# Higher differentiability for a class of problems
under $p,q$ subquadratic growth

E. Mascolo – Antonia Passarelli di Napoli  Dipartimento di Matematica ”U.
Dini”, Università degli Studi di Firenze, viale Morgagni , 41125 Firenze,
Italy  Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Università degli Studi di Napoli
“Federico II” Via Cintia, 80126, Napoli, Italy

###### Abstract.

We study the higher differentiability for nonlinear elliptic equation in
divergence form $\mathcal{A}(x,Du)=b(x)$. The result covers the cases in which
$\mathcal{A}(x,\xi)$ satisfies $p,q$ growth, with $1<p<2$ in $\xi$ and a
Sobolev dependence of with respect to $x$. By means of an a-priori estimate we
ensure the $W^{2,p}_{\mathrm{loc}}(\Omega)$-property for the solution of the
boundary value problem.

###### Key words and phrases:

Higher differentiability, subquadratic problems, non standard growth.

###### 2000 Mathematics Subject Classification:

35J87, 49J40; 47J20

Acknowledgements. The work of the authors is supported by GNAMPA (Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of
INdAM (Istituto Nazionale di Alta Matematica). A. Passarelli di Napoli has
been supported by Università di Napoli $``$Federico II” through the project
FRA–000022-ALTRI-CDA-752021-FRA-PASSARELLI.

##  1\. Introduction

The paper deals with the following boundary value problem for elliptic
equations

$\begin{cases}\mathrm{div}\,(\mathcal{A}(x,Du))=b(x)\qquad\qquad\text{in}\,\,\Omega\cr\cr
u=u_{0}\qquad\qquad\qquad\qquad\qquad\text{on}\,\,\,\partial\Omega\end{cases}$
(1.1)

where $\Omega\subset\mathbb{R}^{n}$ is a bounded open set and the operator
$\mathcal{A}:\Omega\times\mathbb{R}^{n}\to\mathbb{R}$ is a Carathédory map
satisfying the following set of assumptions for a couple of exponents $p,q$
such that $1<p<q$ and $1<p\leq 2$

$\langle\mathcal{A}(x,\xi)-\mathcal{A}(x,\eta),\xi-\eta\rangle\geq\nu|\xi-\eta|^{2}(\mu^{2}+|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}}$
(1.2) $|\mathcal{A}(x,\xi)-\mathcal{A}(x,\eta)|\leq
L|\xi-\eta|\left[(1+|\xi|^{2}+|\eta|^{2})^{\frac{q-2}{2}}+{(\mu^{2}+|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}}}\right]$
(1.3)
$|\mathcal{A}(x,\xi)-\mathcal{A}(y,\xi)|\leq|x-y|(k(x)+k(y))(1+|\xi|^{2})^{\frac{q-1}{2}}$
(1.4)

where $L,\nu$ are fixed constants, $\mu\in[0,1]$ is a parameter and $k\in
L^{r}_{\mathrm{loc}}(\Omega)$ with $r>n$ is a non negative function.
Let us observe that if we assume that

$\mathcal{A}(x,0)=0,$

assumption (1.3) implies

$|\mathcal{A}(x,\xi)|\leq L(1+|\xi|^{2})^{\frac{q-1}{2}}$ (1.5)

Then, setting $f(x)=\mathcal{A}(x,0)\in L^{\infty}$, we have

$|\mathcal{A}(x,\xi)|\leq L(1+|\xi|^{2})^{\frac{q-1}{2}}+f(x)$

To simplify the presentation, from now on, we shall assume that (1.5) is in
force.
Assumption (1.4), by virtue of the characterization of the Sobolev function
due to Hajlasz, [29], implies that the partial map
$x\mapsto\mathcal{A}(x,\xi)$ belongs to the Sobolev space
$W^{1,r}_{\mathrm{loc}}(\Omega)$ and that there exists $\tilde{k}\in
L^{r}_{\mathrm{loc}}(\Omega)$ such that

$|D_{x}\mathcal{A}(x,\xi)|\leq\tilde{k}(x)(1+|\xi|^{2})^{\frac{q-1}{2}}.$
(1.6)

We prove the following existence and higher differentiability result.

######  Theorem 1.1.

Assume that $\mathcal{A}(x,\xi)$ satisfy (1.2)—(1.5) with $1<p<q$ and $1<p\leq
2$ such that

$\frac{q}{p}<1+\min\left\\{\frac{1}{n}-\frac{1}{r},\,\frac{2(p-1)}{p(n-2)}\right\\}\qquad\text{if}\,\,n>2$
(1.7)

or if $n=2$

$\frac{q}{p}<1+\frac{1}{2}-\frac{1}{r}\qquad\qquad\qquad$ (1.8)

If

$u_{0}\in W^{1,\frac{p(q-1)}{p-1}}(\Omega)\,\,\,\text{and}\,\,\,b\in
L^{\frac{p}{p-1}}_{\mathrm{loc}}(\Omega),$

then problem (1.1) admits a solution
$v\in\big{(}u_{0}+W^{1,p}_{0}(\Omega)\big{)}\cap
W^{2,p}_{\mathrm{loc}}(\Omega)$. Moreover setting
$V_{p}(Du)=(\mu^{2}+|Du|^{2})^{\frac{p-2}{4}}Du$, the following estimates

$\displaystyle\int_{B_{\frac{R}{2}}}|D(V_{p}(Dv))|^{2}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\gamma}$
(1.9)

$\displaystyle\int_{B_{\frac{R}{2}}}|D^{2}v|^{p}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\gamma}$
(1.10)

hold for every ball $B_{R}\Subset\Omega$, for an exponent
$\gamma=\gamma(p,q,n)$ and $c=c(\nu,L,n,p,q,R)$.

The novelty of Theorem 1.1 is in two directions First, we consider operators
with $p,q$ growth, which means that the ellipticity of the leading part of
$\xi\mapsto\mathcal{A}(x,\xi)$ has $p$-growth (see (1.2)) while it satisfies a
$q$\- bound (see (1.3)), with $1<p<q$ and $1<p<2$. Roughly speaking, we are
dealing with the subquadratic non standard growth case which means that the
problem under consideration can be singular and degenerate.
In the last few years, the study of the regularity properties of solution and
minimizers of problems with non standard growth conditions has undergone
remarkable developments, motivated in part by the applications. We would
mention the first papers of Marcellini [35, 36, 37] and the recent [38, 39,
40], the result on higher integrability and differentiability in [22, 23, 33,
44] and more recently [5, 6, 11, 12, 13, 16, 17, 18, 19]. For complete details
and references on problems with non standard growth we refer to the recent
surveys [39, 41]
It is well known that a restriction between $p$ and $q$ is necessary by virtue
of the celebrated counterexample by Marcellini (see [36]). The second
principal feature is the Sobolev dependence of $\mathcal{A}(x,\xi)$ with
respect to $x$ (see (1.4)).

Recently, there has been an increasing interest in the study of the regularity
under this assumptions on the function that measures the oscillation of the
operator $\mathcal{A}(x,\xi)$ with respect to the $x$-variable. The degeneracy
of $\mathcal{A}(x,\xi)$ has the same nature of

$\mathrm{div}\big{(}|Du|^{p-2}Du+a(x)|Du|^{q-2}Du\big{)}=0$

related to the double phase functional

$\mathcal{I}(u)=\int_{\Omega}|Du|^{p}+a(x)|Du|^{q}$ (1.11)

which has been intensively studied starting from the papers [7, 8]. For
references on the regularity properties of minimizers to (1.11), see also [4]
and [14].

Actually, it is now completely clear that the weak differentiability of the
partial map $\xi\to\mathcal{A}(x,\xi)$ leads to an higher differentiability of
the gradient of the solutions (see for example [12, 24, 25, 26, 27, 42, 43].

In [18, 19] for integral functionals under $p,q$, growth with Sobolev
coefficients in $L^{r}$, $r>n$, it has been shown that the bound
$\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r}$ permit to obatin the local Lipschitz
continuity of the minimizers.

Moreover, it is worth mentioning that the bound at (1.7) has been already used
in [14] (see also [4]) for the study of problems with subquadratic non
standard growth conditions.

In the standard growth, i.e. $1<p=q<2$ in assumption (1.3)–(1.4), and
Lipschitz continuous coefficients, i.e. $k\in
L^{\infty}_{\mathrm{loc}}(\Omega)$ in assumption (1.4), the regularity is due
to Tolksdorf ([45]) (see also [28, Chapter 8]), for the case of vector-valued
minimizers see [1].

In case of $p,q$ growth with $2\leq p<q$, in [36, 38, 39] Marcellini
established the $W^{2,2}_{\mathrm{loc}}$ regularity for local weak solution
$u\in W^{1,q}(\Omega)$.

More recently, in the $p$-growth case, $1<p<2$ and $k\in
L^{r}_{\mathrm{loc}}(\Omega)$ with $r\geq n$ in (1.4), an higher
differentiability result for local minimizers of integral functionals has been
established in [24, 25].

A first main step in the proof of our main result is an a-priori estimate for
the $W^{2,p}$-norm of the weak local solution $u$ of of the equation

$\mathrm{div}\,(\mathcal{A}(x,Du))=b(x)$

i.e. $u\in W^{1,q}_{\mathrm{loc}}$ such that

$\int\mathcal{A}(x,Du)D\varphi(x)\,dx=\int b(x)\varphi(x)\,dx\qquad\text{for
all}\,\,\varphi\in C^{\infty}_{0}(\Omega).$

We prove that if $Du\in L^{\frac{2^{*}}{2}p}$ then it locally belongs to
$W^{2,p}$ and the norm of its second derivatives can be estimated by the
$L^{p}$ norm of its gradient.
Actually, as far as we know, our a-priori estimate is a first regularity
result for weak local solutions of an equation under $p,q$-growth $1<p<q$,
$1<p<2$, and a Sobolev assumption on the partial map
$x\mapsto\mathcal{A}(x,\xi)$ (see (1.4)).

Then we give an existence and regularity result for the Dirichlet boundary
problem (1.1).

More precisely, the solution to problem (1.1) which belongs to $W^{2,p}$
locally is constructed as a $W^{1,q}$-limit of a sequence of solution to
regular equations with standard $q$-growth of the form

$\mathcal{A}_{\varepsilon}(x,\xi)=\tilde{\mathcal{A}}_{\varepsilon}(x,\xi)+\varepsilon(1+|\xi|^{2})^{\frac{q-2}{2}}\xi,$

where $\tilde{\mathcal{A}}_{\varepsilon}(x,\xi)$ is the regularized of
$\mathcal{A}(x,\xi)$ with respect to the variable $x$, i.e.

$\tilde{\mathcal{A}}_{\varepsilon}(x,\xi)=\int_{B_{1}(0)}\phi(\omega)\mathcal{A}(x+\varepsilon\omega,\xi)\,d\omega,$

with $\phi$ a smooth mollifier.

Each problem has a unique smooth solution $v_{\varepsilon}\in
W^{1,q}_{0}+u_{0}$ in view of the available existence and regularity results.

Then, we apply to the sequence of approximating solutions the a-priori
estimate which takes into account only the assumptions (1.3)–(1.4) and
therefore is independent of $\varepsilon$. Therefore we obtain a uniform
control for the norm of $v_{\varepsilon}$ in $W^{2,p}$. By passing to the
limit we obtain a solution $v$ of (1.1) with higher differentiability
properties.
Let us remark that in subquadratic $p,q$\- growth case, in [32] a higher
integrability result has been established when

$\frac{2n}{n+2}\leq p\leq q\leq 2.$

It is worth to observe that if $\frac{2n}{n+2}\leq p$, the restriction on the
gap (1.7) reduces to

$\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r}$

which is the sharp one found in [22] to establish the local Lipschitz
continuity of minimizers of degenerate functionals with $(p,q)$ growth
conditions and Sobolev coefficients. Indeed

$p\geq\frac{2n}{n+2}\,\,\Longrightarrow\,\,\frac{p-1}{p}\geq\frac{n-2}{2n}\,\,\Longrightarrow\,\,\frac{2(p-1)}{p(n-2)}\geq\frac{1}{n}$

and so

$\min\left\\{\frac{1}{n}-\frac{1}{r},\,\frac{2(p-1)}{p(n-2)}\right\\}=\frac{1}{n}-\frac{1}{r}.$

##  2\. Preliminary results

In this paper we shall denote by $C$ or $c$ a general positive constant that
may vary on different occasions, even within the same line of estimates.
Relevant dependencies will be suitably emphasized using parentheses or
subscripts. In what follows,
$B(x,r)=B_{r}(x)=\\{y\in\mathbb{R}^{n}:\,\,|y-x|<r\\}$ will denote the ball
centered at $x$ of radius $r$. We shall omit the dependence on the center and
on the radius when no confusion arises.

The following lemma has important applications in the so called hole-filling
method. Its proof can be found for example in [28, Lemma 6.1] .

######  Lemma 2.1.

Let $h:[r,R_{0}]\to\mathbb{R}$ be a nonnegative bounded function and
$0<\vartheta<1$, $A,B\geq 0$ and $\beta>0$. Assume that

$h(s)\leq\vartheta h(t)+\frac{A}{(t-s)^{\beta}}+B,$

for all $r\leq s<t\leq R_{0}$. Then

$h(r)\leq\frac{cA}{(R_{0}-r)^{\beta}}+cB,$

where $c=c(\vartheta,\beta)>0$.

We introduce the usual notation

$\displaystyle
2^{*}=\begin{cases}\frac{2n}{n-2}\qquad\qquad\qquad\qquad\qquad\qquad\text{if}\,\,n>2\\\
\text{any finite exponent}\,\,t>2\qquad\,\,\,\text{if}\,\,n=2\end{cases}$

For $p>1$ and $\mu\in[0,1]$, let us define

$V_{p}(\xi)=(\mu^{2}+|\xi|^{2})^{\frac{p-2}{4}}\xi,\qquad\xi\in\mathbb{R}^{n}$
(2.1)

We shall use the following estimates, whose proof can be found in [1] (see
also [34, Step 2]).

######  Lemma 2.2.

Let $1<p<2$. There exists a constant $c=c(n,p)>0$ such that, for any $\xi$,
$\eta\in\mathbb{R}^{n}$, $\xi\not=\eta$, it holds

$c^{-1}\Bigl{(}\mu^{2}+|\xi|^{2}+|\eta|^{2}\Bigr{)}^{\frac{p-2}{2}}\leq\frac{|V_{p}(\xi)-V_{p}(\eta)|^{2}}{|\xi-\eta|^{2}}\leq
c\Bigl{(}\mu^{2}+|\xi|^{2}+|\eta|^{2}\Bigr{)}^{\frac{p-2}{2}}$ (2.2)

######  Lemma 2.3.

Let $1<p<2$ and $u\in W^{1,1}_{\mathrm{loc}}(\Omega)$. Then

$V_{p}(Du)\in L^{2^{*}}_{\textnormal{loc}}(\Omega)\quad\Longrightarrow\quad
Du\in L^{\frac{2^{*}p}{2}}_{\textnormal{loc}}(\Omega).$

###### Proof.

Note that the thesis is obvious if $\mu=0$. In case $\mu>0$, we have

$\displaystyle\int_{B_{R}}|Du|^{\frac{2^{*}p}{2}}\,dx=\int_{B_{R}}\left||Du|^{\frac{p}{2}-1}Du\right|^{2^{*}}\,dx$
$\displaystyle=$ $\displaystyle\int_{\\{x\in
B_{R}:\,|Du|\leq\mu\\}}\left||Du|^{\frac{p}{2}-1}Du\right|^{2^{*}}\,dx+\int_{\\{x\in
B_{R}:\,|Du|>\mu\\}}\left||Du|^{\frac{p}{2}-1}Du\right|^{2^{*}}\,dx$
$\displaystyle=$ $\displaystyle\int_{\\{x\in
B_{R}:\,|Du|\leq\mu\\}}\left||Du|^{\frac{p}{2}-1}Du\right|^{2^{*}}\,dx+\int_{\\{x\in
B_{R}:\,|Du|>\mu\\}}\left|\left(\frac{|Du|^{2}}{2}+\frac{|Du|^{2}}{2}\right)^{\frac{p-2}{4}}|Du|\right|^{2^{*}}\,dx$
$\displaystyle\leq$ $\displaystyle\mu^{\frac{2^{*}p}{2}}|B_{R}|+\int_{\\{x\in
B_{R}:\,|Du|>\mu\\}}\left|\left(\frac{\mu^{2}}{2}+\frac{|Du|^{2}}{2}\right)^{\frac{p-2}{4}}|Du|\right|^{2^{*}}\,dx,$

since $p-2<0$. Therefore, we have

$\displaystyle\int_{B_{R}}|Du|^{\frac{2^{*}p}{2}}\,dx$ $\displaystyle=$
$\displaystyle\leq\mu^{\frac{2^{*}p}{2}}|B_{R}|+\int_{\\{x\in
B_{R}:\,|Du|>\mu\\}}\left|V_{p}(Du)\right|^{2^{*}}\,dx$ (2.3)
$\displaystyle\leq$
$\displaystyle\int_{B_{R}}\left(1+\left|V_{p}(Du)\right|^{2^{*}}\right)\,dx<+\infty,$
(2.5)

where we also used that $\mu\leq 1$. ∎

######  Lemma 2.4.

Let $1<p<2$ and $u\in W^{1,1}_{\mathrm{loc}}(\Omega)$. Then

$V_{p}(Du)\in W^{1,2}_{\mathrm{loc}}(\Omega)\quad\Longrightarrow\quad u\in
W^{2,p}_{\mathrm{loc}}(\Omega)$

###### Proof.

By the use of Hölder’s inequality we get

$\displaystyle\int_{B_{R}}|D^{2}u|^{p}$ $\displaystyle=$
$\displaystyle\int_{B_{R}}|D^{2}u|^{p}(\mu^{2}+|Du|^{2})^{\frac{p(p-2)}{4}}(\mu^{2}+|Du|^{2})^{\frac{p(2-p)}{4}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{B_{R}}|D^{2}u|^{2}(\mu^{2}+|Du|^{2})^{\frac{p-2}{2}}\right)^{\frac{p}{2}}\left(\int_{B_{R}}(\mu^{2}+|Du|^{2})^{\frac{p}{2}}\right)^{\frac{2-p}{2}}$
$\displaystyle\leq$ $\displaystyle
c\left(\int_{B_{R}}|D(V_{p}(Du))|^{2}\right)^{\frac{p}{2}}\left(\int_{B_{R}}(\mu^{2}+|Du|^{2})^{\frac{p}{2}}\right)^{\frac{2-p}{2}}$

∎

###  2.1. Difference quotient

We recall some properties of the finite difference operator needed in the
sequel. We start with the description of some elementary properties that can
be found, for example, in [28].

######  Proposition 2.5.

Let $f$ and $g$ be two functions such that $f,g\in W^{1,p}(\Omega)$, with
$p\geq 1$, and let us consider the set

$\Omega_{|h|}:=\left\\{x\in\Omega:dist(x,\partial\Omega)>|h|\right\\}.$

Then

  * $(d1)$

$\tau_{h}f\in W^{1,p}(\Omega)$ and

$D_{i}(\tau_{h}f)=\tau_{h}(D_{i}f).$

  * $(d2)$

If at least one of the functions $F$ or $G$ has support contained in
$\Omega_{|h|},$ then

$\int_{\Omega}f\,\tau_{h}g\,dx=-\int_{\Omega}g\,\tau_{-h}f\,dx.$

  * $(d3)$

We have

$\tau_{h}(fg)(x)=f(x+h)\tau_{h}g(x)+g(x)\tau_{h}f(x).$

The next result about finite difference operator is a kind of integral version
of Lagrange Theorem.

######  Lemma 2.6.

If $0<\rho<R$, $|h|<\frac{R-\rho}{2}$, $1<p<+\infty$, and $f,Df\in
L^{p}(B_{R})$ then

$\int_{B_{\rho}}|\tau_{h}f(x)|^{p}\ dx\leq
c(n,p)|h|^{p}\int_{B_{R}}|Df(x)|^{p}\ dx.$

Moreover

$\int_{B_{\rho}}|f(x+h)|^{p}\ dx\leq\int_{B_{R}}|f(x)|^{p}\ dx.$

Now, we recall the fundamental Sobolev embedding property.

######  Lemma 2.7.

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{N}$, $f\in L^{p}(B_{R})$ with $1<p<n$.
Suppose that there exist $\rho\in(0,R)$ and $M>0$ such that

$\sum_{s=1}^{n}\int_{B_{\rho}}|\tau_{s,h}f(x)|^{p}\,dx\leq M^{p}|h|^{p},$

for every $h$ with $|h|<\frac{R-\rho}{2}$. Then

$f\in W^{1,p}(B_{\rho})\cap L^{\frac{np}{n-p}}(B_{\rho}).$

Moreover

$||Df||_{L^{p}(B_{\rho})}\leq M$ (2.6)

and

$||f||_{L^{\frac{np}{n-p}}(B_{\rho})}\leq
c\left(M+||f||_{L^{p}(B_{R})}\right)$ (2.7)

with $c=c(n,N,p)$.

For the proof see, for example, [28, Lemma 8.2].

##  3\. Proof of the a-priori estimate

This section is devoted to the proof of an a-priori estimate, which is at the
same time a first regularity result, for local solution to the following
equation

$\mathrm{div}(\mathcal{A}(x,Du))=b(x)\qquad\qquad\text{in}\,\,\,\Omega$ (3.1)

where $\Omega\subset\mathbb{R}^{n}$ is a bounded open set and the operator
$\mathcal{A}:\Omega\times\mathbb{R}^{N\times n}\to\mathbb{R}$ is a
Carathéodory map satisfying the following set of assumptions for a couple of
exponents $1<p<q$ with $1<p\leq 2$

$\langle\mathcal{A}(x,\xi)-\mathcal{A}(x,\eta),\xi-\eta\rangle\geq\nu|\xi-\eta|^{2}(\mu^{2}+|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}}$
(3.2)

$|\mathcal{A}(x,\xi)-\mathcal{A}(x,\eta)|\leq
L|\xi-\eta|\left[(1+|\xi|^{2}+|\eta|^{2})^{\frac{p+q-4}{4}}+(\mu^{2}+|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}}\right]$
(3.3)

$|\mathcal{A}(x,\xi)-\mathcal{A}(y,\xi)|\leq|x-y|(k(x)+k(y))(1+|\xi|^{2})^{\frac{p+q-2}{4}}$
(3.4)

where $c_{1},c_{2},L,\nu>0$ are fixed constants, $\mu\in[0,1]$ is a parameter,
$k\in L^{r}$ with $r>n$ is a non negative function.
We define a local solution to (3.1) as a function $u\in
W^{1,q}_{\mathrm{loc}}(\Omega)$ that satisfies the following integral identity

$\displaystyle\int_{\Omega}\langle\mathcal{A}(x,Du),D\varphi\rangle\,dx=\int_{\Omega}b(x)\varphi(x)\,dx\qquad\text{for
all}\,\,\varphi\in C^{\infty}_{0}(\Omega).$ (3.5)

Our next aim is to prove the following

######  Theorem 3.1.

Let $u\in W^{1,q}_{\mathrm{loc}}(\Omega)$ be a local solution to (3.1) under
the assumptions (3.2)–(3.4) such that

$Du\in L^{\frac{2^{*}p}{2}}_{\mathrm{loc}}(\Omega).$ (3.6)

Assume moreover that

$\frac{q}{p}<1+\min\left\\{2\left(\frac{1}{n}-\frac{1}{r}\right),\,\frac{4(p-1)}{p(n-2)}\right\\}\qquad\text{if}\,\,n>2$
(3.7)

or

$\frac{q}{p}<1+2\left(\frac{1}{n}-\frac{1}{r}\right),\qquad\qquad\qquad\qquad\qquad\text{if}\,\,n=2.$
(3.8)

Then

$u\in W^{2,p}_{\mathrm{loc}}(\Omega),\qquad V_{p}(Du)\in
W^{1,2}_{\mathrm{loc}}(\Omega)$

and the following estimates

$\displaystyle\int_{B_{\frac{R}{2}}}|D(V_{p}(Du))|^{2}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\gamma},$
(3.9)

and

$\displaystyle\int_{B_{\frac{R}{2}}}|D^{2}u|^{p}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\gamma}$
(3.10)

hold for every ball $B_{R}\Subset\Omega$, for and exponent
$\gamma=\gamma(p,q,n,r)$, and $c=c(\nu,L,n,p,q,r,R)$.

###### Proof.

Let $B_{R}\Subset\Omega$ be a ball, fix radii $0<\frac{R}{2}<s<t<R$ and a cut
off function $\eta\in C^{\infty}_{0}(B_{t})$ such that $\eta\equiv 1$ on
$B_{s}$ and $|D\eta|\leq\frac{C}{t-s}$. We choose
$\varphi=\tau_{-h}(\eta^{2}\tau_{h}u)\in W^{1,q}_{\mathrm{loc}}(\Omega)$ as
test function in (3.5) to deduce that

$\int_{\Omega}\langle\mathcal{A}(x,Du),\tau_{-h}(D(\eta^{2}\tau_{h}u))\rangle\,dx=\int_{\Omega}b\tau_{-h}(\eta^{2}\tau_{h}u)\,dx$

i.e.

$\int_{\Omega}\langle\mathcal{A}(x,Du),\tau_{-h}(\eta^{2}\tau_{h}Du)+2\tau_{-h}(\eta
D\eta\tau_{h}u)\rangle\,dx=\int_{\Omega}b\tau_{-h}(\eta^{2}\tau_{h}u)\,dx.$

From previous equality, we get

$\displaystyle\int_{\Omega}\langle\mathcal{A}(x,Du),\tau_{-h}(\eta^{2}\tau_{h}Du)\rangle\,dx$
$\displaystyle=$
$\displaystyle-2\int_{\Omega}\langle\mathcal{A}(x,Du),\tau_{-h}(\eta
D\eta\tau_{h}u)\rangle\,dx$
$\displaystyle\qquad+\int_{\Omega}b\tau_{-h}(\eta^{2}\tau_{h}u)\,dx.$

which, by virtue of (d2) in Proposition 2.5, implies

$\displaystyle\int_{\Omega}\eta^{2}\langle\tau_{h}(\mathcal{A}(x,Du)),\tau_{h}Du\rangle\,dx$
$\displaystyle=$
$\displaystyle-2\int_{\Omega}\langle\mathcal{A}(x,Du),\tau_{-h}(\eta
D\eta\tau_{h}u)\rangle\,dx$
$\displaystyle\qquad+\int_{\Omega}b\tau_{-h}(\eta^{2}\tau_{h}u)\,dx.$

Exploiting the definition of $\tau_{h}(\mathcal{A}(x,Du))$ in previous
equality, we have

$\displaystyle\int_{\Omega}\eta^{2}\langle\mathcal{A}(x+h,Du(x+h))-\mathcal{A}(x+h,Du(x)),\tau_{h}Du\rangle\,dx$
$\displaystyle=$
$\displaystyle-\int_{\Omega}\eta^{2}\langle\mathcal{A}(x+h,Du(x))-\mathcal{A}(x,Du(x)),\tau_{h}Du\rangle\,dx$
$\displaystyle\qquad-2\int_{\Omega}\langle\mathcal{A}(x,Du),\tau_{-h}(\eta
D\eta\tau_{h}u)\rangle\,dx$
$\displaystyle\qquad+\int_{\Omega}b\tau_{-h}(\eta^{2}\tau_{h}u)\,dx.$

Using the ellipticity assumption (3.2) in the left hand side and assumptions
(3.3) and (3.4) in the right hand side of previous estimate, we obtain

$\displaystyle\nu\int_{\Omega}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}\eta^{2}|\mathcal{A}(x+h,Du(x))-\mathcal{A}(x,Du(x))||\tau_{h}Du|\,dx$
$\displaystyle+\quad 2\int_{\Omega}\eta|\mathcal{A}(x,Du(x))||\tau_{-h}(\eta
D\eta\tau_{h}u)|\,dx$
$\displaystyle\qquad+\int_{\Omega}|b||\tau_{-h}(\eta^{2}\tau_{h}u)|\,dx$
$\displaystyle\leq$
$\displaystyle|h|\int_{\Omega}\eta^{2}(k(x+h)+k(x))(1+|Du(x)|^{2})^{\frac{p+q-2}{4}}|\tau_{h}Du|\,dx$
$\displaystyle+c\int_{\Omega}(1+|Du(x)|)^{\frac{p+q-2}{2}}|\tau_{-h}(\eta
D\eta\tau_{h}u)|\,dx$
$\displaystyle\qquad+\int_{\Omega}|b||\tau_{-h}(\eta^{2}\tau_{h}u)|\,dx$

By the use of Young’s inequality in the first integral of the right hand side
we get

$\displaystyle\nu\int_{\Omega}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{\nu}{2}\int_{\Omega}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\quad+c(\nu)|h|^{2}\int_{\Omega}\eta^{2}(k(x+h)+k(x))^{2}(1+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{q}{2}}\,dx$
$\displaystyle+c\int_{\Omega}(1+|Du(x)|)^{\frac{p+q-2}{2}}|\tau_{-h}(\eta
D\eta\tau_{h}u)|\,dx$
$\displaystyle\qquad+\int_{\Omega}|b||\tau_{-h}(\eta^{2}\tau_{h}u)|\,dx$

Reabsorbing the first integral in the right hand side, we get

$\displaystyle\int_{\Omega}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
(3.11) $\displaystyle\leq$ $\displaystyle
c|h|^{2}\int_{\Omega}\eta^{2}(k(x+h)+k(x))^{2}(1+|Du(x)|+|Du(x+h)|)^{q}\,dx$
(3.17)
$\displaystyle+c\int_{\Omega}(1+|Du(x)|)^{\frac{p+q-2}{2}}|\tau_{-h}(\eta
D\eta\tau_{h}u)|\,dx$
$\displaystyle\qquad+\int_{\Omega}|b||\tau_{-h}(\eta^{2}\tau_{h}u)|\,dx=:I_{1}+I_{2}+I_{3}$

The assumption $k\in L^{r}$ with $r>n$ and Hölder’s inequality yields

$\displaystyle I_{1}$ $\displaystyle\leq$ $\displaystyle
c|h|^{2}\left(\int_{B_{R}}k(x)^{r}\,dx\right)^{\frac{2}{r}}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
(3.18) $\displaystyle=:$ $\displaystyle
c|h|^{2}||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}.$
(3.20)

The a-priori assumption $Du\in L^{\frac{np}{n-2}}_{\mathrm{loc}}(\Omega)$ and
again Hölder’s inequality imply

$\displaystyle I_{2}+I_{3}$ $\displaystyle\leq$ $\displaystyle
c\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p}{p-1}\frac{p+q-2}{2}}\,dx\right)^{\frac{p-1}{p}}\left(\int_{B_{t}}|\tau_{-h}(\eta\nabla\eta\tau_{h}u)|^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle\qquad+c\left(\int_{B_{t}}|b(x)|^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}\left(\int_{B_{t}}|\tau_{-h}(\eta^{2}\tau_{h}u)|^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle=$ $\displaystyle
c\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p}{p-1}\frac{p+q-2}{2}}\,dx\right)^{\frac{p-1}{p}}+\left(\int_{B_{t}}|b(x)|^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}\right]$
$\displaystyle\quad\cdot\left[\left(\int_{B_{t}}|\tau_{-h}(\eta
D\eta\tau_{h}u)|^{p}\,dx\right)^{\frac{1}{p}}+\left(\int_{B_{t}}|\tau_{-h}(\eta^{2}\tau_{h}u)|^{p}\,dx\right)^{\frac{1}{p}}\right]$
$\displaystyle\leq$ $\displaystyle
c|h|\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p}{p-1}\frac{p+q-2}{2}}\,dx\right)^{\frac{p-1}{p}}+\left(\int_{B_{R}}|b(x)|^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}\right]$
$\displaystyle\qquad\cdot\left[\left(\int_{B_{t}}|D(\eta
D\eta\tau_{h}u)|^{p}\,dx\right)^{\frac{1}{p}}+\left(\int_{B_{t}}|D(\eta^{2}\tau_{h}u)|^{p}\,dx\right)^{\frac{1}{p}}\right],$

where, in the last inequality, we used the first statement of Lemma 2.6.
Performing the calculations in the last two integrals, using the properties of
$\eta$ and assuming without loss of generality that $0<t-s<R<1$, we get

$\displaystyle I_{2}+I_{3}$ $\displaystyle\leq$ $\displaystyle
c|h|\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p}{p-1}\frac{p+q-2}{2}}\,dx\right)^{\frac{p-1}{p}}+||b||_{L^{\frac{p}{p-1}}(B_{R})}\right]$
(3.23)
$\displaystyle\qquad\cdot\left[\frac{c}{(t-s)^{2}}\left(\int_{B_{t}}|\tau_{h}u|^{p}\,dx\right)^{\frac{1}{p}}+\frac{c}{(t-s)}\left(\int_{B_{t}}\eta^{p}|\tau_{h}Du|^{p}\,dx\right)^{\frac{1}{p}}\right]$
$\displaystyle=$
$\displaystyle\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p}{p-1}\frac{p+q-2}{2}}\,dx\right)^{\frac{p-1}{p}}+||b||_{L^{\frac{p}{p-1}}(B_{R})}\right]$
(3.31)
$\displaystyle\qquad\qquad\cdot\left(\int_{B_{t}}|Du|^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle+\frac{c|h|}{t-s}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p}{p-1}\frac{p+q-2}{2}}\,dx\right)^{\frac{p-1}{p}}+||b||_{L^{\frac{p}{p-1}}(B_{R})}\right]$
$\displaystyle\qquad\qquad\cdot\left(\int_{B_{t}}\eta^{p}|\tau_{h}Du|^{p}\,dx\right)^{\frac{1}{p}}.$

Inserting (3.18) and (3.23) in (3.11) and using the notation
$p^{\prime}=\frac{p}{p-1}$, we obtain

$\displaystyle\int_{B_{t}}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
(3.32) $\displaystyle\leq$ $\displaystyle
c|h|^{2}||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
(3.34) $\displaystyle+$
$\displaystyle\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}\\!\\!+\\!||b||_{L^{p^{\prime}}(B_{R})}\right]\left(\int_{B_{t}}|Du|^{p}\,dx\right)^{\frac{1}{p}}$
(3.36) $\displaystyle+$
$\displaystyle\frac{c|h|}{t-s}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}\\!\\!\\!\\!+\\!||b||_{L^{p^{\prime}}(B_{R})}\right]\\!\\!\left(\int_{B_{t}}\eta^{p}|\tau_{h}Du|^{p}\,dx\right)^{\frac{1}{p}}$
(3.38)

By the assumption $1<p<2$, we have that

$\displaystyle\left(\int_{B_{t}}\eta^{p}|\tau_{h}Du|^{p}\,dx\right)^{\frac{1}{p}}$
(3.39) $\displaystyle=$
$\displaystyle\left(\int_{B_{t}}\eta^{p}|\tau_{h}Du|^{p}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p(p-2)}{4}}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p(2-p)}{4}}\,dx\right)^{\frac{1}{p}}$
(3.41) $\displaystyle\leq$
$\displaystyle\left(\int_{B_{t}}\eta^{2}|\tau_{h}Du|^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}\right)^{\frac{1}{2}}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{2p}},$
(3.43)

and, using (3.39) to estimate the last integral in the right hand side of
(3.32), we arrive at

$\displaystyle\nu\int_{B_{t}}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
(3.44) $\displaystyle\leq$ $\displaystyle
c|h|^{2}||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{qr}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
(3.56)
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}+||b||_{L^{p^{\prime}}(B_{R})}\right]$
$\displaystyle\qquad\qquad\cdot\left(\int_{B_{t}}(1+|Du|)^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle+\frac{c|h|}{(t-s)}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{p-1}{p}}+||b||_{L^{p^{\prime}}(B_{R})}\right]$
$\displaystyle\qquad\left(\int_{B_{t}}\eta^{2}|\tau_{h}Du|^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}\right)^{\frac{1}{2}}$
$\displaystyle\qquad\cdot\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{2p}}$

By the use of Young’s inequality, we get

$\displaystyle\nu\int_{B_{t}}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
(3.57) $\displaystyle\leq$ $\displaystyle
c|h|^{2}||k||^{2}_{L^{r}(B_{R})}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{qr}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
(3.65)
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}+||b||_{L^{p^{\prime}}(B_{R})}\right]\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}+||b||_{L^{p^{\prime}}(B_{R})}\right]^{2}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{p}}$
$\displaystyle\qquad\quad+\frac{\nu}{2}\left(\int_{B_{t}}\eta^{2}|\tau_{h}Du|^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}\right)$

Reabsorbing the last integral in the right hand side, we get

$\displaystyle\frac{\nu}{2}\int_{B_{t}}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
c|h|^{2}||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}+||b||_{L^{p^{\prime}}(B_{R})}\right]\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}+||b||_{L^{p^{\prime}}(B_{R})}\right]^{2}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}.$

By Young’s inequality in the second term in the right hand side of previous
estimate, we have

$\displaystyle\frac{\nu}{2}\int_{B_{t}}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
c|h|^{2}||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}+\frac{c|h|^{2}}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}}$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left[\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{1}{p^{\prime}}}+||b||_{L^{p^{\prime}}(B_{R})}\right]^{2}$
$\displaystyle\qquad\qquad\cdot\left[1+\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{p}}\right].$

So, by the elementary inequality $(a+b)^{2}\leq c(a^{2}+b^{2})$, we get

$\displaystyle\frac{\nu}{2}\int_{B_{t}}\eta^{2}(\mu^{2}+|Du(x)|^{2}+|Du(x+h)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
c|h|^{2}||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{2}{p^{\prime}}}$
$\displaystyle\qquad\qquad\cdot\left[1+\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{p}}\right]$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}||b||_{L^{p^{\prime}}(B_{R})}^{2}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
$\displaystyle+\frac{c|h|^{2}}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}}$

Dividing both sides of previous estimate by $|h|^{2}$ and using Lemma 2.2, we
have

$\displaystyle\int_{B_{t}}\eta^{2}\frac{|\tau_{h}V_{p}(Du)|^{2}}{|h|^{2}}\,dx$
(3.66) $\displaystyle\leq$ $\displaystyle
c||k||_{L^{p^{\prime}}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
(3.76)
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{2}{p^{\prime}}}$
$\displaystyle\qquad\qquad\cdot\left[1+\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{p}}\right]$
$\displaystyle+\frac{c}{(t-s)^{2}}||b||_{L^{p^{\prime}}(B_{R})}^{2}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}}$

Note the right hand side of (3.66) is finite by virtue of the a-priori
assumption $Du\in L^{\frac{2^{*}}{2}p}_{\mathrm{loc}}(\Omega)$ and the bound
at (3.7). Indeed, the bound on the gap at (3.7), implies that

$\displaystyle\begin{cases}p<\frac{qr}{r-2}<\frac{2^{*}}{2}p\\\ \text{and}\\\
p<\frac{p}{p-1}\frac{p+q-2}{2}<\frac{2^{*}}{2}p\end{cases}$ (3.77)

Indeed for $n>2$ we have $\frac{2^{*}}{2}=\frac{n}{n-2}$ and

$\frac{qr}{r-2}<\frac{np}{n-2}\,\,\Longleftrightarrow\,\,\frac{q}{p}<\frac{r-2}{r}\frac{n}{n-2}=1+\frac{2}{n-2}-\frac{2n}{r(n-2)}$

and

$\frac{2}{n-2}-\frac{2n}{r(n-2)}=\frac{2(r-n)}{r(n-2)}>\frac{1}{n}-\frac{1}{r}\,\,\Longleftrightarrow\,\,\frac{2}{n-2}>\frac{1}{n}.$

Moreover

$\frac{p}{p-1}\frac{p+q-2}{2}<\frac{np}{n-2}\,\,\Longleftrightarrow\,\,p+q-2<\frac{2n(p-1)}{n-2}$

i.e.

$q<\frac{2n(p-1)}{n-2}-p+2=p+\frac{2n(p-1)}{n-2}-2(p-1)=p+2(p-1)\left[\frac{n}{n-2}-1\right]$

and so

$\frac{q}{p}<1+\frac{4n(p-1)}{p(n-2)}$

On the other hand for $n=2$, since $\frac{2^{*}}{2}$ is any exponent larger
than $2$ inequalities (3.77) are trivially satisfied choosing $2^{*}$
sufficiently large.

Therefore, estimate (3.66), by the use of Lemma 2.7 with $p=2$, yields

$V_{p}(Du)\in W^{1,2}(B_{s})\cap L^{\frac{2^{*}}{2}}(B_{s}).$

Moreover by (2.3) in Lemma 2.3 and (2.7), we get

$\displaystyle\left(\int_{B_{s}}|Du|^{\frac{2^{*}}{2}p}\,dx\right)^{\frac{2}{2^{*}}}\leq
c(n)\left(\int_{B_{s}}\left(1+|V_{p}(Du)|^{2^{*}}\right)\,dx\right)^{\frac{2}{2^{*}}}$
(3.78) (3.79) $\displaystyle\quad\leq
c||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}\eta^{2}(1+|Du(x)|)^{\frac{rq}{r-2}}\,dx\right)^{\frac{r-2}{r}}$
(3.80) (3.81)
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{2}{p^{\prime}}}$
(3.82) (3.83)
$\displaystyle\qquad\qquad\cdot\left[1+\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\frac{2-p}{p}}\right]$
(3.84) (3.85)
$\displaystyle+\frac{c}{(t-s)^{2}}||b||_{L^{p^{\prime}}(B_{R})}^{2}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
(3.86) (3.87)
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}}$
(3.88)

Our next aim is to establish an estimate independent on the a-priori
assumption $Du\in L^{\frac{2^{*}}{2}p}_{\mathrm{loc}}(\Omega).$ Setting

$\mathbb{J}_{1}=\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{qr}{r-2}}\,dx\right)^{\frac{r-2}{r}}$

and

$\mathbb{J}_{2}=\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{p^{\prime}(p+q-2)}{2}}\,dx\right)^{\frac{2}{p^{\prime}}}$

we can rewrite (3.78) as follows

$\displaystyle\left(\int_{B_{s}}|Du|^{\frac{2^{*}}{2}p}\,dx\right)^{\frac{2}{2^{*}}}\leq
c||k||_{L^{r}(B_{R})}^{2}\,\,\mathbb{J}_{1}$ (3.89) (3.90)
$\displaystyle\quad+\frac{c\,\mathbb{J}_{2}}{(t-s)^{2}}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
(3.91) (3.92)
$\displaystyle\qquad+\frac{c}{(t-s)^{2}}||b||_{L^{p^{\prime}}(B_{R})}^{2}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
(3.93) (3.94)
$\displaystyle\quad\qquad+\frac{c}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}}.$
(3.95)

By the inequalities in (3.77), there exist $\vartheta_{1}\in(0,1)$ and
$\vartheta_{2}\in(0,1)$ such that

$1=\frac{\vartheta_{1}\frac{qr}{r-2}}{\frac{2^{*}p}{2}}+\frac{(1-\vartheta_{1})\frac{qr}{r-2}}{p}$

and

$1=\frac{\vartheta_{2}\frac{p}{p-1}\frac{p+q-2}{2}}{\frac{2^{*}p}{2}}+\frac{(1-\vartheta_{2})\frac{p}{p-1}\frac{p+q-2}{2}}{p}$

It is easy to check that

$\vartheta_{1}=\frac{2^{*}-2}{2}\frac{(q-p)r+2p}{qr}$

and

$\vartheta_{2}=\frac{2^{*}-2}{2}\frac{q-p}{p+q-2}$

Therefore, the interpolation inequality implies

$\displaystyle\mathbb{J}_{1}$ $\displaystyle\leq$
$\displaystyle\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{\vartheta_{1}\frac{qr}{r-2}}{\frac{2^{*}p}{2}}\frac{r-2}{r}}$
(3.98)
$\displaystyle\qquad\cdot\left(\int_{B_{t}}(1+|Du(x)|)^{p}\,dx\right)^{\frac{(1-\vartheta_{1})\frac{qr}{r-2}}{p}\frac{r-2}{r}}$
$\displaystyle=$
$\displaystyle\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2^{*}-2}{2}\frac{(q-p)r+2p}{pr}\frac{2}{2^{*}}}$
(3.102)
$\displaystyle\qquad\cdot\left(\int_{B_{t}}(1+|Du(x)|)^{p}\,dx\right)^{(1-\vartheta_{1})\frac{q}{p}}$

and

$\displaystyle\mathbb{J}_{2}$ $\displaystyle\leq$
$\displaystyle\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{\vartheta_{2}\frac{p^{\prime}(q+p-2)}{2}}{\frac{2^{*}p}{2}}\frac{2}{p^{\prime}}}$
(3.105)
$\displaystyle\qquad\cdot\left(\int_{B_{t}}(1+|Du(x)|)^{p}\,dx\right)^{\frac{(1-\vartheta_{2})\frac{p^{\prime}(q+p-2)}{2}}{p}\frac{2}{p^{\prime}}}$
$\displaystyle=$
$\displaystyle\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2^{*}-2}{2}\frac{q-p}{p}\frac{2}{2^{*}}}$
(3.109)
$\displaystyle\qquad\cdot\left(\int_{B_{t}}(1+|Du(x)|)^{p}\,dx\right)^{\frac{(1-\vartheta_{2})(q+p-2)}{p}},$

where we used the expressions of $\vartheta_{1},\vartheta_{2}$. Using
estimates (3.98) and (3.105) in (3.89) we obtain

$\displaystyle\left(\int_{B_{s}}|Du|^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2}{2^{*}}}$
(3.110) $\displaystyle\leq$ $\displaystyle
c||k||_{L^{r}(B_{R})}^{2}\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2^{*}-2}{2}\frac{(q-p)r+2p}{pr}\frac{2}{2^{*}}}\left(\int_{B_{t}}(1+|Du(x)|)^{p}\,dx\right)^{\frac{(1-\vartheta_{1})q}{p}}$
(3.120)
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2^{*}-2}{2}\frac{q-p}{p}\frac{2}{2^{*}}}\left(\int_{B_{t}}(1+|Du(x)|)^{p}\,dx\right)^{\frac{(1-\vartheta_{2})(q-p+2)}{p}}$
$\displaystyle\qquad\cdot\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
$\displaystyle+\frac{c}{(t-s)^{2}}||b||_{L^{p^{\prime}}(B_{R})}^{2}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}}.$

Let us note that

$\frac{(2^{*}-2)}{2}\frac{(q-p)r+2p}{rp}<1\quad\Longleftrightarrow\quad\frac{q}{p}<1+2\left(\frac{1}{n}-\frac{1}{r}\right)$

which is precisely (3.7), and

$\frac{(2^{*}-2)}{2}\frac{q-p}{p}<1\quad\Longleftrightarrow\quad\frac{q}{p}<1+\frac{2}{n}$

which is less restrictive than (3.7). Therefore by Young’s inequality with
exponents

$\frac{2rp}{(2^{*}-2)[r(q-p)+2p]},\qquad\frac{2rp}{2rp-(2^{*}-2)[r(q-p)+2p]}$

in the first term of the right hand side of (3.110) and with exponents

$\frac{2p}{(2^{*}-2)(q-p)},\qquad\frac{2p}{2p-(2^{*}-2)(q-p)}$

in the second term to get

$\displaystyle\left(\int_{B_{s}}|Du|^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2}{2^{*}}}\leq\frac{1}{2}\left(\int_{B_{t}}(1+|Du(x)|)^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2}{2^{*}}}$
(3.121) (3.122)
$\displaystyle\quad+c||k||_{L^{r}(B_{R})}^{\gamma_{1}}\left(\int_{B_{R}}(1+|Du(x)|)^{p}\,dx\right)^{\gamma_{2}}$
(3.123) (3.124)
$\displaystyle+\frac{c}{(t-s)^{2\gamma_{0}}}\left(\int_{B_{R}}(1+|Du(x)|)^{p}\,dx\right)^{\gamma_{3}}$
(3.125) (3.126)
$\displaystyle+\frac{c}{(t-s)^{2}}||b||_{L^{p^{\prime}}(B_{R})}^{2}\left[1+\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2-p}{p}}\right]$
(3.127) (3.128)
$\displaystyle+\frac{c}{(t-s)^{2}}\left(\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\frac{2}{p}},$
(3.129)

where $\gamma_{i}>1$ only depend on $p,q,r,n$, for $i=0,1,2,3$. The iteration
Lemma 2.1 implies

$\left(\int_{B_{\frac{R}{2}}}|Du|^{\frac{2^{*}p}{2}}\,dx\right)^{\frac{2}{2^{*}}}\leq\frac{c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}}{R^{2\gamma}}\left(1+\int_{B_{R}}(1+|Du|)^{p}\,dx\right)^{\gamma}$
(3.130)

for every ball $B_{R}\Subset\Omega$ for a constant $c=c(n,p,q,L,\nu)$ and with
a positive exponent $\gamma=\gamma(n,p,q,r)$. Finally the integrals
$\mathbb{J}_{1}$ and $\mathbb{J}_{2}$ can be estimated as follows

$\displaystyle\mathbb{J}_{1}\leq
c\left(\int_{B_{R}}(1+|Du(x)|)^{p}\,dx\right)^{\gamma_{4}}$ (3.131)

and

$\displaystyle\mathbb{J}_{2}\leq
c\left(\int_{B_{R}}(1+|Du(x)|)^{p}\,dx\right)^{\gamma_{5}},$ (3.132)

with $c=c(n,R,p,q,L,\nu,||k||_{L^{r}(B_{R})},||b||_{L^{p^{\prime}}(B_{R})})$
and positive exponents $\gamma_{4}=\gamma_{4}(n,p,q,r)$ and
$\gamma_{5}=\gamma_{5}(n,p,q,r)$. Therefore, inserting these estimates in
(3.66), we obtain

$\displaystyle\int_{B_{\frac{R}{2}}}\frac{|\tau_{h}V_{p}(Du)|^{2}}{|h|^{2}}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\vartheta}\left(1+\int_{B_{R}}(1+|Du(x)|)^{p}\,dx\right)^{\vartheta}$
(3.133)

where, again, $c=c(n,R,p,q,L,\nu)$ and $\vartheta=\vartheta(n,p,q,r)>0$.
Denoting by

$\displaystyle M^{2}$ $\displaystyle=$ $\displaystyle
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\vartheta}\left(\int_{B_{R}}(1+|Du(x)|)^{p}\,dx\right)^{\vartheta}$

we conclude, by the use of Lemma 2.7, that

$V_{p}(Du)\in W^{1,2}_{\mathrm{loc}}(\Omega)$

and therefore, by Lemma 2.4,

$D^{2}u\in L^{p}_{\mathrm{loc}}(\Omega).$

Moreover estimates (3.9) and (3.10) hold true. ∎

As a consequence of previous theorem, we deduce the following

######  Theorem 3.2.

Let $u\in W^{1,\frac{2^{*}p}{2}}_{\mathrm{loc}}(\Omega)$ be a solution to
(3.1) under the assumptions (1.2)–(1.4) with $1<p<q$ such that

$\frac{q}{p}<1+\min\left\\{\frac{1}{n}-\frac{1}{r},\,\frac{2(p-1)}{p(n-2)}\right\\}\qquad\text{if}\,\,n>2$
(3.134)

or

$\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r},\qquad\qquad\qquad\qquad\qquad\text{if}\,\,n=2.$
(3.135)

Then estimates (3.9) and (3.10) hold true.

###### Proof.

The proof follows as that of Theorem 5.1 in [36]. Let $t=2q-p$ then (1.2)–
(1.4) are nothing else but (3.2)– (3.4) with $t$ in place of $q$. Moreover

$\frac{t}{p}=\frac{2q}{p}-1$

and so the validity of (3.134) and (3.135) implies the validity of (3.7) and
(3.8), respectively. ∎

##  4\. Proof of Theorem 1.1

Let $\phi\in C_{c}^{\infty}(B_{1}(0))$, $\phi\geq 0$, be such that
$\int_{B_{1}(0)}\phi\,dx=1$ and let $\phi_{\varepsilon}$ be the corresponding
family of mollifiers. Let us set

$\tilde{\mathcal{A}}_{\varepsilon}(x,\xi)=\int_{B_{1}(0)}\phi(\omega)\mathcal{A}(x+\varepsilon\omega,\xi)\,d\omega,$

$\mathcal{A}_{\varepsilon}(x,\xi)=\tilde{\mathcal{A}}_{\varepsilon}(x,\xi)+\varepsilon(1+|\xi|^{2})^{\frac{q-2}{2}}\xi$

and

$b_{\varepsilon}=b\star\phi_{\varepsilon}.$

Let $u_{0}\in W^{1,\frac{p(q-1)}{p-1}}(\Omega)$, fix a ball
$B_{R}\Subset\Omega$ and let $v_{\varepsilon}\in u_{0}+W^{1,q}_{0}(\Omega)$ be
the solution to the Dirichlet problem

$\begin{cases}\mathrm{div}\mathcal{A}_{\varepsilon}(x,Dv_{\varepsilon})=b_{\varepsilon}\qquad\qquad\mathrm{in}\,\,\Omega\cr\cr
v_{\varepsilon}=u_{0}\quad\qquad\qquad\qquad\qquad\mathrm{on}\,\,\partial\Omega\end{cases}$
(4.1)

Note that

$|\mathcal{A}_{\varepsilon}(x,\xi)-\mathcal{A}_{\varepsilon}(x,\eta)|\leq|\xi-\eta|\left[L(\mu^{2}+1+|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}}+c_{q}\varepsilon(1+|\xi|^{2}+|\eta|^{2})^{\frac{q-2}{2}}\right]$
(4.2)

$\langle\mathcal{A}_{\varepsilon}(x,\xi)-\mathcal{A}_{\varepsilon}(x,\eta),\xi-\eta\rangle\geq\nu|\xi-\eta|^{2}(\mu^{2}+|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}}+\varepsilon
c_{q}|\xi-\eta|^{2}(\mu^{2}+|\xi|^{2}+|\eta|^{2})^{\frac{q-2}{2}}$ (4.3)

$|\mathcal{A}_{\varepsilon}(x,\xi)-\mathcal{A}_{\varepsilon}(y,\xi)|\leq|x-y|(k_{\varepsilon}(x)+k_{\varepsilon}(y))(1+|\xi|^{2})^{\frac{q-1}{2}}$
(4.4)

where

$k_{\varepsilon}=k\star\phi_{\varepsilon}.$

Note that assumption (1.5) implies that

$|\mathcal{A}_{\varepsilon}(x,\xi)|\leq(L+c_{q}\varepsilon)|\xi|(1+|\xi|^{2})^{\frac{q-1}{2}}$
(4.5)

Since the operator $\mathcal{A}_{\varepsilon}$ satisfies assumptions
(4.2)–(4.4) and $\frac{p(q-1)}{p-1}\geq q$, there exists a unique solution
$v_{\varepsilon}\in u_{0}+W^{1,q}(\Omega)$ and by the regularity results in
[45], we have that $v_{\varepsilon}\in W^{2,q}_{\mathrm{loc}}(\Omega)$ and
$V_{q}(Dv_{\varepsilon})=(\mu^{2}+|Dv_{\varepsilon}|^{2})^{\frac{q-2}{4}}Dv_{\varepsilon}\in
W^{1,2}_{\mathrm{loc}}(\Omega)$.
Therefore, by Lemma 2.3, $Dv_{\varepsilon}\in
L^{\frac{2^{*}q}{2}}_{\mathrm{loc}}(\Omega)\subset
L^{\frac{2^{*}p}{2}}_{\mathrm{loc}}(\Omega)$ and so we are legitimate to apply
estimates (3.9) and (3.10) to each $v_{\varepsilon}$ to obtain that

$\displaystyle\int_{B_{\frac{r}{2}}}|D^{2}v_{\varepsilon}|^{p}\,dx\leq
c(1+||k_{\varepsilon}||_{L^{r}(B_{R})}+||b_{\varepsilon}||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{r}}(1+|Dv_{\varepsilon}(x)|)^{p}\,dx\right)^{\gamma}$
(4.6)

and

$\displaystyle\int_{B_{\frac{r}{2}}}|D(V_{p}(Dv_{\varepsilon}))|^{2}\,dx\leq
c(1+||k_{\varepsilon}||_{L^{r}(B_{R})}+||b_{\varepsilon}||_{L^{p^{\prime}}(B_{R})})^{\vartheta}\left(1+\int_{B_{r}}(1+|Dv_{\varepsilon}(x)|)^{p}\,dx\right)^{\vartheta},$
(4.7)

for all $B_{r}\Subset B_{R}$. Moreover by assumption (4.3) and since
$v_{\varepsilon}$ solves problem (4.1) we get

$\displaystyle\nu\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{2}(\mu^{2}+|Dv_{\varepsilon}|^{2}+|Du_{0}|^{2})^{\frac{p-2}{2}}\,dx\leq\int_{\Omega}\langle\mathcal{A}_{\varepsilon}(x,Dv_{\varepsilon})-\mathcal{A}_{\varepsilon}(x,Du_{0}),Dv_{\varepsilon}-Du_{0}\rangle\,dx$
(4.8) $\displaystyle=$
$\displaystyle\int_{\Omega}\langle\mathcal{A}_{\varepsilon}(x,Dv_{\varepsilon}),Dv_{\varepsilon}-Du_{0}\rangle\,dx-\int_{\Omega}\langle\mathcal{A}_{\varepsilon}(x,Du_{0}),Dv_{\varepsilon}-Du_{0}\rangle\,dx$
(4.10) $\displaystyle=$ $\displaystyle\int_{\Omega}\langle
b_{\varepsilon},Dv_{\varepsilon}-Du_{0}\rangle\,dx-\int_{\Omega}\langle\mathcal{A}_{\varepsilon}(x,Du_{0}),Dv_{\varepsilon}-Du_{0}\rangle\,dx$
(4.12) $\displaystyle\leq$
$\displaystyle\int_{\Omega}|b_{\varepsilon}||Dv_{\varepsilon}-Du_{0}|\,dx+\int_{\Omega}|\mathcal{A}_{\varepsilon}(x,Du_{0})||Dv_{\varepsilon}-Du_{0}|\,dx.$
(4.14)

As long as $1<p<2$, we can use Young’s inequality with exponents $\frac{2}{p}$
and $\frac{2}{2-p}$ as follows

$\displaystyle\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{p}\,dx$ (4.15)
$\displaystyle=$
$\displaystyle\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{p}(\mu^{2}+|Dv_{\varepsilon}|^{2}+|Du_{0}|^{2})^{\frac{p-2}{2}\frac{p}{2}}(\mu^{2}+|Dv_{\varepsilon}|^{2}+|Du_{0}|^{2})^{\frac{2-p}{2}\frac{p}{2}}\,dx$
(4.17) $\displaystyle\leq$
$\displaystyle\nu\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{2}(\mu^{2}+|Dv_{\varepsilon}|^{2}+|Du_{0}|^{2})^{\frac{p-2}{2}}\,dx$
(4.21)
$\displaystyle\qquad+c(\nu)\int_{\Omega}(\mu^{2}+|Dv_{\varepsilon}|^{2}+|Du_{0}|^{2})^{\frac{p}{2}}\,dx$

Using (4.8) to estimate the first term in the right hand side of (4.15), we
get

$\displaystyle\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{p}\,dx$ (4.22)
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{p}\,dx+c\int_{\Omega}|b_{\varepsilon}|^{\frac{p}{p-1}}+c\int_{\Omega}|\mathcal{A}_{\varepsilon}(x,Du_{0})|^{\frac{p}{p-1}},$
(4.24)

where, in the last line, we used Young’s inequality. Reabsorbing the first
integral in the right hand side by the left hand side and using (4.5) we have

$\displaystyle\int_{\Omega}|Dv_{\varepsilon}-Du_{0}|^{p}\,dx\leq
c\int_{\Omega}|b_{\varepsilon}|^{\frac{p}{p-1}}+c\int_{\Omega}|\mathcal{A}_{\varepsilon}(x,Du_{0})|^{\frac{p}{p-1}}$
$\displaystyle\leq$ $\displaystyle
c\int_{\Omega}|b|^{\frac{p}{p-1}}+c\int_{\Omega}|Du_{0}|^{\frac{p(q-1)}{p-1}}$

and so

$\displaystyle\int_{\Omega}|Dv_{\varepsilon}|^{p}\,dx\leq
c\int_{\Omega}|b|^{\frac{p}{p-1}}+c\int_{B_{R}}|Du_{0}|^{\frac{p(q-1)}{p-1}}$
(4.25)

where we used that $b_{\varepsilon}$ strongly converges to $b$ in
$L^{\frac{p}{p-1}}(\Omega)$. Estimate (4.25) implies that $v_{\varepsilon}$ is
a bounded sequence in $u_{0}+W^{1,p}_{0}(\Omega)$ and therefore there exists
$v\in u_{0}+W^{1,p}_{0}(\Omega)$ such that

$v_{\varepsilon}\rightharpoonup v\qquad\text{weakly in
}\,\,u_{0}+W^{1,p}_{0}(\Omega).$

On the other hand, since $b_{\varepsilon}$ strongly converges to $b$ in
$L^{\frac{p}{p-1}}_{\mathrm{loc}}(\Omega)$ and $k_{\varepsilon}$ strongly
converges to $k$ in $L^{r}_{\mathrm{loc}}(\Omega)$, using (4.25) in (4.6) and
in (4.7), we get

$\displaystyle\int_{B_{\frac{R}{2}}}|D^{2}v_{\varepsilon}|^{p}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{R}}|Du_{0}|^{\frac{p(q-1)}{p-1}}\right)^{\gamma}$
(4.26)

and

$\displaystyle\int_{B_{\frac{R}{2}}}|D(V_{p}(Dv_{\varepsilon}))|^{2}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\vartheta}\left(1+\int_{B_{R}}|Du_{0}|^{\frac{p(q-1)}{p-1}}\right)^{\vartheta},$
(4.27)

for every balls $B_{R}\Subset\Omega$. Estimate (4.26) implies that
$v_{\varepsilon}$ is a bounded sequence in $W^{2,p}_{\mathrm{loc}}(\Omega)$
and therefore

$v_{\varepsilon}\rightharpoonup v\qquad\text{weakly in
}\,\,W^{2,p}_{\mathrm{loc}}(\Omega)$

and so

$v_{\varepsilon}\to v\qquad\text{strongly in
}\,\,W^{1,p}_{\mathrm{loc}}(\Omega).$

Passing to the limit as $\varepsilon\to 0$ in (4.26), by the lower
semicontinuity of the norm, we obtain

$\displaystyle\int_{B_{\frac{r}{2}}}|D^{2}v|^{p}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\gamma}\left(1+\int_{B_{R}}|Du_{0}|^{\frac{p(q-1)}{p-1}}\right)^{\gamma}$
(4.28)

The continuity of the map $DV_{p}(\xi)$ implies that the sequence
$DV_{p}(Dv_{\varepsilon})$ converges, up to a subsequence, a.e. to
$DV_{p}(Dv)$. Therefore by Fatou’s Lemma Passing to the limit as
$\varepsilon\to 0$ in (4.27) we also have

$\displaystyle\int_{B_{\frac{r}{2}}}|D(V_{p}(Dv))|^{2}\,dx\leq
c(1+||k||_{L^{r}(B_{R})}+||b||_{L^{p^{\prime}}(B_{R})})^{\vartheta}\left(1+\int_{B_{R}}|Du_{0}|^{\frac{p(q-1)}{p-1}}\right)^{\vartheta}.$
(4.29)

It remains to prove that $v$ satisfies (4.1). To this aim, we observe that

$\displaystyle\int_{\Omega}\langle\mathcal{A}(x,Dv),D\varphi\rangle\,dx-\int_{\Omega}\langle
b,D\varphi\rangle\,dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}\langle\mathcal{A}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv),D\varphi\rangle\,dx+\int_{\Omega}\langle\tilde{\mathcal{A}}_{\varepsilon}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv_{\varepsilon}),D\varphi\rangle\,dx$
$\displaystyle+\int_{\Omega}\langle\tilde{\mathcal{A}}_{\varepsilon}(x,Dv_{\varepsilon}),D\varphi\rangle\,dx-\int_{B_{R}}\langle
b,D\varphi\rangle\,dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}\langle\mathcal{A}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv),D\varphi\rangle\,dx+\int_{B_{R}}\langle\tilde{\mathcal{A}}_{\varepsilon}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv_{\varepsilon}),D\varphi\rangle\,dx$
$\displaystyle+\int_{\Omega}\langle
b_{\varepsilon}-b,D\varphi\rangle\,dx-\varepsilon\int_{\Omega}\langle(1+|Dv_{\varepsilon}|^{2})^{\frac{q-2}{2}}Dv_{\varepsilon},D\varphi\rangle\,dx,$

for every $\varphi\in C^{1}_{c}(\Omega)$. Therefore, we are left to prove that
the right hand side of previous equality vanishes as $\varepsilon\to 0$. This
will come if we show that

$\lim_{\varepsilon\to 0}I_{1}^{\varepsilon}:=\lim_{\varepsilon\to
0}\left|\int_{\Omega}\langle\mathcal{A}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv),D\varphi\rangle\,dx\right|=0$
$\lim_{\varepsilon\to 0}I_{2}^{\varepsilon}:=\lim_{\varepsilon\to
0}\left|\int_{\Omega}\langle\tilde{\mathcal{A}}_{\varepsilon}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv_{\varepsilon}),D\varphi\rangle\,dx\right|=0$
$\lim_{\varepsilon\to 0}I_{3}^{\varepsilon}:=\lim_{\varepsilon\to
0}\left|\int_{\Omega}\langle b_{\varepsilon}-b,D\varphi\rangle\,dx\right|=0$

and

$\lim_{\varepsilon\to
0}\varepsilon\left|\int_{\Omega}\langle(1+|Dv_{\varepsilon}|^{2})^{\frac{q-2}{2}}Dv_{\varepsilon},D\varphi\rangle\,dx\right|=0$

Last two estimate are obvious since $b_{\varepsilon}\to b$ strongly in
$L^{\frac{p}{p-1}}(\Omega)$ and since

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon\left|\int_{\Omega}\langle(1+|Dv_{\varepsilon}|^{2})^{\frac{q-2}{2}}Dv_{\varepsilon},D\varphi\rangle\,dx\right|$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}\varepsilon||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\mathrm{supp}\varphi}(1+|Dv_{\varepsilon}|^{2})^{\frac{q-1}{2}}\,dx=0$

by virtue of the bound $q<\frac{np}{n-2}$ and of estimate (4.27) and where we
denoted by $\Omega^{\prime}$ a compact set that contains the support of the
test function $\varphi$. For what concerns $I_{1}$ we have by the definition
of $\tilde{\mathcal{A}}_{\varepsilon}$ that

$\displaystyle\lim_{\varepsilon\to 0}I_{1}^{\varepsilon}$ $\displaystyle\leq$
$\displaystyle\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}|\mathcal{A}(x,Dv)-\tilde{\mathcal{A}}_{\varepsilon}(x,Dv)|\,dx$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}\left|\mathcal{A}(x,Dv)-\int_{B_{1}(0)}\phi(\omega)\mathcal{A}(x+\varepsilon\omega,Dv)\,d\omega\right|\,dx$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime}))}\int_{\Omega^{\prime}}\int_{B_{1}(0)}\phi(\omega)\Big{|}\mathcal{A}(x,Dv)-\mathcal{A}(x+\varepsilon\omega,Dv)\Big{|}\,d\omega\,dx$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}\varepsilon||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}\int_{B_{1}(0)}\phi(\omega)k(x)+k(x+\varepsilon\omega)(1+|Dv|^{2})^{\frac{q-1}{2}}\Big{|}\,d\omega\,dx$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}\varepsilon||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}(k(x)+k_{\varepsilon}(x))(1+|Dv|^{2})^{\frac{q-1}{2}}\,dx$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}\varepsilon||D\varphi||_{L^{\infty}(\Omega^{\prime})}\left(\int_{\Omega^{\prime}}(k(x)+k_{\varepsilon}(x))^{r}\right)^{\frac{1}{r}}\left(\int_{\Omega^{\prime}}(1+|Dv|^{2})^{\frac{q-1}{2}\frac{r}{r-1}}\,dx\right)^{\frac{r-1}{r}}=0$

since $k_{\varepsilon}\to k$ strongly in $L^{r}(\Omega)$, by the bound

$\frac{q}{2}\frac{r}{r-1}<\frac{qr}{r-2}<\frac{2^{*}p}{2}$

and again by (4.7). For the estimate of $I_{2}^{\varepsilon}$, we observe that

$\displaystyle\lim_{\varepsilon\to 0}I_{2}^{\varepsilon}$ $\displaystyle\leq$
$\displaystyle\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}|\mathcal{A}_{\varepsilon}(x,Dv)-\mathcal{A}_{\varepsilon}(x,Dv_{\varepsilon})|\,dx$
$\displaystyle\leq$ $\displaystyle\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}\int_{B_{1}(0)}\phi(\omega)\Big{|}\mathcal{A}(x+\varepsilon\omega,Dv)-\mathcal{A}(x+\varepsilon\omega,Dv_{\varepsilon})\Big{|}\,d\omega\,dx$
$\displaystyle\leq$ $\displaystyle L\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}|Dv-
Dv_{\varepsilon}|(\mu^{2}+|Dv|^{2}+|Dv_{\varepsilon}|^{2})^{\frac{p-2}{2}}\,dx$
$\displaystyle+c\varepsilon\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\int_{\Omega^{\prime}}|Dv-
Dv_{\varepsilon}|(1+|Dv|^{2}+|Dv_{\varepsilon}|^{2})^{\frac{q-2}{2}}\,dx$
$\displaystyle\leq$ $\displaystyle L\lim_{\varepsilon\to
0}||D\varphi||_{L^{\infty}(\Omega^{\prime})}\left(\int_{\Omega^{\prime}}|Dv-
Dv_{\varepsilon}|^{p}\,dx\right)^{\frac{1}{p}}$
$\displaystyle\qquad\cdot\left(\int_{\Omega^{\prime}}(1+|Dv|^{2}+|Dv_{\varepsilon}|^{2})^{\frac{q-2}{2}\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}=0,$

since $Dv_{\varepsilon}\to Dv$ strongly in $L^{p}(\mathrm{supp}\varphi)$ and
by virtue of the bound

$\frac{q-2}{2}\frac{p}{p-1}<\frac{q-1}{2}\frac{p}{p-1}<\frac{np}{n-2}$

and again by (4.7). With these estimates at our disposal, we conclude that for
every $\varphi\in C^{1}_{c}(\Omega)$

$\int_{\Omega}\langle\mathcal{A}(x,Dv),D\varphi\rangle\,dx=\int_{\Omega}\langle
b,D\varphi\rangle\,dx,$

then $v\in(u_{0}+W^{1,p}_{0}(\Omega))\cap W^{2,p}_{\mathrm{loc}}(\Omega)$ is a
solution to (4.1).

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