# Singular orthotropic functionals
with nonstandard growth conditions

Pierre Bousquet  ,  Lorenzo Brasco  and  Chiara Leone  Institut de
Mathématiques de Toulouse, CNRS UMR 5219
Université de Toulouse
F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>Dipartimento di Matematica e Informatica
Università degli Studi di Ferrara
Via Machiavelli 35, 44121 Ferrara, Italy<EMAIL_ADDRESS>Dipartimento
di Matematica “R. Caccioppoli”
Università degli Studi di Napoli “Federico II”
Via Cinthia, Complesso Universitario di Monte S. Angelo, 80126 Napoli, Italy
<EMAIL_ADDRESS>
###### Abstract.

We pursue the study of a model convex functional with orthotropic structure
and nonstandard growth conditions, this time focusing on the sub-quadratic
case. We prove that bounded local minimizers are locally Lipschitz. No
restriction on the ratio between the highest and the lowest growth rates are
needed. The result holds also in presence of a non-autonomous lower order
term, under sharp integrability assumptions. Finally, we prove higher
differentiability of bounded local minimizers, as well.

###### Key words and phrases:

Nonstandard growth conditions, singular elliptic equations, Lipschitz
regularity, Sobolev regularity, orthotropic functionals

###### 2010 Mathematics Subject Classification:

35J75, 35B65, 49K20

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Main results
    3. 1.3 Comparison with known results
    4. 1.4 Structure of the proofs
    5. 1.5 Plan of the paper
  2. 2 Preliminaries
    1. 2.1 Some auxiliary functions
    2. 2.2 Regularized problems
  3. 3 Caccioppoli–type inequalities for the gradient
  4. 4 Uniform higher integrability
  5. 5 Uniform Lipschitz bound
  6. 6 Uniform higher differentiability
  7. 7 Proofs of the main results
    1. 7.1 Proof of Proposition 1.1
    2. 7.2 Proof of Theorem L
    3. 7.3 Proof of Theorem S
  8. A A weak maximum principle

##  1\. Introduction

###  1.1. Overview

In this paper we expand on the gradient regularity theory for minimizers of
functionals from the Calculus of Variations, having an orthotropic structure,
in the nonstandard growth case. This may be seen as a follow up of our
previous papers [9] and [12].

Specifically, for an open set $\Omega\subseteq\mathbb{R}^{N}$ and a set of
exponents $1<p_{1}\leq\dots\leq p_{N}$, we take the anisotropic Sobolev space
$W^{1,\mathbf{p}}_{\rm{loc}}(\Omega)$, defined by

$W^{1,\mathbf{p}}_{\rm{loc}}(\Omega)=\Big{\\{}u\in
L^{1}_{\rm{loc}}(\Omega)\,:\,u_{x_{i}}\in
L_{\rm{loc}}^{p_{i}}(\Omega),\,i=1,\cdots,N\Big{\\}}.$

Given a function $f\in L^{1}_{\rm loc}(\Omega)$, we consider the following
functional

$\mathfrak{F}_{\mathbf{p}}(u,\Omega^{\prime}):=\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{\Omega^{\prime}}|u_{x_{i}}|^{p_{i}}\,dx-\int_{\Omega^{\prime}}f\,u\,dx,\qquad\mbox{
for }u\in W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)\
\mbox{ and }\Omega^{\prime}\Subset\Omega.$

In the superquadratic case, i.e. for

$2\leq p_{1}\leq\cdots\leq p_{N}<\infty,$

and for $f\equiv 0$, it has been recently proved in [9] that any local
minimizer $U\in W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}(\Omega)$ is
such that

$\nabla U\in L^{\infty}_{\rm loc}(\Omega)\qquad\mbox{ and
}\qquad|U_{x_{i}}|^{\frac{p_{i}-2}{2}}\,U_{x_{i}}\in W^{1,2}_{\rm
loc}(\Omega),\ \mbox{ for }i=1,\dots,N.$

The main goal of this paper is to address the same kind of regularity issues,
again for bounded local minimizers, this time in the subquadratic case

$1<p_{1}\leq\cdots\leq p_{N}\leq 2.$

However, we will obtain some regularity results which actually hold in the
full range $1<p_{1}\leq\dots\leq p_{N}<\infty$, see the next section for more
details.

We recall that $u\in W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}_{\rm
loc}(\Omega)$ is a local minimizer of $\mathfrak{F}_{\mathbf{p}}$ in $\Omega$
if

$\mathfrak{F}_{\mathbf{p}}(u,\Omega^{\prime})\leq\mathfrak{F}_{\mathbf{p}}(\varphi,\Omega^{\prime}),\qquad\hbox{
for every }\varphi-u\in W^{1,\mathbf{p}}_{0}(\Omega^{\prime})\cap
L^{\infty}(\Omega^{\prime})\mbox{ and every }\Omega^{\prime}\Subset\Omega.$

Here we denote by $W^{1,\mathbf{p}}_{0}(\Omega^{\prime})$ the completion of
$C^{\infty}_{0}(\Omega^{\prime})$ with respect to the norm

$\varphi\mapsto\sum_{i=1}^{N}\|\varphi_{x_{i}}\|_{L^{p_{i}}(\Omega^{\prime})},\qquad\mbox{
for every }\varphi\in C^{\infty}_{0}(\Omega^{\prime}).$

By convexity of $\mathfrak{F}_{p}$, we have that $u$ is a local minimizer if
and only if it is a local weak solution in $W^{1,\mathbf{p}}_{\rm
loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ of the quasilinear equation

$-\sum_{i=1}^{N}\Big{(}|u_{x_{i}}|^{p_{i}-2}\,u_{x_{i}}\Big{)}_{x_{i}}=f,\qquad\mbox{
in }\Omega.$

This can be seen as a particular instance of elliptic equation in the wide
context of Musielak-Orlicz spaces, see [17] for a comprehensive study on the
subject.

We emphasize the fact that in this paper we just consider _bounded_ minimizers
$u$. As a consequence, we discard a priori all the counterexamples to
regularity arising in the literature related to nonstandard growth variational
problems, see [33, 35, 45]. For completeness, we mention that the boundedness
of minimizers in this setting has already been extensively studied, see [32]
for the homogeneous case $f\equiv 0$ and [19, 20] for the non-homogeneous one.
We also ignore the problem of the _existence_ of a minimizer in
$W^{1,\mathbf{p}}$, for which we would also need to assume that $f$ belongs to
a suitable dual Sobolev space. Here instead, we assume a priori to have a
bounded minimizer $U$ and focus on identifying sharp conditions on the
function $f$ needed to obtain its Lipschitz continuity and higher
differentiability.

The main feature of all our regularity results will be that we do not need to
impose any restriction on the ratio $p_{N}/p_{1}$. We refer the reader to our
previous papers [9, 11, 12] for an introduction to the realm of gradient
regularity for minimizers of orthotropic functionals (see also [24] for an
approach based on viscosity methods). We just recall here that already for the
standard growth case $p_{1}=p_{N}=p$, the superquadratic case $p>2$ is much
more involved than the case of the model functional

$\mathfrak{L}(u;\Omega^{\prime})=\int_{\Omega^{\prime}}|\nabla
u|^{p}\,dx,\qquad\mbox{ for }u\in W^{1,p}_{\rm loc}(\Omega)\ \mbox{ and
}\Omega^{\prime}\Subset\Omega,$

as far as the regularity of the gradient of local minimizers is concerned. On
the other hand, the subquadratic case $1<p<2$ is simpler, in a sense: the
Lipschitz continuity is a consequence of a general result due to Fonseca and
Fusco [31, Theorem 2.2], as observed in the introduction of [11].

In particular, it seems natural to try to adapt the techniques used in [31]
since they allow to establish the Lipschitz regularity for the subquadratic
case when $p_{1}=p_{N}=p<2$. However, we stress that in the case
$p_{1}\not=p_{N}$, our functional pertains to the class of variational
problems with nonstandard growth conditions, following the terminology of
Marcellini in [43, 44]. Then it couples in a nontrivial way the difficulties
coming from the two situations: orthotropic structure and nonstandard growth
conditions. Thus, even if we will prove Lipschitz regularity with a proof
inspired from [31, Theorem 2.2], nontrivial adaptations and intermediate
results will be needed.

Finally, it is worth recalling that, in spite of a large number of papers and
contributions on nonstandard growth problems (including for example [5, 7, 15,
16, 29, 30, 36, 37, 38, 47, 50]), a complete gradient regularity theory is
still missing, even for the case of orthotropic structures. Moreover, we
recall that also the case of basic regularity (i.e. $C^{0,\alpha}$ estimates,
Harnack inequalities, and an extension of the De Giorgi’s regularity theory)
is still not fully well-understood (see for example [1, 3] and [42] for some
results) for local minimizers of $\mathfrak{F}_{\mathbf{p}}$.

###  1.2. Main results

Our first result is an higher integrability statement, which is valid without
any restriction on the exponents $p_{i}$. As we will see in a while, this will
be instrumental to the two main regularity results of this paper. In what
follows, we use the notation

(1.1) $\mathcal{G}_{0}(\nabla
u)=\left(\sum_{i=1}^{N}\frac{1}{p_{i}}\,|u_{x_{i}}|^{p_{i}}-1\right)_{+}+1,$

where $(\,\cdot\,)_{+}$ stands for the positive part. This function naturally
arises from the principal part of $\mathfrak{F}_{\mathbf{p}}$. It encodes in a
natural way the full summability informations for each component of the
gradient. A similar idea has been considered for example in the papers [2, 22,
23], dealing with the so-called double phase problems.

######  Proposition 1.1 (Higher integrability: general growth).

Let $1<p_{1}\leq\dots\leq p_{N}<\infty$ and let $f\in L^{\gamma}_{\rm
loc}(\Omega)$, for some $\gamma\geq 2$. Then for every local minimizer $U\in
W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$, we have

$\mathcal{G}_{0}(\nabla U)\in L^{\gamma}_{\rm loc}(\Omega).$

Moreover, for every ball $B_{R}(x_{0})\Subset\Omega$ such that
$B_{4R}(x_{0})\Subset\Omega$, as well, we have

(1.2) $\int_{B_{\frac{R}{2}}(x_{0})}\mathcal{G}_{0}(\nabla
U)^{\gamma}\,dx\leq\left(\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}(x_{0})}\mathcal{G}_{0}(\nabla
U)\,dx\right),$

for two constants $\Gamma_{1},\Gamma_{2}>0$ which depend only on

$N,\,p_{N},\,p_{1},\,\gamma,\,R,\,\|f\|_{L^{\gamma}(B_{4R}(x_{0}))}\mbox{ and
}\|U\|_{L^{\infty}(B_{4R}(x_{0}))}.$

######  Theorem L (Lipschitz regularity: subquadratic growth).

Let $1<p_{1}\leq\dots\leq p_{N}\leq 2$ and let $f\in L^{\gamma}_{\rm
loc}(\Omega)$, for some $\gamma>N$. Then every local minimizer $U\in
W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ is
locally Lipschitz continuous. Moreover, for every ball
$B_{R}(x_{0})\Subset\Omega$ such that $B_{4R}(x_{0})\Subset\Omega$, we have

$\begin{split}\left\|\mathcal{G}_{0}(\nabla
U)\right\|_{L^{\infty}(B_{R/4}(x_{0}))}&\leq
C\,\left[\frac{1}{R^{\frac{N\,\gamma}{\gamma-N}}}\,\left(\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}(x_{0})}\mathcal{G}_{0}(\nabla
U)\,dx\right)^{\frac{N}{\gamma-N}}+\|f\|_{L^{\gamma}(B_{R}(x_{0}))}^{\frac{N\,\gamma}{\gamma-N}}\right]\\\
&\times\left\|\mathcal{G}_{0}(\nabla
U)\right\|_{L^{1}(B_{R}(x_{0}))}.\end{split}$

for some $C=C(N,p_{N},p_{1},\gamma)>0$ and for the same constants
$\Gamma_{1},\Gamma_{2}$ as in (1.2).

Observe that the assumption $\gamma>N$ is sharp (in the scale of Lebesgue
spaces) to obtain the Lipschitz continuity of $U$. Actually, this is already
true when $p_{1}=\dots=p_{N}=2$. It is a remarkable fact that, even in the
orthotropic case with nonstandard growth conditions, this universal assumption
on $f$ still leads to Lipschitz continuity. We refer the reader to [4] for a
wide class of variational problems (not including orthotropic structures,
however) where this same condition is known to guarantee Lipschitz continuity
of local minimizers.

######  Theorem S (Sobolev regularity: subquadratic growth).

Let $1<p_{1}\leq\dots\leq p_{N}\leq 2$ and let

(1.3) $f\in L^{1+\frac{2}{p_{1}}}_{\rm loc}(\Omega).$

Then for every local minimizer $U\in W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap
L^{\infty}_{\rm loc}(\Omega)$ we have

$\mathcal{W}_{i}:=|U_{x_{i}}|^{\frac{p_{i}-2}{2}}\,U_{x_{i}}\in W^{1,2}_{\rm
loc}(\Omega),\ \mbox{ for }i=1,\dots,N\qquad\mbox{ and }\qquad\nabla U\in
W^{1,\mathbf{p}}_{\rm loc}(\Omega).$

Moreover, for every ball $B_{R}(x_{0})\Subset\Omega$ such that
$B_{4R}(x_{0})\Subset\Omega$, we have for every $i=1,\dots,N$

$\begin{split}\sum_{i=1}^{N}\int_{B_{\frac{R}{4}}(x_{0})}\Big{|}\nabla\mathcal{W}_{i}\Big{|}^{2}\,dx&\leq\frac{C}{R^{2}}\,\left(\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}(x_{0})}\mathcal{G}_{0}(\nabla
U)\,dx\right)+C\,R^{\frac{2}{p_{1}}-1}\,\int_{B_{R}(x_{0})}|f|^{1+\frac{2}{p_{1}}}\,dx,\end{split}$

and

$\|\nabla
U_{x_{i}}\|_{L^{p_{i}}(B_{R/4}(x_{0}))}\leq\frac{2}{p_{i}}\,\|U_{x_{i}}\|_{L^{p_{i}}(B_{R/4}(x_{0}))}^{\frac{2-p_{i}}{2}}\,\|\nabla\mathcal{W}_{i}\|_{L^{2}(B_{R/4}(x_{0}))},$

for some $C=C(N,p_{N},p_{1})>0$ and for the same constants
$\Gamma_{1},\Gamma_{2}$ as in (1.2), corresponding to the choice
$\gamma=(p_{1}+2)/p_{1}$.

###  1.3. Comparison with known results

In the homogeneous case $f=0$, Proposition 1.1 can be obtained as a
consequence of [40, Lemma 4.2]. In the superquadratic case $p_{1}\geq 2$ and
still for $f\equiv 0$, an alternate proof can also be found in [9, Proposition
6.1]. We present here a new proof that takes into account the presence of the
forcing term $f$. Our argument is certainly more elementary than the one in
[9], and arguably more natural in our setting than the one in [40], in the
sense that it strongly relies on some tools that will be repeatedly used in
the proofs of the other main results, see the next section for further
comments. Theorem L is the counterpart for the subquadratic case of our
previous result [9, Theorem 1.1], which deals with the superquadratic case. We
shall explain in the next section why the two situations require different
arguments. We point out that experts in the field may recognize Theorem L (and
[9, Theorem 1.1], as well) as a particular case of the main result in [40], at
least in the homogeneous case $f\equiv 0$. However, it turns out that the
proof [40, Proposition 2.1] is affected by a crucial flaw, we refer the reader
to [9, Remark 1.4] for a detailed discussion on this delicate point. In any
case, it is fair to admit that some other parts of Lieberman’s paper [40] have
been an important source of inspiration for the proof of Proposition 1.1.

In [28, Corollary 3.4], the authors proves the local Lipschitz continuity of
local minimizers (not a priori bounded) of the following functional

(1.4)
$\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int(1+|u_{x_{i}}|^{2})^{\frac{p_{i}}{2}}\,dx,\qquad\mbox{
for }\frac{2\,N}{N+2}<p_{1}\leq\dots\leq p_{N}\leq 2.$

Observe that such a functional has an orthotropic structure, with nonstandard
subquadratic growth conditions, exactly as our $\mathfrak{F}_{p}$. However, it
should be noticed that the functional (1.4) is neither degenerate nor singular
: this is the crucial difference with our case. Indeed, the Hessian of the
function

$\widetilde{G}(z):=\sum_{i=1}^{N}\frac{1}{p_{i}}(1+z_{i}^{2})^{\frac{p_{i}}{2}},\qquad\mbox{
for }z=(z_{1},\dots,z_{N})\in\mathbb{R}^{N},$

satisfies

$(p_{1}-1)(1+|z|^{2})^{\frac{p_{1}-2}{2}}\,|\xi|^{2}\leq\langle
D^{2}\widetilde{G}(z)\,\xi,\xi\rangle\leq|\xi|^{2},\qquad\mbox{ for
}z,\xi\in\mathbb{R}^{N}.$

This property fails to be satisfied by our functional, where the integrand is
given by

(1.5) $G(z)=\sum_{i=1}^{N}\frac{1}{p_{i}}|z_{i}|^{p_{i}},\qquad\mbox{ for
}z\in\mathbb{R}^{N}.$

Even worse, in contrast with the general framework of [28], in our situation
there is no continuous functions $h_{1},h_{2}:[0,+\infty)\to[0,+\infty)$ such
that

$h_{1}(|z|)\,|\xi|^{2}\leq\langle D^{2}G(z)\,\xi,\xi\rangle\leq
h_{2}(|z|)\,|\xi|^{2},\qquad\mbox{ for }z,\xi\in\mathbb{R}^{N},$

even for large values of $z$. Indeed, $D^{2}G(z)$ is given by the diagonal
matrix

$D^{2}G(z)=\left[\begin{array}[]{ccc}(p_{1}-1)\,|z_{1}|^{p_{1}-2}&&\\\
&\ddots&\\\ &&(p_{N}-1)\,|z_{N}|^{p_{N}-2}\end{array}\right],$

and each entry on the diagonal blows-up as the corresponding component of $z$
vanishes. As for Theorem S, we observe that this may be seen as a
generalization of the following classical result for the $p-$Laplacian: for
$1<p<2$, any local weak solution $u\in W^{1,p}_{\rm loc}(\Omega)$ of

$-\hbox{div}(|\nabla u|^{p-2}\nabla u)=f,\qquad\mbox{ in }\Omega,$

belong to $W^{2,p}_{\rm loc}(\Omega)$, provided $f\in L^{p^{\prime}}_{\rm
loc}(\Omega)$ (see for example [26]). The reader may notice that for
$p_{1}=p_{N}=p$ our assumption (1.3) boils down to

$f\in L^{\gamma}_{\rm loc}(\Omega),\qquad\mbox{ for }\gamma\geq
1+\frac{2}{p}.$

Since for $1<p<2$ we have $1+2/p<p^{\prime}$, this is a weaker requirement
when compared with the classical result recalled above. This is not
surprising, since we are now assuming that $u$ is a priori bounded. Such an
assumption is responsible for this new feature. In the standard growth case,
this has been recently observed in [21, Theorem 1.2].

Higher differentiability of local minimizers is a well-studied problem: for
the specific case of orthotropic functionals with subquadratic nonstandard
growth, some prior results can be found for example in [5, Theorem 3], [6,
Corollary 1] and [16, Theorem 2].

Finally, in the superquadratic case $p_{1}\geq 2$, as already recalled the
counterpart of Theorem S has been obtained in [9, Corollary 7.1], for $f\equiv
0$. In the case of a right-hand side $f\not\equiv 0$, some results have been
obtained in [14, Theorem 1.1] and [47, Corollary 2].

######  Remark 1.2 (On the $C^{1}$ regularity).

In dimension $N=2$, the $C^{1}$ regularity of a Lipschitz local minimizer
essentially follows from [25, Theorem 1.1], both in the case $p_{1}\leq 2$ and
$p_{1}\geq 2$, provided that $f\equiv 0$. This assertion is detailed in [8],
where the “mixed” case $p_{1}\leq 2\leq p_{2}$ is considered, as well. For a
slightly different approach, see [41] when $p_{1}\geq 2$ and [48] when
$p_{1}=p_{2}<2$: these references still deal with the case $f\equiv 0$. In the
non-homogenous case, the strategy followed in [10] (and originally written for
$p_{1}=p_{2}$ and $f\equiv 0$) could be adapted to more general situations,
provided $f$ satisfies suitable differentiability and summability conditions.
In any case, the $C^{1}$ regularity of local minimizers when $N\geq 3$ is
entirely open, even for $p_{1}=p_{N}$ and $f\equiv 0$.

###  1.4. Structure of the proofs

The proofs of Proposition 1.1, Theorem L and Theorem S are based on a
classical three steps strategy. We first approximate our local minimizer $U\in
W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ by a
sequence of minimizers $\\{u^{\varepsilon}\\}_{\varepsilon>0}$ of regularized
functionals $\mathfrak{F}_{\mathbf{p},\varepsilon}$ having good smoothing
properties. We next obtain uniform a priori estimates on these minimizers.
Finally, we pass to the limit in order to transfer these a priori bounds to
$U$. The first step is usually quite easy, it is sufficient to perturb the
initial functional by adding some uniformly convex $\varepsilon-$perturbation
and possibly smooth out the coefficients, for example by replacing $f$ with
its mollifcation $f^{\varepsilon}$. This regularization strategy allows to
avoid the usual difference quotient method, and the technicalities that go
with its use in the nonstandard growth setting. However, here we face a first
difficulty: remember that we are not assuming $f$ to be in the correct dual
Sobolev space. This also entails that we do not have a good a priori
$L^{\infty}$ estimate at our disposal. Thus, such an approximation has to be
handled with great care. We circumvent this technical difficulty, by adding a
nonlinear lower term in the regularized functional, which forces the
minimizers $u^{\varepsilon}$ to be bounded, with a uniform $L^{\infty}$ bound
which only depends on the local $L^{\infty}$ norm of $U$ (see Lemma 2.4). This
is a technical aspect of the proof, which we believe to have its own interest.
The core of the matter is next to establish the a priori estimates for the
gradient of $u^{\varepsilon}$, the minimizer of the regularized functional
$\mathfrak{F}_{\mathbf{p},\varepsilon}$. As for the estimates leading to
Proposition 1.1 and Theorem L, these are achieved by means of Moser–type
schemes: a slow one and a fast one, respectively.

The cornerstone of these schemes is a Caccioppoli inequality for power
functions of the gradients (see Proposition 3.1). In a simplified way, for
every $\alpha\geq 0$ this reads as

(1.6) $\sum_{i=1}^{N}\int|u_{x_{i}}|^{p_{i}-2}\,\left|\Big{(}G(\nabla
u)^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,dx\lesssim\alpha^{2}\int
G(\nabla u)^{\alpha+2-\frac{2}{p_{N}}}\,dx,$

where $G$ is the same function as in (1.5). For simplicity, we put $f\equiv 0$
and write $u$ in place of $u^{\varepsilon}$.

Observe that on the left-hand side of (1.6) we have a weighted gradient of a
power of $G(\nabla u)$: the weights $|u_{x_{i}}|^{p_{i}-2}$ are the typical
feature of degenerate/singular orthotropic functionals. The main difficulty in
getting regularity results out of this estimate is precisely due to their
presence. In contrast with the Caccioppoli inequality previously obtained in
[9, Lemma 3.1] to handle the superquadratic case $p_{1}\geq 2$, now these
weights do not pop-up on the right-hand side. This is a crucial ingredient of
the estimate: indeed, no control from above would be possible on
$|u_{x_{i}}|^{p_{i}-2}$ if $p_{i}<2$.

Not surprisingly, the proof of (1.6) relies on the differentiated Euler-
Lagrange equation, which is nothing but the equation solved by the components
of $\nabla u^{\varepsilon}$. In a nutshell, the idea to reach such an estimate
not containing the nasty weights $|u_{x_{i}}|^{p_{i}-2}$ on the right-hand
side, is that of using an integration by parts trick: this permits to trade
the presence of the term $D^{2}G$, with the more tractable one $DG$. This idea
is certainly not new in the context of singular variational problems: it goes
back at least to [46], and has then become standard in the field.

However, as natural as this idea may appear, its technical implementation in
our context needs some efforts: in particular, a careful choice of the test
functions for the differentiated equation has to be done. Such a choice must
reflect the algebraic structure of the operator, in a sense. Without entering
too much into the details, we refer to the proof of Proposition 3.1 below. The
choice of the correct test functions here has been suggested to us by [40],
even if our choice seems to be simpler and more natural. The Caccioppoli
inequality (1.6) is first used in the proof of the higher integrability result
of Proposition 1.1. More specifically, it permits to obtain a self-improving
estimate of the type

(1.7) $\int G(\nabla
u)^{\beta+1}\,dx\lesssim\|u\|_{L^{\infty}}^{2}\,\beta^{2}\,\int G(\nabla
u)^{\beta+1-\frac{2}{p_{N}}}\,dx.$

This is the slow Moser’s iteration scheme we were referring to above: by
iterating (1.7) a finite number of times, we can conclude that $G(\nabla u)$
(and thus $\nabla u$ itself) can be estimated in $L^{q}$, for every finite
$q\geq 1$. Observe that the additive integrability gain at each step and the
presence of the factor $\beta^{2}$ on the right-hand side make the previous
scheme not suitable for being iterated infinitely many times. This explains
why we cannot reach the limiting case $\nabla u\in L^{\infty}$ with this
approach.

Estimates like (1.7) are quite typical in the Regularity Theory, both in the
contexts of standard and nonstandard growth problems (among others, see for
example [27, Proposition 3.1] and [18, Theorem 3], respectively). Usually,
they are obtained by coupling an integration by parts, with a Caccioppoli
inequality for the gradient, like the one (1.6) at our disposal. The
$L^{\infty}$ bound on the solution is used to treat the solution itself as a
constant in the estimates.

We stress here that in this part of the proofs we do not need the restriction
$p_{N}\leq 2$. Thus, in particular we can extend and simplify the higher
integrability result we previously obtained111There is however a subtle detail
here: the result in [9] was obtained through a complicate self-improving
iterative scheme (inspired from that of [7, Theorem 1.1]), which was not of
Moser–type. Actually, this was much more sophisticated and could be roughly
described as follows: improvement of integrability of $N-1$ components of the
gradient entails that the missing one improves its integrability, as well. in
[9, Proposition 4.3]. With the aid of (1.7), in the case $p_{N}\leq 2$ we can
transpose to our situation the typical absorption trick which lies at the
basis of the Lipschitz estimate for the standard $p-$Laplacian (see for
example [27, Section 3]). Up to some nontrivial technical issues, this
consists in observing that when $G(\nabla u)\geq 1$, we have

$|u_{x_{i}}|^{p_{i}-2}\gtrsim G(\nabla u)^{\frac{p_{1}-2}{p_{1}}},\qquad\mbox{
since }p_{i}-2\leq 0,$

and thus

(1.8) $\sum_{i=1}^{N}\int|u_{x_{i}}|^{p_{i}-2}\,\left|\Big{(}G(\nabla
u)^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,dx\gtrsim\sum_{i=1}^{N}\int\left|\left(G(\nabla
u)^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\right)_{x_{i}}\right|^{2}\,dx.$

The weights $|u_{x_{i}}|^{p_{i}-2}$ have then been absorbed into a suitable
power function of the gradient. In this sense, in the case $p_{N}\leq 2$, the
presence of the weights $|u_{x_{i}}|^{p_{i}-2}$ on the left-hand side of (1.7)
helps, more than it hurts.

At this point, by joining (1.8) and (1.7), the orthotropic nature of the
problem completely disappears and we simply fall into the realm of nonstandard
growth problems. A standard application of Sobolev inequality makes then
possible to launch a standard Moser’s iterative scheme (i.e. a fast one, with
a multiplicative gain of integrability at each step). This permits to reach an
$L^{\infty}-L^{q}$ estimate on $G(\nabla u)$, after infinitely many
iterations. This is not the end of the story. Indeed, we still have to pay
attention to a detail which is quite typical of the nonstandard growth case:
the exponent $q$ in this a priori estimate could be too large. However, this
preliminary estimate can be “rectified” by combining the higher integrability
result of Proposition 1.1 together with an interpolation trick which decreases
the initial integrability requirement on $G(\nabla u)$. We then finally get a
$L^{\infty}-L^{1}$ estimate on $G(\nabla u)$, as desired. In contrast to
Proposition 1.1 and Theorem L, the proof of Theorem S does not rely on the
Caccioppoli inequality of Proposition 3.1. The proof follows the same idea as
in the case of the result for the familiar $p-$Laplacian, for the case
$1<p<2$: we test the differentiated equation with the gradient itself
$u_{x_{k}}$ and perform an integration by parts as in Naumann’s trick [46].
Again, this permits to avoid using the undesired upper bound on the Hessian
$D^{2}G$. In order to conclude, one has to control from above terms of the
form

$u_{x_{k}x_{k}}\,|u_{x_{i}}|^{p_{i}-1}.$

Observe that for every $k\not=i$, the two terms are completely decoupled.
However, when $p_{i}\leq 2$, we can simply estimate this term from above by
Young’s inequality

$u_{x_{k}x_{k}}\,|u_{x_{i}}|^{p_{i}-1}\lesssim|u_{x_{k}}|^{p_{k}-2}\,|u_{x_{k}x_{k}}|^{2}+|u_{x_{k}}|^{2-p_{k}}\,|u_{x_{i}}|^{2\,(p_{i}-1)}.$

The first term is absorbed on the left-hand side, while the second term can be
estimated from above by means of an integrability estimate (here we rely again
on the information provided by Proposition 1.1). This explains why we require
$p_{N}\leq 2$ in the statement of Theorem S.

###  1.5. Plan of the paper

In Section 2, we present the approximation scheme and some basic material used
all along the paper. Section 3 contains the crucial Caccioppoli-type
inequality for the gradient (Proposition 3.1). The latter is exploited in
Section 4 to perform the slow Moser iteration leading to the higher
integrability estimate needed in Proposition 1.1. The Lipschitz bound related
to Theorem L is proved in Section 5, while Section 6 is devoted to the proof
of the higher differentiability estimates corresponding to Theorem S. Then, in
Section 7, we eventually prove our three main results by passing to the limit
in the approximation scheme. Finally, for completeness, we include in Appendix
A the proof of a maximum principle ensuring the uniform boundedness of the
approximating sequence.

######  Acknowledgments.

This work has been finalized during a staying of P. B. and L. B. at the
Institute of Applied Mathematics and Mechanics of the University of Warsaw, in
July 2022. Iwona Chlebicka and Anna Zatorska-Goldstein are gratefully
acknowledged for their kind invitation and the nice working atmosphere
provided during the whole staying.

C. L. is a member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM). The three authors gratefully acknowledge the financial
support of the projects FAR 2019 and FAR 2020 of the University of Ferrara.

##  2\. Preliminaries

In this section, we fix

$1<p_{1}\leq p_{2}\leq\dots\leq p_{N}<\infty.$

###  2.1. Some auxiliary functions

For every $i=1,\dots,N$ and $\varepsilon>0$, we define

(2.1)
$g_{i,\varepsilon}(t)=\frac{1}{p_{i}}\,(\varepsilon+t^{2})^{\frac{p_{i}}{2}},\qquad\mbox{
for every }t\in\mathbb{R}.$

######  Lemma 2.1 (Sub-quadratic case).

Let $1<p_{i}\leq 2$. Then for every $t\in\mathbb{R}$, we have

(2.2) $(p_{i}-1)\,(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\leq
g^{\prime\prime}_{i,\varepsilon}(t)\leq(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}},$
(2.3)
$g_{i,\varepsilon}(t)\leq\frac{1}{p_{i}}\,\left(\varepsilon^{\frac{p_{i}}{2}}+g_{i,\varepsilon}^{\prime}(t)\,t\right),$
(2.4)
$|g_{i,\varepsilon}^{\prime}(t)|^{2}\leq\frac{p_{i}}{p_{i}-1}\,g_{i,\varepsilon}^{\prime\prime}(t)\,g_{i,\varepsilon}(t).$

###### Proof.

The second derivative of $g_{i,\varepsilon}$ is given by

$\begin{split}g^{\prime\prime}_{i,\varepsilon}(t)&=(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}+(p_{i}-2)\,(\varepsilon+t^{2})^{\frac{p_{i}-4}{2}}\,t^{2}=(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,\left(1+(p_{i}-2)\,\frac{t^{2}}{\varepsilon+t^{2}}\right).\end{split}$

In particular, by using that $0<p_{i}-1\leq 1$, we easily get (2.2). We also
observe that

$\begin{split}g_{i,\varepsilon}(t)=\frac{1}{p_{i}}\,(\varepsilon+t^{2})^{\frac{p_{i}}{2}}&=\frac{1}{p_{i}}\,(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,\varepsilon+\frac{1}{p_{i}}\,(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,t^{2}\leq\frac{\varepsilon^{\frac{p_{i}}{2}}}{p_{i}}+\frac{g_{i,\varepsilon}^{\prime}(t)}{p_{i}}\,t,\end{split}$

which proves (2.3). Finally, (2.4) follows by writing

$\begin{split}|g^{\prime}_{i,\varepsilon}(t)|^{2}&=(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,t^{2}\leq(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,(\varepsilon+t^{2})^{\frac{p_{i}}{2}},\end{split}$

and then using the definition of $g_{i,\varepsilon}$ and the lower bound in
(2.2). ∎

######  Lemma 2.2 (Super-quadratic case).

Let $p_{i}>2$. Then for every $t\in\mathbb{R}$, we have

(2.5) $(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\leq
g^{\prime\prime}_{i,\varepsilon}(t)\leq(p_{i}-1)\,(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}},$
(2.6)
$g_{i,\varepsilon}(t)\leq\frac{2^{\frac{p_{i}-2}{2}}}{p_{i}}\,\left(\varepsilon^{\frac{p_{i}}{2}}+g_{i,\varepsilon}^{\prime}(t)\,t\right),$
(2.7) $|g_{i,\varepsilon}^{\prime}(t)|^{2}\leq
p_{i}\,g_{i,\varepsilon}^{\prime\prime}(t)\,g_{i,\varepsilon}(t).$

###### Proof.

The proofs of (2.5) and (2.7) are similar to those of (2.2) and (2.4)
respectively and we omit them. In order to prove (2.6), we use the convexity
of the map $\tau\mapsto|\tau|^{p_{i}/2}$. This implies

(2.8)
$g_{i,\varepsilon}(t)=\frac{1}{p_{i}}\,(\varepsilon+t^{2})^{\frac{p_{i}}{2}}\leq\frac{2^{\frac{p_{i}-2}{2}}}{p_{i}}\,\left(\varepsilon^{\frac{p_{i}}{2}}+|t|^{p_{i}}\right).$

We then observe that

$|t|^{p_{i}}=|t|^{p_{i}-2}\,t^{2}\leq(\varepsilon+t^{2})^{\frac{p_{i}-2}{2}}\,t^{2}=g_{i,\varepsilon}^{\prime}(t)\,t.$

By combining the last two inequalities, we get (2.6). ∎

We also define the function

(2.9)
$G_{\varepsilon}(z):=\sum_{i=1}^{N}g_{i,\varepsilon}(z_{i})=\sum_{i=1}^{N}\frac{1}{p_{i}}(\varepsilon+z_{i}^{2})^{\frac{p_{i}}{2}},$

which will play a crucial role in our estimates. The next result holds without
any restriction on $p_{i}$.

######  Lemma 2.3.

For every $z=(z_{1},\dots,z_{N})\in\mathbb{R}^{N}$ and every $i=1,\dots,N$, we
have

(2.10) $|g^{\prime}_{i,\varepsilon}(z_{i})|\leq
p_{N}^{\frac{p_{N}-1}{p_{N}}}\,G_{\varepsilon}(z)^{\frac{p_{i}-1}{p_{i}}}.$

###### Proof.

By recalling the definition of both $g_{i,\varepsilon}$ and $G_{\varepsilon}$,
we have

$\begin{split}|g^{\prime}_{i,\varepsilon}(z_{i})|\leq(\varepsilon+|z_{i}|^{2})^{\frac{p_{i}-1}{2}}&=\big{(}p_{i}\,g_{i,\varepsilon}(z_{i})\big{)}^{\frac{p_{i}-1}{p_{i}}}\leq\left(\sum_{k=1}^{N}p_{k}\,g_{k,\varepsilon}(z_{k})\right)^{\frac{p_{i}-1}{p_{i}}}\leq
p_{N}^{\frac{p_{i}-1}{p_{i}}}\,G_{\varepsilon}(z)^{\frac{p_{i}-1}{p_{i}}}.\end{split}$

By using that $p_{N}>1$ and that $p_{i}\leq p_{N}$, we can estimate the last
term from above as claimed. ∎

###  2.2. Regularized problems

We will use an approximation scheme which is similar to that already used in
our previous papers, starting from [11, Section 2]. We want to consider local
minimizers of the following convex integral functional

$\mathfrak{F}_{\mathbf{p}}(u;\Omega^{\prime})=\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{\Omega^{\prime}}|u_{x_{i}}|^{p_{i}}\,dx-\int_{\Omega^{\prime}}f\,u\,dx,\qquad
u\in W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega),\
\Omega^{\prime}\Subset\Omega.$

The function $f$ is taken to belong to $L^{1}_{\rm loc}(\Omega)$. In the rest
of the paper, we fix $U\in W^{1,\mathbf{p}}_{\rm loc}(\Omega)\cap
L^{\infty}_{\rm loc}(\Omega)$ a bounded local minimizer of
$\mathfrak{F}_{\mathbf{p}}$. We also fix a ball

$B\Subset\Omega\quad\mbox{ such that }\quad 2\,B\Subset\Omega\mbox{ as well}.$

Here by $\lambda\,B$ we denote the ball concentric with $B$, scaled by a
factor $\lambda>0$. We set

$\varepsilon_{0}=\min\left\\{1,\frac{\mbox{ radius of }B}{2}\right\\}>0.$

For every $0<\varepsilon\leq\varepsilon_{0}$ and every $x\in\overline{B}$, we
then define

$U^{\varepsilon}(x)=U\ast\varrho_{\varepsilon}(x)\qquad\mbox{ and }\qquad
f^{\varepsilon}(x)=f\ast\varrho_{\varepsilon}(x).$

As usual, we denote by $\varrho_{\varepsilon}$ the usual family of Friedrichs
mollifiers, supported in a ball of radius $\varepsilon$ centered at the
origin. Finally, we set

$M:=\|U\|_{L^{\infty}(2\,B)},$

and take $\zeta^{\varepsilon}$ to be a $C^{\infty}$ function of one variable,
such that

$0\leq(\zeta^{\varepsilon})^{\prime}\leq 1\qquad\mbox{ and
}\qquad\zeta^{\varepsilon}(t)=\left\\{\begin{array}[]{cc}M,&\mbox{ if }t\geq
M+\varepsilon,\\\ -M,&\mbox{ if }t\leq-M-\varepsilon,\end{array}\right.$

and such that

$\lim_{\varepsilon\to
0}\|\zeta^{\varepsilon}-\zeta\|_{L^{\infty}(\mathbb{R})}=0,\qquad\mbox{ where
}\zeta(t)=\max\Big{\\{}\min\\{t,M\\},-M\Big{\\}},\mbox{ for }t\in\mathbb{R}.$

By recalling the definition (2.1) of $g_{i,\varepsilon}$, we then define the
regularized functional

$\mathfrak{F}_{\mathbf{p},\varepsilon}(v;B)=\sum_{i=1}^{N}\int_{B}g_{i,\varepsilon}(v_{x_{i}})\,dx-\int_{B}f^{\varepsilon}\,\zeta^{\varepsilon}(v)\,dx.$

######  Lemma 2.4 (Existence and regularity of a minimum for
$\mathfrak{F}_{\mathbf{p},\varepsilon}$).

For every $0<\varepsilon\leq\varepsilon_{0}$, the problem

(2.11)
$\min\left\\{\mathfrak{F}_{\mathbf{p},\varepsilon}(v;B)\,:\,v-U^{\varepsilon}\in
W^{1,\mathbf{p}}_{0}(B)\right\\},$

admits a solution $u^{\varepsilon}$, which belongs to
$C^{\infty}(\overline{B})$. Moreover, for every
$0<\varepsilon\leq\varepsilon_{0}$, we have

$\|u^{\varepsilon}\|_{L^{\infty}(B)}\leq M+\varepsilon.$

###### Proof.

We first show that we can apply [49, Theorem 9.2] and get existence of a
solution to

(2.12)
$\min\left\\{\mathfrak{F}_{\mathbf{p},\varepsilon}(v;B)\,:\,v-U^{\varepsilon}\in
W^{1,\infty}_{0}(B)\right\\}.$

For this, we check the required assumptions. We first claim that for every
$z,\xi\in\mathbb{R}^{N}$

(2.13)
$\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime\prime}(z_{i})\,\xi_{i}^{2}\geq\nu_{\varepsilon}\,(1+|z|^{2})^{\tau}\,|\xi|^{2},\qquad\mbox{
with
}\left\\{\begin{array}[]{l}\nu_{\varepsilon}:=\min\left\\{p_{1}-1,\varepsilon^{\frac{p_{N}-2}{2}}\right\\}>0,\\\
\\\
\tau:=\min\left\\{0,\dfrac{p_{1}-2}{2}\right\\}>-\dfrac{1}{2}.\end{array}\right.$

From (2.2) and (2.5) we get

$\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime\prime}(z_{i})\,\xi_{i}^{2}\geq\sum_{i=1}^{N}\min\\{p_{i}-1,1\\}\,(\varepsilon+z_{i}^{2})^{\frac{p_{i}-2}{2}}\,\xi_{i}^{2}.$

Then we observe that if $p_{i}\leq 2$, then $p_{1}\leq 2$ and we can write:

$\min\\{p_{i}-1,1\\}\,(\varepsilon+z_{i}^{2})^{\frac{p_{i}-2}{2}}\geq(p_{1}-1)\,(1+|z|^{2})^{\frac{p_{1}-2}{2}}=(p_{1}-1)\,(1+|z|^{2})^{\tau}.$

If instead $p_{i}>2$, then

$\min\\{p_{i}-1,1\\}\,(\varepsilon+z_{i}^{2})^{\frac{p_{i}-2}{2}}\geq\varepsilon^{\frac{p_{N}-2}{2}}\geq\,\varepsilon^{\frac{p_{N}-2}{2}}\,(1+|z|^{2})^{\tau}.$

Thus, in both cases, (2.13) holds.

As for the lower order term, observe that the smooth function
$h^{\varepsilon}(x,u):=f^{\varepsilon}(x)\,\zeta^{\varepsilon}(u)$ verifies

$\frac{\partial h^{\varepsilon}}{\partial u}(x,u)=0,\qquad\mbox{ for every
}(x,u)\in B\times(\mathbb{R}\setminus(-M-\varepsilon,M+\varepsilon)).$

Finally, the uniform convexity of $B$ and the smoothness of $U^{\varepsilon}$
entail that the latter satisfies the bounded slope condition.

Then [49, Theorem 9.2] yields the existence of a solution to (2.12). Since all
the data are smooth, [49, Theorem 9.3] implies that $u^{\varepsilon}\in
C^{\infty}(\overline{B})$. We claim that $u^{\varepsilon}$ is a solution of
(2.11), as well. Indeed, by using [13, Theorem 1.1], for every $v\in
U^{\varepsilon}+W^{1,\mathbf{p}}_{0}(B)$ we can infer the existence of
$\\{v^{k}\\}_{k\in\mathbb{N}}\subset U^{\varepsilon}+W^{1,\infty}_{0}(B)$
converging to $v$ in $W^{1,p_{1}}(B)$ and such that

$\lim_{k\to\infty}\sum_{i=1}^{N}\int_{B}g_{i,\varepsilon}(v^{k}_{x_{i}})\,dx=\sum_{i=1}^{N}\int_{B}g_{i,\varepsilon}(v_{x_{i}})\,dx.$

By the dominated convergence theorem and the uniform boundedness of
$\zeta^{\varepsilon}$, we also have

$\lim_{k\to\infty}\int_{B}f^{\varepsilon}\,\zeta^{\varepsilon}(v^{k})\,dx=\lim_{k\to\infty}\int_{B}f^{\varepsilon}\,\zeta^{\varepsilon}(v)\,dx.$

This proves that there is no Lavrentiev phenomenon for
$\mathfrak{F}_{\mathbf{p},\varepsilon}$, that is

$\mathfrak{F}_{\mathbf{p},\varepsilon}(u^{\varepsilon};B)=\min_{U^{\varepsilon}+W^{1,\infty}_{0}(B)}\mathfrak{F}_{\mathbf{p},\varepsilon}=\inf_{U^{\varepsilon}+W^{1,\mathbf{p}}_{0}(B)}\mathfrak{F}_{\mathbf{p},\varepsilon}.$

Thus, we get that $u^{\varepsilon}$ solves (2.11), as well. Finally, the
claimed $L^{\infty}$ estimate readily follows from Lemma A.1 in the Appendix.
∎

The smooth minimizer $u^{\varepsilon}$ satisfies the Euler-Lagrange equation

(2.14) $\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\varphi_{x_{i}}\,dx=\int
f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\varphi\,dx,\qquad\mbox{
for every }\varphi\in W^{1,\mathbf{p}}_{0}(B).$

For every $k=1,\dots,N$, one can insert test functions of the form
$\varphi_{x_{k}}$, with $\varphi\in C^{2}$ compactly supported in $B$. By
integrating by parts, we then get the equation for the partial derivatives of
$u^{\varepsilon}$

(2.15) $\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\varphi_{x_{i}}\,dx=\int\big{(}f^{\varepsilon}\,(\zeta^{\varepsilon})^{\prime}\,(u^{\varepsilon})\big{)}_{x_{k}}\,\varphi\,dx,\qquad\mbox{
for every }k=1,\dots,N.$

As usual, by a density argument, the equation can be tested by any $\varphi\in
W^{1,\mathbf{p}}_{0}(B)$.

The first ingredient of our recipe is a simple a priori estimate, which is
essentially the same as in [9, Lemma 2.1]: the only difference is the presence
of the non-autonomous and nonlinear term
$f^{\varepsilon}\,\zeta^{\varepsilon}(v)$, together with a slight modification
of the function $g_{i,\varepsilon}$.

######  Lemma 2.5 (Basic energy estimate).

For every $0<\varepsilon\leq\varepsilon_{0}$, the following uniform estimate
holds

$\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|u^{\varepsilon}_{x_{i}}|^{p_{i}}\,dx\leq
C\,\left(\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{2\,B}|U_{x_{i}}|^{p_{i}}\,dx+\varepsilon^{\frac{p_{1}}{2}}+\|f\|_{L^{1}(2\,B)}\right),$

for some $C=C(N,B,\mathbf{p},M)>0$.

###### Proof.

By testing the minimality of $u^{\varepsilon}$ against $U^{\varepsilon}$, we
obtain

$\begin{split}\sum_{i=1}^{N}\int_{B}g_{i,\varepsilon}(u^{\varepsilon}_{x_{i}})\,dx\leq\sum_{i=1}^{N}\int_{B}g_{i,\varepsilon}(U^{\varepsilon}_{x_{i}})\,dx-\int_{B}f^{\varepsilon}\,\big{(}\zeta^{\varepsilon}(U^{\varepsilon})-\zeta^{\varepsilon}(u^{\varepsilon})\big{)}\,dx.\end{split}$

The convexity of the function $g_{i,\varepsilon}$ allows to apply Jensen’s
inequality in connection with the fact that $U^{\varepsilon}$ is defined by a
convolution. This gives

$\int_{B}g_{i,\varepsilon}(U^{\varepsilon}_{x_{i}})\,dx\leq\int_{B}g_{i,\varepsilon}(U_{x_{i}})\ast\varrho_{\varepsilon}\,dx\leq\int_{2\,B}g_{i,\varepsilon}(U_{x_{i}})\,dx.$

By using also that $g_{i,\varepsilon}(t)\geq|t|^{p_{i}}/p_{i}$ and the
$1-$Lipschitz character of $\zeta^{\varepsilon}$, we get

$\begin{split}\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|u^{\varepsilon}_{x_{i}}|^{p_{i}}\,dx&\leq\sum_{i=1}^{N}\int_{2\,B}g_{i,\varepsilon}(U_{x_{i}})\,dx+\int_{B}|f^{\varepsilon}|\,|U^{\varepsilon}-u^{\varepsilon}|\,dx\\\
&\leq\sum_{i=1}^{N}\int_{2\,B}g_{i,\varepsilon}(U_{x_{i}})\,dx+C_{M}\,\|f\|_{L^{1}(2B)},\end{split}$

where $C_{M}$ is a positive constant which only depends on $M$. We finally
rely on (2.8) when $p_{i}>2$ or the subadditivity of
$t\mapsto|t|^{\frac{p_{i}}{2}}$ when $p_{i}\leq 2$ to obtain

(2.16)
$g_{i,\varepsilon}(t)\leq\frac{\max\left\\{1,2^{\frac{p_{i}-2}{2}}\right\\}}{p_{i}}\,({\varepsilon}^{\frac{p_{i}}{2}}+|t|^{p_{i}}).$

This concludes the proof. ∎

In view of our scopes, it is mandatory to have a convergence result for the
minimizers $\\{u^{\varepsilon}\\}_{0<\varepsilon<\varepsilon_{0}}$. This is
the content of the next lemma, which is an extension of [9, Lemma 2.2].

######  Lemma 2.6 (Convergence to a minimizer).

With the same notation as above, we have

(2.17) $\lim_{\varepsilon\to
0}\left[\|u^{\varepsilon}-U\|_{L^{q}(B)}+\sum_{i=1}^{N}\|(u^{\varepsilon}-U)_{x_{i}}\|_{L^{p_{i}}(B)}\right]=0,$

for every $1\leq q<\infty$.

###### Proof.

The proof goes as in [9, Lemma 2.2]. We repeat the argument, since this gives
us the occasion to fix some missing details in [9]. By using the uniform
estimate of Lemma 2.5 and the definition of $U^{\varepsilon}$, we get that
$\\{u^{\varepsilon}-U^{\varepsilon}\\}_{0<\varepsilon\leq\varepsilon_{0}}$ is
a bounded family in $W^{1,\mathbf{p}}_{0}(B)$. Thanks to Lemma 2.4, we also
have that
$\\{u^{\varepsilon}-U^{\varepsilon}\\}_{0<\varepsilon\leq\varepsilon_{0}}$ is
a bounded family in $L^{\infty}(B)$. From those two facts, we can infer the
existence of an infinitesimal sequence
$\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}\subset(0,\varepsilon_{0}]$ such that
$\\{u^{\varepsilon_{k}}-U^{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ converges
weakly in $W_{0}^{1,\mathbf{p}}(B)$ and almost everywhere to a function
$\phi\in W_{0}^{1,\mathbf{p}}(B)\cap L^{\infty}(B)$, in the sense that

$\lim_{k\to\infty}\int_{B}(u^{\varepsilon_{k}}-U^{\varepsilon_{k}})\,\varphi\,dx=\int_{B}\phi\,\varphi\,dx,\qquad\mbox{
for every }\varphi\in L^{1}(B),$

and

$\lim_{k\to\infty}\int_{B}(u^{\varepsilon_{k}}-U^{\varepsilon_{k}})_{x_{i}}\,\varphi\,dx=\int_{B}\phi_{x_{i}}\,\varphi\,dx,\qquad\mbox{
for every }\varphi\in L^{p_{i}^{\prime}}(B),i=1,\dots,N.$

By recalling that $U^{\varepsilon_{k}}$ has been constructed by convolution,
we also have that it converges strongly in $W^{1,\mathbf{p}}(B)$ and almost
everywhere to $U$. This permits to conclude that
$\\{u^{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ converges weakly and almost
everywhere to $u:=\phi+U$. We need to prove that actually $u=U$. With this
aim, we test the minimality of each $u^{\varepsilon_{k}}$ against the function
$U^{\varepsilon_{k}}$. Thus, by lower semicontinuity of the $L^{p_{i}}$ norms,
we can infer

(2.18)
$\begin{split}\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|u_{x_{i}}|^{p_{i}}\,dx&\leq\liminf_{k\to\infty}\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|u^{\varepsilon_{k}}_{x_{i}}|^{p_{i}}\,dx\\\
&\leq\liminf_{k\to\infty}\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}g_{i,\varepsilon_{k}}(u^{\varepsilon_{k}}_{x_{i}})\,dx\\\
&\leq\lim_{k\to\infty}\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}g_{i,\varepsilon_{k}}(U^{\varepsilon_{k}}_{x_{i}})\,dx-\int_{B}f^{\varepsilon_{k}}\,\big{(}\zeta^{\varepsilon_{k}}(U^{\varepsilon_{k}})-\zeta^{\varepsilon_{k}}(u^{\varepsilon_{k}})\big{)}\,dx\\\
&=\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|U_{x_{i}}|^{p_{i}}\,dx-\int_{B}f\,(\zeta(U)-\zeta(u))\,dx.\end{split}$

Observe that for the convergence of the lower order term, we used that
$f^{\varepsilon_{k}}$ converges strongly in $L^{1}(B)$, that
$U^{\varepsilon_{k}}$ and $u^{\varepsilon_{k}}$ are equibounded in
$L^{\infty}(B)$ and converge almost everywhere to $U$ and $u$ respectively and
that $\zeta^{\varepsilon_{k}}$ converges uniformly to the Lipschitz function
$\zeta$. This shows that

$\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|u_{x_{i}}|^{p_{i}}\,dx-\int_{\Omega^{\prime}}f\,\zeta(u)\,dx\leq\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}|U_{x_{i}}|^{p_{i}}\,dx-\int_{\Omega^{\prime}}f\,\zeta(U)\,dx.$

The $L^{\infty}$-boundedness of $u^{\varepsilon}$ proved in Lemma 2.4, gives
that $\|u\|_{L^{\infty}(B)}\leq M$. A similar estimate holds for $U$ by
assumption. Since $\zeta(t)=t$ for every $t\in[-M,M]$, one gets

$\mathfrak{F}_{\mathbf{p}}(u;B)\leq\mathfrak{F}_{\mathbf{p}}(U;B).$

By the strict convexity of the functional $\mathfrak{F}_{\mathbf{p}}$, the
minimizer must be unique and thus we get $u=U$, as desired.

In order to prove (2.17), we can adapt the argument of [9, Lemma 2.2]. By
(2.18) we get

(2.19)
$\lim_{k\to\infty}\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}\left|u^{\varepsilon_{k}}_{x_{i}}\right|^{p_{i}}\,dx=\sum_{i=1}^{N}\frac{1}{p_{i}}\,\int_{B}\left|U_{x_{i}}\right|^{p_{i}}\,dx.$

For every $i=1,\dots,N$, we rely on the lower semicontinuity of the
$L^{p_{i}}$ norm to get

$\liminf_{k\to\infty}\int_{B}\left|u^{\varepsilon_{k}}_{x_{i}}\right|^{p_{i}}\,dx\geq\int_{B}\left|U_{x_{i}}\right|^{p_{i}}\,dx,$

In connection with (2.19), this implies that

$\lim_{k\to\infty}\int_{B}\left|u^{\varepsilon_{k}}_{x_{i}}\right|^{p_{i}}\,dx=\int_{B}\left|U_{x_{i}}\right|^{p_{i}}\,dx.$

The convergence of the norms, in conjunction with the weak convergence,
permits to infer that $(u^{\varepsilon_{k}}_{x_{i}})_{k\in\mathbb{N}}$
converges to $U_{x_{i}}$ in $L^{p_{i}}(B)$ for every $i=1,\dots,N$ (see for
example [39, Theorem 2.11]).

Moreover, since $\\{u^{{\varepsilon}_{k}}\\}_{k\in\mathbb{N}}$ is bounded by
$M$ and converges almost everywhere in $B$ to $U$, the dominated convergence
theorem implies that $(u^{\varepsilon_{k}})_{k\in\mathbb{N}}$ converges to $U$
in $L^{q}(B)$ for every $1\leq q<\infty$.

Finally, we observe that we can repeat this argument with any subsequence of
the original family $\\{u^{\varepsilon}\\}_{\varepsilon>0}$. Thus the above
limit holds true for the whole family
$\\{u^{\varepsilon}\\}_{0<\varepsilon\leq\varepsilon_{0}}$ instead of
$\\{u^{\varepsilon_{k}}\\}_{k\in\mathbb{N}}$ and (2.17) follows. ∎

The following technical result is classical in the Regularity Theory. This is
taken from [34, Lemma 6.1] and we state it here for the reader’s convenience.

######  Lemma 2.7.

Let $0<r<R$ and let $Z:[r,R]\to[0,\infty)$ be a bounded function. Assume that
for $r\leq s<t\leq R$ we have

$Z(s)\leq\frac{\mathcal{A}}{(t-s)^{\alpha_{0}}}+\frac{\mathcal{B}}{(t-s)^{\beta_{0}}}+\mathcal{C}+\vartheta\,Z(t),$

with $\mathcal{A},\mathcal{B},\mathcal{C}\geq 0$, $\alpha_{0}\geq\beta_{0}>0$
and $0\leq\vartheta<1$. Then we have

$Z(r)\leq\left(\frac{1}{(1-\lambda)^{\alpha_{0}}}\,\frac{\lambda^{\alpha_{0}}}{\lambda^{\alpha_{0}}-\vartheta}\right)\,\left[\frac{\mathcal{A}}{(R-r)^{\alpha_{0}}}+\frac{\mathcal{B}}{(R-r)^{\beta_{0}}}+\mathcal{C}\right],$

where $\lambda$ is any number such that
$\vartheta^{\frac{1}{\alpha_{0}}}<\lambda<1$.

##  3\. Caccioppoli–type inequalities for the gradient

Throughout this section, we will assume that $1<p_{1}\leq\dots\leq p_{N}$,
without any further restriction. In what follows, we will use the following
function

(3.1)
$\mathcal{G}_{\varepsilon}(z)=\Big{(}(G_{\varepsilon}(z)-1)_{+}+1\Big{)},\qquad\mbox{
for every }z\in\mathbb{R}^{N},$

where $G_{\varepsilon}$ is the same function as in (2.9).

######  Proposition 3.1 (Caccioppoli inequality for power functions of the
gradient).

Let $1<p_{1}\leq p_{2}\leq\dots\leq p_{N}<\infty$. For every $\alpha\geq 0$
and every non-negative $\eta\in C^{2}_{0}(B)$, we have

(3.2) $\begin{split}\sum_{i=1}^{N}\int
g^{\prime\prime}_{i,\varepsilon}(u_{x_{i}}^{\varepsilon})&\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx\\\
&\leq C\,(\alpha+1)^{2}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,(\alpha+1)^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\eta^{2}\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$.

###### Proof.

We are going to use a trick based on integration by parts, taken from [46,
Theorem 1] (see also [31]). This permits to circumvent the use of the upper
bound on the Hessian of the function $G_{\varepsilon}$. We start by fixing
$k\in\\{1,\dots,N\\}$ and inserting in (2.15) the test function

$\varphi=F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2},$

where $F$ is a non-negative $C^{1}$ monotone non-decreasing function, that
will be specified later on. This is a feasible test function, thanks to the
regularity of $u^{\varepsilon}$. Thus we get

(3.3) $\begin{split}\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}&\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{i}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&+\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}_{x_{i}}\,\eta^{2}\,dx\\\
&=-2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta\,\eta_{x_{i}}\,dx\\\
&+\int(f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon}))_{x_{k}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx.\end{split}$

We observe that

$g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}=\Big{(}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\Big{)}_{x_{k}}.$

Then by integrating by parts on the right-hand side222This is the trick in
[46] (and [31]) mentioned above. of (3.3), we obtain

(3.4) $\begin{split}\sum_{i=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{i}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&+\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}_{x_{i}}\,\eta^{2}\,dx\\\
&=2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta\,\eta_{x_{i}}\,dx\\\
&+2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}_{x_{k}}\,\eta\,\eta_{x_{i}}\,dx\\\
&+2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})\,dx\\\
&+\int(f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon}))_{x_{k}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx.\end{split}$

This is valid for every $k=1,\dots,N$, we then take the sum over $k$.

On the left-hand side, the first term then becomes

$\begin{split}&\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{i}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&=\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{i}}\,\Big{(}g_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}_{x_{i}}\,\eta^{2}\,dx\\\
&=\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{i}}\,\left(\sum_{k=1}^{N}g_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\right)_{x_{i}}\,\eta^{2}\,dx\\\
&=\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{i}}\,\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\,\eta^{2}\,dx\\\ &=\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx.\end{split}$

For the second term of the left-hand side in (3.4), we observe that

$\begin{split}\sum_{i,k=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}_{x_{i}}\,\eta^{2}\,dx\\\
&=\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx,\end{split}$

and this is non-negative, since each $g_{k,\varepsilon}$ is convex and $F\geq
0$. We thus obtain

(3.5) $\begin{split}\sum_{i=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx\\\
&+\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&=2\,\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta\,\eta_{x_{i}}\,dx\\\
&+2\,\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}_{x_{k}}\,\eta\,\eta_{x_{i}}\,dx\\\
&+2\,\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})\,dx\\\
&+\sum_{k=1}^{N}\int(f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon}))_{x_{k}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx.\end{split}$

By (2.14), the second term of the right-hand side can be written as

(3.6) $-2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\eta\,\eta_{x_{i}}\,dx.$

By an integration by parts and (2.14) again, the last term on the right-hand
side of (3.5) is equal to

$\begin{split}-\sum_{k=1}^{N}&\int
f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&+\int|f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\eta^{2}\,dx\\\ &-2\,\sum_{k=1}^{N}\int
f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta\,\eta_{x_{k}}\,dx.\end{split}$

We observe that the third term in the above sum is equal to the quantity in
(3.6). Hence, (3.5) is equivalent to

(3.7) $\begin{split}\mathcal{I}_{1}+\mathcal{I}_{2}&:=\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx\\\
&+\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&=2\,\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta\,\eta_{x_{i}}\,dx\\\
&-4\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\eta\,\eta_{x_{i}}\,dx\\\
&+2\,\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})\,dx\\\
&-\sum_{k=1}^{N}\int
f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&+\int|f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\eta^{2}\,dx=:2\,\mathcal{I}_{3}+4\,\mathcal{F}_{1}+2\,\mathcal{I}_{4}+\mathcal{F}_{2}+\mathcal{F}_{3}.\end{split}$

We first estimate $\mathcal{I}_{3}$: by Young’s inequality, we have for every
$\tau>0$

$\begin{split}|\mathcal{I}_{3}|&\leq\int\sum_{k=1}^{N}\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)\,\left(F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{k}}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\right|\right)\,\eta\,|\nabla\eta|\,dx\\\
&\leq\frac{1}{2\,\tau}\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left(\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\right)|\nabla\eta|^{2}\\\
&+\frac{\tau}{2}\,\int\sum_{k=1}^{N}F^{\prime}(G(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{k}}\right|^{2}\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&\leq\frac{1}{2\,\tau}\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left(\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\right)|\nabla\eta|^{2}\,dx+\frac{\tau}{2}\,\mathcal{I}_{1}.\end{split}$

By taking $\tau=1/2$, we can absorb the term $\mathcal{I}_{1}$ on the right-
hand side and obtain from (3.7)

(3.8) $\begin{split}\frac{1}{2}\,\mathcal{I}_{1}+\mathcal{I}_{2}\leq
2\,&\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left(\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\right)|\nabla\eta|^{2}\,dx\\\
&+4\,\mathcal{F}_{1}+2\,\mathcal{I}_{4}+\mathcal{F}_{2}+\mathcal{F}_{3}.\end{split}$

The term

$\mathcal{I}_{4}=\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})\,dx,$

is easier to handle: we simply have

$\begin{split}|\mathcal{I}_{4}|\leq\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx.\end{split}$

In conclusion, from (3.8) we get

(3.9) $\begin{split}\mathcal{I}_{1}+2\,\mathcal{I}_{2}&\leq
4\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left(\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\right)|\nabla\eta|^{2}\,dx\\\
&+8\,\mathcal{F}_{1}+4\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+2\,\mathcal{F}_{2}+2\,\mathcal{F}_{3}.\end{split}$

We now treat the three terms containing $f^{\varepsilon}$: we start from

$\begin{split}\mathcal{F}_{1}&=-\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\eta\,\eta_{x_{i}}\,dx\\\
&\leq\int\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,|f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})|\,\eta\,|\nabla\eta|\,dx\\\
&\leq\frac{1}{2}\,\int\left(\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,|\nabla\eta|^{2}\,dx+\frac{1}{2}\,\int|f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\eta^{2}\,dx.\end{split}$

The last term coincides with $\frac{1}{2}\mathcal{F}_{3}$ while the first term
is bounded from above (up to a multiplicative constant) by the third term on
the right-hand side of (3.9) . Using also that
$\|(\zeta^{\varepsilon})^{\prime}\|_{L^{\infty}(\mathbb{R})}\leq 1$, we thus
get from (3.9)

(3.10) $\begin{split}\mathcal{I}_{1}+2\,\mathcal{I}_{2}&\leq
4\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left(\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\right)|\nabla\eta|^{2}\,dx\\\
&+8\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+6\,\int|f^{\varepsilon}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\eta^{2}\,dx+2\,\mathcal{F}_{2}.\end{split}$

The last term $\mathcal{F}_{2}$ contains second order derivatives of
$u^{\varepsilon}$ that should be absorbed on the left-hand side. We proceed
similarly as for $\mathcal{I}_{3}$ and estimate it as follows

$\begin{split}\mathcal{F}_{2}&=-\sum_{k=1}^{N}\int
f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\Big{(}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&\leq\sum_{k=1}^{N}\int|f^{\varepsilon}|\,\left(F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{k}}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\right|\right)\,\eta^{2}\,dx\\\
&\leq\frac{1}{2\,\tau}\,\sum_{k=1}^{N}\int|f^{\varepsilon}|^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\frac{\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\,\eta^{2}\,dx\\\
&+\frac{\tau}{2}\,\sum_{k=1}^{N}\int F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{k}}\right|^{2}\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&=\frac{1}{2\,\tau}\,\sum_{k=1}^{N}\int|f^{\varepsilon}|^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\frac{\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\,\eta^{2}\,dx+\frac{\tau}{2}\mathcal{I}_{1}.\end{split}$

thanks to Young’s inequality. Here as always $\tau>0$ is arbitrary. By
inserting this estimate in (3.10) and choosing $\tau=1/2$, we obtain

$\begin{split}\mathcal{I}_{1}+4\,\mathcal{I}_{2}&\leq
8\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left(\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\right)|\nabla\eta|^{2}\,dx\\\
&+16\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+12\,\int|f^{\varepsilon}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\eta^{2}\,dx+4\,\sum_{k=1}^{N}\int|f^{\varepsilon}|^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\frac{\Big{(}g^{\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\,\eta^{2}\,dx.\end{split}$

We observe that if we set

(3.11)
$\delta_{k}=\max\left\\{\frac{p_{k}}{p_{k}-1},p_{k}\right\\}\qquad\mbox{ and
}\qquad\overline{\delta}=\max_{k=1,\dots,N}\delta_{k}=\max\left\\{\frac{p_{1}}{p_{1}-1},p_{N}\right\\},$

by (2.4) or (2.7), we have

(3.12)
$\begin{split}\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\leq\sum_{k=1}^{N}\delta_{k}\,g_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})&\leq\overline{\delta}\,G_{\varepsilon}(\nabla
u^{\varepsilon}).\end{split}$

Thus we have obtained

(3.13) $\begin{split}\mathcal{I}_{1}+4\,\mathcal{I}_{2}&\leq
8\,\overline{\delta}\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,G_{\varepsilon}(\nabla
u^{\varepsilon})\,|\nabla\eta|^{2}\,dx\\\
&+16\,\int\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+12\,\int|f^{\varepsilon}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\eta^{2}\,dx+4\,\overline{\delta}\,\int|f^{\varepsilon}|^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,G_{\varepsilon}(\nabla
u^{\varepsilon})\,\eta^{2}\,dx.\end{split}$

We now use Lemma 2.3 to estimate from above the right-hand side. Thus, from
(3.13), we get

(3.14) $\begin{split}\mathcal{I}_{1}+4\mathcal{I}_{2}&\leq
8\,\overline{\delta}\,\left(p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\int\left(\sum_{i=1}^{N}G_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{i}-1}{p_{i}}}\right)^{2}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,G_{\varepsilon}(\nabla
u^{\varepsilon})\,|\nabla\eta|^{2}\,dx\\\
&+16\,\left(p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\int\left(\sum_{i=1}^{N}G_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{i}-1}{p_{i}}}\right)^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+12\,\overline{\delta}\,\int|f^{\varepsilon}|^{2}\,\Big{[}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))+F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{]}\,\eta^{2}\,dx.\end{split}$

By recalling the definition (3.1) of $\mathcal{G}_{\varepsilon}$, we observe
that $\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\geq{G}_{\varepsilon}(\nabla u^{\varepsilon})$ and
$\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})\geq 1$. Thus in particular,
we get

$\sum_{i=1}^{N}G_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{i}-1}{p_{i}}}\leq
N\,\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})^{\frac{p_{N}-1}{p_{N}}},$

and from (3.14), we obtain

(3.15) $\begin{split}\mathcal{I}_{1}+4\mathcal{I}_{2}&\leq
8\,\overline{\delta}\,\left(N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{2\,\frac{p_{N}-1}{p_{N}}+1}\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,|\nabla\eta|^{2}\,dx\\\
&+16\,\left(N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{2\,\frac{p_{N}-1}{p_{N}}}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+12\,\overline{\delta}\,\int|f^{\varepsilon}|^{2}\,\Big{[}F(G_{\varepsilon}(\nabla
u^{\varepsilon}))+F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{]}\,\eta^{2}\,dx.\end{split}$

In order to conclude, we now make the choice

$F(t)=\big{(}(t-1)_{+}+1\big{)}^{\alpha},\qquad\mbox{ with }\alpha>0.$

We observe that

(3.16) $\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}\leq\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2},$

thanks to the definition of $\mathcal{G}_{\varepsilon}$.

It follows that

$\begin{split}F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}&\geq\alpha\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha-1}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}=\frac{4\,\alpha}{(\alpha+1)^{2}}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2},\end{split}$

and thus

$\begin{split}\mathcal{I}_{1}&=\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,F^{\prime}(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx\\\
&\geq\frac{4\,\alpha}{(\alpha+1)^{2}}\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx.\end{split}$

From (3.15) we get

(3.17)
$\begin{split}\frac{4\,\alpha}{(\alpha+1)^{2}}\,\sum_{i=1}^{N}\int&g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx+4\,\mathcal{I}_{2}\\\
&\leq C\,(\alpha+1)\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2\,\frac{p_{N}-1}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,(\alpha+1)\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\eta^{2}\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$. We are only left with estimating
$\mathcal{I}_{2}$ from below: recall that we have

$\begin{split}\mathcal{I}_{2}&=\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,F(G_{\varepsilon}(\nabla
u^{\varepsilon}))\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx\\\
&=\sum_{i,k=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,\eta^{2}\,dx.\end{split}$

We now observe that by (3.16) and through some lengthy though elementary
computations, we get

$\begin{split}\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{2}}\Big{)}_{x_{i}}\right|^{2}=\frac{1}{4\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon}\Big{)}_{x_{i}}\right|^{2}&\leq\frac{1}{4\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\,\left|\Big{(}G_{\varepsilon}(\nabla
u^{\varepsilon}\Big{)}_{x_{i}}\right|^{2}\\\
&=\frac{1}{4\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\,\left|\left(\sum_{k=1}^{N}g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\right)_{x_{i}}\right|^{2}\\\
&=\frac{1}{4\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\,\left|\sum_{k=1}^{N}g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,u^{\varepsilon}_{x_{k}\,x_{i}}\right|^{2}\\\
&\leq\frac{N}{4\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\,\sum_{k=1}^{N}|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|^{2}\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}.\end{split}$

We then apply (2.4) or (2.7) on the last term, so to get

$\begin{split}\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{2}}\Big{)}_{x_{i}}\right|^{2}&\leq\frac{N}{4\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\,\sum_{k=1}^{N}\delta_{k}\,g_{k,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{k}})\,g_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2},\end{split}$

where $\delta_{k}$ is the same quantity defined in (3.11). We further observe
that

$\frac{g_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}{\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})}\leq\frac{G_{\varepsilon}(\nabla
u^{\varepsilon})}{\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})}\leq 1.$

This discussion leads us to

$\frac{4}{N}\,\frac{1}{\overline{\delta}}\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{2}}\Big{)}_{x_{i}}\right|^{2}\leq\sum_{k=1}^{N}g_{k,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{k}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}.$

By inserting this inequality in $\mathcal{I}_{2}$, we get

$\begin{split}\mathcal{I}_{2}&\geq\frac{4}{N}\,\frac{1}{\overline{\delta}}\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{1}{2}}\Big{)}_{x_{i}}\right|^{2}\eta^{2}\,dx\\\
&=\frac{4}{(\alpha+1)^{2}}\,\frac{1}{N\,\overline{\delta}}\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx.\end{split}$

Finally, we can use this estimate in (3.17), so as to get the desired
conclusion for $\alpha>0$. The limit case $\alpha=0$ can now be simply
obtained by taking the limit $\alpha$ goes to $0$ in the previously obtained
estimate, since the relevant constant remains bounded. ∎

######  Proposition 3.2 (Slow Moser’s iteration).

Let $1<p_{1}\leq p_{2}\leq\dots\leq p_{N}<\infty$. For every $\vartheta\geq
2/p_{N}^{\prime}$ and every non-negative function $\eta\in C^{2}_{0}(B)$, we
have

(3.18) $\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta+\frac{2}{p_{N}}}\,\eta^{2}\,dx&\leq\frac{C^{\vartheta}}{\vartheta}\,\int\eta^{2}\,dx+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\vartheta^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C^{\vartheta}\,\vartheta^{\vartheta}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{\vartheta+\frac{2}{p_{N}}}\,\int|f^{\varepsilon}|^{\vartheta+\frac{2}{p_{N}}}\,\eta^{2}\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$.

###### Proof.

We start by taking $\beta\geq 1$ and writing

$\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&=\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\big{(}(G_{\varepsilon}(\nabla
u^{\varepsilon})-1)_{+}+1\big{)}\,\eta^{2}\,dx\\\
&\leq\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx+\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\sum_{k=1}^{N}g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,\eta^{2}\,dx.\end{split}$

We observe that if we set

$\sigma_{k}=\max\left\\{1,2^{\frac{p_{k}-2}{2}}\right\\}\qquad\mbox{ and
}\qquad\overline{\sigma}=\max_{k=1,\dots,N}\sigma_{k}=\max\left\\{1,2^{\frac{p_{N}-2}{2}}\right\\},$

using (2.3) or (2.6) on the second integral, we get

$\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx+\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\sum_{k=1}^{N}\sigma_{k}\frac{\varepsilon^{\frac{p_{k}}{2}}}{p_{k}}\,\eta^{2}\,dx\\\
&+\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\sum_{k=1}^{N}\sigma_{k}\frac{g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})}{p_{k}}\,u_{x_{k}}^{\varepsilon}\,\eta^{2}\,dx.\\\
\end{split}$

By recalling that $g_{k,\varepsilon}^{\prime}(t)\,t\geq 0$ and using that
$0<\varepsilon\leq 1$, we get

$\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx+\overline{\sigma}\,\frac{N}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\\\
&+\frac{\overline{\sigma}}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\sum_{k=1}^{N}g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,u_{x_{k}}^{\varepsilon}\,\eta^{2}\,dx.\\\
\end{split}$

On the last term, using product rule and equation (2.14), we obtain

$\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\left(1+\overline{\sigma}\,\frac{N}{p_{1}}\right)\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx-\frac{\overline{\sigma}}{p_{1}}\,\sum_{k=1}^{N}\int\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\Big{)}_{x_{k}}\,g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,u^{\varepsilon}\,\eta^{2}\,dx\\\
&-\frac{2\,\overline{\sigma}}{p_{1}}\,\sum_{k=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,u^{\varepsilon}\,\eta\,\eta_{x_{k}}\,dx\\\
&+\frac{\overline{\sigma}}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,u^{\varepsilon}\,\eta^{2}\,dx.\end{split}$

By using that $u^{\varepsilon}$ is bounded and that
$0\leq(\zeta^{\varepsilon})^{\prime}\leq 1$, we get

(3.19) $\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\left(1+\overline{\sigma}\,\frac{N}{p_{1}}\right)\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\\\
&+\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\sum_{k=1}^{N}\int\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\Big{)}_{x_{k}}\right|\,|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|\,\eta^{2}\,dx\\\
&+\frac{2\,\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\sum_{k=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|\,\eta\,|\eta_{x_{k}}|\,dx\\\
&+\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx.\end{split}$

Now, (2.10) together with the fact that $\mathcal{G}_{\varepsilon}\geq
G_{\varepsilon}$ and $\mathcal{G}_{\varepsilon}\geq 1$ entail

$\begin{split}&\frac{2\,\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\sum_{k=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|\,\eta\,|\eta_{x_{k}}|\,dx\\\
&\leq\frac{2\,\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{1}{p_{N}}}\,\eta\,|\nabla\eta|\,dx\\\
&\leq\tau\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx+\frac{1}{\tau}\,\left(N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,|\nabla\eta|^{2}\,dx,\end{split}$

where in the last inequality we applied Young’s inequality. By choosing
$\tau=1/2$, we can absorb the term containing
$\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})^{\beta+1}$ and get from
(3.19)

(3.20) $\begin{split}\frac{1}{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\left(1+\frac{\overline{\sigma}\,N}{p_{1}}\right)\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\\\
&+\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\sum_{k=1}^{N}\int\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\Big{)}_{x_{k}}\right|\,|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|\,\eta^{2}\,dx\\\
&+2\,\left(N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\left(\overline{\sigma}\,\frac{\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,|\nabla\eta|^{2}\,dx\\\
&+\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx.\end{split}$

For the second term on the right-hand side, we use Young’s inequality: for
every $\tau>0$,

$\begin{split}\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\sum_{k=1}^{N}\int&\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\Big{)}_{x_{k}}\right|\,|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|\,\eta^{2}\,dx\\\
&=\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\sum_{k=1}^{N}\beta\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta-1}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{k}}\right|\,|g^{\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})|\,\eta^{2}\,dx\\\
&\leq\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\frac{\beta^{2}}{2\,\tau}\,\sum_{k=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta-2}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}_{x_{k}}\right|^{2}\,g^{\prime\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,\eta^{2}\,dx\\\
&+\frac{\tau}{2}\,\sum_{k=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\,\eta^{2}\,dx\\\
&=\frac{2}{\tau}\,\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\sum_{k=1}^{N}\int\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\beta}{2}}\Big{)}_{x_{k}}\right|^{2}\,g^{\prime\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,\eta^{2}\,dx\\\
&+\frac{\tau}{2}\,\sum_{k=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\,\eta^{2}\,dx.\end{split}$

We also notice that by (3.12) we have

$\sum_{k=1}^{N}\frac{\Big{(}g_{k,\varepsilon}^{\prime}(u_{x_{k}}^{\varepsilon})\Big{)}^{2}}{g^{\prime\prime}_{k,\varepsilon}(u^{\varepsilon}_{x_{k}})}\leq\overline{\delta}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon}).$

Thus from (3.20), we get

$\begin{split}\frac{1}{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\left(1+\frac{\overline{\sigma}\,N}{p_{1}}\right)\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\\\
&+\frac{2}{\tau}\,\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\sum_{k=1}^{N}\int\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\beta}{2}}\Big{)}_{x_{k}}\right|^{2}\,g^{\prime\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,\eta^{2}\,dx\\\
&+\frac{\tau\,\overline{\delta}}{2}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\\\
&+2\,\left(N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,|\nabla\eta|^{2}\,dx\\\
&+\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx.\end{split}$

By choosing $\tau=1/(2\,\overline{\delta})$, we can absorb again the term
containing $\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})^{\beta+1}$ on
the right-hand side and obtain

$\begin{split}\frac{1}{4}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\left(1+\frac{\overline{\sigma}\,N}{p_{1}}\right)\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\\\
&+4\,\overline{\delta}\,\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\sum_{k=1}^{N}\int\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\beta}{2}}\Big{)}_{x_{k}}\right|^{2}\,g^{\prime\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,\eta^{2}\,dx\\\
&+2\,\left(N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\right)^{2}\,\left(\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\right)^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,|\nabla\eta|^{2}\,dx\\\
&+\frac{\overline{\sigma}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}}{p_{1}}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx.\end{split}$

On the right-hand side, we now use the Caccioppoli inequality (3.2) with
$\alpha=\beta-1\geq 0$, so to get

$\begin{split}\sum_{k=1}^{N}\int
g^{\prime\prime}_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\beta}{2}}\Big{)}_{x_{k}}\right|^{2}\,\eta^{2}\,dx&\leq
C\,\beta^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,\beta^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta-1}\,\eta^{2}\,dx,\\\ \end{split}$

for some $C=C(N,p_{1},p_{N})>0$. This finally gives

$\begin{split}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq
C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\beta^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\beta^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta-1}\,\eta^{2}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$. On the first term on the right-hand side, we
can use Young’s inequality

$C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,\eta^{2}\,dx\leq\frac{\tau\,\beta}{\beta+1}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx+\frac{C^{\beta+1}}{\tau^{\beta}\,(\beta+1)}\,\int\eta^{2}\,dx.$

By choosing $\tau=1/2$, we can re-absorb the term
$\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})^{\beta+1}$. This gives

$\begin{split}\frac{1}{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx&\leq\frac{2^{\beta}\,C^{\beta+1}}{\beta+1}\,\int\eta^{2}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\beta^{2}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\beta^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta-1}\,\eta^{2}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx.\end{split}$

We proceed in a similar way for the two terms containing $f^{\varepsilon}$. By
using Young’s inequality with exponents
$\displaystyle{\frac{\beta+1}{\beta-1},\ \frac{\beta+1}{2}}$ and
$\displaystyle{\frac{\beta+1}{\beta},\ \beta+1}$, respectively, we get

$\begin{split}C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\beta^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta-1}\,\eta^{2}\,dx&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta}\,|f^{\varepsilon}|\,\eta^{2}\,dx\\\
&\leq\tau\,\frac{\beta-1}{\beta+1}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx\\\
&+\frac{2\,\tau^{\frac{1-\beta}{2}}\,C^{\frac{\beta+1}{2}}}{\beta+1}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{\beta+1}\,\beta^{\beta+1}\,\int|f^{\varepsilon}|^{\beta+1}\eta^{2}\,dx\\\
&+\tau\,\frac{\beta}{\beta+1}\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\beta+1}\,\eta^{2}\,dx\\\
&+\frac{C^{\beta+1}}{\tau^{\beta}\,(\beta+1)}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{\beta+1}\,\int|f^{\varepsilon}|^{\beta+1}\,\eta^{2}\,dx.\end{split}$

By choosing $\tau=1/8$, we can absorb again the terms containing the power
$\beta+1$ of $\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})$. This finally
leads to the estimate (3.18), up to renaming $\vartheta=\beta+1-2/p_{N}$. This
concludes the proof. ∎

##  4\. Uniform higher integrability

In this section, we establish a higher integrability estimate for $\nabla
u^{\varepsilon}$, which will eventually lead to the result of Proposition 1.1.
We assume throughout the section that $1<p_{1}\leq\dots\leq p_{N}$, without
any further restriction.

######  Proposition 4.1.

Let $1<p_{1}\leq p_{2}\leq\dots\leq p_{N}<\infty$. Then for every
$B_{r}\Subset B_{R}\Subset B$ and every $\gamma\geq 2$, we have

$\begin{split}\int_{B_{r}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx&\leq\Gamma_{1}+\Gamma_{2}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx,\end{split}$

for two constants $\Gamma_{1},\Gamma_{2}>0$ depending on

$N,\,p_{N},\,p_{1},\,\gamma,\,R,\,R-r,\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}\mbox{
and }\|u^{\varepsilon}\|_{L^{\infty}(B)}.$

###### Proof.

We take $\gamma\geq 2$ and define the sequence of exponents

$\vartheta_{0}=\frac{2}{p_{N}^{\prime}},\qquad\vartheta_{i+1}=\vartheta_{i}+\frac{2}{p_{N}}=\frac{2}{p_{N}^{\prime}}+(i+1)\,\frac{2}{p_{N}}=2+i\,\frac{2}{p_{N}},\
\mbox{ for }i\in\mathbb{N}.$

We set

(4.1)
$i_{0}=\max\left\\{i\in\mathbb{N}\,:\,i\leq\frac{p_{N}}{2}\,\left(\gamma-2\right)\right\\}.$

This in particular implies that

$\vartheta_{i_{0}+1}\leq\gamma<\vartheta_{i_{0}+2}.$

We now need to distinguish various cases, according to the values of $p_{N}$
and $\gamma$. Case A.1. Here we assume that

(4.2) $p_{N}\leq 2\qquad\mbox{ and
}\qquad\frac{p_{N}}{2}\,(\gamma-2)\in\mathbb{N}.$

This is the simplest case: we get the estimate by iterating Proposition 3.2
with exponents $\vartheta=\vartheta_{i}$ and a suitable sequence of shrinking
balls.

More precisely, we fix $B_{r}$ and $B_{R}$ as in the statement and define the
sequence of decreasing radii

$r_{i}=R-i\,\frac{R-r}{i_{0}+1},\qquad\mbox{ for }i=0,\dots,i_{0}+1.$

Accordingly, we take a cut-off function $\eta_{i}\in C^{2}_{0}(B_{r_{i}})$ for
$i=0,\dots,i_{0}$, such that

$0\leq\eta_{i}\leq 1,\qquad\eta_{i}\equiv 1\mbox{ on
}B_{r_{i+1}},\qquad|\nabla\eta_{i}|^{2}+|D^{2}\eta_{i}|\leq\frac{C\,(i_{0}+1)^{2}}{(R-r)^{2}}.$

By applying (3.18) with $\vartheta=\vartheta_{i}$, $\eta=\eta_{i}$ and using
the properties of the cut-off function, we get

$\begin{split}\int_{B_{r_{i+1}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i+1}}\,dx&\leq\frac{C^{\vartheta_{i}}}{\vartheta_{i}}\,|B_{R}|+\frac{C\,\vartheta_{i}^{2}\,(i_{0}+1)^{2}}{(R-r)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\,\int_{B_{r_{i}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i}}\,dx\\\
&+C^{\vartheta_{i}}\,\vartheta_{i}^{\vartheta_{i}}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{\vartheta_{i+1}}\,\int_{B_{r_{i}}}|f^{\varepsilon}|^{\vartheta_{i+1}}\,dx,\end{split}$

for a constant $C=C(N,p_{1},p_{N})>0$. By using Hölder’s inequality on the
right-hand side, we also get

(4.3) $\begin{split}\int_{B_{r_{i+1}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i+1}}\,dx&\leq\frac{C^{\vartheta_{i}}}{\vartheta_{i}}\,|B_{R}|+\frac{C\,\vartheta_{i}^{2}\,(i_{0}+1)^{2}}{(R-r)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\,\int_{B_{r_{i}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i}}\,dx\\\
&+C^{\vartheta_{i}}\,\vartheta_{i}^{\vartheta_{i}}\,|B_{R}|^{1-\frac{\vartheta_{i+1}}{\gamma}}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{\vartheta_{i+1}}\,\left(\int_{B_{R}}|f^{\varepsilon}|^{\gamma}\,dx\right)^{\frac{\vartheta_{i+1}}{\gamma}}.\end{split}$

Starting from $i=0$ and iterating (4.3) from $0$ to $i_{0}$, we get

(4.4) $\begin{split}\int_{B_{r}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i_{0}+1}}\,dx&\leq
M+D_{0}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx.\end{split}$

where we set for notational simplicity for every natural number $0\leq k\leq
i_{0}$

$D_{k}=\left[\prod_{i=k}^{i_{0}}\frac{C\,\vartheta_{i}^{2}\,(i_{0}+1)^{2}}{(R-r)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\right]=\left[\frac{C\,(i_{0}+1)^{2}}{(R-r)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\right]^{i_{0}-k+1}\,\prod_{i=k}^{i_{0}}\vartheta_{i}^{2},$

while

$M=\sum_{i=0}^{i_{0}}\left(\frac{C^{\vartheta_{i}}}{\vartheta_{i}}\,|B_{R}|+C^{\vartheta_{i}}\,\vartheta_{i}^{\vartheta_{i}}\,|B_{R}|^{1-\frac{\vartheta_{i+1}}{\gamma}}\,\left(\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\|f\|_{L^{\gamma}(B_{R})}\right)^{\vartheta_{i+1}}\right)\,D_{i+1},$

with the notational agreement that $D_{i_{0}+1}=1$. Their precise expression
is not very important, but we point out that $D_{0}$ and $M$ depend only on

$N,p_{N},p_{1},\gamma,R,R-r,\|f_{\varepsilon}\|_{L^{\gamma}(B_{R})}\qquad\mbox{
and }\qquad\|u_{\varepsilon}\|_{L^{\infty}(B)}.$

Thanks to the assumption (4.2) and recalling that
$\mathcal{G}_{\varepsilon}\geq 1$, we get

$\vartheta_{i_{0}+1}=\gamma\qquad\mbox{ and
}\qquad\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx\leq\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx,$

thus the desired conclusion follows from (4.4). Case A.2. Here we assume that

$p_{N}\leq 2\qquad\mbox{ and
}\qquad\frac{p_{N}}{2}\,(\gamma-2)\not\in\mathbb{N}.$

In light of the second assumption, we have

$\vartheta_{i_{0}+1}<\gamma<\vartheta_{i_{0}+2},$

where $i_{0}$ is the index defined in (4.1). Thus in this case, we need an
extra step of the iteration, by suitably adapting the choice of the exponent
$\vartheta$. We take $B_{r}\Subset B_{\varrho}\Subset B_{R}$ where

$\varrho=\frac{R+r}{2},$

and define the sequence of decreasing radii

$r_{i}=R-i\,\frac{R-\varrho}{i_{0}+1},\qquad\mbox{ for }i=0,\dots,i_{0}+1.$

We take a cut-off function $\eta_{i}\in C^{2}_{0}(B_{r_{i}})$ for
$i=0,\dots,i_{0}$, such that

$0\leq\eta_{i}\leq 1,\qquad\eta_{i}\equiv 1\mbox{ on
}B_{r_{i+1}},\qquad|\nabla\eta_{i}|^{2}+|D^{2}\eta_{i}|\leq\frac{C\,(i_{0}+1)^{2}}{(R-\rho)^{2}}.$

By proceeding as above, we now get

(4.5) $\begin{split}\int_{B_{\varrho}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i_{0}+1}}\,dx&\leq
M+D_{0}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx.\end{split}$

The last term can be estimated from above, by using again that
$2/p_{N}^{\prime}\leq 1$. However, in order to reach the desired exponent
$\gamma$, we need to apply (3.18) once more. We take a cut-off function
$\eta\in C^{2}_{0}(B_{\varrho})$ such that

$0\leq\eta\leq 1,\qquad\eta\equiv 1\mbox{ on
}B_{r},\qquad|\nabla\eta|^{2}+|D^{2}\eta|\leq\frac{C\,(i_{0}+1)^{2}}{(\varrho-r)^{2}}.$

By applying (3.18) with333Observe that such a choice is feasible, since
$\gamma-\frac{2}{p_{N}}\geq\frac{2}{p_{N}^{\prime}}\qquad\Longleftrightarrow\qquad\gamma\geq
2.$

$\vartheta=\gamma-\frac{2}{p_{N}},$

and the cut-off function above, we get

$\begin{split}\int_{B_{r}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx&\leq
C\,|B_{R}|^{N}+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\int_{B_{\varrho}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma-\frac{2}{p_{N}}}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{\gamma}\,\int_{B_{R}}|f^{\varepsilon}|^{\gamma}\,dx.\end{split}$

On the right hand side, the term containing $\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})$ is under control, since by construction

$\gamma-\frac{2}{p_{N}}<\vartheta_{i_{0}+2}-\frac{2}{p_{N}}=\vartheta_{i_{0}+1}.$

Thus we have

$\int_{B_{\varrho}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma-\frac{2}{p_{N}}}\,dx\leq\int_{B_{\varrho}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\vartheta_{i_{0}+1}}\,dx,$

and the last term can be estimated by (4.5). Case B. Here we assume that
$p_{N}>2$. The proof goes exactly as before, so, for every $r<R$ with
$B_{R}\Subset B$ we certainly have

$\begin{split}\int_{B_{r}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\gamma}\,dx&\leq\Gamma_{1}+\Gamma_{2}\,\int_{B_{\frac{R+r}{2}}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx,\end{split}$

but now the major difference is that we need to estimate the integral on the
right hand side. Indeed, in this case $2/p_{N}^{\prime}>1$ and we can not
directly assure that this term is bounded, uniformly in $\varepsilon$. We need
to use an interpolation trick to get a reverse $L^{2/p_{N}^{\prime}}-L^{1}$
estimate on $\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})$. We denote
$\varrho=(R+r)/2$ and we observe that

$1<\frac{2}{p_{N}^{\prime}}<2,$

thus by interpolation in Lebesgue spaces, we get for every $\varrho\leq
s<t\leq R$

(4.6) $\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx\leq\left(\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{2}\,dx\right)^{\frac{p_{N}-2}{p_{N}}}.$

The $L^{2}$ norm on the right-hand side can in turn be estimated by means of
(3.18), observing that

$2=\frac{2}{p_{N}^{\prime}}+\frac{2}{p_{N}}.$

By taking $\vartheta=2/p_{N}^{\prime}$ and a cut-off function $\eta\in
C^{2}_{0}(B_{t})$ such that

$0\leq\eta\leq 1,\qquad\eta\equiv 1\mbox{ on
}B_{s},\qquad|\nabla\eta|^{2}+|D^{2}\eta|\leq\frac{C}{(t-s)^{2}},$

we thus get from (3.18)

$\begin{split}\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{2}\,dx&\leq
C\,|B_{R}|+\frac{C}{(t-s)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\,\int_{B_{t}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx\\\
&+C\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\int_{B_{R}}|f^{\varepsilon}|^{2}\,dx\\\
&\leq
C\,|B_{R}|+\frac{C}{(t-s)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\,\int_{B_{t}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx\\\
&+C\,|B_{R}|^{1-\frac{2}{\gamma}}\,\|u^{\varepsilon}\|_{L^{\infty}(B)}^{2}\,\left(\int_{B_{R}}|f^{\varepsilon}|^{\gamma}\,dx\right)^{\frac{2}{\gamma}}.\end{split}$

for $C=C(N,p_{1},p_{N})>0$. We spend this information into (4.6) and use the
subadditivity of $\tau\mapsto\tau^{(p_{N}-2)/p_{N}}$, so to get

$\begin{split}\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx&\leq\left(\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(C\,|B_{R}|\right)^{\frac{p_{N}-2}{p_{N}}}\\\
&+\left(\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(\frac{C}{(t-s)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\right)^{\frac{p_{N}-2}{p_{N}}}\,\left(\int_{B_{t}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx\right)^{\frac{p_{N}-2}{p_{N}}}\\\
&+\left(\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(C\,|B_{R}|^{1-\frac{2}{\gamma}}\,\left(\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}\right)^{2}\right)^{\frac{p_{N}-2}{p_{N}}}.\end{split}$

For the second term on the right-hand side, we apply Young’s inequality with
conjugate exponents $p_{N}/2$ and $p_{N}/(p_{N}-2)$. We get

$\begin{split}\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx&\leq\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(C\,|B_{R}|\right)^{\frac{p_{N}-2}{p_{N}}}\\\
&+\frac{2}{p_{N}}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\,\left(\frac{C}{(t-s)^{2}}\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\right)^{\frac{p_{N}-2}{2}}\\\
&+\frac{p_{N}-2}{p_{N}}\,\int_{B_{t}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx\\\
&+\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(C\,|B_{R}|^{1-\frac{2}{\gamma}}\,\left(\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}\right)^{2}\right)^{\frac{p_{N}-2}{p_{N}}}.\end{split}$

We now use Lemma 2.7 with the choices

$Z(s)=\int_{B_{s}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx,\qquad\vartheta=\frac{p_{N}-2}{p_{N}},\qquad\mathcal{A}=\frac{2}{p_{N}}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\,\left(C\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\right)^{\frac{p_{N}-2}{2}},$
$\alpha_{0}=p_{N}-2,\qquad\mathcal{B}=0,$

and

$\mathcal{C}=\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left[\left(C\,|B_{R}|\right)^{\frac{p_{N}-2}{p_{N}}}+\left(C\,|B_{R}|^{1-\frac{2}{\gamma}}\,\left(\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}\right)^{2}\right)^{\frac{p_{N}-2}{p_{N}}}\right],$

in order to absorb the penultimate integral. This permits to conclude that

$\begin{split}\int_{B_{\varrho}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2}{p_{N}^{\prime}}}\,dx&\leq\widetilde{C}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(C\,|B_{R}|\right)^{\frac{p_{N}-2}{p_{N}}}\\\
&+\widetilde{C}\,\frac{2}{p_{N}}\,\frac{1}{(R-r)^{p_{N}-2}}\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\,\left(C\,\|u^{\varepsilon}\|^{2}_{L^{\infty}(B)}\right)^{\frac{p_{N}-2}{2}}\\\
&+\widetilde{C}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,dx\right)^{\frac{2}{p_{N}}}\,\left(|B_{R}|^{1-\frac{2}{\gamma}}\,\left(\|u^{\varepsilon}\|_{L^{\infty}(B)}\,\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}\right)^{2}\right)^{\frac{p_{N}-2}{p_{N}}}.\end{split}$

This gives the claimed $L^{2/p_{N}^{\prime}}-L^{1}$ estimate on
$\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})$. The desired conclusion
now easily follows. We leave the details to the reader. ∎

######  Remark 4.2 (Quality of the constants).

For future references, it is important to notice that the two constants
$\Gamma_{1}$ and $\Gamma_{2}$ in the previous statement are uniformly bounded
from above, whenever there exists a constant $C\geq 1$ such that

$\|f^{\varepsilon}\|_{L^{\gamma}(B_{R})}+\|u^{\varepsilon}\|_{L^{\infty}(B)}\leq
C,\qquad\mbox{ for every }0<\varepsilon\leq\varepsilon_{0},$

and

$R-r\geq\frac{1}{C}.$

On the contrary, we see from the proof above that

$\lim_{\gamma\to\infty}\Gamma_{i}=+\infty,\qquad\mbox{ for }i=1,2,$

with an exponential rate of divergence.

##  5\. Uniform Lipschitz bound

We now establish a local $L^{\infty}$ estimate for $\nabla u^{\varepsilon}$:
this time, this will lead to Theorem L.

######  Proposition 5.1.

Let $1<p_{1}\leq\dots\leq p_{N}\leq 2$ and $0<\varepsilon\leq\varepsilon_{0}$.
Then for every pair of concentric balls $B_{r}\Subset B_{R}\Subset B$ and
every $q>N$, we have

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{r})}&\leq
C\,\left[\frac{1}{(R-r)^{\frac{N\,q}{q-N}}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{q}\,dx\right)^{\frac{N}{q-N}}+\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{\frac{N\,q}{q-N}}\right]\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})},\end{split}$

for some $C=C(N,p_{N},p_{1},q)>0$.

###### Proof.

We will use a Moser’s iteration scheme, in order to get the claimed estimate.
By (2.2), for every $i=1,\dots,N$ we have

$\begin{split}g^{\prime\prime}_{i,\varepsilon}(u_{x_{i}}^{\varepsilon})\geq(p_{i}-1)\,(\varepsilon+|u^{\varepsilon}_{x_{i}}|^{2})^{\frac{p_{i}-2}{2}}&=(p_{i}-1)\,\Big{(}p_{i}\,g_{i,\varepsilon}(u^{\varepsilon}_{x_{i}})\Big{)}^{\frac{p_{i}-2}{p_{i}}}\\\
&\geq(p_{i}-1)\,\Big{(}p_{i}\,G_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}^{\frac{p_{i}-2}{p_{i}}}\geq(p_{i}-1)\,\Big{(}p_{i}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\Big{)}^{\frac{p_{i}-2}{p_{i}}},\end{split}$

thanks to the fact that $p_{i}\leq 2$, for every $i=1,\dots,N$. We can further
estimate the last term from below as follows

$g^{\prime\prime}_{i,\varepsilon}(u_{x_{i}}^{\varepsilon})\geq(p_{1}-1)\,(p_{N})^{\frac{p_{1}-2}{p_{1}}}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{1}-2}{p_{1}}}.$

By using this lower bound in (3.2), we get

(5.1) $\begin{split}\sum_{i=1}^{N}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{1}-2}{p_{1}}}&\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}\,\eta^{2}\,dx\\\
&\leq C\,(\alpha+1)^{2}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,(\alpha+1)^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\eta^{2}\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$. With simple algebraic manipulations, for
every $\alpha\geq 0$ we have

$\begin{split}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{1}-2}{p_{1}}}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+1}{2}}\Big{)}_{x_{i}}\right|^{2}&=\left(\frac{\alpha+1}{2}\right)^{2}\,\frac{1}{\left(\dfrac{\alpha+2}{2}-\dfrac{1}{p_{1}}\right)^{2}}\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}_{x_{i}}\right|^{2}\\\
&\geq\,\left|\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}_{x_{i}}\right|^{2}.\end{split}$

Thus from (5.1) we obtain

(5.2) $\begin{split}\int\left|\nabla\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}\right|^{2}\,\eta^{2}\,dx&\leq
C\,(\alpha+1)^{2}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,(\alpha+1)^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\eta^{2}\,dx,\end{split}$

for some $C=C(N,p_{1},p_{N})>0$. By adding on both sides the term

$\int\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}^{2}\,\left|\nabla\eta\right|^{2}\,dx,$

and using again that $\mathcal{G}_{\varepsilon}\geq 1$, we then obtain from
(5.2)

(5.3)
$\begin{split}\int\left|\nabla\left(\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}\,\eta\right)\right|^{2}\,dx&\leq
C\,(\alpha+1)^{2}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,(\alpha+1)^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\eta^{2}\,dx,\end{split}$

possibly for a different constant $C=C(N,p_{1},p_{N})>0$. Let us suppose for
simplicity that $N\geq 3$. The case $N=2$ can be treated with minor
modifications. On the left-hand side of (5.3), we then use Sobolev’s
inequality in $W^{1,2}(\mathbb{R}^{N})$. This gives

(5.4) $\begin{split}\left(\int\left(\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\,\eta\right)^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}&\leq
C\,(\alpha+1)^{2}\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,(\alpha+1)^{2}\,\int|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,\eta^{2}\,dx,\end{split}$

with some new constant $C=C(N,p_{1},p_{N})>0$. We choose $\eta\in
C^{2}_{0}(B_{R})$ to be a cut-off function such that

$0\leq\eta\leq 1,\qquad\eta\equiv 1\mbox{ on
}B_{r},\qquad|\nabla\eta|^{2}+|D^{2}\eta|\leq\frac{C}{(R-r)^{2}}.$

Thus we obtain from (5.4)

$\begin{split}\left(\int_{B_{r}}\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}&\leq
C\,\frac{(\alpha+1)^{2}}{(R-r)^{2}}\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,dx\\\
&+C\,(\alpha+1)^{2}\,\int_{B_{R}}|f^{\varepsilon}|^{2}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha}\,dx.\end{split}$

We now take an exponent $q>N$. By Hölder’s inequality on the last term, we
deduce that

(5.5) $\begin{split}\left(\int_{B_{r}}\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{\alpha+2}{2}-\frac{1}{p_{1}}}\Big{)}^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}&\leq
C\,\frac{(\alpha+1)^{2}}{(R-r)^{2}}\,\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,dx\\\
&+C\,(\alpha+1)^{2}\,\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{2}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha\,\frac{q}{q-2}}\,dx\right)^{\frac{q-2}{q}}.\end{split}$

Before proceeding further, we rely on Hölder’s inequality to get

$\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{N}}}\,dx\leq\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha\,\frac{q}{q-2}}\,dx\right)^{\frac{q-2}{q}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}.$

Moreover, by recalling that $\mathcal{G}_{\varepsilon}\geq 1$, we have

$\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha\,\frac{q}{q-2}}\leq\Big{(}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\alpha+2-\frac{2}{p_{1}}}\Big{)}^{\frac{q}{q-2}}.$

By using these two facts in (5.5), we obtain

(5.6) $\begin{split}\left(\int_{B_{r}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\left(\alpha+2-\frac{2}{p_{1}}\right)\,\frac{2^{*}}{2}}\,dx\right)^{\frac{2}{2^{*}}}&\leq
C\,(\alpha+1)^{2}\\\
&\times\,\left[\frac{1}{(R-r)^{2}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}+\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{2}\right]\\\
&\times\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\left(\alpha+2-\frac{2}{p_{1}}\right)\,\frac{q}{q-2}}\,dx\right)^{\frac{q-2}{q}}.\end{split}$

We now set

$\theta=\alpha+2-\frac{2}{p_{1}},$

so that from (5.6), we get

(5.7) $\begin{split}\left(\int_{B_{r}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2^{*}}{2}\theta}\,dx\right)^{\frac{2}{2^{*}\,\theta}}&\leq\left(C\theta^{2}\right)^{\frac{1}{\theta}}\,\left[\frac{1}{(R-r)^{2}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}+\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{2}\right]^{\frac{1}{\theta}}\\\
&\times\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\theta\frac{q}{q-2}}\,dx\right)^{\frac{q-2}{q\,\theta}}.\end{split}$

We define the sequence of exponents through the following recursive relation

$\theta_{0}=1,\qquad\theta_{i+1}\,\frac{q}{q-2}=\frac{2^{*}}{2}\,\theta_{i},\qquad\mbox{
for }i\in\mathbb{N};$

that is444Observe that
$\frac{2\,q}{2^{*}\,(q-2)}<1\qquad\Longleftrightarrow\qquad q>N,$ and the
latter holds true, in view of our assumption.,

$\theta_{i+1}=\left(\frac{2^{*}}{2}\,\frac{q-2}{q}\right)\,\theta_{i}=\left(\frac{2^{*}}{2}\,\frac{q-2}{q}\right)^{i+1},\qquad\mbox{
for }i\in\mathbb{N}.$

We also define the classical sequence of shrinking radii

$r_{i}=r+\frac{R-r}{2^{i}},\qquad\mbox{ for }i\in\mathbb{N}.$

With this notation, from (5.7) we get

(5.8) $\begin{split}\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\frac{q}{q-2}\,\theta_{i+1}}(B_{r_{i+1}})}&\leq\left[\frac{C}{(R-r)^{2}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}+C\,\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{2}\right]^{\frac{1}{\theta_{i}}}\\\
&\times(4^{i}\,\theta_{i}^{2})^{\frac{1}{\theta_{i}}}\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\frac{q}{q-2}\,\theta_{i}}(B_{r_{i}})}.\end{split}$

By starting from $i=0$ and iterating (5.8) $n$ times, we get

(5.9) $\begin{split}\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\frac{q}{q-2}\,\theta_{n+1}}(B_{r_{n+1}})}&\leq\left[\frac{C}{(R-r)^{2}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}+C\,\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{2}\right]^{\sum\limits_{i=0}^{n}\frac{1}{\theta_{i}}}\\\
&\times\prod_{i=0}^{n}(4^{i}\,\theta_{i}^{2})^{\frac{1}{\theta_{i}}}\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\frac{q}{q-2}}(B_{R})}.\end{split}$

By observing that

$\lim_{n\to\infty}\prod_{i=0}^{n}(4^{i}\,\theta_{i}^{2})^{\frac{1}{\theta_{i}}}=:C_{N,q}<+\infty,$

and

$\lim_{n\to\infty}\sum\limits_{i=0}^{n}\frac{1}{\theta_{i}}=\sum_{i=0}^{\infty}\left(\frac{2\,q}{2^{*}\,(q-2)}\right)^{i}=\frac{N}{2}\,\frac{q-2}{q-N},$

if we take the limit as $n$ goes to $\infty$ in (5.9), we end up with

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{r})}&\leq
C\,\left[\frac{1}{(R-r)^{2}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}+\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{2}\right]^{\frac{N}{2}\,\frac{q-2}{q-N}}\\\
&\times\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\frac{q}{q-2}}(B_{R})},\end{split}$

for some $C=C(N,p_{1},p_{N},q)>0$. The previous estimate holds for every $r<R$
such that $B_{R}\Subset B$. Thus, we can now use a standard interpolation
trick to rectify it and replace the $L^{q/(q-2)}$ norm on the right-hand side
by the $L^{1}$ norm.

This goes as follows: we first observe that

$\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\frac{q}{q-2}}(B_{R})}\leq\left(\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\infty}(B_{R})}\right)^{\frac{2}{q}}\,\left(\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})}\right)^{\frac{q-2}{q}}.$

Then by using Young’s inequality with exponents $q/2$ and $q/(q-2)$, we get

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{r})}&\leq\frac{q-2}{q}\,C^{\frac{q}{q-2}}\,\left[\frac{1}{(R-r)^{2}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{2}{q}}+\|f\|_{L^{q}(B_{R})}^{2}\right]^{\frac{N}{2}\frac{q}{q-N}}\\\
&\times\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})}+\frac{2}{q}\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\infty}(B_{R})}.\end{split}$

We now take $s,t$ such that $r\leq s<t\leq R$. The previous estimate is valid
by replacing $r$ with $s$ and $R$ with $t$. Thus, with some simple algebraic
manipulations, we get

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{s})}\leq\frac{q-2}{q}\,\frac{C^{\frac{q}{q-2}}}{2^{\frac{q(N-2)+2N}{2(N-q)}}}&\left[\frac{1}{(t-s)^{\frac{N\,q}{q-N}}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{N}{q-N}}+\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{\frac{N\,q}{q-N}}\right]\\\
&\times\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})}+\frac{2}{q}\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{\infty}(B_{R})}.\end{split}$

By relying once again on Lemma 2.7, from the last estimate we get

$\begin{split}\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\|_{L^{\infty}(B_{r})}&\leq\widetilde{C}\,\left[\frac{1}{(R-r)^{\frac{N\,q}{q-N}}}\,\left(\int_{B_{R}}\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\,dx\right)^{\frac{N}{q-N}}+\|f^{\varepsilon}\|_{L^{q}(B_{R})}^{\frac{N\,q}{q-N}}\right]\,\left\|\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\right\|_{L^{1}(B_{R})}.\end{split}$

By finally using that

$\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{q}{p_{N}^{\prime}}}\leq\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{q},$

we eventually conclude the proof. ∎

##  6\. Uniform higher differentiability

At last, we prove a Sobolev–type regularity result for (some nonlinear
function of) $\nabla u^{\varepsilon}$, which eventually will permit to
establish Theorem S.

######  Proposition 6.1.

Let $1<p_{1}\leq\dots\leq p_{N}\leq 2$. For $0<\varepsilon\leq\varepsilon_{0}$
and $i=1,\dots,N$, we set

(6.1)
$\mathcal{V}_{i,\varepsilon}=V_{i,\varepsilon}(u^{\varepsilon}_{x_{i}}),\qquad\mbox{
with
}V_{i,\varepsilon}(t)=\int_{0}^{t}\sqrt{g_{i,\varepsilon}^{\prime\prime}(\tau)}\,d\tau.$

Then for every non-negative $\eta\in C^{2}_{0}(B)$ and every $\gamma\geq 2$,
we have

(6.2)
$\begin{split}\sum_{i=1}^{N}\int\Big{|}\nabla\mathcal{V}_{i,\varepsilon}\Big{|}^{2}\eta^{2}\,dx&\leq
C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx\\\
&+C\,\left(\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2-p_{1}}{p_{1}}\frac{\gamma}{\gamma-2}}\,\eta^{2}\,dx\right)^{\frac{\gamma-2}{\gamma}}\,\left(\int|f^{\varepsilon}|^{\gamma}\,\eta^{2}\,dx\right)^{\frac{2}{\gamma}},\end{split}$

for some $C=C(N,p_{N},p_{1})>0$.

When $\gamma=2$, the last term is simply
$C\,\|f^{\varepsilon}\eta\|_{L^{2}}^{2}$.

###### Proof.

We start by fixing $k\in\\{1,\dots,N\\}$ and inserting in the differentiated
equation (2.15) the test function $\varphi=u^{\varepsilon}_{x_{k}}\,\eta^{2}$.
Thus we get

$\begin{split}\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,\eta^{2}\,dx&=-2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}\,u^{\varepsilon}_{x_{k}}\,\eta\,\eta_{x_{i}}\,dx\\\
&+\int(f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon}))_{x_{k}}\,u^{\varepsilon}_{x_{k}}\,\eta^{2}\,dx.\end{split}$

For the first term of the right-hand side, we use the same trick as in the
proof of Proposition 3.1: we observe that

$g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,u^{\varepsilon}_{x_{k}\,x_{i}}=\big{(}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\big{)}_{x_{k}},$

and then integrate by parts. We integrate by parts the term
$(f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon}))_{x_{k}}$, as
well. This yields

(6.3) $\begin{split}\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,\eta^{2}\,dx&=2\,\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,(u^{\varepsilon}_{x_{k}}\,\eta\,\eta_{x_{i}})_{x_{k}}-\int
f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,(u^{\varepsilon}_{x_{k}}\,\eta^{2})_{x_{k}}\,dx\\\
&=\int
u_{x_{k}\,x_{k}}^{\varepsilon}\left(2\,\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\eta\,\eta_{x_{i}}-f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\eta^{2}\right)\,dx\\\
&+2\,\int
u^{\varepsilon}_{x_{k}}\,\left(\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})-\eta\,\eta_{x_{k}}\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\right)\,dx.\end{split}$

By Young’s inequality and the fact that
$0\leq(\zeta^{\varepsilon})^{\prime}\leq 1$, we can estimate the first term of
the right-hand side as follows:

$\begin{split}\int
u_{x_{k}\,x_{k}}^{\varepsilon}&\left(2\,\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\eta\,\eta_{x_{i}}-f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\eta^{2}\right)\,dx\leq\frac{1}{2}\,\int
g_{k,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{k}})\,|u^{\varepsilon}_{x_{k}\,x_{k}}|^{2}\,\eta^{2}\,dx\\\
&+4\,\int\frac{1}{g_{k,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{k}})}\,\left(\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}|\nabla\eta|^{2}+|f^{\varepsilon}|^{2}\,\eta^{2}\right)\,dx.\end{split}$

We use (2.2) to estimate $1/g_{k,\varepsilon}^{\prime\prime}$ on the right-
hand side. On account of this inequality, we get

$\begin{split}\int
u_{x_{k}\,x_{k}}^{\varepsilon}\,&\left(2\,\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,\eta\,\eta_{x_{i}}-f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\,\eta^{2}\right)\,dx\leq\frac{1}{2}\,\int
g_{k,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{k}})\,|u^{\varepsilon}_{x_{k}\,x_{k}}|^{2}\,\eta^{2}\,dx\\\
&+\frac{4}{p_{k}-1}\,\int\left(\varepsilon+(u^{\varepsilon}_{x_{k}})^{2}\right)^{\frac{2-p_{k}}{2}}\,\left(\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}|\nabla\eta|^{2}+|f^{\varepsilon}|^{2}\,\eta^{2}\right)\,dx.\end{split}$

Inserting the above inequality into (6.3) and absorbing the Hessian term of
the right-hand side into the left-hand side, one gets

(6.4) $\begin{split}\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})&\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,\eta^{2}\,dx\\\
&\leq\frac{8}{p_{k}-1}\,\int\left(\varepsilon+(u^{\varepsilon}_{x_{k}})^{2}\right)^{\frac{2-p_{k}}{2}}\,\left(\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}|\nabla\eta|^{2}+|f^{\varepsilon}|^{2}\,\eta^{2}\right)\,dx\\\
&+4\,\int
u^{\varepsilon}_{x_{k}}\,\left(\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})-\eta\,\eta_{x_{k}}\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\right)\,dx.\end{split}$

We now estimate the last term as follows:

$\begin{split}\left|\int
u^{\varepsilon}_{x_{k}}\right.&\left.\left(\sum_{i=1}^{N}g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\,(\eta_{x_{k}}\,\eta_{x_{i}}+\eta\,\eta_{x_{i}\,x_{k}})-\eta\,\eta_{x_{k}}\,f^{\varepsilon}(\zeta^{\varepsilon})^{\prime}(u^{\varepsilon})\right)\,dx\right|\\\
&\leq\int|u^{\varepsilon}_{x_{k}}|\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|\,dx+\int|u^{\varepsilon}_{x_{k}}|\,\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx.\end{split}$

Observe that we used again that $0\leq(\zeta^{\varepsilon})^{\prime}\leq 1$.
We apply Young’s inequality on the last term, so to obtain

$\begin{split}\int|u^{\varepsilon}_{x_{k}}|\,\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx&\leq\frac{1}{2}\int|u^{\varepsilon}_{x_{k}}|^{2-p_{k}}\\!\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})\big{|}\right)^{2}\\!\\!\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|)\Big{)}\,dx\\\
&+\frac{1}{2}\int|u^{\varepsilon}_{x_{k}}|^{p_{k}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx.\end{split}$

By further using that $p_{k}\leq 2$, we have

$|u^{\varepsilon}_{x_{k}}|^{2-p_{k}}\leq(\varepsilon+(u^{\varepsilon}_{x_{k}})^{2})^{\frac{2-p_{k}}{2}}=p_{k}^{\frac{2-p_{k}}{p_{k}}}\,g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})^{\frac{2-p_{k}}{p_{k}}}.$

It follows from the above inequality and (6.4) that

$\begin{split}\sum_{i=1}^{N}\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|u^{\varepsilon}_{x_{k}\,x_{i}}|^{2}\,\eta^{2}\,dx&\leq
C\,\int
g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})^{\frac{2-p_{k}}{p_{k}}}\,\left(\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}|\nabla\eta|^{2}+|f^{\varepsilon}|^{2}\,\eta^{2}\right)\,dx\\\
\
&+C\,\int|u^{\varepsilon}_{x_{k}}|^{p_{k}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+C\,\int|u^{\varepsilon}_{x_{k}}|\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|\,dx,\end{split}$

for some $C=C(p_{1},p_{N})>0$. Then take the sum over $k=1,\dots,N$. This
gives

(6.5) $\begin{split}\sum_{i=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|\nabla
u^{\varepsilon}_{x_{i}}|^{2}\,\eta^{2}\,dx\\\ &\leq
C\,\int\sum_{k=1}^{N}g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})^{\frac{2-p_{k}}{p_{k}}}\,\left(\left(\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u^{\varepsilon}_{x_{i}})|\right)^{2}|\nabla\eta|^{2}+|f^{\varepsilon}|^{2}\,\eta^{2}\right)\,dx\\\
\
&+C\,\int\sum_{k=1}^{N}|u^{\varepsilon}_{x_{k}}|^{p_{k}}\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+C\,\int\sum_{k=1}^{N}|u^{\varepsilon}_{x_{k}}|\,|f^{\varepsilon}|\,\eta\,|\nabla\eta|\,dx.\end{split}$

By Lemma 2.3, we have

$\sum_{i=1}^{N}|g_{i,\varepsilon}^{\prime}(u_{x_{i}}^{\varepsilon})|\leq
p_{N}^{\frac{p_{N}-1}{p_{N}}}\,\sum_{i=1}^{N}G_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{i}-1}{p_{i}}}\leq
N\,p_{N}^{\frac{p_{N}-1}{p_{N}}}\,\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{p_{N}-1}{p_{N}}}.$

Moreover, by the definitions of $g_{k,\varepsilon},\,G_{\varepsilon}$, and
$\mathcal{G}_{\varepsilon}$, it is easily seen that

$\sum_{k=1}^{N}g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})^{\frac{2-p_{k}}{p_{k}}}\leq
C\,\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})^{\frac{2-p_{1}}{p_{1}}},$
$\sum_{k=1}^{N}|u^{\varepsilon}_{x_{k}}|^{p_{k}}\leq\sum_{k=1}^{N}p_{k}\,g_{k,\varepsilon}(u_{x_{k}}^{\varepsilon})\leq
C\,\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})\qquad\mbox{ and
}\qquad\sum_{k=1}^{N}|u^{\varepsilon}_{x_{k}}|\leq
C\,\mathcal{G}_{\varepsilon}(\nabla u^{\varepsilon})^{\frac{1}{p_{1}}},$

where all the constants depend only on $N,p_{1}$ and $p_{N}$. From (6.5), we
get

$\begin{split}\sum_{i=1}^{N}&\int
g_{i,\varepsilon}^{\prime\prime}(u^{\varepsilon}_{x_{i}})\,|\nabla
u^{\varepsilon}_{x_{i}}|^{2}\,\eta^{2}\,dx\\\ &\leq
C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{1+2\,\left(\frac{1}{p_{1}}-\frac{1}{p_{N}}\right)}\,|\nabla\eta|^{2}\,dx+C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})^{\frac{2-p_{1}}{p_{1}}}\,|f^{\varepsilon}|^{2}\,\eta^{2}\,dx\\\
\ &+C\,\int\mathcal{G}_{\varepsilon}(\nabla
u^{\varepsilon})\,\Big{(}|\nabla\eta|^{2}+\eta\,|D^{2}\eta|\Big{)}\,dx+C\,\int\mathcal{G}_{\varepsilon}(\nabla