# Matrix weights and regularity for degenerate elliptic equations

Giuseppe Di Fazio, Maria Stella Fanciullo, Dario Daniele Monticelli, Scott
Rodney, Pietro Zamboni

###### Abstract.

We prove local boundedness, Harnack inequality and local regularity for weak
solutions of quasilinear degenerate elliptic equations in divergence form.
Degeneracy is via a non negative, symmetric, measurable matrix-valued function
$Q(x)$ and two suitable non negative weight functions. We setup an axiomatic
approach in terms of suitable geometric conditions and local Sobolev –
Poincaré inequalities. Data integrability is close to $L^{1}$ and it is
exploited in terms of suitable version of Stummel-Kato class that in some
cases is also necessary to the regularity.

##  1\. Introduction

The aim of this note is to study the local behaviour of solutions to
quasilinear degenerate elliptic equations of the following kind

(1) $\displaystyle-\textrm{div}\left(h(x)\left|\sqrt{Q(x)}\nabla
u(x)\right|^{p-2}Q(x)\nabla u(x)\right)=m(x)|f(x)|^{p-2}f(x)$

that admits two different types of degeneracy.

The first one is given by a non negative definite, symmetric, measurable
matrix-valued function $Q(x)$ associated with the principal part of the
differential operator. The other one is expressed in terms of two non negative
weight functions $h$ and $m$ that can vanish or blow up.

We assume that $f$ lies in a natural 2-weight generalization of the Stummel-
Kato class (see [1] and [3]) and prove local boundedness and Harnack
inequality for weak solutions of equation (1) (see Theorems 5.4 and 5.7). We
accomplish this using Moser’s iteration technique (see [42]) and a Fefferman-
Phong inequality adapted to our setting. As a consequence, we obtain local
Hölder continuity of weak solutions (see Theorem 5.8 and Remark 5.9).

Each of our results is in the spirit of [32] and [33] where the authors
investigate quasilinear degenerate elliptic equations of the type

(2) $\text{\rm div }A(x,u,\nabla u)+B(x,u,\nabla u)=0\,$

for functions
$A:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
and $B:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
that satisfy Trudinger - Serrin type structural conditions. In the case
$h=m=1$, our equation (1) is also of this form. However, in [32] and [33], the
data function $f$ is required to satisfy minimal $L^{p}$ or Morrey type
restrictions stated in terms of the gain in a Sobolev-type inequality. As
such, our work generalizes [32] and [33] for the special case of equation (1)
in two ways. First, we use an approach that allows the incorporation of the
weights $h,m$, and second, we allow a new non-$L^{p}$ restriction on the data
function $f$.

Concerning the study of the regularity properties of solutions of quasilinear
elliptic equations of the kind (2), the first paper where non - $L^{p}$
conditions appear is [37], where a simplified form of (2) was studied with
right hand side being a measure in a Morrey space. Local Hölder continuity of
weak solutions is proved using De Giorgi’s methods, as adapted by
Ladyzhenskaya and Ural’tseva in [28] to quasilinear equations. Improvements of
[37] can be found in [36] and [29] where $L^{p}$ assumptions are weakened or
replaced by Morrey space restrictions. Similar investigations were made in
[43] with Stummel-Kato type requirements on the lower order coefficients
generalizing [1] and [3] to quasilinear elliptic equations.

Aside from minimal data restrictions, many authors have also investigated the
regularity of weak solutions of (1) where degeneracy is encoded by the weights
$h,m$ and the matrix function $Q$. For example, when $Q(x)$ has non negative
minimum eigenvalue given by an appropriate $A_{\infty}$ weight, local
regularity of solutions was studied by Fabes, Kenig, and Serapioni in [20]
with further progress in [5] among others. In each of these works, continuity
results are obtained using a Sobolev inequality adapted to $Q$ and an
iterative argument of Moser type. Even when the minimum eigenvalue is not
$A_{\infty}$, some progress has been made in [27] where they allow the matrix
$Q$ to vanish to inifinite order at an interior point of a domain. Similarily,
[19] and [11] obtain similar results for data in appropriate weighted Stummel-
Kato type classes.

Many regularity results for weak solutions of elliptic equations have also
been generalized to the Carnot Caratheodory (CC) spaces where a metric is
generated using sub-unit curves associated to a system of non-commuting vector
fields, and these vector fields define the matrix $Q$. In this direction we
quote [2], [9], [10], [17] and [18] where the above mentioned results are
obtained in the subelliptic setting through Morrey and Stummel-Kato type
assumptions on lower order terms.

The first attempt to consider together the several types of degeneracy
described above has been exploited by Franchi, Gutierrez and Wheeden in [21]
and [22], where they proved Harnack inequality for non negative weak solutions
of equation

$\text{\rm div }(w(z)\nabla_{\lambda}u(z))=0$

where $w$ is a suitable power of a strong $A_{\infty}$ weight, the $\lambda$
gradient is $\nabla_{\lambda}u(z)=(\nabla_{x}u(z)$,
$\lambda(x)\nabla_{y}u(z))$ and $z=(x,y)$. We remark that equations studied in
[21] and [22] do not contain lower order terms. We also note that [14] and
[15] extend the results contained in [21] and [22] to more general equations
with lower order terms in suitable Stummel-Kato classes.

The paper is organized as follows: in Section 2 we collect the axiomatic
assumptions under which we prove our results concerning weak solutions of
equation (1) and we provide some preliminary results; in Section 3 we describe
our functional setting; in Section 4 we introduce the Stummel-Kato class
adapted to our setting and we prove a Fefferman-Phong inequality; lastly, in
Section 5 we state and prove our local regularity results and the Harnack
inequality for weak solutions of equation (1).

##  2\. Preliminaries

Fix a bounded domain $\Omega$ of $\mathbb{R}^{n}$ and assume it is endowed
with a metric $\rho$ that generates a topology equivalent to the Euclidean
one. We will denote respectively by

$B_{r}(x)=B(x,r)=\\{y\in\Omega\,|\,\rho(x,y)<r\\}$

and $D_{r}(x)=D(x,r)$ respectively the $\rho$-metric ball and the Euclidean
ball centred at $x\in\Omega$ with radius $r>0$. By our assumptions, every
metric ball $B$ is an open set; moreover, for every small enough radius $r$,
the Euclidean closure of the metric ball $B(x,r)$ is contained in $\Omega$.

Indeed, we have

######  Lemma 2.1.

Let $E,E_{0}$ be bounded open domains such that $\overline{E}\subset E_{0}$,
$\overline{E}_{0}\subset\Omega$. Then there exists $\delta>0$ such that for
every $x\in E$, $0<r<\delta$ one has

(3) $\overline{B(x,r)}\subseteq\\{y\in\Omega\,|\,\rho(x,y)\leq r\\}\subset
E_{0}$

###### Proof.

Arguing by contradiction, if the result does not hold one can find two
sequences $x_{n}$, $y_{n}$ such that for every $n\in\mathbb{N}$ one has
$x_{n}\in E$, $y_{n}\in\partial E_{0}$ and $\rho(x_{n},y_{n})<\frac{1}{n}$.
Since $\overline{E}$ is compact, up to a subsequence we have that $x_{n}$
converges to some $\bar{x}\in\overline{E}$, in the Euclidean sense. Since the
topologies are equivalent we see that $\rho(\bar{x},x_{n})$ tends to $0$, and
thus by the triangle inequality we deduce that also $\rho(\bar{x},y_{n})$
tends to $0$. Hence, again by the equivalence of the topologies, we have that
$y_{n}$ converges to $\bar{x}$, so that $\bar{x}\in\partial E_{0}$. Then
$\overline{E}\cap\partial E_{0}\neq\emptyset$, a contradiction. ∎

We will write $r=r(B)$ to denote the radius of a given ball $B$. For
simplicity of presentation, given any positive constant $c$, we denote by $cB$
the ball concentric to $B$ with radius $cr(B)$. Given any two balls
$B_{1}=B(x,r_{1})$, $B_{2}=B(y,r_{2})$ with non-empty intersection and radii
$r_{1}\geq r_{2}$, we remind the reader that one has $B_{2}\subset 3B_{1}$,
the standard swallowing result for metric balls.

With a view to studying regularity for degenerate elliptic operators of second
order with rough coefficients, let $\mu$ be a given measure and
$Q:\Omega\rightarrow S_{n}$ be a $\mu$-measurable matrix function taking
values in $S_{n}$, the collection of all non negative definite $n\times n$
symmetric matrices. We will always assume that the $p^{th}$ power of the
pointwise operator norm of $\sqrt{Q}$ is a weight - that is,
$v=\|Q(x)\|_{op}^{p/2}\in L^{1}_{loc}(\Omega;\mu)$, where

$\|Q(x)\|_{op}:=\sup_{\xi\in\mathbf{R}^{n},|\xi|=1}\langle\xi,Q(x)\xi\rangle.$

As an application, we consider principal second order equations of the form

(4) $\displaystyle-\textrm{div}\left(h(x)\left|\sqrt{Q(x)}\nabla
u(x)\right|^{p-2}Q(x)\nabla u(x)\right)=m(x)|f(x)|^{p-2}f(x)$

for $x\in\Omega$ and $f$ in the Stummel-Kato class of functions defined below.

We will always assume that the non negative weight functions $h$ and $m$ are
in $L^{1}_{loc}({\mathbb{R}}^{n})$.

We will be working in an axiomatic setting where we will assume the validity
of geometric and analytic assumptions that we list below.

  * (H1)

Segment property. For every $x,y\in\Omega$ there exists a continuous curve
$\gamma:I\subset\mathbb{R}\rightarrow\Omega$ connecting $x,y$ such that
$\rho(\gamma(s),\gamma(t))=|s-t|$ for every $s,t\in I$.

  * (H2)

The $\sigma$-finite measures $\mu=h\,dx$, $\nu=m\,dx$ are both doubling for
balls $B$ with $2B\Subset\Omega$. Hence, there are constants
$A_{\nu},A_{\mu}>0$ so that for any such ball

(5) $\displaystyle\nu(2B)\leq A_{\nu}\nu(B)\qquad\mu(2B)\leq A_{\mu}\mu(B).$

We also assume that $\mu$ is reverse-doubling of order 1, i.e. there exists a
constant $C>0$ such that for every pair of balls $\tilde{B}\subset
B\Subset\Omega$ one has

$\mu(B)\geq C\left(\frac{r(B)}{r(\tilde{B})}\right)\mu(\tilde{B}).$

  * (H3)

We require that $d\mu\ll d\nu$, where $d\mu=h\,dx$ and $d\nu=m\,dx$. Moreover,
we also assume that $\nu(E)>0$ for every Lebesgue measurable set
$E\subset\Omega$ having positive Lebesgue measure. This is equivalent to
assume that $m>0$ Lebesgue-a.e. Finally we require that there exists $C>0$
such that for every compact subset $E\subset\Omega$ we have

(6) $\int_{E}v~{}d\mu\leq C\int_{E}~{}d\nu$

where we recall that $v=\|Q\|^{p/2}_{op}$. Note that this is equivalent to
$vh\leq Cm$ Lebesgue-a.e.

  * (H4)

Local Poincaré inequality. There exists a constant $C_{0}>0$ so that given any
ball $B\Subset\Omega$ one has

(7) $\frac{1}{\nu(B)}\int_{B}|u-u_{B}|~{}d\nu\leq
C_{0}\frac{r(B)}{\mu(B)}\int_{B}|\sqrt{Q}\nabla u|~{}d\mu$

for every $u\in Lip_{loc}(\Omega)$, the collection of all locally Lipschitz
functions defined on $\Omega$, where $u_{B}=\frac{1}{\nu(B)}\int_{B}u\,d\nu$.

  * (H5)

Local Sobolev inequality. For any $p\geq 1$ there exists constants $C_{1}>0$,
$k>1$ so that given any ball $B\Subset\Omega$ one has

(8) $\left(\frac{1}{\nu(B)}\int_{B}|u|^{kp}~{}d\nu\right)^{\frac{1}{kp}}\leq
C_{1}r(B)\left(\frac{1}{\mu(B)}\int_{B}|\sqrt{Q}\nabla
u|^{p}~{}d\mu\right)^{\frac{1}{p}}$

for every $u\in Lip_{0}(B)$, the collection of all Lipschitz functions
compactly supported in $B$.

  * (H6)

Accumulating sequences of Lipschitz cut-off functions.

Given any metric ball $B=B(x,r)$ such that $B\Subset\Omega$ there exists a
sequence of functions $\varphi_{j}\in Lip_{0}(B)$ so that for every
$j\in\mathbb{N}$

    1. (1)

$0\leq\varphi_{j}(x)\leq 1$,

    2. (2)

$\varphi_{j}(x)=1$ for $x\in\tau B$,

    3. (3)

$\operatorname{supp}\varphi_{j+1}\subset\\{y\in B\,:\,\varphi_{j}(y)=1\\}$,

    4. (4)

$\displaystyle\left\|\frac{|\sqrt{Q}\nabla\varphi_{j}|}{v^{\frac{1}{p}}}\right\|_{\infty}\leq\displaystyle\frac{\tilde{C}T^{j}}{r^{\gamma}}$.

where $\gamma,\tilde{C}>0,T\geq 1$, $0<\tau<1$ are positive constants.

######  Remark 2.2.

Since $\nu$, $\mu$ are doubling measures on the collection of (open) metric
balls by (H2), it is easy to see that no open set can have measure zero.

Concerning condition (H6)-(4), we note that, by the definition of $v$, one
always has

$\left\|\frac{|\sqrt{Q}\nabla\varphi|}{v^{\frac{1}{p}}}\right\|_{\infty}\leq\|\nabla\varphi\|_{\infty}$

for every $\varphi\in Lip(\Omega)$.

Note that, since $d\nu$ is doubling for balls $B$ with closure in $\Omega$, we
also have that $(\Omega,\rho)$ is geometrically doubling for $\varrho$ \-
balls. That is, there exists a constant $M\geq 1$ such that if $B$ is a ball
satisfying $2B\Subset\Omega$, then it may contain at most $M$ centres of
disjoint metric balls of radius $r(B)/2$. The connection between doubling
properties for measures and geometric doubling for the collection of balls
$\\{B\\}$ whose doubles have closure in $\Omega$ can be further explored in
[25, 26].

This property is useful on a scale of radii that we present in the following
lemma.

######  Lemma 2.3.

If $(\Omega,\rho)$ is a geometrically doubling metric space and the ball
$B=B(x,r)$ satisfies $2B\Subset\Omega$, then it may contain at most $Ms^{-Q}$
centers of disjoint balls of radius $sr$, with $0<s<1$, where $Q=\log_{2}(M)$
and $M$ is as above.

The following covering lemma will play an important role in the proof of
Corollary 4.7. We note that this is a simplified version of [4, Lemma 3.12]
but we will prove it for the reader’s convenience.

######  Lemma 2.4.

Fix any $c^{*}\geq 1$. Then given any compact subset $E$ of $\Omega$ for every
$r>0$ small enough there exists a finite collection of metric balls
$\\{B_{j}\\}_{j=1}^{N}=\\{B(z_{j},r)\\}_{j=1}^{N}$ centred in $E$, with
$N=N(r)$, with closure contained in $\Omega$ and such that

  1. (1)

$E\subset\bigcup_{j}B_{j}$,

  2. (2)

$\left\\{\frac{1}{3}B_{j}\right\\}_{j}$ is a pairwise disjoint collection,

  3. (3)

$\sum_{j}\chi_{c^{*}B_{j}}(x)\leq(18c^{*})^{Q}$ for every $x\in\Omega$, with
$Q$ as in Lemma 2.3 (uniformly finite overlap).

###### Proof.

Fix a compact $E\subset\Omega$, and let $r>0$ be small enough so that
$B(x,6c^{*}r)\Subset\Omega$ for every $x\in E$. Since metric balls are open
sets, cover $E$ with a finite collection of balls
$\\{\tilde{B}_{j}\\}_{j=1}^{m}=\\{\tilde{B}(z_{j},\frac{r}{3})\\}_{j=1}^{m}$.
We extract a pairwise disjoint subcollection $\\{\hat{B}_{k}\\}_{k=1}^{N}$ of
these via the following process. Set $\hat{B}_{1}=\tilde{B}_{1}$. From
$\\{\tilde{B}_{2},...,\tilde{B}_{m}\\}$, select $j$ so that
$\hat{B}_{1}\cap\tilde{B}_{j}=\emptyset$ and set $\hat{B}_{2}=\tilde{B}_{j}$.
If there is no such $j$, stop the process. Continue as directed until no ball
can be found which has empty intersection with each of the balls previously
found. From this new collection,
$\\{\hat{B}_{k}\\}_{k=1}^{N}\subset\\{\tilde{B}_{j}\\}_{j=1}^{m}$, we set
$B_{k}=3\hat{B}_{k}\supseteq\hat{B}_{k}$ and now claim that
$\\{B_{k}\\}_{k=1}^{N}$ satisfies items $1-3$. Item 2 is clear since the balls
$\frac{1}{3}{B}_{k}=\hat{B}_{k}$ form a pairwise disjoint family by
construction. To see item $1$, let $x\in E$ and choose $1\leq j\leq m$ so that
$x\in\tilde{B}_{j}$. If $\tilde{B}_{j}=\hat{B}_{k}$ for some $1\leq k\leq N$
we are finished. If not, then there exists a $1\leq k\leq N$ so that
$\tilde{B}_{j}\bigcap\hat{B}_{k}\neq\emptyset$. Then we see that $x\in
3\hat{B}_{k}=B_{k}$. We conclude in either situation that
$x\in\bigcup_{j}B_{j}$ proving item $1$.

To address item $3$, suppose $\\{k_{s}\\}_{s=1}^{\Gamma}$ is a sub collection
of $\\{1,..,N\\}$ with the property that

$\bigcap_{s=1}^{\Gamma}c^{*}B_{k_{s}}\neq\emptyset.$

Then $c^{*}B_{k_{s}}\subset 3c^{*}B_{k_{1}}$ for each $s$. Since
$6c^{*}B_{k_{1}}\Subset\Omega$ and $\\{\frac{1}{3}{B}_{k_{s}}\\}_{s}$ is
pairwise disjoint, Lemma 2.3 shows there are at most
$\left(18c^{*}\right)^{Q}$ such balls showing that
$\Gamma\leq\tilde{M}=\left(18c^{*}\right)^{Q}$. This proves item $3$ and
completes the proof of this lemma. ∎

Since $d\nu$ is doubling, we also have the following result.

######  Lemma 2.5.

Let $B=B(x_{0},r)$ be such that $B(x_{0},\frac{11}{2}r)\Subset\Omega$. Then

(9) $1-\frac{\nu(B)}{\nu(2B)}\geq\frac{1}{1+A_{\nu}^{3}}>0.$

###### Proof.

Let $x\in\Omega$ be such that $\rho(x,x_{0})=\frac{3}{2}r$. Let
$\tilde{B}=B(x,\frac{r}{2})$, then $\tilde{B}\subset 2B\setminus B$ and thus

(10) $\nu(2B)\geq\nu(B)+\nu(\tilde{B}).$

Now note that $B\subset 8\tilde{B}\subset
B(x_{0},\frac{11}{2}r)\Subset\Omega$, hence by doubling

$\nu(B)\leq\nu(2^{3}\tilde{B})\leq A_{\nu}^{3}\nu(\tilde{B}).$

From (10) we deduce that

$\nu(2B)\geq\left(1+\frac{1}{A_{\nu}^{3}}\right)\nu(B),$

whence (9) follows. ∎

##  3\. Functional setting

######  Definition 3.1.

Let $p>1$. We denote by $QW^{1,p}_{\nu,\mu}(\Omega)=QW^{1,p}(\Omega)$ and
$QW^{1,p}_{0,\nu,\mu}(\Omega)=QW^{1,p}_{0}(\Omega)$ the respective completions
of $Lip_{loc}(\Omega)$ and $Lip_{0}(\Omega)$ subject to the norm

(11) $\displaystyle\|u\|_{QW^{1,p}(\Omega)}$
$\displaystyle=\left(\int_{\Omega}|u|^{p}~{}d\nu\right)^{1/p}+\left(\int_{\Omega}\langle\nabla
u,Q\nabla u\rangle^{p/2}~{}d\mu\right)^{\frac{1}{p}}$
$\displaystyle=\left(\int_{\Omega}|u|^{p}~{}d\nu\right)^{1/p}+\left(\int_{\Omega}|\sqrt{Q}\nabla
u|^{p}~{}d\mu\right)^{\frac{1}{p}}.$

While $QW^{1,p}(\Omega)$ $\left(QW^{1,p}_{0}(\Omega)\right)$ is defined as a
collecton of equivalence classes of sequences of $Lip_{loc}(\Omega)$
$\left(Lip_{0}(\Omega)\right)$ functions Cauchy with respect to the norm (11),
it is well known (see [6, 7, 8, 31, 32, 33, 34]) that it is isometrically
isomorphic to a collection of pairs $(u,\vec{g})\in
L^{p}_{\nu}(\Omega)\times\mathcal{L}^{p}_{\mu}(Q;\Omega)$ endowed with the
norm

(12)
$\displaystyle\|(u,\vec{g})\|_{QW^{1,p}(\Omega)}=\left(\int_{\Omega}|u|^{p}~{}d\nu\right)^{1/p}+\left(\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p}~{}d\mu\right)^{\frac{1}{p}},$

where we denote by $\mathcal{L}^{p}_{\mu}(Q;\Omega)$ the space of measurable
vector valued functions $\vec{g}$ such that

$\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p}~{}d\mu<\infty.$

While the generalized derivative $\vec{g}$ of $u$ may not depend on $u$
itself, we will abuse notation and denote a pair $(u,\vec{g})\in
QW^{1,p}(\Omega)$ by $\vec{\bf u}=(u,\nabla u)$. We will sometimes abuse
notation even further, writing $u$ in place of $\vec{\bf u}=(u,\nabla u)$ for
elements of $QW^{1,p}(\Omega)$. As a result of this notational abuse, we will
often write

$\|\vec{\bf
u}\|_{QW^{1,p}(\Omega)}=\|(u,\vec{g})\|_{QW^{1,p}(\Omega)}=\|u\|_{QW^{1,p}(\Omega)}.$

######  Remark 3.2.

Using a simple density argument, one may show that if (H4) and (H5) hold, then
(7) and (8) also hold for any $\vec{\bf w}=(w,\vec{h})\in QW^{1,p}(B)$ and
$\vec{\bf u}=(u,\vec{g})\in QW^{1,p}_{0}(B)$ respectively, where
$B\Subset\Omega$ is any metric ball. That is, for any such pairs,

$\frac{1}{\nu(B)}\int_{B}|w-w_{B}|d\nu\leq
C_{0}r(B)\frac{1}{\mu(B)}\int_{B}|\sqrt{Q}\vec{h}|d\mu,\textrm{ and}$
$\left(\frac{1}{\nu(B)}\int_{B}|u|^{kp}~{}d\nu\right)^{\frac{1}{kp}}\leq
Cr(B)\left(\frac{1}{\mu(B)}\int_{B}|\sqrt{Q}~{}\vec{g}|^{p}~{}d\mu\right)^{\frac{1}{p}}.$

We also explicitly note that if $\Omega_{0}$ is a compact domain such that
$\Omega_{0}\Subset\Omega$ and if $\vec{\bf u}=(u,\vec{g})\in
QW^{1,p}_{0}(\Omega_{0})$, then extending $\vec{\bf u}=(u,\vec{g})$ to $0$
outside $\Omega_{0}$ we obtain an element of $QW^{1,p}_{0}(\Omega)$ which we
will still denote by $\vec{\bf u}=(u,\vec{g})$ and which satisfies

$\|u\|_{QW^{1,p}(\Omega)}=\|u\|_{QW^{1,p}(\Omega_{0})}.$

In this case we will say that the support of such function is contained in
$\Omega_{0}$.

As a result of our definition, there is a natural product rule available for
products of Lipschitz functions with compact support with elements of
$QW^{1,p}(\Omega)$. We present this in the following proposition and also note
that it will play an important role in the proof of our main result.

######  Proposition 3.3.

Assume condition (H3), let $u\in QW^{1,p}(\Omega)$, $\varphi\in
Lip_{0}(\Omega)$, $\varphi\geq 0$, and $\theta\geq 1$. Then
$\varphi^{\theta}u\in QW_{0}^{1,p}(\Omega_{0})$, where $\Omega_{0}$ is any
bounded open set containing $\operatorname{supp}\varphi$ and such that
$\Omega_{0}\Subset\Omega$, with

$\sqrt{Q}\nabla(\varphi^{\theta}u)=\varphi^{\theta}\sqrt{Q}\nabla
u+\theta\varphi^{\theta-1}u\sqrt{Q}\nabla\varphi\quad\text{a.e. in }\Omega.$

###### Proof.

Let $u_{j}$ be a sequence of functions in $Lip_{\text{loc}}(\Omega)$ such that

$\displaystyle(i)~{}$ $\displaystyle u_{j}\rightarrow u\quad\text{in
}L^{p}(\Omega,d\nu)\text{ and }\nu-a.e.$ $\displaystyle(ii)~{}$
$\displaystyle\sqrt{Q}\nabla u_{j}\rightarrow\sqrt{Q}\nabla u\quad\text{in
}(L^{p}(\Omega,d\mu))^{n}\text{ and }\mu-a.e.$

If $\Omega_{0}$ is any bounded open set containing
$\operatorname{supp}\varphi$ and such that
$\overline{\Omega}_{0}\subset\Omega$, then $v_{j}:=\varphi^{\theta}u_{j}\in
Lip_{0}(\Omega_{0})$. Since

$\int_{\Omega_{0}}|v_{j}-\varphi^{\theta}u|^{p}\,d\nu\leq
C\int_{\Omega}|u_{j}-u|^{p}\,d\nu$

we immediately have that $v_{j}$ converges to $\varphi^{\theta}u$ in
$L^{p}(\Omega_{0},d\nu)$ and $\nu$-a.e.

Since $v_{j}$ is differentiable Lebesgue-a.e. with

$\sqrt{Q}\nabla v_{j}=\varphi^{\theta}\sqrt{Q}\nabla
u_{j}+\theta\varphi^{\theta-1}u_{j}\sqrt{Q}\nabla\varphi$

Lebesgue-a.e., by (H3) we see that

$\sqrt{Q}\nabla v_{j}\rightarrow\varphi^{\theta}\sqrt{Q}\nabla
u+\theta\varphi^{\theta-1}u\sqrt{Q}\nabla\varphi$

$\mu$-a.e. Moreover, using (H3) again and recalling that $v=\|Q\|^{p/2}_{op}$

$\displaystyle\int_{\Omega_{0}}|\sqrt{Q}\nabla
v_{j}-(\varphi^{\theta}\sqrt{Q}\nabla
u+\theta\varphi^{\theta-1}u\sqrt{Q}\nabla\varphi)|^{p}\,d\mu$
$\displaystyle\leq C\int_{\Omega_{0}}|\varphi^{\theta}(\sqrt{Q}\nabla
u_{j}-\sqrt{Q}\nabla
u)|^{p}\,d\mu+C\int_{\Omega_{0}}|\varphi^{\theta-1}(u_{j}-u)\sqrt{Q}\nabla\varphi|^{p}\,d\mu$
$\displaystyle\leq C\int_{\Omega_{0}}|\sqrt{Q}\nabla u_{j}-\sqrt{Q}\nabla
u|^{p}\,d\mu+C\int_{\Omega_{0}}|u_{j}-u|^{p}\,vd\mu$ $\displaystyle\leq
C\int_{\Omega_{0}}|\sqrt{Q}\nabla u_{j}-\sqrt{Q}\nabla
u|^{p}\,d\mu+C\int_{\Omega_{0}}|u_{j}-u|^{p}\,d\nu.$

Hence

$\sqrt{Q}\nabla v_{j}\rightarrow\varphi^{\theta}\sqrt{Q}\nabla
u+\theta\varphi^{\theta-1}u\sqrt{Q}\nabla\varphi$

in $(L^{p}(\Omega_{0},d\mu))^{n}$ and the proof is complete. ∎

The following propositions provide useful calculus results in the degenerate
Sobolev space $QW^{1,p}(\Omega)$. Similar results can be found in Lemmas 4.1
and 4.2 in [33], in case $\mu$ and $\nu$ are Lebesgue measure. Since the only
change required here is the incorporation of the measures $\mu$ and $\nu$ and
since the proofs go through taking into account the absolute continuity
assumption in (H3), we will omit them.

######  Proposition 3.4.

Assume condition (H3). Let $(u,\nabla u)\in QW^{1,p}(\Omega)$ and $\phi\in
C^{1}(\mathbb{R})$ with $\phi^{\prime}\in L^{\infty}(\mathbb{R})$. Then
$(\phi(u),\nabla\phi(u))\in QW^{1,p}(\Omega)$ with
$\displaystyle{\sqrt{Q}\nabla\big{(}\phi(u)\big{)}=\phi^{\prime}(u)\sqrt{Q}\nabla
u,\ \mu-a.e.in\ \Omega.}$

######  Proposition 3.5.

Assume condition (H3). Let $u\in QW^{1,p}(\Omega)$ and $k\in{\mathbb{R}}$.
Then we have $(\bar{u},\nabla\bar{u})\in QW^{1,p}(\Omega)$ with
$\bar{u}=|u|+k$ and

(13) $\sqrt{Q(x)}\nabla\bar{u}(x)=\begin{cases}\,\sqrt{Q(x)}\nabla
u(x)\,\,\,\qquad\text{if }u(x)\geq 0,\\\ -\sqrt{Q(x)}\nabla u(x)\qquad\text{if
}u(x)<0\end{cases}\quad\mu-\text{a.e. in }\Omega.$

######  Proposition 3.6.

Assume (H3), (H5), (H6). Then for every open subset $\Omega_{0}$ with
$\Omega_{0}\Subset\Omega$ there exists $C>0$ such that

$\left(\int_{\Omega_{0}}|u|^{kp}\,d\nu\right)^{\frac{1}{kp}}\leq
C\|u\|_{QW^{1,p}(\Omega)}$

for every $u\in QW^{1,p}(\Omega)$.

###### Proof.

We choose a small enough $r>0$ and use Lemma 2.4 to cover the set $\Omega_{0}$
with a finite number of balls $B_{\tau r}(x_{1}),\ldots,B_{\tau r}(x_{K})$,
with $\tau$ as in (H6). For each of the balls
$B_{r}(x_{1}),\ldots,B_{r}(x_{K}),$ by (H6) we can find a Lipschitz function
which is identically one on the ball of radius $\tau r$ and with support in
the ball of radius $r$. Then we use (H5) in each of the balls and we conclude
using Proposition 3.3 with $\theta=1$ and (H3):

$\displaystyle\left(\int_{\Omega_{0}}|u|^{kp}\,d\nu\right)^{\frac{1}{kp}}$
$\displaystyle\leq\left(\sum_{j=1}^{K}\int_{B_{r}(x_{j})}|\varphi_{j}u|^{kp}\,d\nu\right)^{\frac{1}{kp}}$
$\displaystyle\leq
C\sum_{j=1}^{K}\left(\int_{B_{r}(x_{j})}|\varphi_{j}u|^{kp}\,d\nu\right)^{\frac{1}{kp}}$
$\displaystyle\leq
C\sum_{j=1}^{K}\left(\int_{B_{r}(x_{j})}|\sqrt{Q}\nabla(\varphi_{j}u)|^{p}\,d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq
C\sum_{j=1}^{K}\left(\int_{B_{r}(x_{j})}|\varphi_{j}\sqrt{Q}\nabla
u|^{p}\,d\mu+\int_{B_{r}(x_{j})}|u\sqrt{Q}\nabla\varphi_{j}|^{p}\,d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq C\sum_{j=1}^{K}\left(\int_{B_{r}(x_{j})}|\sqrt{Q}\nabla
u|^{p}\,d\mu+\int_{B_{r}(x_{j})}|u|^{p}v\,d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq C\sum_{j=1}^{K}\left(\int_{B_{r}(x_{j})}|\sqrt{Q}\nabla
u|^{p}\,d\mu+\int_{B_{r}(x_{j})}|u|^{p}\,d\nu\right)^{\frac{1}{p}}$
$\displaystyle\leq C\|u\|_{QW^{1,p}(\Omega)}$

∎

##  4\. Fefferman-Phong embedding

The main tools to be exploited in our proofs of local boundedness and
continuity of weak solutions to equations of the type (4) are a
Subrepresentaiton inequality and also a Fefferman-Phong inequality that we
will develop now. We recall the following

######  Theorem 4.1 ([30], Theorem $A^{*}$).

If (H1), (H2), and (H4) hold, then for any ball $B\Subset\Omega$ one has

(14) $\displaystyle\left|u(x)-u_{B}\right|$ $\displaystyle\leq$ $\displaystyle
C\displaystyle\int_{B}\frac{\rho(x,y)}{\mu(B(x,\rho(x,y)))}|\sqrt{Q}\nabla
u(y)|\,d\mu(y)$

for $d\nu$-a.e. $x\in B$, for every $u\in QW^{1,p}(\Omega)$.

######  Remark 4.2.

Theorem 4.1 relies on the construction of special chains of metric balls,
which can be carried out assuming condition (H1). Indeed one can show that the
following property is a consequence of (H1), see Theorem D in [30]: let
$B_{0}\Subset\Omega$ be a metric ball, then for any $x\in B_{0}$ there exists
a chain of balls $\\{B_{k}\\}_{k\geq 1}$ such that

  1. (1)

$B_{k}\subset B_{0}$ and $\rho(B_{k},x)\rightarrow 0$ as $k\rightarrow\infty$;

  2. (2)

$r(B_{1})\approx r(B_{0})$ and $r(B_{k})\rightarrow 0$ as $k\rightarrow\infty$

  3. (3)

if $y\in B_{k}$, then $\rho(x,y)\approx r(B_{k})$

  4. (4)

$B_{k}\cap B_{k-1}$ contains a ball $S_{k}$ with $r(S_{k})\approx
r(B_{k})\approx r(B_{k+1})$

  5. (5)

if $j<k$, then $B_{k}\subset cB_{j}$

  6. (6)

$\\{B_{k}\\}_{k\geq 1}$ has bounded overlap, i.e. $\sum_{k}\chi_{B_{k}}(y)\leq
c$ for all $y$

where the constants of equivalence in (2), (3), (4) and the constants $c$ in
(5) and (6) are independent of $x,k,j,B_{0}$, but the chain of balls
$\\{B_{k}\\}_{k\geq 1}$ depends on $x$.

We explicitly note that one could drop condition (H1), replacing it with the
axiomatic assumption

  * (H1′)

given any metric ball $B_{0}\Subset\Omega$, for every $x\in B_{0}$ there
exists a chain of balls $\\{B_{k}\\}_{k\geq 1}$ that satisfy conditions
(1)–(6).

This subrepresentation formula is useful in developing a weighted Fefferman-
Phong inequality where we exploit the following class of functions.

######  Definition 4.3 (Stummel-Kato).

Let $V\in L^{1}_{loc}(\Omega)$, $p>1$, $r>0$ we set the function $\Phi_{V}(r)$
as

$\sup_{x\in\Omega}\left(\int_{B(x,r)}\\!\\!\frac{\rho(x,y)}{\mu(B(x,\rho(x,y)))}\left(\int_{B(x,r)}\\!\\!\\!\\!|V(z)|\frac{\rho(z,y)d\nu(z)}{\mu(B(z,\rho(z,y)))}\,\right)^{\frac{1}{p-1}}\\!\\!\\!\\!d\mu(y)\right)^{p-1}$

We say that

  1. (1)

$V\in\tilde{M}_{p}(\Omega)$ if $\Phi_{V}(r)$ is finite for any $r>0$;

  2. (2)

$V\in M_{p}(\Omega)$ if $V\in\tilde{M}_{p}(\Omega)$ and $\lim_{r\rightarrow
0^{+}}\Phi_{V}(r)=0$

  3. (3)

$V\in M^{\prime}_{p}(\Omega)$ if $V\in M_{p}(\Omega)$ and

$\int_{0}^{\rho}\frac{(\Phi_{V}(r))^{\frac{1}{p-1}}}{r}\,dr<\infty$

for some $\rho>0$.

######  Remark 4.4.

We remark that in the case $p=2$, $\mu=\nu=1$ and $Q=I$ we obtain the
classical Stummel-Kato class introduced by Aizenman and Simon in [1] (see also
[38] and [39]).

######  Remark 4.5.

Note that $M^{\prime}_{p}(\Omega)\subset
M_{p}(\Omega)\subset\tilde{M}_{p}(\Omega)$ are vector spaces. If
$V\in\tilde{M}_{p}(\Omega)\setminus\\{0\\}$ then $\Phi_{V}$ is defined in
$(0,+\infty)$, it is positive, nondecreasing and continuous. If we set
$\Phi_{V}(0)=\lim_{r\rightarrow 0^{+}}\Phi_{V}(r)$ then the function is
defined and continuous also at $0$. If $V\in M_{p}(\Omega)\setminus\\{0\\}$
then $\Phi_{V}(0)=0$ and we set

$\Phi^{-1}_{V}(t)=\min\\{r\geq 0\,|\,\Phi_{V}(r)=t\\},$

for every $t\in[0,b)$, with $b=\sup\Phi_{V}$. Then $\Phi^{-1}_{V}$ is
increasing, with

$\displaystyle\Phi_{V}^{-1}(t)\leq r\qquad\forall r\geq 0\textrm{ such that
}\Phi_{V}(r)=t,$ $\displaystyle\Phi_{V}(\Phi_{V}^{-1}(t))=t\qquad\forall
t\in[0,b),$ $\displaystyle\Phi_{V}^{-1}(0)=0=\lim_{t\rightarrow
0^{+}}\Phi_{V}^{-1}(t),\quad\Phi_{V}^{-1}(t)>0\qquad\forall t\in(0,b).$

######  Theorem 4.6.

(Fefferman-Phong Embedding) Let $1<p<\infty$ and let $V$ belongs to
$\tilde{M}_{p}(\Omega)$. Assume (H1), (H2) and (H4). Then, there exists a
constant $C>0$ (independent of $V$) so that for every metric ball $B\subset
4B\Subset\Omega$ one has

(15) $\displaystyle\int_{B}\left|V(x)\right|~{}|u(x)-u_{B}|^{p}~{}d\nu$
$\displaystyle\leq C\Phi_{V}(2r)\int_{B}\left|\sqrt{Q}\nabla
u\right|^{p}~{}d\mu$

for any $u\in Lip(B)$, where $r=r(B)$.

###### Proof.

Fix a ball $B\subset 4B\Subset\Omega$. For any $x,y\in\Omega,y\neq x$, set

$R(x,y)=\frac{\rho(x,y)}{\mu(B(x,\rho(x,y)))}.$

Using Theorem 4.1, Fubini-Tonelli Theorem and Hölder inequality we get

$\displaystyle\int_{B}|V(x)|~{}|u(x)-u_{B}|^{p}~{}d\nu(x)$ $\displaystyle\leq
C\int_{B}|V(x)||u(x)-u_{B}|^{p-1}\left(\int_{B}|\sqrt{Q}\nabla
u(y)|R(x,y)~{}d\mu(y)\right)d\nu(x)$ $\displaystyle=C\int_{B}|\sqrt{Q}\nabla
u(y)|\left(\int_{B}|V(x)||u(x)-u_{B}|^{p-1}R(x,y)\,d\nu(x)\right)\,d\mu(y)$
$\displaystyle\leq C\left(\int_{B}|\sqrt{Q}\nabla
u(y)|^{p}\,d\mu(y)\right)^{\frac{1}{p}}$
$\displaystyle\quad\quad\times\left[\int_{B}\left(\int_{B}|V(x)||u(x)-u_{B}|^{p-1}R(x,y)\,d\nu(x)\right)^{\frac{p}{p-1}}\,d\mu(y)\right]^{\frac{p-1}{p}}$
$\displaystyle\leq C\left(\int_{B}|\sqrt{Q}\nabla
u(y)|^{p}\,d\mu(y)\right)^{\frac{1}{p}}$
$\displaystyle\quad\quad\times\left[\int_{B}\left(\int_{B}|V(x)||u(x)-u_{B}|^{p}R(x,y)\,d\nu(x)\right)\right.$
$\displaystyle\hskip
119.24506pt\times\left.\left(\int_{B}|V(z)|R(z,y)\,d\nu(z)\right)^{\frac{1}{p-1}}\,d\mu(y)\right]^{\frac{p-1}{p}}$

Reversing the order of integration again, we find this last line to be
identical to

$\displaystyle C\left(\int_{B}|\sqrt{Q}\nabla
u(y)|^{p}\,d\mu(y)\right)^{\frac{1}{p}}$
$\displaystyle\quad\quad\times\left[\int_{B}|V(x)||u(x)-u_{B}|^{p}\right.$
$\displaystyle\quad\quad\quad\quad\times\left.\left(\int_{B}R(x,y)\left(\int_{B}|V(z)|R(z,y)\,d\nu(z)\right)^{\frac{1}{p-1}}\,d\mu(y)\right)\,d\nu(x)\right]^{\frac{p-1}{p}}.$

Before we apply Definition 4.3 to the last term above we note that the
integration above is over the ball $B$ whose centre is of course not the same
as $x$, the centre of $B(x,\rho(x,y))$ in the definition of $R(x,y)$.

However, using the swallowing property of metric balls, given any
$x\in\Omega$, $B\subset 2B(x,r)\subset 4B\Subset\Omega$. Using this idea,
Definition 4.3 gives

$\displaystyle\int_{B}|V(x)|~{}|u(x)-u_{B}|^{p}~{}d\nu(x)$ $\displaystyle\leq
C\left(\int_{B}|\sqrt{Q}\nabla
u(y)|^{p}\,d\mu(y)\right)^{\frac{1}{p}}\left[(\Phi_{V}(2r))^{\frac{1}{p-1}}\int_{B}|V(x)||u(x)-u_{B}|^{p}\,d\nu(x)\right]^{\frac{p-1}{p}}.$

Reordering terms then gives (15) concluding the proof. ∎

######  Corollary 4.7.

Let $1<p<\infty$ and let $V$ be any function in $\tilde{M}_{p}(\Omega)$.
Assume (H1), (H2), (H4). Then, there exists a constant $C>0$ (independent of
$V$) such that for every metric ball $B$ with $8B\Subset\Omega$ one has

(16) $\displaystyle\int_{B}\left|V(x)\right||u(x)|^{p}~{}d\nu\leq
C\Phi_{V}(4r)\int_{B}|\sqrt{Q}\nabla u|^{p}~{}d\mu$

for any $u\in Lip_{0}(B)$, where $r=r(B)$.

###### Proof.

Let $u$ be a Lipschitz function compactly supported in a ball $B$ with radius
$r$. By Theorem 4.6 we have

(17) $\displaystyle\left(\int_{B}|V(x)||u(x)|^{p}d\nu\right)^{\frac{1}{p}}$
$\displaystyle\quad\leq\left(\int_{B}|V(x)||u(x)-u_{B}|^{p}~{}d\nu\right)^{\frac{1}{p}}+\left(\int_{B}|V(x)||u_{B}|^{p}~{}d\nu\right)^{\frac{1}{p}}$
$\displaystyle\quad\leq C\left(\Phi_{V}(2r)\int_{B}|\sqrt{Q}\nabla
u|^{p}~{}d\nu\right)^{\frac{1}{p}}+\left(\int_{B}|V(x)||u_{B}|^{p}~{}d\nu\right)^{\frac{1}{p}}$

Using that $u$ has compact support in $B$,

$\left(1-\frac{\nu(B)}{\nu(2B)}\right)u_{B}=u_{B}-u_{2B}$

and hence by Theorem 4.6

$\displaystyle\left(1-\frac{\nu(B)}{\nu(2B)}\right)\left(\int_{B}|V(x)||u_{B}|^{p}~{}d\nu\right)^{\frac{1}{p}}$
$\displaystyle\quad=\left(\int_{B}|V(x)||u_{B}-u_{2B}|^{p}~{}d\nu\right)^{\frac{1}{p}}$
$\displaystyle\quad\leq\left(\int_{B}|V(x)||u-u_{B}|^{p}~{}d\nu\right)^{\frac{1}{p}}+\left(\int_{2B}|V(x)||u-u_{2B}|^{p}~{}d\nu\right)^{\frac{1}{p}}$
$\displaystyle\quad\leq
C\left(\left(\Phi_{V}(2r)\right)^{\frac{1}{p}}+\left(\Phi_{V}(4r)\right)^{\frac{1}{p}}\right)\left(\int_{B}|\sqrt{Q}\nabla
u|^{p}~{}d\mu\right)^{\frac{1}{p}}$

since $\operatorname{supp}u\subset B$. As $\frac{11}{2}B\subset
8B\Subset\Omega$, Lemma 2.5 provides a constant $c=c(A_{\nu})>0$ such that
$\left(1-\frac{\nu(B)}{\nu(2B)}\right)\geq c>0$. Dividing by this factor,
since $V\in\tilde{M}_{p}(\Omega)$ and since $\Phi_{V}(r)$ is obviously
nondecreasing, we obtain

(18) $\left(\int_{B}|V(x)||u_{B}|^{p}~{}d\nu\right)^{\frac{1}{p}}\leq
C\left(\Phi_{V}(4r)\int_{B}|\sqrt{Q}\nabla
u|^{p}~{}d\mu,\right)^{\frac{1}{p}}.$

We get the result by (18), (17) and monotonicity of $\Phi_{V}$. ∎

######  Corollary 4.8.

Let $1<p<\infty$ and let $V$ be any function in
$M_{p}(\Omega)\setminus\\{0\\}$. Assume (H1), (H2), (H3), (H4), (H6). Then
there exists $C>0$ such that for every small enough $\varepsilon>0$ and every
Lipschitz function $u$ with compact support contained in $\Omega$ one has

(19) $\displaystyle\int_{\Omega}|V|\,|u|^{p}\,d\nu$
$\displaystyle\leq\,C\varepsilon\int_{\Omega}|\sqrt{Q}\nabla
u|^{p}\,d\mu+C\omega_{V}(\varepsilon)\int_{\Omega}|u|^{p}\,d\nu,$

where $\omega_{V}(\varepsilon)=\varepsilon r_{\varepsilon}^{-\gamma p}$,
$r_{\varepsilon}>0$ is any number such that
$\Phi_{V}(r_{\varepsilon})\leq\varepsilon$ and $\gamma>0$ is as in (H6).

###### Proof.

Since $V$ is not $0$ a.e. in $\Omega$, $b:=\sup\Phi_{V}>0$. Let
$\varepsilon\in(0,b)$ and $E$ be any bounded domain such that
$\operatorname{supp}u\Subset E\Subset\Omega$. Let $r$ be a positive number to
be chosen later. By Lemma 2.4 there exists $N=N(r)$ such that

$E\subset\bigcup_{j=1}^{N}B(x_{j},\tau r)$

where $\tau$ is as in $(H6)$ and $x_{j}\in E$ for all $j=1,\ldots,N.$ We also
have

$\sum_{j=1}^{N}\chi_{B(x_{j},r)}(x)\leq\left(\frac{18}{\tau}\right)^{Q}=M.$

From $(H6)$ there exists $\varphi_{j}\in Lip_{0}(B(x_{j},r))$ such that
$0\leq\varphi_{j}\leq 1$, and $\varphi_{j}=1$ on $B(x_{j},\tau r)$,
$j=1,\ldots,N.$ By Corollary 4.7 we have

$\displaystyle\int_{\Omega}V|u|^{p}\,d\nu$
$\displaystyle=\int_{\bigcup_{j=1}^{N}B(x_{j},\tau r)}V|u|^{p}\,d\nu$
$\displaystyle\leq\sum_{j=1}^{N}\int_{B(x_{j},r)}V|u|^{p}\varphi_{j}^{p}\,d\nu$
$\displaystyle\leq
C\sum_{j=1}^{N}\Phi_{V}(4r)\int_{B(x_{j},r)}|\sqrt{Q}\nabla(u\varphi_{j})|^{p}\,d\mu$
$\displaystyle\leq
C\Phi_{V}(4r)\left(\sum_{j=1}^{N}\int_{B(x_{j},r)}|\sqrt{Q}\nabla
u|^{p}\varphi_{j}^{p}\,d\mu\right.$ $\displaystyle\hskip
99.58464pt\left.+\sum_{j=1}^{N}\int_{B(x_{j},r)}|\sqrt{Q}\nabla\varphi_{j}|^{p}|u|^{p}\,d\mu\right)$

for some positive constant $C$. Note that

$\displaystyle\sum_{j=1}^{N}\int_{B(x_{j},r)}|\sqrt{Q}\nabla
u|^{p}\varphi_{j}^{p}\,d\mu$ $\displaystyle\leq\int_{\Omega}|\sqrt{Q}\nabla
u|^{p}\sum_{j=1}^{N}\chi_{B(x_{j},r)}\,d\mu$ $\displaystyle\leq
M\int_{\Omega}|\sqrt{Q}\nabla u|^{p}\,d\mu.$

By condition (H6)-(4),

$\displaystyle\sum_{j=1}^{N}\int_{B(x_{j},r)}|\sqrt{Q}\nabla\varphi_{j}|^{p}|u|^{p}\,d\mu$
$\displaystyle\leq Cr^{-\gamma
p}\sum_{j=1}^{N}\int_{B(x_{j},r)}|u|^{p}v\,d\mu$ $\displaystyle\leq
Cr^{-\gamma p}\int_{\Omega}|u|^{p}v\sum_{j=1}^{N}\chi_{B(x_{j},r)}\,d\mu$
$\displaystyle\leq MCr^{-\gamma p}\int_{\Omega}|u|^{p}v\,d\mu.$

Taking $t=4r$ we get

$\displaystyle\int_{\Omega}|V||u|^{p}\,d\nu$ $\displaystyle\leq
MC\Phi_{V}(t)\int_{\Omega}|\sqrt{Q}\nabla u|^{p}\,d\mu$
$\displaystyle\,\,\,\,\,+MC\Phi_{V}(t)Ct^{-\gamma
p}\int_{\Omega}|u|^{p}v\,d\mu.$

Let $t_{\varepsilon}$ be such that $\Phi_{V}(t_{\varepsilon})\leq\varepsilon$.
Condition (H3) yields, we get (19) where $\omega_{V}(\varepsilon)=\varepsilon
t_{\varepsilon}^{-\gamma p}$. ∎

######  Remark 4.9.

Note that $r_{\varepsilon}=\Phi_{V}^{-1}(\varepsilon)$ in (19) is an
admissible choice.

######  Remark 4.10.

By a density argument and Fatou’s Lemma, one can easily see that (19) holds
also for every function $u\in QW^{1,p}_{0}(E)$, where $E$ is any bounded
domain such that $E\Subset\Omega$.

##  5\. Local boundedness and Harnack inequality

In this section we prove local boundedness and continuity of weak solutions to
equation (4), where $|f|^{p-1}$ belongs to the Stummel-Kato class (see
Definition 4.3).

######  Definition 5.1.

A function $\vec{\bf u}=(u,\vec{g})\in QW^{1,p}(\Omega)$ is a solution of (4)
if

(20) $\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p-2}\nabla\phi\cdot
Q~{}\vec{g}~{}hdx+\int_{\Omega}|f|^{p-2}f\phi~{}mdx=0$

for every $\phi\in Lip_{0}(\Omega)$.

We can enlarge the class of test functions in Definition 5.1 to include all
pairs $(\phi,\vec{\psi})\in QW^{1,p}_{0}(\Omega)$ such that
$(\phi,\vec{\psi})\in QW^{1,p}_{0}(\Omega_{0})$ for some subdomain
$\Omega_{0}\Subset\Omega$ (such that both $\phi$ and $\vec{\psi}$ vanish
outside of $\Omega_{0}$).

######  Lemma 5.2.

Let $Q(x)$ be symmetric, non negative definite matrix valued function for
which $v(x)=\|Q(x)\|^{p/2}_{op}\in L^{1}_{loc}(\Omega)$ and assume (H3), (H5),
(H6). Assume moreover that $f\in
L^{p-\frac{p(k-1)}{pk-1}}_{\text{loc}}(\Omega,d\nu)$, where $k$ is as in (H5).
Let $(u,\vec{g})$ be a weak solution of (4). Then

$\displaystyle\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p-2}\vec{\psi}\cdot
Q~{}\vec{g}~{}hdx+\int_{\Omega}|f|^{p-2}f\varphi~{}mdx=0$

for any compact subdomain $\Omega_{0}\Subset\Omega$ and any
$(\varphi,\vec{\psi})\in QW^{1,p}_{0}(\Omega_{0})$.

###### Proof.

For every $\varphi\in Lip_{0}(\Omega_{0})$ (20) holds. By Hölder inequality

$\left|\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p-2}\nabla\varphi\cdot
Q~{}\vec{g}~{}hdx\right|\leq\|u\|_{QW^{1,p}(\Omega)}^{p-1}\|\varphi\|_{QW^{1,p}(\Omega_{0})}$

Proposition 3.6 yields

$\displaystyle\left|\int_{\Omega}|f|^{p-2}f\varphi~{}mdx\right|$
$\displaystyle\leq\|f\|_{L^{p-\frac{p(k-1)}{pk-1}}(\Omega_{0};d\nu)}^{p-1}\|\varphi\|_{L^{pk}(\Omega_{0},d\nu)}$
$\displaystyle\leq
C\|f\|_{L^{p-\frac{p(k-1)}{pk-1}}(\Omega_{0};d\nu)}^{p-1}\|\varphi\|_{QW^{1,p}(\Omega_{0})}.$

Then the map

$(\varphi,\nabla\varphi)\mapsto\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p-2}\nabla\varphi\cdot
Q~{}\vec{g}~{}hdx+\int_{\Omega}|f|^{p-2}f\varphi~{}mdx$

is bounded in $QW^{1,p}(\Omega_{0})$-norm, and can thus be extended by
continuity on $QW^{1,p}_{0}(\Omega_{0})$. By density formula (20) holds for
every $(\varphi,\vec{\psi})\in QW^{1,p}_{0}(\Omega_{0})$. ∎

We also state here a Lemma, that we will need in the proof of our main
Theorems (see [35] and [16]).

######  Lemma 5.3.

Let $\Phi:(0,\infty)\rightarrow(0,\infty)$ be a continuous nondecreasing
function such that $\lim_{r\rightarrow 0}\Phi(r)=0$, let $C>0$ be such that
$C^{-1}\in\operatorname{Im}\Phi$ and let $k,p>1$. Then the series

$\sum_{j=0}^{\infty}\frac{1}{k^{j}}\log\Phi^{-1}\left(\frac{1}{Ck^{(p-1)j}}\right)$

converges if and only if

$\int_{0}^{R}\frac{(\Phi(r))^{\frac{1}{p-1}}}{r}\,dr$

converges for some $R>0$. In this case, if we choose $R_{0}>0$ such that
$\Phi(R_{0})=C^{-1}$, we have

$\displaystyle\frac{1}{C^{\frac{1}{p-1}}}\left(-\frac{k-1}{k}\sum_{j=0}^{\infty}\frac{1}{k^{j}}\log\Phi^{-1}\left(\frac{1}{Ck^{(p-1)j}}\right)+\log
R_{0}\right)$
$\displaystyle\quad\leq\int_{0}^{R_{0}}\frac{(\Phi(r))^{\frac{1}{p-1}}}{r}\,dr$
$\displaystyle\quad\quad\leq\frac{1}{C^{\frac{1}{p-1}}}\left(-(k-1)\sum_{j=0}^{\infty}\frac{1}{k^{j}}\log\Phi^{-1}\left(\frac{1}{Ck^{(p-1)j}}\right)+k\log
R_{0}\right).$

######  Theorem 5.4 (Local boundedness).

Let $1<p<\infty$. Assume (H1)-(H6). Suppose that $u\in QW^{1,p}(\Omega)$ is a
solution of (4) where $|f|^{p-1}\in M_{p}^{\prime}(\Omega)$ and $f\in
L^{p-\frac{p(k-1)}{pk-1}}_{\text{loc}}(\Omega,d\nu)$, where $k$ as in (H5).
Then, there exists a constant $C>0$ such that for every ball $B\equiv B_{r}$
such that $8B\Subset\Omega$ we have

(21) $\|u\|_{L^{\infty}(B_{\tau r})}\leq\\\ \leq Ce^{\frac{\gamma
k}{k-1}M_{1}}\\!\\!\left(r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)^{\frac{k}{p(k-1)}}\\!\\!\left(\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|u|^{p}\,d\nu\right)^{\frac{1}{p}}\\!\\!\\!+(\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}\right),$

where $\gamma$ is as in (H6) and $M_{1}>0$ is any constant satisfying

$\int_{0}^{r}\frac{\left(\Phi_{|f|^{p-1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds\leq
M_{1}\left(\Phi_{|f|^{p-1}}(r)\right)^{\frac{1}{p-1}}.$

###### Proof.

We set $\bar{u}=|u|+\lambda$, with $\lambda>0$ arbitrary. Fix $q\geq 1$,
$l>\lambda$ and let

$F(\bar{u})=\begin{cases}\bar{u}^{q}\qquad\qquad&\quad\text{if}\quad\lambda\leq\bar{u}\leq
l\cr ql^{q-1}(\bar{u}-l)+l^{q}&\quad\text{if}\quad\bar{u}\geq l\,.\end{cases}$

Set

$G(u)=\hbox{\rm
sign}(u)\left(F(\bar{u})[F^{\prime}(\bar{u})]^{p-1}-q^{p-1}\lambda^{\beta}\right)\quad
u\in\left(-\infty,+\infty\right)\,,$

where $\beta$ satisfies $pq=p+\beta-1$.

Let $B$ be a ball satisfying $8B\Subset\Omega$; for the rest of the proof,
$r=r(B)$ will be fixed. Let $\phi=\varphi^{p}G(u)$, where $\varphi$ is a
Lipschitz function such that $0\leq\varphi\leq 1$, $\varphi\equiv 1$ in
$B_{\tau r}$, compactly supported in $B_{r}$, see (H6). Arguing as in
Subsection 2.2 in [33], taking into account the presence of the measures
$d\mu$ and $d\nu$, it is easy to see that there exists a sequence of $l$’s
monotonically diverging at $+\infty$ such that the corresponding function
$\phi\in QW^{1,p}_{0}(B_{r})$ is a feasible test function in (20). We will
always and only be working with $l$’s in that sequence, but in order to keep
the notation simple we will not specify the sequence when using $\phi$. Then
using Young’s inequality, and noting that $|G(u)|\leq
F(\bar{u})(F^{\prime}(\bar{u}))^{p-1}$, we obtain the following for any
$\varepsilon\in(0,1)$

(22) $\displaystyle\int_{B_{r}}$
$\displaystyle\varphi^{p}G^{\prime}(u)|\sqrt{Q}\nabla u|^{p}\,hdx$
$\displaystyle=\int_{B_{r}}\varphi^{p}G^{\prime}(u)|\sqrt{Q}\nabla
u|^{p-2}\nabla u\cdot Q\nabla u\,hdx$
$\displaystyle=-p\int_{B_{r}}\varphi^{p-1}G(u)|\sqrt{Q}\nabla
u|^{p-2}\nabla\varphi\cdot Q\nabla u\,hdx$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\;\;\,\,\,\,\,\,-\int_{B_{r}}|f|^{p-2}f\varphi^{p}G(u)\,mdx$
$\displaystyle\leq p\int_{B_{r}}\varphi^{p-1}|G(u)||\sqrt{Q}\nabla
u|^{p-1}|\sqrt{Q}\nabla\varphi|\,hdx$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\;\;+\int_{B_{r}}|f|^{p-1}\varphi^{p}|G(u)|\,mdx$
$\displaystyle\leq\varepsilon\int_{B_{r}}\varphi^{p}(F^{\prime}(\bar{u}))^{p}|\sqrt{Q}\nabla
u|^{p}\,hdx+C(\varepsilon)\int_{B_{r}}\left(F(\bar{u})\right)^{p}|\sqrt{Q}\nabla\varphi|^{p}\,hdx$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\;\;\;\,\,\,\,\,\,+\int_{B_{r}}|f|^{p-1}\varphi^{p}|G(u)|\,mdx$

Then, set $\mathcal{U}=F(\bar{u})$ and
$f_{1}=\left(\frac{|f|}{\lambda}\right)^{p-1}$, we get $|f|^{p-1}\leq
f_{1}\bar{u}^{p-1}$.

We have $f_{1}\in M^{\prime}_{p}(\Omega)$ where

$\Phi_{f_{1}}(t)=\frac{1}{\lambda^{p-1}}\Phi_{|f|^{p-1}}(t)\qquad\forall t>0.$

By Propositions 3.4 and 3.5 we have $\mathcal{U}\in QW^{1,p}(\Omega)$. Note
that, since

$F^{\prime}(\bar{u})^{p}\leq G^{\prime}(u)\leq p(F^{\prime}(\bar{u}))^{p},$

we can absorb the first term in the last line into the first term in the first
line. Moreover we also have

$|G(u)|\leq
F(\bar{u})(F^{\prime}(\bar{u}))^{p-1},\qquad\bar{u}F^{\prime}(\bar{u})\leq
qF(\bar{u}),$

and then we obtain

(23) $\displaystyle\int_{B_{r}}$
$\displaystyle\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}hdx$
$\displaystyle\leq
C\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}\,hdx+C\int_{B_{r}}\varphi^{p}f_{1}|\bar{u}|^{p-1}(F^{\prime}(\bar{u}))^{p-1}\mathcal{U}\,mdx$
$\displaystyle\leq
C\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}\,hdx+Cq^{p-1}\int_{B_{r}}\varphi^{p}f_{1}|\mathcal{U}|^{p}\,mdx.$

By Corollary 4.8 on $\varphi\,\mathcal{U}$ (with the choice
$r_{\varepsilon}=(\Phi_{f_{1}}^{-1}(\varepsilon))$), see also Remarks 4.9 and
4.10, and we obtain

$\displaystyle\int_{B_{r}}\varphi^{p}$
$\displaystyle|\sqrt{Q}\nabla\mathcal{U}|^{p}hdx$ $\displaystyle\leq
C\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}hdx+Cq^{p-1}\varepsilon\int_{B_{r}}|\sqrt{Q}\nabla(\varphi\mathcal{U})|^{p}h\,dx$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+Cq^{p-1}\omega_{f_{1}}(\varepsilon)\int_{B_{r}}|\varphi\mathcal{U}|^{p}m\,dx$
$\displaystyle\leq
C(1+q^{p-1}\varepsilon)\\!\\!\\!\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}hdx+\hat{C}q^{p-1}\varepsilon\\!\\!\\!\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}hdx$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+Cq^{p-1}\omega_{f_{1}}(\varepsilon)\int_{B_{r}}|\varphi\mathcal{U}|^{p}m\,dx.$

for every $0<\varepsilon<\sup\Phi_{f_{1}}$. Since we are assuming $f$ is not
$0$ a.e. in $\Omega$, we choose
$\lambda=(2\hat{C}\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}>0$. In particular
$\Phi_{f_{1}}(r)=\frac{1}{2\hat{C}}$ and
$\sup\Phi_{f_{1}}>\frac{1}{2\hat{C}}$. We can choose
$\varepsilon=\frac{1}{2\hat{C}q^{p-1}}$ and then

(24) $\displaystyle\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}hdx$
$\displaystyle\quad\leq
C\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}hdx+Cq^{p-1}\omega_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)\int_{B_{r}}|\varphi\mathcal{U}|^{p}m\,dx.$

Since $d\nu=mdx$, $d\mu=hdx$, $vh\leq Cm$, by Remark 3.2 and (24) we have

$\displaystyle\left(\int_{B_{r}}|\varphi\mathcal{U}|^{kp}d\nu\right)^{\frac{1}{k}}$
$\displaystyle\leq
Cr^{p}\frac{(\nu(B_{r}))^{\frac{1}{k}}}{\mu(B_{r})}\int_{B_{r}}|\sqrt{Q}\nabla(\varphi\mathcal{U})|^{p}d\mu$
$\displaystyle\leq
Cr^{p}\frac{(\nu(B_{r}))^{\frac{1}{k}}}{\mu(B_{r})}\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}d\mu$
$\displaystyle\qquad\qquad+Cq^{p-1}\omega_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)r^{p}\frac{(\nu(B_{r}))^{\frac{1}{k}}}{\mu(B_{r})}\int_{B_{r}}|\varphi\mathcal{U}|^{p}d\nu.$

We specialize our choice of test function $\varphi$, using the sequence of
Lipschitz cutoff functions $\\{\varphi_{j}\\}$ provided by (H6), relative to
the ball $B$. We denote by $S_{j}:=\operatorname{supp}\varphi_{j}$,
$S_{0}:=B$.

$\displaystyle\left(\int_{S_{j+1}}|\mathcal{U}|^{kp}d\nu\right)^{\frac{1}{k}}\leq\left(\int_{S_{j}}|\varphi_{j}\mathcal{U}|^{kp}d\nu\right)^{\frac{1}{k}}$
$\displaystyle\leq
Cr^{p(1-\gamma)}\frac{(\nu(B_{r}))^{\frac{1}{k}}}{\mu(B_{r})}T^{pj}\int_{S_{j}}|\mathcal{U}|^{p}\,vd\mu$
$\displaystyle\qquad\qquad+Cq^{p-1}\omega_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)r^{p}\frac{(\nu(B_{r}))^{\frac{1}{k}}}{\mu(B_{r})}\int_{S_{j}}|\mathcal{U}|^{p}d\nu$
$\displaystyle\leq
Cr^{p}\frac{(\nu(B_{r}))^{\frac{1}{k}}}{\mu(B_{r})}\left(r^{-p\gamma}T^{pj}+q^{p-1}\omega_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)\right)\int_{S_{j}}|\mathcal{U}|^{p}d\nu$

Letting $l\to\infty$ along the sequence of $l$’s we are using, by monotone
convergence we have

(25)
$\displaystyle\left(\int_{S_{j+1}}|\bar{u}|^{kqp}d\nu\right)^{\frac{1}{kqp}}$
$\displaystyle\leq
C^{\frac{1}{pq}}r^{\frac{1-\gamma}{q}}\frac{(\nu(B_{r}))^{\frac{1}{kqp}}}{(\mu(B_{r}))^{\frac{1}{qp}}}\left(T^{pj}+r^{p\gamma}q^{p-1}\omega_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)\right)^{\frac{1}{qp}}\left(\int_{S_{j}}|\bar{u}|^{qp}d\nu\right)^{\frac{1}{qp}}.$

Since $\Phi_{f_{1}}(r)=\frac{1}{2\hat{C}}$ and $\Phi_{f_{1}}^{-1}$ is monotone
we have

$r^{-1}\Phi_{f_{1}}^{-1}\left(\frac{1}{2\hat{C}q^{p-1}}\right)\leq
r^{-1}\Phi_{f_{1}}^{-1}\left(\frac{1}{2\hat{C}}\right)\leq 1$

and

$r^{p\gamma}q^{p-1}\omega_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)=r^{p\gamma}\frac{1}{2\hat{C}}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\hat{C}q^{p-1}}\right)\right)^{-\gamma
p}\geq\frac{1}{2\hat{C}}.$

Thus we have

$\displaystyle\|\bar{u}\|_{L^{kpq}_{\nu}(S_{j+1})}$ $\displaystyle\leq
C^{\frac{1}{pq}}r^{\frac{1}{q}}\frac{(\nu(B_{r}))^{\frac{1}{kqp}}}{(\mu(B_{r}))^{\frac{1}{pq}}}T^{\frac{j}{q}}\left(\Phi^{-1}_{f_{1}}\left(\frac{1}{2\hat{C}q^{p-1}}\right)\right)^{-\frac{\gamma}{q}}\|\bar{u}\|_{L^{pq}_{\nu}(S_{j})}.$

We choose $q=q_{j}=k^{j}$, set $\alpha_{j}=pk^{j}$ and we obtain

$\|\bar{u}\|_{L^{\alpha_{j+1}}_{\nu}(S_{j+1})}\leq(Cr^{p})^{\frac{1}{\alpha_{j}}}\frac{(\nu(B_{r}))^{\frac{1}{k\alpha_{j}}}}{(\mu(B_{r}))^{\frac{1}{\alpha_{j}}}}T^{\frac{pj}{\alpha_{j}}}\left(\Phi^{-1}_{f_{1}}\left(\frac{p^{p-1}}{2\hat{C}\alpha_{j}^{p-1}}\right)\right)^{-\frac{p\gamma}{\alpha_{j}}}\|\bar{u}\|_{L^{\alpha_{j}}_{\nu}(S_{j})}.$

We iterate this inequality, sending $j$ to $\infty$, and we find

(26) $\displaystyle\|\bar{u}\|_{L^{\infty}(B_{\tau r})}$ $\displaystyle\leq
Cr^{\frac{k}{k-1}}\frac{(\nu(B_{r}))^{\frac{1}{p(k-1)}}}{(\mu(B_{r}))^{\frac{k}{p(k-1)}}}\prod_{j=0}^{\infty}\left(\Phi^{-1}_{f_{1}}\left(\frac{p^{p-1}}{2\hat{C}\alpha_{j}^{p-1}}\right)\right)^{-\frac{p\gamma}{\alpha_{j}}}\|\bar{u}\|_{L^{p}_{\nu}(B_{r})}$
$\displaystyle=C\left(\exp\left(\frac{1}{(2\hat{C})^{\frac{1}{p-1}}}\sum_{j=0}^{\infty}\frac{-p\gamma}{\alpha_{j}}\log\left(\Phi^{-1}_{f_{1}}\left(\frac{p^{p-1}}{2\hat{C}\alpha_{j}^{p-1}}\right)\right)\right)\right)^{(2\hat{C})^{\frac{1}{p-1}}}\times$
$\displaystyle\quad\quad\times\left(r^{p}\frac{\nu(B_{r})}{\mu(B_{r})}\right)^{\frac{k}{p(k-1)}}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{p}\,d\nu\right)^{\frac{1}{p}}.$

Now we note that

$\displaystyle\frac{1}{(2\hat{C})^{\frac{1}{p-1}}}\sum_{j=0}^{\infty}\frac{-p\gamma}{\alpha_{j}}\log\left(\Phi^{-1}_{f_{1}}\left(\frac{p^{p-1}}{2\hat{C}\alpha_{j}^{p-1}}\right)\right)$
$\displaystyle\quad=-\gamma\frac{1}{(2\hat{C})^{\frac{1}{p-1}}}\sum_{j=0}^{\infty}\frac{1}{k^{j}}\log\left(\Phi^{-1}_{f_{1}}\left(\frac{1}{2\hat{C}k^{(p-1)j}}\right)\right)$

and hence the series converges by Lemma 5.3 since
$\Phi_{f_{1}}(r)=\frac{1}{2\hat{C}}$ and thus

$\displaystyle-\gamma\frac{1}{(2\hat{C})^{\frac{1}{p-1}}}\sum_{j=0}^{\infty}\frac{1}{k^{j}}\log\left(\Phi^{-1}_{f_{1}}\left(\frac{1}{2\hat{C}k^{(p-1)j}}\right)\right)$
$\displaystyle\qquad\leq\frac{\gamma
k}{k-1}\left(\int_{0}^{r}\frac{\left(\Phi_{f_{1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds-\frac{1}{(2\hat{C})^{\frac{1}{p-1}}}\log
r\right)$ $\displaystyle\qquad=\frac{\gamma
k}{k-1}\frac{1}{(2\hat{C})^{\frac{1}{p-1}}}\left(\frac{1}{\left(\Phi_{|f|^{p-1}}(r)\right)^{\frac{1}{p-1}}}\int_{0}^{r}\frac{\left(\Phi_{|f|^{p-1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds-\log
r\right)<\infty$

by our definition of $f_{1}$. Now if

$\frac{1}{\left(\Phi_{|f|^{p-1}}(r)\right)^{\frac{1}{p-1}}}\int_{0}^{r}\frac{\left(\Phi_{|f|^{p-1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds\leq
M_{1}$

from (26) we have

(27) $\|\bar{u}\|_{L^{\infty}(B_{\tau r})}\leq Ce^{\frac{\gamma
k}{k-1}M_{1}}\left(r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)^{\frac{k}{p(k-1)}}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{p}\,d\nu\right)^{\frac{1}{p}}.$

This concludes the proof if $\lambda=(\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}>0$,
i.e. if $f$ does not vanish almost everywhere. If
$(\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}=0$ then $f_{1}=0$ a.e. and (23) yields

$\displaystyle\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}hdx\leq
C\int_{B_{r}}|\mathcal{U}|^{p}|\sqrt{Q}\nabla\varphi|^{p}\,hdx.$

Proceeding as in the previous case, for any $\lambda>0$ one gets

$\displaystyle\left(\int_{S_{j+1}}|\bar{u}|^{kqp}d\nu\right)^{\frac{1}{kqp}}\leq
C^{\frac{1}{pq}}r^{\frac{1-\gamma}{q}}\frac{(\nu(B_{r}))^{\frac{1}{kqp}}}{(\mu(B_{r}))^{\frac{1}{qp}}}T^{\frac{j}{q}}\left(\int_{S_{j}}|\bar{u}|^{qp}d\nu\right)^{\frac{1}{qp}},$

formally as in (25) with $\omega_{f_{1}}=0$. Choosing again $q=q_{j}=k^{j}$
and setting $\alpha_{j}=pk^{j}$ we obtain

$\|\bar{u}\|_{L^{\alpha_{j+1}}_{\nu}(S_{j+1})}\leq\left(Cr^{p(1-\gamma)}\right)^{\frac{1}{\alpha_{j}}}\frac{(\nu(B_{r}))^{\frac{1}{k\alpha_{j}}}}{(\mu(B_{r}))^{\frac{1}{\alpha_{j}}}}T^{\frac{pj}{\alpha_{j}}}\|\bar{u}\|_{L^{\alpha_{j}}_{\nu}(S_{j})}.$

Iterating the above inequality and letting $j$ to infinity we deduce that

(28) $\displaystyle\|\bar{u}\|_{L^{\infty}(B_{\tau r})}$ $\displaystyle\leq
Cr^{\frac{k(1-\gamma)}{k-1}}\frac{(\nu(B_{r}))^{\frac{1}{p(k-1)}}}{(\mu(B_{r}))^{\frac{k}{p(k-1)}}}\|\bar{u}\|_{L^{p}_{\nu}(B_{r})}$
$\displaystyle=C\left(r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)^{\frac{k}{p(k-1)}}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{p}\,d\nu\right)^{\frac{1}{p}}$

and sending $\lambda$ to $0$ we conclude. ∎

######  Remark 5.5.

If $\nu\leq C\mu$ for some constant $C>0$ and $\gamma=1$, then constant in
(21) does not depend on the metric balls but it depends on $r$ and $f$ only
through $M_{1}$.

######  Corollary 5.6.

Let $1<p<\infty$. Assume conditions (H1)-(H6). Suppose that $u\in
QW^{1,p}(\Omega)$ is a solution of (4) where $|f|^{p-1}\in
M_{p}^{\prime}(\Omega)$ and $f\in
L^{p-\frac{p(k-1)}{pk-1}}_{\text{loc}}(\Omega,d\nu)$, with $k$ as in (H5). Let
$E,E_{0}$ be bounded open domains such that $E\Subset E_{0}\Subset\Omega$ and
let $0<r<\frac{1}{8}\rho(E,\partial E_{0})$ be such that
$\Phi_{|f|^{p-1}}(r)<\sup\Phi_{|f|^{p-1}}$, if $f$ does not vanish almost
everywhere. Then there exists $C>0$ depending on $E,E_{0},r$ such that

$\|u\|_{L^{\infty}(E)}\leq Ce^{\frac{\gamma
k}{k-1}M_{1}}\left\\{\left(\int_{E_{0}}|u|^{p}\,d\nu\right)^{\frac{1}{p}}+(\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}\right\\},$

where $M_{1}>0$ is any constant such that

$\int_{0}^{r}\frac{\left(\Phi_{|f|^{p-1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds\leq
M_{1}\left(\Phi_{|f|^{p-1}}(r)\right)^{\frac{1}{p-1}}.$

###### Proof.

Covering $E$ with a finite number of balls $B_{\tau r}(x_{1})$,
$\ldots,B_{\tau r}(x_{N})$ for suitably small $r>0$, with centers in $E$ and
such that the closures of $B_{r}(x_{1}),\ldots,B_{r}(x_{N})$ are contained in
$E_{0}$, the result follows by applying Theorem 5.4 on each $B_{\tau
r}(x_{j})$. ∎

######  Theorem 5.7.

Let $1<p<\infty$, (H1)-(H6) hold true and $u\in QW^{1,p}(\Omega)$ be a non
negative solution of (4), where $|f|^{p-1}\in M_{p}^{\prime}(\Omega)$, $f\in
L^{p-\frac{p(k-1)}{pk-1}}_{\text{loc}}(\Omega,d\nu)$ and $k$ is as in (H5).

Assume there exist $r^{*},C>0$ such that

(29) $\sup r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\leq C$

where the supremum is computed over all balls such that ${8}B\Subset\Omega$
and $r\leq r^{*}$, with $\gamma$ as in (H6).

Then, there exist constants $C_{2},C_{3}>0$ such that for every ball $B=B_{r}$
with $\frac{8}{\tau}B\Subset\Omega$, $0<r\leq r^{*}$ such that
$\Phi_{|f|^{p-1}}\left(\frac{r}{\tau}\right)<\sup\Phi_{|f|^{p-1}}$ if $f$ does
not vanish a.e. we have

$\displaystyle\sup_{B_{\tau r}}u\leq C_{2}e^{C_{3}M_{1}}\left(\inf_{B_{\tau
r}}u+(\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}\right)$

where $\tau$ is as in (H6) and $M_{1}>0$ is any constant satisfying

(30)
$\int_{0}^{r}\frac{\left(\Phi_{|f|^{p-1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds\leq
M_{1}\left(\Phi_{|f|^{p-1}}(r)\right)^{\frac{1}{p-1}}.$

###### Proof.

We start as in Theorem 5.4, setting $\bar{u}=u+\lambda$, where $\lambda>0$ is
specified below. We note that we can apply Theorem 5.4 on
$B_{\frac{r}{\tau}}$, hence $\|u\|_{L^{\infty}(B_{r})}$ is finite. Now let
$\varphi$ be a non negative Lipschitz function compactly supported in $B_{r}$,
then for any $\beta\in{\mathbb{R}}\setminus\\{0\\}$ we can take
$\phi=\varphi^{p}\bar{u}^{\beta}$ as a test function in (20).

We find

(31) $\displaystyle\int_{B_{r}}\varphi^{p}\bar{u}^{\beta-1}|\sqrt{Q}\nabla
u|^{p}d\mu$ $\displaystyle\leq
C\left\\{|\beta|^{-p}\int_{B_{r}}\bar{u}^{\beta+p-1}|\sqrt{Q}\nabla\varphi|^{p}d\mu\right.$
$\displaystyle\qquad\,\,\,\,\left.+|\beta|^{-1}\int_{B_{r}}f_{1}\varphi^{p}\bar{u}^{\beta+p-1}\,d\nu\right\\},$

where we set $f_{1}=\left(\frac{|f|}{\lambda}\right)^{p-1}$, such that we have
$|f|^{p-1}\leq f_{1}\bar{u}^{p-1}$ and $f_{1}\in M^{\prime}_{p}(\Omega)$. Now
set

$\displaystyle\mathcal{U}(x)=\begin{cases}\bar{u}^{q}(x)\quad\quad\hbox{where
}pq=p+\beta-1,\hbox{ if }\beta\neq 1-p\\\ \log\bar{u}(x)\quad\hbox{if
}\beta=1-p\end{cases}$

Then from (31) we have

(32)
$\displaystyle\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\leq
C|q|^{p}\left\\{|\beta|^{-p}\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+|\beta|^{-1}\int_{B_{r}}f_{1}\varphi^{p}\mathcal{U}^{p}\,d\nu\right\\}\,\quad{\rm
if}\beta\neq 1-p,$

while

(33) $\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu\\\ \leq
C\left\\{\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\,d\mu+\int_{B_{r}}f_{1}\varphi^{p}\,d\nu\right\\}\quad\hbox{if}\
\beta=1-p\,.$

From now on we consider the case when $f$ is not $0$ almost everywhere, such
that $\Phi_{|f|^{p-1}}(r)>0$. We set

(34) $\lambda=(2\sigma\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}},$

where $\sigma\geq 1$ is a suitable constant to be chosen later. In particular
$\Phi_{f_{1}}(r)=\frac{1}{2\sigma}$.

We start considering (33). We use Corollary 4.8, with the choice
$\varepsilon=\frac{1}{2\sigma}$ and $r_{\varepsilon}=r$, and by condition (H5)
we have

$\displaystyle\int_{B_{r}}f_{1}\varphi^{p}\,d\nu$
$\displaystyle\leq\frac{C}{\sigma}\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\,d\mu+\frac{C}{\sigma}r^{-\gamma
p}\int_{B_{r}}\varphi^{p}\,d\nu$ $\displaystyle\leq
C\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\,d\mu+Cr^{-\gamma
p}\nu(B_{r})\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\varphi^{kp}\,d\nu\right)^{\frac{1}{k}}$
$\displaystyle\leq
C\left(1+r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\,d\mu$

Then, if $\beta=1-p$

(35) $\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}d\mu\leq
C\left(1+r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}d\mu.$

Now we choose $\varphi$ to be the first function in the sequence provided by
condition (H6). From (H4) and (35) we get

$\displaystyle\left(\frac{1}{\nu(B_{\tau r})}\int_{B_{\tau
r}}|\mathcal{U}-\mathcal{U}_{B_{\tau r};\nu}|\,d\nu\right)^{p}$
$\displaystyle\qquad\leq C\left(\frac{r}{\mu(B_{\tau r})}\int_{B_{\tau
r}}|\sqrt{Q}\nabla\mathcal{U}|\,d\mu\right)^{p}$ $\displaystyle\qquad\leq
C\frac{r^{p}}{\mu(B_{\tau
r})}\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\qquad\leq C\frac{r^{p}}{\mu(B_{\tau
r})}\left(1+r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}d\mu$
$\displaystyle\qquad\leq C\frac{r^{p(1-\gamma)}}{\mu(B_{\tau
r})}\left(1+r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right)\int_{B_{r}}d\nu$
$\displaystyle\qquad=Cr^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{\tau
r})}\left(1+r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\right).$

By the doubling assumption on $\nu$ and by condition (29) we have that

$\sup\left(\frac{1}{\nu(B_{s})}\int_{B_{s}}|\mathcal{U}-\mathcal{U}_{B_{s};\nu}|\,d\nu\right)$

is finite, where the supremum is computed over all balls with radius small
enough and such that $\frac{8}{\tau}B_{s}\Subset\Omega$. Then $\mathcal{U}$
locally belongs to $BMO_{\nu}$. Thus, arguing as in [34][Proposition 5.1 and
Corollary 5.3], we see that there are positive constants $p_{0},C$,
independent of $\sigma\geq 1$, such that for $R$ sufficiently small and
$\frac{8}{\tau}B_{R}\Subset\Omega$ the John-Nirenberg inequality

(36)
$\left(\frac{1}{\nu(B_{R})}\int_{B_{R}}e^{p_{0}\mathcal{U}}\,d\nu\right)\left(\frac{1}{\nu(B_{R})}\int_{B_{R}}e^{-{p_{0}\mathcal{U}}}\,d\nu\right)\leq
C\,,$

holds, where we recall that here $\mathcal{U}=\log\bar{u}$. We explicitly note
for later use that $p_{0}>0$ small can be chosen independent of $\sigma\geq 1$
and in such a way that

(37) $p_{0}k^{j}-p+1\neq 0$

for every $j\in\mathbb{N}$, with $k$ given by (H5). In particular we have

$\min_{j\in\mathbb{N}}|p_{0}k^{j}-p+1|>0.$

Now we consider the case $\beta>1-p$, see (32). For every $j\in\mathbb{N}$ we
choose $\beta_{j}=p_{0}k^{j}-p+1$ such that
$q_{j}=\frac{\beta_{j}+p-1}{p}=\frac{p_{0}}{p}k^{j}>0$. Using (32) and
Corollary 4.8 we obtain for every $0<\varepsilon<\sup\Phi_{f_{1}}$

$\displaystyle\int_{B_{r}}$
$\displaystyle\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\leq
C|q_{j}|^{p}|\beta_{j}|^{-p}\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+C|q_{j}|^{p}|\beta_{j}|^{-1}\varepsilon\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\,\,\,\,+C|q_{j}|^{p}|\beta_{j}|^{-1}\varepsilon\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+C|q_{j}|^{p}|\beta_{j}|^{-1}\omega_{f_{1}}(\varepsilon)\int_{B_{r}}\varphi^{p}\mathcal{U}^{p}\,d\nu.$

By our choice of $\beta_{j},q_{j}$ and by condition (37) we see that

(38) $\displaystyle\int_{B_{r}}$
$\displaystyle\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\leq
C\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+\tilde{C}k^{j(p-1)}\varepsilon\int_{B_{r}}\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\,\,\,\,+\tilde{C}k^{j(p-1)}\varepsilon\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+Ck^{j(p-1)}\omega_{f_{1}}(\varepsilon)\int_{B_{r}}\varphi^{p}\mathcal{U}^{p}\,d\nu.$

Up to choosing an even larger constant, we can assume for later use that
$\tilde{C}\geq 1$. Now we choose
$\varepsilon=\varepsilon_{j}=\frac{1}{2\tilde{C}k^{j(p-1)}}$, and we note that
by our assumptions we can fix the constant $\sigma\geq 1$ introduced in (34)
and set $\sigma=\tilde{C}$. With this choice of $\sigma$ we have
$\Phi_{f_{1}}(r)=\frac{1}{2\sigma}\geq\varepsilon,$ and hence $\varepsilon$ is
a feasible choice in (38). Thus we have

$\displaystyle\int_{B_{r}}$
$\displaystyle\varphi^{p}|\sqrt{Q}\nabla\mathcal{U}|^{p}\,d\mu$
$\displaystyle\leq
C\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+Ck^{j(p-1)}\omega_{f_{1}}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\int_{B_{r}}\varphi^{p}\mathcal{U}^{p}\,d\nu.$

From the Sobolev inequality (8) we find,

(39)
$\displaystyle\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|\varphi\mathcal{U}|^{kp}d\nu\right)^{\frac{1}{k}}$
$\displaystyle\quad\leq
C\frac{r^{p}}{\mu(B_{r})}\left\\{\int_{B_{r}}|\sqrt{Q}\nabla\varphi|^{p}\mathcal{U}^{p}\,d\mu+k^{j(p-1)}\omega_{f_{1}}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\int_{B_{r}}|\varphi\mathcal{U}|^{p}d\nu\right\\}.$

We specialize our choice of cutoff function $\varphi$ to be $\varphi_{j}$ as
provided by condition (H6) relative to the ball $B_{r}$. From (39) and (H6) we
obtain with $S_{j}=\operatorname{supp}(\varphi_{j})$, $S_{0}=B_{r}$,

(40)
$\displaystyle\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|\mathcal{U}|^{kp}\chi_{S_{j+1}}d\nu\right)^{\frac{1}{k}}$
$\displaystyle\leq
Cr^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\left\\{T^{jp}+k^{j(p-1)}r^{\gamma
p}\omega_{f_{1}}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right\\}\\!\\!\\!\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|\mathcal{U}|^{p}\chi_{S_{j}}d\nu\right).$

Now recall that
$\Phi_{f_{1}}(r)=\frac{1}{2\sigma}\geq\frac{1}{2\tilde{C}k^{j(p-1)}}$, hence

$r^{-1}\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\leq
r^{-1}\Phi_{f_{1}}^{-1}\left(\frac{1}{2\sigma}\right)\leq 1.$

Thus

$k^{j(p-1)}r^{\gamma
p}\omega_{f_{1}}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)=\frac{1}{2\tilde{C}}r^{\gamma
p}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{-\gamma
p}\geq\frac{1}{2\tilde{C}}.$

From (40) then we have

$\displaystyle\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|\mathcal{U}|^{kp}\chi_{S_{j+1}}d\nu\right)^{\frac{1}{k}}$
$\displaystyle\leq
Cr^{p}\frac{\nu(B_{r})}{\mu(B_{r})}T^{jp}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{-\gamma
p}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|\mathcal{U}|^{p}\chi_{S_{j}}d\nu\right).$

Now for $j\in\mathbb{N}$, set

$\Phi(s,j)=\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}|\bar{u}|^{s}\chi_{S_{j}}d\nu\right)^{1/s}.$

Thus, our previous inequality can be rewritten using
$\mathcal{U}=\bar{u}^{q_{j}}$ and condition (29) as

(41) $\displaystyle\Phi(kpq_{j},j+1)^{pq_{j}}\leq Cr^{\gamma
p}T^{jp}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{-\gamma
p}\Phi(pq_{j},j)^{pq_{j}},$

that is

$\displaystyle\Phi(k^{j+1}p_{0},j+1)$ $\displaystyle\leq(CT^{pj}r^{\gamma
p})^{\frac{1}{p_{0}k^{j}}}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{-\frac{\gamma
p}{p_{0}k^{j}}}\Phi(k^{j}p_{0},j).$

We now iterate the inequality for $j\in\mathbb{N}$, using Lemma 5.3 and noting
that $\Phi_{f_{1}}(r)=\frac{1}{2\tilde{C}}$, arguing as in the proof of (27).
We obtain

(42) $\displaystyle\sup_{B_{\tau r}}\bar{u}$ $\displaystyle\leq
Cr^{\frac{\gamma
kp}{p_{0}(k-1)}}\prod_{j=0}^{\infty}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{-\frac{\gamma
p}{p_{0}k^{j}}}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{p_{0}}\,d\nu\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq Ce^{\frac{\gamma
kp}{p_{0}(k-1)}M_{1}}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{p_{0}}\,d\nu\right)^{\frac{1}{p_{0}}},$

with $M_{1}>0$ satisfying (30).

We proceed in a similar way in case $\beta<1-p$, choosing
$\beta_{j}=-p_{0}k^{j}-p+1$ for any $j\in\mathbb{N}$ such that
$q_{j}=\frac{\beta_{j}+p-1}{p}=-\frac{p_{0}}{p}k^{j}<0$, with $p_{0}$ as in
(36). Up to choosing an even larger constant $\tilde{C}\geq 1$ we again obtain
(41), this time for negative $q_{j}$’s. Then we deduce

$\displaystyle\Phi(-k^{j+1}p_{0},j+1)\geq$ $\displaystyle(CT^{pj}r^{\gamma
p})^{-\frac{1}{p_{0}k^{j}}}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{\frac{\gamma
p}{p_{0}k^{j}}}\Phi(-k^{j}p_{0},j).$

Iterating the inequality for $j\in\mathbb{N}$ we obtain

$\displaystyle\inf_{B_{\tau r}}\bar{u}$ $\displaystyle\geq Cr^{-\frac{\gamma
kp}{p_{0}(k-1)}}\prod_{j=0}^{\infty}\left(\Phi_{f_{1}}^{-1}\left(\frac{1}{2\tilde{C}k^{j(p-1)}}\right)\right)^{\frac{\gamma
p}{p_{0}k^{j}}}\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{-p_{0}}\,d\nu\right)^{-\frac{1}{p_{0}}}.$

Hence using Lemma 5.3 we have

(43) $\displaystyle Ce^{\frac{\gamma kp}{p_{0}(k-1)}M_{1}}\inf_{B_{\tau
r}}\bar{u}\geq\left(\frac{1}{\nu(B_{r})}\int_{B_{r}}\bar{u}^{-p_{0}}\,d\nu\right)^{-\frac{1}{p_{0}}}.$

Now using (36), from (42) and (43) we deduce that

$\sup_{B_{\tau r}}\bar{u}\leq Ce^{\frac{2\gamma
kp}{p_{0}(k-1)}M_{1}}\inf_{B_{\tau r}}\bar{u}.$

Therefore, recalling the definition of $\bar{u}$,

$\sup_{B_{\tau r}}u\leq C_{2}e^{C_{3}M_{1}}\left(\inf_{B_{\tau
r}}u+\lambda\right).$

If $\Phi_{|f|^{p-1}}(r)>0$ the proof is complete. The proof in case
$\Phi_{|f|^{p-1}}(r)=0$ follows along the same lines, and therefore we will
omit it. ∎

Now, as a simple consequence of the Harnack inequality, we get some regularity
results for weak solutions of (4). The proof is a standard consequence of the
Harnack inequality, see e.g. [42], so we will omit it.

######  Theorem 5.8.

Let $1<p<\infty$. Assume condition (H1)-(H6). Suppose that $u\in
QW^{1,p}(\Omega)$ is a solution of (4) where $|f|^{p-1}\in
M_{p}^{\prime}(\Omega)$ and $f\in
L^{p-\frac{p(k-1)}{pk-1}}_{\text{loc}}(\Omega,d\nu)$, with $k$ as in (H5).

Assume also that there exists $r^{*},C>0$ such that

(44) $\sup r^{p(1-\gamma)}\frac{\nu(B_{r})}{\mu(B_{r})}\leq C$

where the supremum is computed over all balls such that ${8}B\Subset\Omega$
and with radius $r\leq r^{*}$, with $\gamma$ as in (H6).

Suppose moreover that there exists $M_{1}>0$ such that condition

$\int_{0}^{r}\frac{\left(\Phi_{|f|^{p-1}}(s)\right)^{\frac{1}{p-1}}}{s}\,ds\leq
M_{1}\left(\Phi_{|f|^{p-1}}(r)\right)^{\frac{1}{p-1}}.$

is satisfied for every small enough $r>0$ and that

$(\Phi_{|f|^{p-1}}(r))^{\frac{1}{p-1}}\leq Cr^{\alpha}$

for some positive constants $C,\alpha$. Then $u$ is locally Hölder continuous
with respect to the metric $\rho$.

######  Remark 5.9.

We explicitly note that, as a consequence of Theorem 5.8, $u$ is also
continuous with respect to the Euclidean topology. Moreover, if for every
subdomain $\Omega^{\prime}\Subset\Omega$ there are positive constants
$c_{0},\delta$ such that for every $x,x_{0}\in\Omega^{\prime}$ one has

$d(x,x_{0})\leq c_{0}|x-x_{0}|^{\delta},$

then $u$ is locally Hölder continuous also with respect to the Euclidean
distance.

Acknowledgement. This work has been supported by Università degli Studi di
Catania, ”Piano PIA.CE.RI”, upb 53722122154.

G. Di Fazio, M.S. Fanciullo, D.D. Monticelli and P. Zamboni are also members
of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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