\biggr{)}^{\frac{1}{n-1}}\,dt\biggr{)}^{\frac{n-1}{n}}\geq\biggl{(}\int_{t_{1}}^{s}\biggl{(}\frac{t}{\widehat{A}(t)}\biggr{)}^{\frac{1}{n-1}}\,dt\biggr{)}^{\frac{n-1}{n}}=\biggl{(}\int_{t_{1}}^{s}\biggl{(}\frac{t}{A(t)}\biggr{)}^{\frac{1}{n-1}}\,dt\biggr{)}^{\frac{n-1}{n}}\geq\frac{1}{2}H_{n}(s).$

Observe that the last inequality holds, for large $s$, thanks to assumption
(2.5). Hence, $\widehat{H}_{n}^{-1}(t)\leq H_{n}^{-1}(2t)$ for sufficiently
large $t$ and thereby

$\widehat{A}_{n}(t)=\widehat{A}(\widehat{H}_{n}^{-1}(t))=A(\widehat{H}_{n}^{-1}(t))\leq
A(H_{n}^{-1}(2t)))=A_{n}(2t)\quad\text{for $t\geq t_{2}$,}$

provided that $t_{2}$ is sufficiently large. Inequality (4.19) is thus
established.

_Step 2._ _Construction of $\widehat{B}$_. First, consider the case when (2.8)
and (2.9) hold. Since $B$ is a Young function, there exists $t_{3}\geq t_{2}$,
where $t_{2}$ is the number from Step 2, such that $B(t_{3})>A_{n-1}(t_{1})$.
Define the Young function $\widehat{B}$ as

$\widehat{B}(t)=\begin{cases}\widehat{A}_{n-1}(t)&\mbox{if $0\leq t<t_{2}$}\\\
\frac{t_{3}-t}{t_{3}-t_{2}}\widehat{A}_{n-1}(t_{2})+\frac{t-t_{2}}{t_{3}-t_{2}}B(t_{3})&\mbox{if
$t_{2}\leq t<t_{3}$}\\\ B(t)&\mbox{if $t\geq t_{3}$.}\end{cases}$

We claim that inequality (4.12) holds with this choice of $\widehat{B}$,
provided that $\widehat{L}$ is large enough. If $t\in[0,t_{2})$, the
inequality in question is trivially satisfied with $\widehat{L}=1$. If
$t\in[t_{2},t_{3})$, then

$\widehat{B}(t)\leq\widehat{B}(t_{3})=B(t_{3})\leq
A_{n-1}(Lt_{3})\leq\widehat{A}_{n-1}(Lt_{3})\leq\widehat{A}_{n-1}((Lt_{3}/t_{2})t),$

where the third inequality holds thanks to (4.18). Finally, if $t>t_{3}$, then

$\widehat{B}(t)=B(t)\leq A_{n-1}(Lt)\leq\widehat{A}_{n-1}(Lt).$

Altogether, inequality (4.12) is fulfilled with
$\widehat{L}=\max\big{\\{}1,\frac{Lt_{3}}{t_{2}}\big{\\}}$.
In order to establish inequality (4.11), it suffices to show that
$\widehat{B}$ satisfies the $\Delta_{2}$-condition globally. Since
$\widehat{B}$ is a Young function, this condition is in turn equivalent to the
fact that there exists a constant $c$ such that

(4.20) $\frac{t\widehat{B}^{\prime}(t)}{\widehat{B}(t)}\leq c\quad\text{for
a.e. $t>0$.}$

Since $B$ is a Young function satisfying the $\Delta_{2}$-condition near
infinity, and $\widehat{B}(t)=B(t)$ for large $t$, condition (4.20) certainly
holds for large $t$. On the other hand, since

$\lim_{t\to 0+}\frac{t\widehat{B}^{\prime}(t)}{\widehat{B}(t)}=\lim_{t\to
0+}\frac{t\widehat{A}_{n-1}^{\prime}(t)}{\widehat{A}_{n-1}(t)}=\frac{n-1}{n-2},$

condition (4.20) also holds for $t$ close to $0$. Hence, it holds for every
$t>0$.
Next, consider the case when (2.10) holds. The $\Delta_{2}$-condition near
infinity for $B$ implies that there exist constants $q>1$, $t_{4}>1$ and $c>0$
such that with $B(t)\leq ct^{q}$ for all $t\geq t_{4}$. Since $t_{4}>1$, we
may suppose, without loss of generality, that $q>n$. Since
$B(t)\leq\widehat{L}(t^{q}+1)$ for $t\geq 0$, provided that $\widehat{L}$ is
sufficiently large, the choice $\widehat{B}(t)=\widehat{L}t^{q}$ makes
inequalities (4.11) and (4.13) true.

_Step 3._ Construction of $\widehat{E}$. We define $\widehat{E}$ analogously
to $\widehat{B}$, by replacing $B$ with $E$ and $\widehat{A}_{n-1}$ with
$\widehat{A}_{n}$. The same argument as in Step 2 tells us that inequalities
(4.9) and (4.10) hold for a suitable choice of the constant $\widehat{L}$.

_Step 4._ Conclusion. Since

$\displaystyle f(x,t,\xi)\leq
B(|\xi|)+E(|t|)+L\leq\widehat{B}(|\xi|)+\widehat{E}(|\xi|)+B(t_{3})+E({t_{3}})+L$

and

$f(x,t,\xi)\geq
A(|\xi|)-E(|t|)-L\geq\widehat{A}(|\xi|)-\widehat{E}(|\xi|)-A(t_{1})-E({t_{3}})-L,$

for a.e. $x\in\Omega$, and for every $t\in\mathbb{R}$ and
$\xi\in\mathbb{R}^{n}$, equation (4.6) follows, provided that $\widetilde{L}$
is chosen sufficiently large. ∎

We conclude this section by recalling the following classical lemma – see e.g.
[Giu, Lemma 6.1]

######  Lemma 4.4.

Let $Z:[\rho,\sigma]\to[0,\infty)$ be a bounded function. Assume that there
exist constants $a,b\geq 0$, $\alpha>0$ and $\theta\in[0,1)$ such that

$Z(r)\leq\theta Z(s)+(s-r)^{-\alpha}a+b\quad\text{if $\rho\leq
r<s\leq\sigma$.}$

Then,

$Z(r)\leq c\big{(}(s-r)^{-\alpha}a+b\big{)}\quad\text{if $\rho\leq
r<s\leq\sigma$,}$

for some constant $c=c(\alpha,\theta)>1$.

##  5\. Proof of Theorem 2.2

We shall limit ourselves to proving Theorem 2.2, since the content of Theorem
2.1 is just a special case of the former. A key ingredient is provided by
Lemma 5.1 below. In the statement, $\Phi_{q}:[0,\infty)\to[0,\infty)$ denotes
the function defined for $q\geq 1$ as

(5.1) $\Phi_{q}(t)=\begin{cases}t&\mbox{if $0\leq t<1$}\\\ t^{q}&\mbox{if
$t\geq 1$.}\end{cases}$

One can verify that

(5.2) $\Phi_{q}(\lambda t)\leq\lambda^{q}\Phi_{q}(t)\qquad\text{for
$\lambda\geq 1$ and $t\geq 0$.}$

Moreover, given a function $u\in W^{1}K^{A}(\mathbb{B}_{1})$, we set

(5.3)
$F(u,\rho,\sigma)=\int_{\mathbb{B}_{\sigma}\setminus\mathbb{B}_{\rho}}A(|u|)+A(|\nabla
u|)\,dx$

for $0<\rho<\sigma<1$.

######  Lemma 5.1.

Let $A$ and $B$ be Young functions and $0<\rho<\sigma<1$.

  * (i)

Suppose that condition (2.8) is in force. Assume that there exist constants
$L\geq 1$ and $q>1$ such that

(5.4) $B(t)\leq A_{n-1}(Lt)\quad\mbox{and}\quad B(\lambda
t)\leq\lambda^{q}B(t)\quad\mbox{for $t\geq 0$ and $\lambda\geq 1$}.$

Then, for every $u\in W^{1}K^{A}(\mathbb{B}_{1})$ there exists a function
$\eta\in W_{0}^{1,\infty}(\mathbb{B}_{1})$ satisfying

(5.5) $0\leq\eta\leq 1\,\mbox{in $\mathbb{B}_{1}$},\quad\eta=1\,\mbox{in
$\mathbb{B}_{\rho}$},\quad\eta=0\,\mbox{in
$\mathbb{B}_{1}\setminus\mathbb{B}_{\sigma}$}\quad\|\nabla\eta\|_{L^{\infty}(\mathbb{B}_{1})}\leq\frac{2}{\sigma-\rho},$

and such that

(5.6) $\displaystyle\int_{\mathbb{B}_{1}}B(|u\nabla\eta(x)|)\,dx\leq$
$\displaystyle c\,\Phi_{q}\biggl{(}\frac{\kappa
F(u,\rho,\sigma)^{\frac{1}{n-1}}}{(\sigma-\rho)^{\frac{n}{n-1}}\rho}\biggr{)}F(u,\rho,\sigma)$

for some constant $c=c(n,q,L)\geq 1$. Here, $\kappa$ denotes the constant
appearing in inequality (3.25).

  * (ii)

Suppose that condition (3.26) is in force. Assume that there exist constants
$L\geq 1$ and $q>n$ such that

(5.7) $B(t)\leq Lt^{q}\qquad\text{ for all $t\geq 0$.}$

Then, for every $u\in W^{1}K^{A}(\mathbb{B}_{1})$ there exists a function
$\eta\in W_{0}^{1,\infty}(\mathbb{B}_{1})$ satisfying conditions (5.5), such
that

(5.8) $\displaystyle\int_{\mathbb{B}_{1}}B(|u\nabla\eta(x)|)\,dx\leq$
$\displaystyle\frac{c\kappa^{q}F(u,\rho,\sigma)^{\frac{q}{n-1}}}{(\sigma-\rho)^{q-1+\frac{q}{n-1}}\rho^{q-(n-1)}}$

for some constant $c=c(n,q,L)\geq 1$. Here, $\kappa$ denotes the constant
appearing in inequality (3.27).

###### Proof.

Let $u\in W^{1}K^{A}(\mathbb{B}_{1})$. Define, for $r\in[0,1]$, the function
$u_{r}:\mathbb{S}^{n-1}\to\mathbb{R}$ as $u_{r}(z)=u(rz)$ for
$z\in\mathbb{S}^{n-1}$. By classical properties of restrictions of Sobolev
functions to $(n-1)$-dimensional concentric spheres, one has that $u_{r}$ is a
weakly differentiable function for a.e. $r\in[0,1]$. Hence, by Fubini’s
theorem, there exists a set $N\subset[0,1]$ such that $|N|=0$, and $u_{r}\in
W^{1}K^{A}(\mathbb{S}^{n-1})$ for every $r\in[0,1]\setminus N$. Set

(5.9) $U_{1}=\biggl{\\{}r\in[\rho,\sigma]\setminus
N\,:\,\int_{\mathbb{S}^{n-1}}A(|\nabla_{\mathbb{S}}u_{r}(z)|)\,d\mathcal{H}^{n-1}(z)\leq\frac{4}{(\sigma-\rho)r^{n-1}}\int_{\mathbb{B}_{\sigma}\setminus\mathbb{B}_{\rho}}A(|\nabla
u|)\,\mathrm{d}x\biggr{\\}}.$

From Fubini’s Theorem, the inequality
$|\nabla_{\mathbb{S}}u_{r}(z)|\leq|\nabla u(rz)|$ for $\mathcal{H}^{n-1}$-a.e.
$z\in\mathbb{S}^{n-1}$, and the very definition of the set $U_{1}$ we infer
that

$\displaystyle\int_{\mathbb{B}_{\sigma}\setminus\mathbb{B}_{\rho}}A(|\nabla
u|)\,dx=$
$\displaystyle\int_{\rho}^{\sigma}r^{n-1}\int_{\mathbb{S}^{n-1}}A(|\nabla
u(rz)|)\,d\mathcal{H}^{n-1}(z)\,dr$ $\displaystyle\geq$
$\displaystyle\int_{(\rho,\sigma)\setminus
U_{1}}r^{n-1}\int_{\mathbb{S}^{n-1}}A(|\nabla_{\mathbb{S}}u_{r}(z)|)\,d\mathcal{H}^{n-1}(z)\,dr$
$\displaystyle>$
$\displaystyle\frac{4((\sigma-\rho)-|U_{1}|)}{(\sigma-\rho)}\int_{\mathbb{B}_{\sigma}\setminus\mathbb{B}_{\rho}}A(|\nabla
u|)\,dx.$

Hence, $|U_{1}|\geq\frac{3}{4}(\sigma-\rho)$. An analogous computation ensures
that the set

(5.10) $U_{2}=\biggl{\\{}r\in[\rho,\sigma]\setminus
N\,:\,\int_{\mathbb{S}^{n-1}}A(|u_{r}(z)|)\,d\mathcal{H}^{n-1}(z)\leq\frac{4}{(\sigma-\rho)r^{n-1}}\int_{\mathbb{B}_{\sigma}\setminus\mathbb{B}_{\rho}}A(|u|)\,dx\biggr{\\}}$

has the property that $|U_{2}|\geq\frac{3}{4}(\sigma-\rho)$. Thereby, if we
define the set

$U=U_{1}\cap U_{2},$

then

(5.11) $|U|\geq|(\rho,\sigma)|-|(\rho,\sigma)\setminus
U_{1}|-|(\rho,\sigma)\setminus U_{2}|\geq\frac{1}{2}(\sigma-\rho).$

Next, define the function $\eta:\mathbb{B}_{1}\to[0,1]$ as

$\eta(x)=\begin{cases}1&\mbox{if $0\leq|x|<\rho$}\\\
\displaystyle\frac{1}{|U|}\int_{|x|}^{\sigma}\chi_{U}(s)\,ds&\mbox{if
$\rho\leq|x|\leq\sigma$}\\\ 0&\mbox{if $\sigma<|x|\leq 1$.}\end{cases}$

One has that $0\leq\eta\leq 1$, $\eta=1$ in $\mathbb{B}_{\rho}$, $\eta=0$ in
$\mathbb{B}_{1}\setminus\mathbb{B}_{\sigma}$, $\eta\in
W_{0}^{1,\infty}(\mathbb{B}_{1})$ and

(5.12) $|\nabla\eta(rz)|=\begin{cases}0&\mbox{for a.e. $r\notin U$}\\\
\frac{1}{|U|}&\mbox{for a.e. $r\in U$,}\end{cases}$

and for $z\in\mathbb{S}^{n-1}$. Hence, the function $\eta$ satisfies the
properties claimed in (5.5).
Next, set, for $r\in[0,1]\setminus N$,

(5.13)
$F_{r}(u)=\int_{\mathbb{S}^{n-1}}A(|u_{r}(z)|)+A(|\nabla_{\mathbb{S}}u_{r}(z)|)\,d\mathcal{H}^{n-1}(z).$

By the definition of the set $U$,

(5.14)
$F_{r}(u)\leq\frac{4}{(\sigma-\rho)r^{n-1}}F(u,\rho,\sigma)\quad\text{for a.e.
$r\in U$.}$

We have now to make use of different inequalities, depending on whether we
deal we case (i) or (ii).
Case (i). Owing to inequality (3.1) and to the second inequality in (5.4),

(5.15) $B(\lambda t)\leq\Phi_{q}(\lambda)B(t)\qquad\quad\text{for $\lambda\geq
0$ and $t\geq 0$.}$

The following chain holds:

(5.16) $\displaystyle\int_{\mathbb{B}_{1}}B(|u\nabla\eta(x)|)\,dx\leq$
$\displaystyle\int_{U}r^{n-1}\int_{\mathbb{S}^{n-1}}B\biggl{(}\biggl{|}\frac{2}{(\sigma-\rho)}u_{r}(z)\biggr{|}\biggr{)}\,d\mathcal{H}^{n-1}(z)\,dr$
$\displaystyle=$
$\displaystyle\int_{U}r^{n-1}\int_{\mathbb{S}^{n-1}}B\biggl{(}\biggl{|}\frac{2\kappa
u_{r}(z)F_{r}(u)^{\frac{1}{n-1}}}{\kappa(\sigma-\rho)F_{r}(u)^{\frac{1}{n-1}}}\biggr{|}\biggr{)}\,d\mathcal{H}^{n-1}(z)\,dr$
$\displaystyle\leq$
$\displaystyle\int_{U}r^{n-1}\Phi_{q}\biggl{(}\biggl{|}\frac{2L\kappa
F_{r}(u)^{\frac{1}{n-1}}}{(\sigma-\rho)}\biggr{|}\biggr{)}\int_{\mathbb{S}^{n-1}}A_{n-1}\biggl{(}\biggl{|}\frac{u_{r}(z)}{\kappa
F_{r}(u)^{\frac{1}{n-1}}}\biggr{|}\biggr{)}\,d\mathcal{H}^{n-1}(z)\,dr$
$\displaystyle\leq$
$\displaystyle\int_{U}r^{n-1}\Phi_{q}\biggl{(}\biggl{|}\frac{2L\kappa
F_{r}(u)^{\frac{1}{n-1}}}{(\sigma-\rho)r}\biggr{|}\biggr{)}F_{r}(u)\,dr$
$\displaystyle\leq$ $\displaystyle\Phi_{q}\biggl{(}\biggl{|}\frac{2L\kappa
4^{\frac{1}{n-1}}F(u,\rho,\sigma)^{\frac{1}{n-1}}}{(\sigma-\rho)^{1+\frac{1}{n-1}}\rho}\biggr{|}\biggr{)}4F(u,\rho,\sigma),$

where the second inequality holds by inequality (5.15) and the first
inequality in (5.4), the third inequality follows from the Sobolev inequality
(3.25), and the last inequality relies upon inequality (5.14) and the fact
that $|U|\leq(\sigma-\rho)$. Clearly, inequality (5.6) follows from (5.16).
Case (ii). The following chain holds:

(5.17) $\displaystyle\int_{\mathbb{B}_{1}}B(|u\nabla\eta(x)|)\,dx\leq$
$\displaystyle
L\int_{U}r^{n-1}\int_{\mathbb{S}^{n-1}}\biggl{|}\frac{2}{(\sigma-\rho)}u_{r}(z)\biggr{|}^{q}\,d\mathcal{H}^{n-1}(z)\,dr$
$\displaystyle\leq$
$\displaystyle\frac{L2^{q}c_{n}\kappa^{q}}{(\sigma-\rho)^{q}}\int_{U}r^{n-1}F_{r}(u)^{\frac{q}{n-1}}\,dr$
$\displaystyle\leq$
$\displaystyle\frac{L2^{q}c_{n}\kappa^{q}}{(\sigma-\rho)^{q}}\int_{U}r^{n-1}\biggl{(}\frac{4F(u,\rho,\sigma)}{(\sigma-\rho)r^{n-1}}\biggr{)}^{\frac{q}{n-1}}\,dr$
$\displaystyle\leq$
$\displaystyle\frac{L2^{q}4^{\frac{q}{n-1}}c_{n}\kappa^{q}}{(\sigma-\rho)^{q-1+\frac{q}{n-1}}{\rho^{q-(n-1)}}}F(u,\rho,\sigma)^{\frac{q}{n-1}},$

where $c_{n}$ is given by (3.29), the first inequality holds by inequality
(5.7), the second one by inequality (3.27), the third one by inequality
(5.14), and the last one since $|U|\leq(\sigma-\rho)$. Inequality (5.8)
follows via (5.17). ∎

We are now in a position to accomplish the proof of our main result.

###### Proof of Theorem 2.2.

Owing to Lemma 4.3, without loss of generality we can assume that the
functions $A$, $B$ and $E$ also satisfy the properties stated for the
functions $\widehat{A}$, $\widehat{B}$ and $\widehat{E}$ in the lemma. When we
refer to properties in the statement of this lemma, we shall mean that they
are applied directly to $A$, $B$ and $E$. In particular, $q$ denotes the
exponent appearing in the statement of the lemma. Moreover, $Q$ is the
constant from the definition of quasi-minimizer.
We also assume that $\mathbb{B}_{1}\Subset\Omega$ and prove that $u$ is
bounded in $\mathbb{B}_{\frac{1}{2}}$. The general case follows via a standard
scaling and translation argument. For ease of presentation, we split the proof
in steps.

_Step 1._ _Basic energy estimate._
Set, for $r>0$ and $l>0$,

(5.18) $\mathcal{A}_{l,r}=\mathbb{B}_{r}\cap\\{x\in\Omega\,:\,u(x)>l\\}$

and

(5.19) $J(l,r)=\int_{\mathbb{B}_{r}}A((u-l)_{+})+A(|\nabla(u-l)_{+}|)\,dx.$

Here, the subscript “$+$” stands for the positive part.
If assumption (2.8) holds, then we claim that there exists a constant
$c=c(n,q,L,Q)\geq 1$ such that

(5.20) $\displaystyle\int_{\mathbb{B}_{\rho}}A(|\nabla(u-k)_{+}|)\,dx\leq$
$\displaystyle c\biggl{(}\frac{\Phi_{q}(\kappa
J(k,\sigma)^{\frac{1}{n-1}})}{(\sigma-\rho)^{\frac{qn}{n-1}}}J(k,\sigma)+\int_{\mathcal{A}_{k,\sigma}}(E(|u|)+1)\,dx\biggr{)}$

for $k\geq 0$ and $\frac{1}{2}\leq\rho<\sigma<1$, where $\kappa$ denotes the
constant from inequality (3.25)
If assumption (2.10) holds, then we claim that there exists a constant
$c=c(n,q,L,Q)\geq 1$ such that

(5.21) $\displaystyle\int_{\mathbb{B}_{\rho}}A(|\nabla(u-k)_{+}|)\,dx\leq$
$\displaystyle
c\biggl{(}\frac{\kappa^{q}}{(\sigma-\rho)^{\frac{qn}{n-1}}}J(k,\sigma)^{\frac{q}{n-1}}+\int_{\mathcal{A}_{k,\sigma}}(E(|u|)+1)\,dx\biggr{)}$

for $k\geq 0$ and $\frac{1}{2}\leq\rho<\sigma<1$, where $\kappa$ denotes the
contstant from inequality (3.27).
We shall first establish inequalities (5.20) and (5.21) under assumption
(2.11).
Given $k\geq 0$ and $\frac{1}{2}\leq\rho<\sigma\leq 1$, let $\eta\in
W_{0}^{1,\infty}(\mathbb{B}_{1})$ be as in the statement of Lemma 5.1, applied
with $u$ replaced with $(u-k)_{+}$. Choose the function
$\varphi=-\eta^{q}(u-k)_{+}$ in the definition of quasi-minimizer for $u$.
From this definition and the first property in (2.11) one infers that

$\displaystyle\int_{\mathcal{A}_{k,\sigma}}f(x,u,\nabla u)\,dx$
$\displaystyle\leq
Q\int_{\mathcal{A}_{k,\sigma}}f(x,u+\varphi,\nabla(u+\varphi))\,dx$
$\displaystyle=Q\int_{\mathcal{A}_{k,\sigma}}f(x,u+\varphi,(1-\eta^{q})\nabla
u-{q}\eta^{{q}-1}\nabla\eta(u-k))\,dx$ $\displaystyle\leq
Q\int_{\mathcal{A}_{k,\sigma}}(1-\eta^{q})f(x,u+\varphi,\nabla
u)+\eta^{q}f\biggl{(}x,u+\varphi,-\frac{{q}\nabla\eta}{\eta}(u-k)\biggr{)}\,dx\,.$

Hence, since $0\leq u+\varphi\leq u$ on $\mathcal{A}_{k,\sigma}$, the second
property in (2.11), the upper bound in (2.13), and the monotonicity of the
function $E$ ensure that

(5.22) $\displaystyle\int_{\mathcal{A}_{k,\sigma}}f(x,u,\nabla u)\,dx\leq
Q\int_{\mathcal{A}_{k,\sigma}}(1-\eta^{q})\big{(}Lf(x,u,\nabla
u)+E(u)+L\big{)}+\eta^{q}\bigg{(}B\biggl{(}\frac{{q}|\nabla\eta|}{\eta}(u-k)\biggr{)}+E(u)+L\bigg{)}\,dx.$

Inasnnuch as $0\leq\eta\leq 1$ and $\eta=1$ in $\mathbb{B}_{\rho}$, the use of
inequality (4.11) on the right-hand side of (5.22) yields:

(5.23) $\displaystyle\int_{\mathcal{A}_{k,\sigma}}f(x,u,\nabla u)\,dx\leq
QL\int_{\mathcal{A}_{k,\sigma}\setminus\mathbb{B}_{\rho}}f(x,u,\nabla
u)\,dx+Q\int_{\mathcal{A}_{k,\sigma}}{q}^{q}B\big{(}|\nabla\eta|(u-k)\big{)}+E(|u|)+L\,dx.$

Now, suppose that assumption (2.8) holds. Combining inequality (5.23) with
estimate (5.6) (applied to $(u-k)_{+}$) tells us that

(5.24) $\displaystyle\int_{\mathcal{A}_{k,\rho}}f(x,u,\nabla u)\,dx\leq$
$\displaystyle
QL\int_{\mathcal{A}_{k,\sigma}\setminus\mathbb{B}_{\rho}}f(x,u,\nabla
u)\,dx+cQ\Phi_{q}\biggl{(}\frac{2\kappa
J(k,\sigma)^{\frac{1}{n-1}}}{(\sigma-\rho)^{\frac{n}{n-1}}}\biggr{)}J(k,\sigma)+Q\int_{\mathcal{A}_{k,\sigma}}(E(u)+L)\,dx$

for some constant $c=c(n,{q},L)\geq 1$. Observe that in deriving inequality
(5.24), we have exploited the inequalities $\frac{1}{2}\leq\rho$ and
$F((u-k)_{+},\rho,\sigma)\leq J(k,\sigma)$.
Adding the expression $QL\int_{\mathcal{A}_{k,\rho}}f(x,u,\nabla u)\,dx$ to
both sides of inequality (5.24) and using inequality (5.2) enable one to
deduce that

$\displaystyle\int_{\mathcal{A}_{k,\rho}}f(x,u,\nabla
u)\,dx\leq\frac{QL}{QL+1}\int_{\mathcal{A}_{k,\sigma}}f(x,u,\nabla
u)\,dx+c\biggl{(}\frac{\Phi_{q}\big{(}\kappa
J(k,\sigma)^{\frac{1}{n-1}}\big{)}}{(\sigma-\rho)^{\frac{{q}n}{n-1}}}J(k,\sigma)+\int_{\mathcal{A}_{k,\sigma}}(E(u)+1)\,dx\biggr{)},$

for some constant $c=c(n,{q},L,Q)\geq 1$. Estimate (5.20) follows via Lemma
4.4 and the lower bound in (2.13).
Assume next that assumtpion (2.10) holds. Hence, the full assumption (3.26)
holds, thanks to equation (4.8). One can start again from (5.23), make use of
inequality (5.8), and argue as above to obtain inequality (5.21). The fact
that

$\frac{1}{(\sigma-\rho)^{q-1+\frac{q}{n-1}}}\leq\frac{1}{(\sigma-\rho)^{\frac{qn}{n-1}}},$

since $\sigma-\rho\leq 1$, is relevant in this argument.
It remains to prove inequalities (5.20) and (5.21) under the alternative
structure condition (2.12).
Let $\varphi$ be as above, and observe that
$u+\varphi=\eta^{q}k+(1-\eta^{q})u$ on $\mathcal{A}_{k,\sigma}$. Hence, by
property (2.12),

$\displaystyle\int_{\mathcal{A}_{k,\sigma}}f(x,u,\nabla u)\,dx\leq$
$\displaystyle
Q\int_{\mathcal{A}_{k,\sigma}}f(x,u+\varphi,\nabla(u+\varphi))\,dx$
$\displaystyle=$ $\displaystyle
Q\int_{\mathcal{A}_{k,\sigma}}f\big{(}x,(1-\eta^{q})u+\eta^{q}k,(1-\eta^{q})\nabla
u-{q}\eta^{{q}-1}\nabla\eta(u-k)\big{)}\,dx$ $\displaystyle\leq$
$\displaystyle Q\int_{\mathcal{A}_{k,\sigma}}(1-\eta^{q})f(x,u,\nabla
u)+\eta^{q}f\biggl{(}x,k,-\frac{{q}\nabla\eta}{\eta}(u-k)\biggr{)}\,dx.$

Thanks to assumption (2.13) and the monotonicity of $E$, which guarantees that
$E(k)\leq E(u)$ in $\mathcal{A}_{k,\sigma}$, we obtain that

(5.25) $\displaystyle\int_{\mathcal{A}_{k,\sigma}}f(x,u,\nabla u)\,dx\leq$
$\displaystyle Q\int_{\mathcal{A}_{k,\sigma}}(1-\eta^{q})f(x,u,\nabla
u)+\eta^{q}(L+E(u)+B\biggl{(}\frac{{q}|\nabla\eta|}{\eta}(u-k)\biggr{)}\,dx.$

A replacement of inequality (5.22) with (5.25) and an analogous argument as
above yields the same conclusions.

_Step 2._ _One-step improvement._
Let us set

$c_{B}=\max\\{\kappa,1\\},$

where $\kappa$ denotes a constant, depending only on $n$, such that inequality
(3.21) holds for every $r\in[\tfrac{1}{2},1]$. We claim that, if $h>0$ is such
that

(5.26) $c_{B}LJ(h,\sigma)^{\frac{1}{n}}\leq 1,$

then

(5.27) $\displaystyle J(k,\rho)\leq$ $\displaystyle
c\biggl{(}\frac{1}{(\sigma-\rho)^{\frac{qn}{n-1}}}+\frac{1}{(k-h)^{\frac{n}{n-1}}}+L^{\log_{2}(\frac{k}{k-h})}\biggr{)}J(h,\sigma)^{1+\frac{1}{n}}\qquad\text{if
$k>h$,}$

for a suitable constant $c=c(n,q,L,Q,A)\geq 1$.
To this purpose, fix $h>0$ such that inequality (5.26) holds. We begin by
showing that there exists a constant $c=c(n,L)$ such that

(5.28) $|\mathcal{A}_{k,\sigma}|\leq
c\frac{J(h,\sigma)^{\frac{n+1}{n}}}{(k-h)^{\frac{n}{n-1}}}\qquad\text{if
$k>h$.}$

Inequality (5.28) is a consequence of the following chain:

(5.29) $\displaystyle|\mathcal{A}_{k,\sigma}|A_{n}(k-h)=$
$\displaystyle\int_{\mathcal{A}_{k,\sigma}}A_{n}(k-h)\,dx\leq\int_{\mathcal{A}_{h,\sigma}}A_{n}(u-h)\,dx$
$\displaystyle\leq$
$\displaystyle\int_{\mathcal{A}_{h,\sigma}}A_{n}\biggl{(}\frac{c_{B}(u-h)J(h,\sigma)^{\frac{1}{n}}}{c_{B}J(h,\sigma)^{\frac{1}{n}}}\biggr{)}\,dx\leq
c_{B}J(h,\sigma)^{\frac{1}{n}}\int_{\mathcal{A}_{h,\sigma}}A_{n}\biggl{(}\frac{u-h}{c_{B}J(h,\sigma)^{\frac{1}{n}}}\biggr{)}\,dx.$

Notice that the last inequality holds thanks to inequality (3.1), applied with
$A$ replaced with $A_{n}$, and to assumption (5.26). Coupling inequality
(5.29) with inequality (3.21) enables us to deduce that

$\displaystyle|\mathcal{A}_{k,\sigma}|\leq$
$\displaystyle\frac{c_{B}J(h,\sigma)^{\frac{n+1}{n}}}{A_{n}(k-h)}.$

Hence inequality (5.28) follows, via (4.7).
Next, by the monotonicity of $E$ and assumption (4.9),

(5.30) $\displaystyle\int_{\mathcal{A}_{k,\sigma}}E(u)\,dx=$
$\displaystyle\int_{\mathcal{A}_{k,\sigma}}E((u-k)+k)\,dx\leq\int_{\mathcal{A}_{k,\sigma}}E(2(u-k))+E(2k)\,dx$
$\displaystyle\leq$ $\displaystyle
L\int_{\mathcal{A}_{k,\sigma}}E(u-k)+E(k)\,dx\quad\text{for $k>0$.}$

From inequality (3.1) applied to $A_{n}$ and assumption (5.26) one infers that

(5.31) $\displaystyle\int_{\mathcal{A}_{k,\sigma}}E(u-k)\,dx\leq$
$\displaystyle\int_{\mathcal{A}_{h,\sigma}}E(u-h)\,dx\leq\int_{\mathcal{A}_{h,\sigma}}A_{n}(L(u-h))\,dx$
$\displaystyle\leq$ $\displaystyle
c_{B}LJ(h,\sigma)^{\frac{1}{n}}\int_{\mathcal{A}_{h,\sigma}}A_{n}\biggl{(}\frac{u-h}{c_{B}J(h,\sigma)^{\frac{1}{n}}}\biggr{)}\,dx\leq
c_{B}LJ(h,\sigma)^{1+\frac{1}{n}}\,\quad\text{if $k>h$.}$

Owing to assumption (4.9) and chain (5.31),

(5.32) $\displaystyle\int_{\mathcal{A}_{k,\sigma}}E(k)=$ $\displaystyle
E\biggl{(}\frac{k}{k-h}(k-h)\biggr{)}|\mathcal{A}_{k,\sigma}|\leq
E\biggl{(}2^{\lfloor\log_{2}\frac{k}{k-h}\rfloor+1}(k-h)\biggr{)}|\mathcal{A}_{k,\sigma}|$
$\displaystyle\leq$ $\displaystyle
L^{\log_{2}(\frac{k}{k-h})+1}E(k-h)|\mathcal{A}_{k,\sigma}|\leq
L^{\log_{2}(\frac{k}{k-h})+1}\int_{\mathcal{A}_{h,\sigma}}E(u-h)\,dx$
$\displaystyle\leq$ $\displaystyle
L^{\log_{2}(\frac{k}{k-h})+1}c_{B}LJ(h,\sigma)^{1+\frac{1}{n}}\quad\text{if
$k>h$,}$

where $\lfloor\,\cdot\,\rfloor$ stands for integer part. Combining
inequalities (5.30)–(5.32) yields:

(5.33) $\int_{\mathcal{A}_{k,\sigma}}E(u)\,dx\leq
cL^{\log_{2}(\frac{k}{k-h})}J(h,\sigma)^{\frac{n+1}{n}}\quad\text{if $k>h$,}$

for some constant $c=c(n,L)$.
From this point, the argument slightly differs depending on whether condition
(2.8) or (3.26) holds.
Assume first that (2.8) is in force. Assumption (5.26) implies that there
exists a constant $c=c(n,q,L)$ such that

(5.34) $\Phi_{q}(\kappa J(k,\sigma)^{\frac{1}{n-1}})\leq
cJ(k,\sigma)^{\frac{1}{n-1}}\quad\text{if $k>h$,}$

where $\kappa$ is the constant from inequality (3.25). Making use of
inequalities (5.28), (5.33) and (5.34) to estimate the right-hand side of
(5.20) results in the following bound for its left-hand side:

(5.35) $\displaystyle\int_{\mathbb{B}_{\rho}}A(|\nabla(u-k)_{+}|)\,dx$
$\displaystyle\leq
c\Bigg{(}\frac{J(h,\sigma)^{\frac{n}{n-1}}}{(\sigma-\rho)^{\frac{qn}{n-1}}}+\frac{J(h,\sigma)^{\frac{n+1}{n}}}{(k-h)^{\frac{n}{n-1}}}+L^{\log_{2}(\frac{k}{k-h})}J(h,\sigma)^{\frac{n+1}{n}}\Bigg{)}$
$\displaystyle\leq
c^{\prime}\Bigg{(}\frac{1}{(\sigma-\rho)^{\frac{qn}{n-1}}}+\frac{1}{(k-h)^{\frac{n}{n-1}}}+L^{\log_{2}(\frac{k}{k-h})}\Bigg{)}J(h,\sigma)^{\frac{n+1}{n}}\quad\text{if
$k>h$,}$

for suitable constants $c=c(n,q,L,Q)\geq 1$ and
$c^{\prime}=c^{\prime}(n,q,L,Q)\geq 1$. From inequality (4.1) we infer that

(5.36)
$\displaystyle\int_{\mathbb{B}_{\rho}}A((u-k)_{+})\,dx\leq\int_{\mathbb{B}_{\rho}}A_{n}((u-k)_{+})\,dx+c|\mathcal{A}_{k,\rho}|\leq\int_{\mathbb{B}_{\sigma}}A_{n}((u-h)_{+})\,dx+c|\mathcal{A}_{k,\sigma}|\quad\text{if
$k>h$,}$

for some constant $c=c(n,A)$. A combination of the latter inequality with
(5.28) and (5.29) tells us that

(5.37) $\displaystyle\int_{\mathbb{B}_{\rho}}A((u-k)_{+})\,dx\leq
cJ(h,\sigma)^{1+\frac{1}{n}}+c\frac{J(h,\sigma)^{1+\frac{1}{n}}}{(k-h)^{\frac{n}{n-1}}}\quad\text{if
$k>h$,}$

for some constant $c=c(n,L,A)$. Coupling inequaliy (5.35) with (5.37) yields
(5.27).
Assume now that condition (3.26) holds. Assumption (5.26) and the inequality
$q>n$ guarantee that there exists a constant $c=c(n,q,L)$ such that

(5.38) $J(k,\sigma)^{\frac{q}{n-1}}\leq
cJ(k,\sigma)^{\frac{n+1}{n}}\quad\text{if $k>h$.}$

From inequalities (5.28), (5.33) and (5.38) one obtains (5.35) also in this
case. Inequality (5.27) again follows via (5.35) and (5.37).

_Step 3._ _Iteration._
Given $K\geq 1$ and $\ell\in\mathbb{N}\cup\\{0\\}$, set

(5.39)
$k_{\ell}=K(1-2^{-(\ell+1)}),\quad\sigma_{\ell}=\frac{1}{2}+\frac{1}{2^{\ell+2}},\quad\mbox{and}\quad
J_{\ell}=J(k_{\ell},\sigma_{\ell}).$

Thanks to inequality (5.27), if $\ell\in\mathbb{N}$ is such that

(5.40) $c_{B}LJ_{\ell}^{\frac{1}{n}}\leq 1,$

then

(5.41) $\displaystyle J_{\ell+1}\leq$ $\displaystyle
c\biggl{(}2^{\ell\frac{qn}{n-1}}+K^{-\frac{n}{n-1}}2^{\ell\frac{n}{n-1}}+L^{\ell}\biggr{)}J_{\ell}^{1+\frac{1}{n}}$

for a suitable constant $c=c(n,q,L,Q,A)\geq 1$. Clearly, inequality (5.41)
implies that

(5.42) $\displaystyle J_{\ell+1}\leq$ $\displaystyle
c_{2}2^{\gamma\ell}J_{\ell}^{1+\frac{1}{n}}$

where $\gamma=\max\\{q\frac{n}{n-1},\log_{2}L\\}$ and
$c_{2}=c_{2}(n,q,L,Q,A)\geq 1$ is a suitable constant. Let
$\tau=\tau(n,q,L,Q,A)\in(0,1)$ be such that

(5.43) $c_{2}2^{\gamma}\tau^{\frac{1}{n}}=1.$

Set

$\varepsilon_{0}=\min\\{(c_{B}L)^{-n},\tau^{n}\\}.$

We claim that, if

(5.44) $J_{0}\leq\varepsilon_{0},$

then

(5.45) $J_{\ell}\leq\tau^{\ell}J_{0}\qquad\mbox{for every
$\ell\in\mathbb{N}\cup\\{0\\}$}.$

We prove this claim by induction. The case $\ell=0$ is trivial. Suppose that
inequality (5.45) holds for some $\ell\in\mathbb{N}$. Assumption (5.44)
entails that

$c_{B}LJ_{\ell}^{\frac{1}{n}}\leq c_{B}L(\tau^{\ell}J_{0})^{\frac{1}{n}}\leq
c_{B}L\varepsilon_{0}^{\frac{1}{n}}\leq 1.$

Therefore, thanks to equations (5.42), (5.45), and (5.43),

(5.46) $\displaystyle J_{\ell+1}\leq$ $\displaystyle
c_{2}2^{\gamma\ell}J_{\ell}^{1+\frac{1}{n}}\leq
c_{2}(2^{\gamma}\tau^{\frac{1}{n}})^{\ell}J_{0}^{\frac{1}{n}}(\tau^{\ell}J_{0})\leq
c_{2}^{1-\ell}\varepsilon_{0}^{\frac{1}{n}}\tau^{\ell}J_{0}\leq\tau^{\ell+1}J_{0}.$

Notice that the last inequality holds thanks to the inequalities $c_{2}\geq
1$, $\ell\geq 1$, and $\varepsilon_{0}\leq\tau^{n}$. Inequality (5.45), with
$\ell$ replaced with $\ell+1$, follows from (5.46).

_Step 4._ _Assumption ( 5.44) holds for large $K$._
Since

$J_{0}=J(K/2,\mathbb{B}_{\frac{3}{4}}),$

inequality (5.44) will follow, for sufficiently large $K$, if we show that

(5.47) $\lim_{k\to\infty}J(k,\mathbb{B}_{\frac{3}{4}})=0.$

Inasmuch as $u\in V^{1}_{\rm loc}K^{A}(\Omega)$, from inclusion (3.20) we
infer that $\lim_{k\to\infty}|\mathcal{A}_{k,\frac{3}{4}}|=0$. Hence, the
dominated convergence theorem guarantees that

(5.48)
$\lim_{k\to\infty}\int_{\mathbb{B}_{\frac{3}{4}}}A(|\nabla(u-k)_{+}|)\,dx=\lim_{k\to\infty}\int_{\mathcal{A}_{k,\frac{3}{4}}}A(|\nabla(u-k)_{+}|)\,dx=0.$

It thus suffices to show that

(5.49) $\lim_{k\to\infty}\int_{\mathbb{B}_{\frac{3}{4}}}A(|(u-k)_{+}|)\,dx=0.$

To this purpose, note that, by inequality (4.1) and the monotonicity of
$A_{n}$,

(5.50) $\displaystyle\int_{\mathbb{B}_{\frac{3}{4}}}A(|(u-k)_{+}|)\,dx$
$\displaystyle\leq
c|\mathcal{A}_{k,\frac{3}{4}}|+\int_{\mathbb{B}_{\frac{3}{4}}}A_{n}(|(u-k)_{+}|)\,dx$
$\displaystyle\leq
c|\mathcal{A}_{k,\frac{3}{4}}|+\int_{\mathbb{B}_{\frac{3}{4}}}A_{n}\bigg{(}2\bigg{|}(u-k)_{+}-\fint_{\mathbb{B}_{\frac{3}{4}}}(u-k)_{+}dy\bigg{|}\bigg{)}\,dx+\int_{\mathbb{B}_{\frac{3}{4}}}A_{n}\bigg{(}2\bigg{|}\fint_{\mathbb{B}_{\frac{3}{4}}}(u-k)_{+}dy\bigg{|}\bigg{)}\,dx$

for some constant $c=c(n,A)$. Moreover,

$\lim_{k\to\infty}\mathcal{A}_{k,\frac{3}{4}}=0,$

and

$\lim_{k\to\infty}\int_{\mathbb{B}_{\frac{3}{4}}}A_{n}\biggl{(}2\biggl{|}\fint_{\mathbb{B}_{\frac{3}{4}}}(u-k)_{+}\biggr{|}\biggr{)}\,dx\leq\lim_{k\to\infty}|\mathbb{B}_{\frac{3}{4}}|A_{n}\biggl{(}\frac{2\|(u-k)_{+}\|_{L^{1}(\mathbb{B}_{\frac{3}{4}})}}{|\mathbb{B}_{\frac{3}{4}}|}\biggr{)}=0.$

It remains to prove that the second addend on the rightmost side of chain
(5.50) vanishes when $k\to\infty$. Thanks to the limit in (5.48), for every
$\delta>0$ there exists $k_{\delta}\in\mathbb{N}$ such that

(5.51)
$\int_{\mathbb{B}_{\frac{3}{4}}}A(|\nabla(u-k)_{+}|)\,dx\leq\delta\qquad\mbox{if
$k\geq k_{\delta}$.}$

Choose $\delta$ in (5.51) such that $2c_{B}\delta^{\frac{1}{n}}\leq 1$.
Property (3.1) applied to $A_{n}$, and the Sobolev-Poincaré inequality in
Orlicz spaces (3.19) applied to the function $(u-k)_{+}$ ensure that, if
$k>k_{\delta}$, then

$\displaystyle\int_{\mathbb{B}_{\frac{3}{4}}}A_{n}\bigg{(}2\bigg{|}(u-k)_{+}-\fint_{\mathbb{B}_{\frac{3}{4}}}(u-k)_{+}\bigg{|}\bigg{)}\,dx\leq$
$\displaystyle
2c_{B}\delta^{\frac{1}{n}}\int_{\mathbb{B}_{\frac{3}{4}}}A_{n}\Bigg{(}\frac{\big{|}(u-k)_{+}-\fint_{\mathbb{B}_{\frac{3}{4}}}(u-k)_{+}dy\big{|}}{c_{B}\big{(}\int_{\mathbb{B}_{\frac{3}{4}}}A(|\nabla(u-k)_{+}|)dy\big{)}^{\frac{1}{n}}}\Bigg{)}\,dx$
$\displaystyle\leq$ $\displaystyle
2c_{B}\delta^{\frac{1}{n}}\int_{\mathbb{B}_{\frac{3}{4}}}A(|\nabla(u-k)_{+}|)dx.$

Since the last integral tends to $0$ as $k\to\infty$, equation (5.49) is
establsihed.

_Step 5._ _Conclusion._
Inequality (5.45) tells us that $\inf_{\ell\in\mathbb{N}}J_{\ell}=0$. Hence,
from the definitions of $J_{\ell}$ and $J(h,\sigma)$ we deduce that

$\int_{\mathbb{B}_{\frac{1}{2}}}A((u-K)_{+})\,dx\leq
J(K,\mathbb{B}_{\frac{1}{2}})\leq\inf_{\ell\in\mathbb{N}}J_{\ell}=0.$

Therefore, $u\leq K$ a.e. in $\mathbb{B}_{\frac{1}{2}}$.
In order to prove a parallel lower bound for $u$, observe that the function
$-u$ is a quasiminimizer of the functional defined as in (1.1), with the
integrand $f$ replaced with the integral $\widetilde{f}$ given by

$\widetilde{f}(x,t,\xi)=f(x,-t,-\xi)\quad\text{ for
$(x,t,\xi)\in\Omega\times\mathbb{R}\times\mathbb{R}^{n}$.}$

The structure conditions (2.11) and (2.12) and the growth condition (2.13) on
the function $f$ are inherited by the function $\widetilde{f}$. An application
of the above argument to the function $-u$ then tells us that there exists a
constant $K^{\prime}>0$ such that $-u\leq K^{\prime}$ a.e. in
$\mathbb{B}_{\frac{1}{2}}$. The proof is complete. ∎

## Compliance with Ethical Standards

Funding. This research was partly funded by:
(i) GNAMPA of the Italian INdAM - National Institute of High Mathematics
(grant number not available) (A. Cianchi);
(ii) Research Project of the Italian Ministry of Education, University and
Research (MIUR) Prin 2017 “Direct and inverse problems for partial
differential equations: theoretical aspects and applications”, grant number
201758MTR2 (A. Cianchi);

Conflict of Interest. The authors declare that they have no conflict of
interest.

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(5.48) $\left|\nabla u\right|\leq\frac{C\delta}{r^{\alpha}}\text{ \ on
}M\backslash B_{p}\left(R_{2}\right),$

where $C$ depends only on $k.$ For $0\leq\alpha<1,$ integrating (5.48) along
minimal geodesics we obtain

$u\leq\frac{C\delta}{1-\alpha}r^{1-\alpha}\text{ \ on }M\backslash
B_{p}\left(R_{3}\right)$

for some $R_{3}$ sufficiently large, which proves that

$\displaystyle B_{p}\left(R\right)\backslash B_{p}\left(R_{3}\right)$
$\displaystyle\subset$ $\displaystyle
L\left(1,\frac{C\delta}{1-\alpha}R^{1-\alpha}\right)$ $\displaystyle=$
$\displaystyle
L\left(1,\frac{CA}{\left(1-\alpha\right)\Gamma}R^{1-\alpha}\right),$

where in the last line we have used (5.46). The gradient of $u$ is bounded,
therefore, by Corollary 5.4

$\displaystyle\int_{B_{p}\left(R\right)\backslash
B_{p}\left(R_{3}\right)}S|\nabla u|$ $\displaystyle\leq$
$\displaystyle\int_{L\left(1,\frac{CA}{\left(1-\alpha\right)\Gamma}\right)}S\left|\nabla
u\right|$ $\displaystyle\leq$
$\displaystyle\frac{CA}{\left(1-\alpha\right)\Gamma}R^{1-\alpha}+\Upsilon,$

for some constant $\Upsilon$. However, using (5.43) that

$S\geq\frac{A}{2}r^{-\alpha}\text{ \ on }M\backslash B_{p}\left(R_{3}\right)$

and (5.42) we obtain from the co-area formula that

$\displaystyle\int_{B_{p}\left(R\right)\backslash
B_{p}\left(R_{3}\right)}S|\nabla u|$ $\displaystyle\geq$
$\displaystyle\int_{R_{3}}^{R}\left(\frac{A}{2}r^{-\alpha}\right)\left(\int_{\partial
B_{p}\left(r\right)}\frac{\partial u}{\partial r}\right)dr$ $\displaystyle=$
$\displaystyle\frac{A}{2\left(1-\alpha\right)}\left(R^{1-\alpha}-R_{3}^{1-\alpha}\right).$

In conclusion, from (5) and (5),

$\frac{CA}{\left(1-\alpha\right)\Gamma}R^{1-\alpha}+\Upsilon\geq\frac{A}{2\left(1-\alpha\right)}\left(R^{1-\alpha}-R_{3}^{1-\alpha}\right)$

for all $R>R_{3}$ with the constant $C>0$ depending only on $k$ and $\nu.$
This shows that $\Gamma\leq C$ when $0\leq\alpha<1.$ A similar argument also
works for $\alpha=1.$ This proves the result. ∎

The volume doubling property (5.32) obviously holds for a manifold with
nonnegative Ricci curvature. So the above theorem implies that

$\mathrm{V}_{p}\left(R_{i}\right)\leq C\,R_{i}^{\alpha+1}$

for a sequence $R_{i}\rightarrow\infty$ if the scalar curvature

$S\left(x\right)\geq\frac{C}{r^{\alpha}\left(x\right)+1}$

for some $\alpha\in\left[0,1\right].$ In the special case that $\alpha=0$ or
$S\geq C>0,$ a standard volume comparison argument then implies
$\mathrm{V}_{p}(R)\leq c\,R.$ This fact also follows from

$\liminf_{x\rightarrow\infty}S\left(x\right)<\frac{C_{0}}{\mathrm{V}_{p}\left(1\right)}$

by a scaling argument.

Finally, we point out that Lemma 5.3 also holds for positive harmonic
functions converging to zero at infinity along an end and apply it to rule out
the existence of proper positive Green’s functions under a suitable scalar
curvature assumption. One may wish to compare the following result with
Theorem 1.3.

######  Theorem 5.7.

For complete three-dimensional manifold $M$ with Ricci curvature bounded from
below by a constant and scalar curvature satisfying

(5.51)
$\liminf_{x\rightarrow\infty}\left(S\left(x\right)r\left(x\right)\right)\geq
C_{0},$

if it has finitely many ends and finite first Betti number, then it does not
admit any positive Green’s function $G$ satisfying

$\lim_{x\rightarrow\infty}G\left(x\right)=0.$

###### Proof.

The proof is by contradiction and uses an analogous monotonicity formula as
Lemma 5.3 for

$w\left(t\right)=\int_{l\left(t\right)}\left|\nabla G\right|^{2}.$

In particular,

$\displaystyle\frac{dw}{dt}(t)-3\frac{w(t)}{t}+4\pi t$ $\displaystyle\geq$
$\displaystyle\frac{dw}{ds}(s)-3\frac{w(s)}{s}+4\pi
s+\frac{1}{2}\int_{L(s,t)}S|\nabla G|$

for all $0<s<t\leq t_{0},$ where $t_{0}>0$ is a fixed small number. Let
$s\rightarrow 0.$ Lemma 2.2 implies that

$\displaystyle\frac{w(s)}{s}$ $\displaystyle\leq$ $\displaystyle C$
$\displaystyle\liminf_{s\rightarrow 0}\frac{dw}{ds}(s)$ $\displaystyle\leq$
$\displaystyle C.$

Hence, we conclude that

$\int_{M}S|\nabla G|\leq C.$

As in Theorem 5.1, this leads to a contradiction by (5.51). ∎

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