# A comparison estimate for singular $p$-Laplace equations and its
consequences

Quoc-Hung Nguyen and Nguyen Cong Phuc E-mail address<EMAIL_ADDRESS>Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing, 100190, China.E-mail address<EMAIL_ADDRESS>Department of
Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA
70803, USA.

###### Abstract

Comparison estimates are an important technical device in the study of
regularity problems for quasilinear possibly degenerate elliptic and parabolic
equations. Such tools have been employed indispensably in many papers of
Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum
problems to obtain pointwise bounds for solutions and their full or fractional
derivatives in terms of appropriate linear or nonlinear potentials. However, a
comparison estimate for $p$-Laplace type elliptic equations with measure data
is still unavailable in the strongly singular case $1<p\leq\frac{3n-2}{2n-1}$,
where $n\geq 2$ is the dimension of the ambient space. This issue will be
completely resolved in this work by proving a comparison estimate in a
slightly larger range $1<p<3/2$. Applications include a ‘sublinear’ Poincaré
type inequality, pointwise bounds for solutions and their derivatives by
Wolff’s and Riesz’s potentials, respectively. Some global pointwise and
weighted estimates are also obtained for bounded domains, which enable us to
treat a quasilinear Riccati type equation with possibly sublinear growth in
the gradient.

MSC2020: primary: 35J92, 35J70, 35J62; secondary: 35J75, 35J66, 31B15.

Keywords: quasilinear equation; $p$-Laplace equation; measure data; pointwise
estimate; comparison estimate; weighted estimate; Riccati type equation;
capacity.

###### Contents

  1. 1 Introduction and main results
  2. 2 Comparison estimates
  3. 3 Proof of Theorem 1.4
  4. 4 Proof of Theorem 1.10

##  1 Introduction and main results

In this paper, we are concerned with the quasilinear elliptic equation with
measure data

$-\operatorname{div}(A(x,\nabla u))=\mu$ (1.1)

in a bounded open subset $\Omega$ of $\mathbb{R}^{n}$, $n\geq 2$. Here $\mu$
is a finite signed measure in $\Omega$ and the nonlinearity
$A=(A_{1},\dots,A_{n}):\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is
a vector valued function.
Our main assumptions on $A$ are as follows. There exist $\Lambda\geq 1$ and
$p\in(1,n)$ such that for every $x\in\mathbb{R}^{n}$ and every
$(\xi,\eta)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\backslash\\{(0,0)\\}$,

$|A(x,\xi)|\leq\Lambda|\xi|^{p-1},\quad|D_{\xi}A(x,\xi)|\leq\Lambda|\xi|^{p-2},$
(1.2) $\langle
D_{\xi}A(x,\xi)\eta,\eta\rangle\geq\Lambda^{-1}|\xi|^{p-2}|\eta|^{2}.$ (1.3)

A typical model for (1.1) is obviously given by the $p$-Laplace equation with
measure data

$-\Delta_{p}\,u:=-{\rm div}(|\nabla u|^{p-2}\nabla
u)=\mu\quad\text{in~{}}\Omega.$

For a reason that will be explained momentarily, unless otherwise stated, in
this paper we only consider the ‘singular’ case

$1<p<3/2.$

An important technical tool in the study of regularity problems for equation
(1.1) is a comparison estimate that connects the solution of measure datum
problem to a solution of a homogeneous problem. To describe it, let
$Q_{r}(x_{0})$ denote the open cube $Q_{r}(x_{0}):=x_{0}+(-r,r)^{n}$ with
center $x_{0}\in\mathbb{R}^{n}$ and side-length $2r$. Let $u\in W_{\rm
loc}^{1,p}(\Omega)$ be a solution of (1.1). For a cube
$Q_{r}=Q_{r}(x_{0})\Subset\Omega$, we then consider the unique solution $w\in
W_{0}^{1,p}(Q_{r}(x_{0}))+u$ to the local interior problem

$\left\\{\begin{array}[]{rcl}-\operatorname{div}\left({A(x,\nabla
w)}\right)&=&0\quad\text{in}\quad Q_{r}(x_{0}),\\\
w&=&u\quad\text{on}\quad\partial Q_{r}(x_{0}).\end{array}\right.$ (1.4)

For $p>\frac{3n-2}{2n-1}$, this comparison estimate reads as follows.

######  Lemma 1.1 ([8, 18, 20])

Assume that $p>\frac{3n-2}{2n-1}$. Then we have

$\displaystyle\left(\fint_{Q_{r}}|\nabla u-\nabla
w|^{\gamma}dx\right)^{\frac{1}{\gamma}}\leq
C\left[\frac{|\mu|(Q_{r})}{r^{n-1}}\right]^{\frac{1}{p-1}}+C\frac{|\mu|(Q_{r})}{r^{n-1}}\left(\fint_{Q_{r}}|\nabla
u|^{\gamma}\right)^{\frac{2-p}{\gamma}}dx.$ (1.5)

Here $\gamma=1$ in the case $p>2-1/n$ and when $p\geq 2$ the second term on
the right can be dropped (see [8], [18]). For $\frac{3n-2}{2n-1}<p\leq
2-\frac{1}{n}$, $\gamma\in(\frac{n}{2n-1},\frac{n(p-1)}{n-1})$ (see [20]).

In fact, as demonstrated in our recent work [23, Lemma 2.1], the comparison
bound (1.5) also holds for all $1<p\leq\frac{3n-2}{2n-1}$, as long as we
replace the quantity $(\fint_{Q_{r}}|\nabla u|^{\gamma}dx)^{(2-p)/\gamma}$ on
the right-hand side with the quantity $\fint_{Q_{r}}|\nabla u|^{2-p}dx$. The
power $|\nabla u|^{2-p}$ arrives naturally from the structure of $p$-Laplace
type equations when $p<2$, (see (2.6)). However, the quantity
$\fint_{Q_{r}}|\nabla u|^{2-p}dx$ is useful only when $2-p<\frac{n(p-1)}{n-1}$
as $\frac{n(p-1)}{n-1}$ is the optimal (weak) integrability of the gradient of
fundamental solution. This shows that the quantity $\fint_{Q_{r}}|\nabla
u|^{2-p}dx$ is ‘super-critical’ in the singular case
$1<p\leq\frac{3n-2}{2n-1}$.

The main result of this paper is the extension of the comparison estimate to
the remaining open case $1<p\leq\frac{3n-2}{2n-1}$. As
$\frac{3n-2}{2n-1}<3/2$, it is enough to consider the range $1<p<3/2$
mentioned above. In what follows, we shall fix a constant $\kappa$ defined by

$\kappa=(p-1)^{2}/2.$

###### Theorem 1.2

Suppose that $n\geq 2$ and $Q_{\Sigma\,r}(x_{0})\subset\Omega$ for $r>0$ and
$\Sigma\in(1,2]$. Let $u\in W_{\rm loc}^{1,p}(\Omega)$ be a solution of (1.1)
and let $w$ be as (1.4). Then

$\displaystyle\left(\fint_{Q_{r}(x_{0})}|\nabla(u-w)|^{\kappa}\right)^{\frac{1}{\kappa}}+\frac{1}{r}\left(\fint_{Q_{r}(x_{0})}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\qquad\lesssim\left(\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-1}}\left(\fint_{Q_{\Sigma r}(x_{0})}|\nabla
u|^{\kappa}\right)^{\frac{2-p}{\kappa}},$ (1.6)

and for any $\lambda\in\mathbb{R}$,

$\displaystyle\left(\fint_{Q_{r}(x_{0})}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\lesssim\left(\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-p}}\right)^{\frac{1}{p-1}}+\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-p}}\left(\fint_{Q_{\Sigma
r}(x_{0})}|u-\lambda|^{\kappa}\right)^{\frac{2-p}{\kappa}}.$

A boundary version of Theorem 1.2, where $x_{0}\in\partial\Omega$ and $u\in
W^{1,p}_{0}(\Omega)$, is also available; see Theorem 2.8 below. Theorem 1.2
follows from Lemmas 2.5, 2.6, and Corollary 2.4 below. Notice that we use the
larger cube $Q_{\Sigma r}(x_{0})$ on the right-hand side of the bounds in
Theorem 1.2, which makes it slightly different from Lemma 1.1. But this is
harmless in applications. As $\kappa<\frac{n(p-1)}{n-1}$ when $1<p<2$, we see
that the quantity $(\fint_{Q_{\Sigma r}(x_{0})}|\nabla
u|^{\kappa})^{1/\kappa}$ is sub-critical in this case. In fact, as the proof
goes, in the first step (see Lemma 2.5) we obtain an inequality similar to
(1.2) but with a mixed norm quantity

$\left(\fint_{x_{0}^{\prime}+(-r,r)^{n-1}}\left(\fint_{x_{0n}+(-r,r)}|\nabla
u(x^{\prime},x_{n})|^{2-p}dx_{n}\right)^{\frac{p-1}{3-p}}dx^{\prime}\right)^{\frac{3-p}{(p-1)(2-p)}}$
(1.7)

in place of $(\fint_{Q_{\Sigma r}(x_{0})}|\nabla u|^{\kappa})^{1/\kappa}$, and
no enlargement of the cube is needed in this step. Note that this is a
substantial improvement of [23, Lemma 2.1] as the exponent $2-p$ is used only
in one direction and the exponent for the other directions goes to zero as $p$
approaches $1$. Another way to look at this is to consider the fundamental
solution $v(x)=c\,|x|^{(p-n)/(p-1)}$, $1<p<n$, of the $p$-Laplace equation.
Then $|\nabla v|\in L^{2-p}_{\rm loc}$ if and only if $p>\frac{3n-2}{2n-1}$,
whereas the mixed norm above for $\nabla v$ is finite for all $p>1$. We indeed
show in Lemma 2.6 (see also Corollary 2.4) that the latter also holds for the
gradient of any solution to (1.1) by using the first step and a suitable
reverse Hölder type inequality for gradients of solutions of the homogeneous
equation ${\rm div}(A(x,\nabla w))=0$. Moreover, this enables us to control
the quantity in (1.7) by $(\fint_{Q_{\Sigma r}(x_{0})}|\nabla
u|^{\kappa})^{1/\kappa}$ in some way to complete the proof.

For the rest of this section, we will present some selected consequences of
Theorem 1.2. More consequences of Theorem 1.2 will be presented elsewhere in
our future work.

First, we have a Poincaré type inequality with low integrability and with a
‘correction’ term. It is well-known that Poincaré inequality generally fails
in the ‘sublinear’ range (see [4]).

###### Corollary 1.3

Suppose that $n\geq 2$ and $Q_{\Sigma\,r}(x_{0})\subset\Omega$ for $r>0$ and
$\Sigma\in(1,2]$. Let $u\in W^{1,p}_{\rm loc}(\Omega)$ be a solution of (1.1).
Then for any $\epsilon>0$ we have

$\displaystyle\left(\inf_{q\in\mathbb{R}}\fint_{Q_{r}(x_{0})}|u-q|^{\epsilon}\right)^{\frac{1}{\epsilon}}\lesssim\left(\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-p}}\right)^{\frac{1}{p-1}}+r\left(\fint_{Q_{\Sigma
r}(x_{0})}|\nabla u|^{\epsilon}\right)^{\frac{1}{\epsilon}}.$

Second, we have a result on potential estimates for solutions of (1.1). In
particular, Theorem 1.4 below extends [14, Theorems 1.1 and 1.10] to all
smaller values of $p$. For nonnegative measures $\mu$, (1.8) is a fundamental
result due to Kilpeläinen and Malý [13] (see also [29]). Hereafter, the
(truncated) Havin-Maz’ya-Wolff’s potential (often called Wolff’s potential)
${\bf W}_{\gamma,p}^{R}(\nu)$, $R,\gamma>0$, of a nonnegative measure $\nu$ is
defined by

${\bf
W}_{\gamma,p}^{R}(\nu)(x):=\int_{0}^{R}\left[\frac{\nu(B_{t}(x))}{t^{n-\gamma
p}}\right]^{\frac{1}{p-1}}\frac{dt}{t},\quad x\in\mathbb{R}^{n}.$

On the other hand, the truncated Riesz’s potential of order $\gamma>0$ is
defined as

${\bf I}^{R}_{\gamma}(\nu):={\bf
W}_{\gamma/2,2}^{R}(\nu)(x)=\int_{0}^{R}\frac{\nu(B_{t}(x))}{t^{n-\gamma}}\frac{dt}{t},\quad
x\in\mathbb{R}^{n}.$

###### Theorem 1.4

Suppose that $u\in C^{0}(\Omega)\cap W^{1,p}_{\rm loc}(\Omega)$ solves (1.1)
for a finite measure $\mu$ in $\Omega$. Then under (1.2)–(1.3), for any
$B_{R}(x_{0})\subset\Omega$ we have

$|u(x_{0})|\lesssim{\bf
W}_{1,p}^{R}(|\mu|)(x_{0})+\Big{(}\fint_{B_{R}(x_{0})}|u(z)|^{\kappa}dz\Big{)}^{\frac{1}{\kappa}}.$
(1.8)

Third, we discuss potential estimates for gradients of solutions to (1.1).
Pointwise estimates for the gradient were obtained in [9, 8, 15, 16] for the
case $p>2-1/n$. For the case $\frac{3n-2}{2n-1}<p\leq 2-1/n$, see the recent
work [7] (see also [21]). The next theorem covers all of the remaining range
of $p$.

###### Theorem 1.5

Assume that $A({x,\xi})$ satisfies the following Dini condition:

$\int_{0}^{1}\omega(\rho)\frac{d\rho}{\rho}<+\infty$ (1.9)

for a non-decreasing function $\omega:[0,1]\rightarrow[0,\infty)$ such that
$\lim_{\rho\rightarrow 0^{+}}\omega(\rho)=\omega(0)=0$ and

$|A(x,\xi)-A(y,\xi)|\leq\omega(|x-y|)|\xi|^{p-1},\qquad\forall
x,y,\xi\in\mathbb{R}^{n},|x-y|\leq 1.$

Suppose that $u\in C^{1}(\Omega)$ solves (1.1) for a finite measure $\mu$ in
$\Omega$. Then for any ball $B_{R}(x_{0})\subset\Omega$ we have

$|\nabla u(x_{0})|\lesssim\Big{[}{\bf
I}_{1}^{R}(|\mu|)(x_{0})\Big{]}^{\frac{1}{p-1}}+\Big{(}\fint_{B_{R}(x_{0})}|\nabla
u(y)|^{\kappa}dy\Big{)}^{\frac{1}{\kappa}}.$

The proof of Theorem 1.5 is based mainly on Theorem 1.2 and the pioneering
idea of [8] (see also [14]). However, some modifications are needed due to the
fact that the gradient of solution may not belong to $L^{1}_{{\rm
loc}}(\Omega)$. Such modifications have been carried out in [7] for
$\frac{3n-2}{2n-1}<p\leq 2-1/n$ (see also [21]). The main point here is to
replace the mean oscillations such as

$\fint_{B_{\rho}}|\nabla u-(\nabla u)_{B_{\rho}}|dy,\qquad(\nabla
u)_{B_{\rho}}:=\fint_{B_{\rho}}\nabla udz,$

with the quantity

$\left(\inf_{\mathbf{q}\in\mathbb{R}^{n}}\fint_{B_{\rho}}|\nabla
u-\mathbf{q}|^{\kappa}dy\right)^{\frac{1}{\kappa}}.$

Thus we shall omit the details of the proof of Theorem 1.5.

Fourth, we discuss some global gradient estimates for renormalized solutions
to the Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{rcl}-{\rm div}(A(x,\nabla
u))&=&\mu\quad\text{in}~{}\Omega,\\\
u&=&0\quad\text{on}~{}\partial\Omega.\end{array}\right.$ (1.12)

We refer to the paper [5] for several equivalent definitions of renormalized
solutions along with their stability property. Here we mention that
truncations of a renormalized solution $u$ of (1.12) are stable even near the
boundary in the sense that if $u_{k}=T_{k}(u)$, $k>0$, where

$T_{k}(s)=\max\\{\min\\{s,k\\},-k\\},\qquad s\in\mathbb{R},$ (1.13)

then $u_{k}\in W^{1,p}_{0}(\Omega)$ is the unique solution of (1.12) with
measure datum $\mu_{k}$ in place of $\mu$ such that $\mu_{k}$ converges in the
narrow topology of measures to $\mu$, i.e.,

$\lim_{k\rightarrow\infty}\int_{\Omega}\varphi\,d\mu_{k}=\int_{\Omega}\varphi\,d\mu,$

for every bounded and continuous function $\varphi$ on $\Omega$.

Pointwise gradient estimates up to the boundary for renormalized solutions of
(1.12) can be deduced from [7] in the case $p\in(\frac{3n-2}{2n-1},2)$. Here
$A(x,\xi)$ is assumed to have the special structure
$A(x,\xi)=a(x)|\xi|^{p-2}\xi$ for any $x$ near $\partial\Omega$, and the
boundary $\partial\Omega$ is assumed to be of class $C^{1,\,{\rm Dini}}$ (see
[7]). With Theorem 1.2 and its boundary counterpart (Theorem 2.8) at hand, the
method of [7] also applies to the case $1<p<3/2$. For that reason, we shall
not present the details of the proof.

###### Theorem 1.6

Assume that $A({x,\xi})$ satisfies the Dini condition (1.9), and that
$\partial\Omega$ is of class $C^{1,\alpha}$ or even $C^{1,\,{\rm Dini}}$.
Assume also that $A(x,\xi)=a(x)|\xi|^{p-2}\xi$ for any $x$ near
$\partial\Omega$. Then for any renormalized solution $u$ of (1.12) we have

$|\nabla u(x)|\leq C_{0}\Big{[}{\bf I}_{1}^{2{\rm
diam}(\Omega)}(|\mu|)(x)\Big{]}^{\frac{1}{p-1}}\qquad{\rm a.e.~{}}x\in\Omega.$
(1.14)

On the other hand, in many applications it is enough to use a weighted
integral estimate of Muckenhoupt-Wheeden type for the gradient, which we shall
describe next. For such weighted integral bounds, we can work with equation
(1.12) under a much weaker condition on the coefficients and the domain.

As for the coefficients, we recall the following definition from [25].

###### Definition 1.7

Given $\delta,R_{0}>0$, we say that $A({x,\xi})$ satisfies the
$(\delta,R_{0})$-BMO condition if

$\sup_{y\in\mathbb{R}^{n},\,0<r\leq
R_{0}}\fint_{B_{r}(y)}\Upsilon(A,B_{r}(y))(x)dx\leq\delta,$

where

$\Upsilon(A,B_{r}(y))(x):=\sup_{\xi\in\mathbb{R}^{n}\setminus\\{0\\}}\frac{|A({x,\xi})-\fint_{B_{r}(y)}A(z,\xi)dz|}{|\xi|^{p-1}}.$

As for the domain, we shall use the notion of Reifenberg flat domain (see
[28]). This includes $C^{1}$ and Lipschitz domains (with sufficiently small
Lipschitz constants), as well as certain fractal domains.

###### Definition 1.8

Given $\delta\in(0,1)$ and $R_{0}>0$, we say that $\Omega$ is a
$(\delta,R_{0})$-Reifenberg flat domain if for every $x\in\partial\Omega$ and
every $r\in(0,R_{0}]$, there exists a system of coordinates
$\\{z_{1},z_{2},\dots,z_{n}\\}$, which may depend on $r$ and $x$, so that in
this coordinate system $x=0$ and that

$B_{r}(0)\cap\\{z_{n}>\delta r\\}\subset B_{r}(0)\cap\Omega\subset
B_{r}(0)\cap\\{z_{n}>-\delta r\\}.$

Our weighted integral estimate is obtained for the class of
$\mathbf{A}_{\infty}$ weights. This class consists of nonnegative functions
$w\in L^{1}_{\text{loc}}(\mathbb{R}^{n})$ such that there are two positive
constants $C$ and $c$ such that

$w(E)\leq C\left(\frac{|E|}{|B|}\right)^{c}w(B),$

for all balls $B$ and all measurable subsets $E$ of $B$. The pair $(C,c)$ is
called the $\mathbf{A}_{\infty}$ constants of $w$ and is denoted by
$[w]_{\mathbf{A}_{\infty}}$.

The following theorem extends the results of [25, 20] (see also [23]) to the
remaining case $1<p\leq\frac{3n-2}{2n-1}$.

###### Theorem 1.9

For any $w\in\mathbf{A}_{\infty}$ and $0<q<\infty$, there exists
$\delta=\delta(n,p,\Lambda,q,[w]_{\mathbf{A}_{\infty}})\in(0,1)$ such that if
$A(x,\xi)$ is $(\delta,R_{0})$-BMO and $\Omega$ is $(\delta,R_{0})$-Reifenberg
flat for some $R_{0}>0$, then for any renormalized solution $u$ of (1.12), we
have

$\|\nabla u\|_{L^{q}_{w}(\Omega)}\leq
C\|[\mathbf{M}_{1}(\mu)]^{\frac{1}{p-1}}\|_{L^{q}_{w}(\Omega)}.$ (1.15)

Here the constant
$C=C(n,p,\Lambda,q,[w]_{\mathbf{A}_{\infty}},diam(\Omega)/R_{0})$, and
$\mathbf{M}_{1}(\mu)$ is a fractional maximal function $\mu$ defined by

$\mathbf{M}_{1}(\mu)(x):=\sup_{r>0}\frac{|\mu|(B_{t}(x))}{t^{n-1}},\qquad
x\in\mathbb{R}^{n}.$

We remark that the weighted bound (1.15) also holds if we replace the weighted
space $L^{q}_{w}(\Omega)$ with the weighted Lorentz space
$L^{q,s}_{w}(\Omega)$ for any $0<s\leq\infty$.

A proof of Theorem 1.9 in the case $p>2-1/n$ and the case
$\frac{3n-2}{2n-1}<p\leq 2-1/n$ was presented in [25] and [20], respectively.
Now with the availability of the interior and boundary comparison estimates
(Theorems 1.2 and 2.8), the proof also works in the present case $1<p<3/2$.
For that, we shall not repeat the proof here. We mention that, following the
approach of [20], a ‘good-$\lambda$’ type inequality involving
$\mathbf{M}_{1}(\mu)^{1/(p-1)}$ and ${\bf M}(|\nabla u|^{\kappa})^{1/\kappa}$,
where ${\bf M}$ is the Hardy-Littlewood maximal function, can also be obtained
as in [20, Theorem 1.5].

Finally, as an application of Theorem 1.6 we obtain a sharp existence result
for a quasilinear Riccati type equation with measure data:

$\displaystyle\left\\{\begin{array}[]{rcl}-{\rm div}(A(x,\nabla u))&=&|\nabla
u|^{q}+\mu\quad\text{in}~{}\Omega,\\\
u&=&0\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{on}~{}\partial\Omega.\end{array}\right.$
(1.18)

To this end, we shall need the notion of capacity associated to the Sobolev
space $W^{1,s}(\mathbb{R}^{n})$, $1<s<+\infty$, defined for each compact set
$K\subset\mathbb{R}^{n}$ by

${\rm
Cap}_{1,s}(K)=\inf\Big{\\{}\int_{\mathbb{R}^{n}}(|\nabla\varphi|^{s}+\varphi^{s})dx:0\leq\varphi\in
C^{\infty}_{0}(\mathbb{R}^{n}),\varphi\geq 1{\rm~{}on~{}}K\Big{\\}}.$

###### Theorem 1.10

Let $q>p-1$. Assume that (1.9) holds and $\partial\Omega$ is of class
$C^{1,\,{\rm Dini}}$. Also, assume that $A(x,\xi)=a(x)|\xi|^{p-2}\xi$ for any
$x$ near the boundary $\partial\Omega$. Then there exists a constant $c_{0}>0$
such that if the measure $\mu$ satisfies

$|\mu|(K)\leq c_{0}\,{\rm Cap}_{1,\,\frac{q}{q-p+1}}(K)$ (1.19)

for all compact sets $K\subset\Omega$, then there exists a renormalized
solution $u$ to the Riccati type equation (1.18) such that

$|\nabla u(x)|\lesssim\Big{[}{\bf I}_{1}^{2{\rm
diam}(\Omega)}(|\mu|)(x)\Big{]}^{\frac{1}{p-1}}\qquad{\rm a.e.~{}}x\in\Omega.$

We remark that condition (1.19) is sharp at least for nonnegative measures
with compact support in $\Omega$ (see [12, 24]). It is worth mentioning that
Theorem 1.10 completely solve a problem raised in [3, pages 13–14]. On the
other hand, it is new mainly in the case $\frac{n(p-1)}{n-1}\leq q<1$ as the
case $p-1<q<\frac{n(p-1)}{n-1}$ and the case $q\geq 1$ have been treated in
[11] and [23], respectively. For other earlier work on equation (1.18), we
refer to the papers [12, 24, 25, 26, 17, 22] and the references therein.

##  2 Comparison estimates

In what follows, for $x\in\mathbb{R}^{n}$ we write
$x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$. With this, we will
use the notation
$\|f\|_{L^{s_{2}}_{x^{\prime}}L^{s_{1}}_{x_{n}}(Q_{\rho}(x_{0}))}$ for
$s_{1},s_{2}>0$ and $\rho>0$, to indicate the mixed norm

$\|f\|_{L^{s_{2}}_{x^{\prime}}L^{s_{1}}_{x_{n}}(Q_{\rho}(x_{0}))}=\left(\int_{x_{0}^{\prime}+(-\rho,\rho)^{n-1}}\left(\int_{x_{0n}+(-\rho,\rho)}|f(x^{\prime},x_{n})|^{s_{1}}dx_{n}\right)^{\frac{s_{2}}{s_{1}}}dx^{\prime}\right)^{\frac{1}{s_{2}}}.$

On the other hand, we write
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s_{2}}_{x^{\prime}}L^{s_{1}}_{x_{n}}(Q_{\rho}(x_{0}))}$
for the normalized mixed norm

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s_{2}}_{x^{\prime}}L^{s_{1}}_{x_{n}}(Q_{\rho}(x_{0}))}=\left(\fint_{x_{0}^{\prime}+(-\rho,\rho)^{n-1}}\left(\fint_{x_{0n}+(-\rho,\rho)}|f(x^{\prime},x_{n})|^{s_{1}}dx_{n}\right)^{\frac{s_{2}}{s_{1}}}dx^{\prime}\right)^{\frac{1}{s_{2}}}.$

We begin with the following lemma.

###### Lemma 2.1

Let $u\in W^{1,p}_{\rm loc}(\Omega)$ be a solution of (1.1). Then for any
function $\phi\in C_{0}^{\infty}(\Omega)$, $\phi\geq 0$, and $\epsilon>0$, we
have

$\displaystyle\int_{\Omega}\left|\nabla\left[(1+|u|)^{\frac{p-1-\epsilon}{p}}\phi\right]\right|^{p}dx$
$\displaystyle\lesssim\int_{\Omega}\phi^{p}d|\mu|+\int_{\Omega}(1+|u|)^{(p-1)(1+\epsilon)}|\nabla\phi|^{p}dx.$

Proof. Let $\phi$ be a function in $C_{0}^{\infty}(\Omega)$ such that
$\phi\geq 0$. Then using

${\rm sign}(u)[1-(1+|u|)^{-\epsilon}]\phi^{p},\qquad\epsilon>0,$

as a test function in (1.1) we have

$\displaystyle\epsilon\int_{\Omega}\frac{|\nabla
u|^{p}}{(1+|u|)^{1+\epsilon}}\phi^{p}dx\lesssim\int\phi^{p}d|\mu|+p\int_{\Omega}|\nabla
u|^{p-1}\phi^{p-1}|\nabla\phi|dx$
$\displaystyle=\int_{\Omega}\phi^{p}d|\mu|+p\int_{\Omega}|\nabla
u|^{p-1}\phi^{p-1}(1+|u|)^{(-\epsilon-1)\frac{p-1}{p}}(1+|u|)^{(\epsilon+1)\frac{p-1}{p}}|\nabla\phi|dx.$

Thus by Young’s inequality we obtain

$\int_{\Omega}\frac{|\nabla
u|^{p}}{(1+|u|)^{1+\epsilon}}\phi^{p}dx\lesssim\int_{\Omega}\phi^{p}d|\mu|+\int_{\Omega}(1+|u|)^{(p-1)(1+\epsilon)}|\nabla\phi|^{p}dx.$

Now in view of the identity

$\nabla\left[(1+|u|)^{\frac{p-1-\epsilon}{p}}\phi\right]=\tfrac{p-1-\epsilon}{p}(1+|u|)^{\frac{-1-\epsilon}{p}}\nabla
u\,{\rm sign}(u)\phi+(1+|u|)^{\frac{p-1-\epsilon}{p}}\nabla\phi,$

we obtain the lemma.

Lemma 2.1 yields the following reverse Hölder type inequality for the
solution.

###### Lemma 2.2

Let $u\in W^{1,p}_{\rm loc}(\Omega)$ be a solution of (1.1). Then for any cube
$Q_{R}(x_{0})\subset\Omega$ and any $q>q_{1}>0$ we we have

$\displaystyle\left(\fint_{Q_{\sigma
R}(x_{0})}|u|^{q}dx\right)^{\frac{1}{q}}\lesssim\left(\frac{|\mu|(Q_{R}(x_{0}))}{R^{n-p}}\right)^{\frac{1}{p-1}}+\left(\fint_{Q_{R}(x_{0})}|u|^{q_{1}}dx\right)^{\frac{1}{q_{1}}},$

provided $q<\frac{n(p-1)}{n-p}$ and $\sigma\in(0,1)$.

Proof. By translation and scale invariance, we may assume that
$x_{0}=(0,\dots,0)$, $Q_{R}(x_{0})=(-1,1)^{n}$, and

$|\mu|((-1,1)^{n})^{\frac{1}{p-1}}+\|u\|_{L^{q_{1}}((-1,1)^{n})}\lesssim 1.$
(2.1)

Moreover, we need to show that for any $\sigma<1$,

$\|u\|_{L^{q}((-\sigma,\sigma)^{n})}\lesssim 1;$

see [8, Remark 4.1] for details.
Let $\epsilon>0$ and let $\phi$ be a function in $C_{0}^{\infty}(Q_{r}(0))$
such that $0\leq\phi\leq 1$, $\phi\equiv 1$ on $Q_{s}(0)$, $1\geq r>s>0$, and
$|\nabla\phi|\leq C/(r-s).$ By Lemma 2.1 and Sobolev embedding theorem we have

$\|(1+|u|)^{\frac{p-1-\epsilon}{p}}\phi\|^{p}_{L^{p^{*}}(\Omega)}\lesssim\int_{\Omega}\phi^{p}d|\mu|+\int_{\Omega}(1+|u|)^{(p-1)(1+\epsilon)}|\nabla\phi|^{p}dx,$

where $p^{*}=\frac{np}{n-p}$. This gives, in view of (2.1),

$\left(\int_{Q_{s}(0)}(1+|u|)^{\frac{(p-1-\epsilon)p^{*}}{p}}dx\right)^{\frac{p}{p^{*}}}\lesssim
1+\frac{1}{(r-s)^{p}}\int_{Q_{r}(0)}(1+|u|)^{(p-1)(1+\epsilon)}dx.$

Let

$q_{\epsilon}:=\frac{(p-1-\epsilon)p^{*}}{p},\qquad\tilde{q}_{\epsilon}:=(p-1)(1+\epsilon),$

and choose $\epsilon$ sufficiently small so that
$q_{\epsilon}\in\left[q,\frac{n(p-1)}{n-p}\right)$, and
$\tilde{q}_{\epsilon}<q_{\epsilon}.$
Then we can rewrite the above inequality as

$\int_{Q_{s}(0)}(1+|u|)^{q_{\epsilon}}dx\lesssim
1+\frac{1}{(r-s)^{p^{*}}}\left(\int_{Q_{r}(0)}(1+|u|)^{\tilde{q}_{\epsilon}}dx\right)^{p^{*}/p}.$

Also, we may assume that $q_{1}<p-1<\tilde{q}_{\epsilon}$ and then by Hölder’s
inequality and (2.1) we have

$\displaystyle\left(\int_{Q_{r}(0)}(1+|u|)^{\tilde{q}_{\epsilon}}dx\right)^{\frac{1}{\tilde{q}_{\epsilon}}}$
$\displaystyle\leq\left(\int_{Q_{r}(0)}(1+|u|)^{q_{\epsilon}}dx\right)^{\frac{\theta}{q_{\epsilon}}}\left(\int_{Q_{r}(0)}(1+|u|)^{q_{1}}dx\right)^{\frac{1-\theta}{q_{1}}}$
$\displaystyle\leq\left(\int_{Q_{r}(0)}(1+|u|)^{q_{\epsilon}}dx\right)^{\frac{\theta}{q_{\epsilon}}},$

where
$\theta=\frac{(\tilde{q}_{\epsilon}-q_{1})q_{\epsilon}}{(q_{\epsilon}-q_{1})\tilde{q}_{\epsilon}}.$
We next further restrict $\epsilon<\frac{q_{1}}{n}$ so that

$\frac{\theta\tilde{q}_{\epsilon}}{q_{\epsilon}}\frac{p^{*}}{p}=\frac{\tilde{q}_{\epsilon}-q_{1}}{q_{\epsilon}-q_{1}}\frac{n}{n-p}<1.$

Then by Young’s inequality with exponents
$\frac{pq_{\epsilon}}{\theta\tilde{q}_{\epsilon}p^{*}},$ and
$\frac{pq_{\epsilon}}{pq_{\epsilon}-\theta\tilde{q}_{\epsilon}p^{*}},$ we get

$\int_{Q_{s}(0)}(1+|u|)^{q_{\epsilon}}dx\leq
C+\frac{1}{2}\int_{Q_{r}(0)}(1+|u|)^{q_{\epsilon}}dx+\frac{C}{(r-s)^{\mathcal{H}_{0}}},$

where

$\mathcal{H}_{0}=\frac{p^{*}pq_{\epsilon}}{pq_{\epsilon}-\theta\tilde{q}_{\epsilon}p^{*}}=\frac{np(q_{\epsilon}-q_{1})}{n(q_{\epsilon}-\tilde{q}_{\epsilon})-p(q_{\epsilon}-q_{1})}>0.$

Now we may apply Lemma 6.1 of [10] to obtain

$\int_{Q_{\sigma}(0)}(1+|u|)^{q_{\epsilon}}dx\leq C,$

which of course yields the desired result.
We next give a mixed norm estimate for the gradient.

###### Lemma 2.3

Let $Q_{r}(x_{0})\subset\Omega$ and $1-\frac{n-1}{p}<\alpha<1/p$, $\alpha>0$,
$n\geq 2$. Then for any $\sigma\in(0,1)$ and $q_{1}>0$ we have

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{p_{2}}_{x^{\prime}}L_{x_{n}}^{p_{1}}(Q_{\sigma
r}(x_{0}))}\lesssim\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+\frac{1}{r}\left(\fint_{Q_{r}(x_{0})}|u|^{q_{1}}dx\right)^{\frac{1}{q_{1}}},$

where $\gamma\in(1,\min\\{p,n/(n-p)\\})$, and

$p_{1}=\frac{p-\gamma}{1-\alpha\gamma},\qquad
p_{2}=\frac{(n-1)(p-\gamma)}{n-1-(1-\alpha)\gamma}.$

Proof. By translating and scaling considerations, we may assume that
$x_{0}=(0,\dots,0)$, $Q_{r}(x_{0})=Q_{1}(0)=(-1,1)^{n}$, and (2.1) holds.
Moreover, we need to show that for any $\sigma<1$

$\|\nabla u\|_{L^{p_{2}}_{x^{\prime}}L_{x_{n}}^{p_{1}}(Q_{\sigma}(0))}\lesssim
1.$

By Lemma 2.1, Lemma 2.2, and (2.1), we find

$\int_{Q_{\sigma}(0)}\frac{|\nabla u|^{p}}{(1+|u|)^{\gamma}}dx\leq
C|\mu|(Q_{1}(0))+C\int_{Q_{1}(0)}(1+|u|)^{(p-1)\gamma}dx\leq C,$ (2.2)

provided $\gamma\in(1,\min\\{p,n/(n-p)\\})$. Here $\sigma\in(0,1)$ and the
constant $C$ may depend on $\sigma$.
Also, by Lemma 2.1 and Sobolev embedding theorem for Lebesgue spaces of mixed
norm (see [2, 1]), for any $\alpha$ satisfying $1-\frac{n-1}{p}<\alpha<1/p$,
$\alpha>0$, we have

$\|(1+|u|)^{\frac{p-\gamma}{p}}\phi\|^{p}_{L_{x^{\prime}}^{\frac{(n-1)p}{n-1-(1-\alpha)p}}L_{x_{n}}^{\frac{p}{1-\alpha
p}}}\lesssim\int_{\Omega}\phi^{p}d|\mu|+\int_{\Omega}(1+|u|)^{(p-1)\gamma}|\nabla\phi|^{p}dx,$

where $\phi\in C_{0}^{\infty}(Q_{1}(0))$, $\phi\geq 0$, and
$1<\gamma<\min\\{p,n/(n-p)\\}$. By Lemma 2.2 and (2.1), this implies in
particular that

$\left\|1+|u|\right\|^{p-\gamma}_{L_{x^{\prime}}^{\frac{(n-1)(p-\gamma)}{n-1-(1-\alpha)p}}L_{x_{n}}^{\frac{p-\gamma}{1-\alpha
p}}(Q_{\sigma}(0))}\lesssim 1.$ (2.3)

On the other hand, by Hölder’s inequality we have

$\|\nabla
u\|_{L^{p_{2}}_{x^{\prime}}L_{x_{n}}^{p_{1}}(Q_{\sigma}(0))}\leq\Big{\|}\frac{\nabla
u}{{(1+|u|)^{\frac{\gamma}{p}}}}\Big{\|}_{L^{p}(Q_{\sigma}(0))}\|1+|u|\|^{\frac{\gamma}{p}}_{L^{\frac{\gamma
q_{2}}{p}}_{x^{\prime}}L_{x_{n}}^{\frac{\gamma q_{1}}{p}}(Q_{\sigma}(0))},$
(2.4)

where

$\frac{1}{p_{1}}=\frac{1}{p}+\frac{1}{q_{1}},\quad\frac{1}{p_{2}}=\frac{1}{p}+\frac{1}{q_{2}},\quad
p_{1},p_{2},q_{1},q_{2}>0.$

We now choose

$q_{1}=\frac{p(p-\gamma)}{\gamma(1-\alpha p)},\qquad
q_{2}=\frac{p(n-1)(p-\gamma)}{\gamma[n-1-(1-\alpha)p]},$

which forces

$p_{1}=\frac{p-\gamma}{1-\alpha\gamma},\qquad
p_{2}=\frac{(n-1)(p-\gamma)}{n-1-(1-\alpha)\gamma}.$

At this point, we combine (2.2), (2.3), and (2.4) to deduce

$\|\nabla u\|_{L^{p_{2}}_{x^{\prime}}L_{x_{n}}^{p_{1}}(Q_{\sigma}(0))}\lesssim
1.$

This completes the proof of the lemma.

We observe that as $\gamma\downarrow 1$, we have
$p_{1}\uparrow\frac{p-1}{1-\alpha}$ and
$p_{2}\uparrow\frac{(n-1)(p-1)}{n-2+\alpha}$. Thus by Hölder’s inequality we
obtain the following result.

###### Corollary 2.4

Let $Q_{r}(x_{0})\subset\Omega$ and $1<p<n$, $n\geq 2$. Then for any
$\sigma\in(0,1)$ and $q_{1}>0$, we have

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{\sigma
r}(x_{0}))}\lesssim\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+\frac{1}{r}\left(\fint_{Q_{r}(x_{0})}|u|^{q_{1}}dx\right)^{\frac{1}{q_{1}}},$

provided

$\frac{p-1}{n-1}<s_{1}<p,\qquad
0<s_{2}<\frac{s_{1}(n-1)(p-1)}{s_{1}(n-1)-p+1}.$

We now obtain a preliminary version of the comparison estimate.

###### Lemma 2.5

Suppose that $n\geq 2$ and $Q_{r}(x_{0})\Subset\Omega$. Let $u\in W_{\rm
loc}^{1,p}(\Omega)$ be a solution of (1.1) and let $w$ be as (1.4). Then for
$\kappa=(p-1)^{2}/2$ and $1<p<3/2$ we have

$\displaystyle\left(\fint_{Q_{r}(x_{0})}|\nabla(u-w)|^{\kappa}\right)^{\frac{1}{\kappa}}+\frac{1}{r}\left(\fint_{Q_{r}(x_{0})}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\qquad\qquad\qquad\lesssim\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2-p}_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{r}(x_{0}))}.$

Proof. Again by translating and scaling, we may assume that
$x_{0}=(0,\dots,0)$, $Q_{r}(x_{0})=(-1,1)^{n}$, and

$A:=|\mu|((-1,1)^{n})^{\frac{1}{p-1}}+|\mu|((-1,1)^{n})\|\nabla
u\|^{2-p}_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}((-1,1)^{n})}\lesssim
1.$ (2.5)

Moreover, we just need to show that

$\|\nabla u-\nabla
w\|_{L^{\kappa}((-1,1)^{n})}+\|u-w\|_{L^{\kappa}((-1,1)^{n})}\lesssim 1.$

For any given $k>0$ we now set

$E_{k}=(-1,1)^{n}\cap\\{k<|u-w|<2k\\},\quad\quad
F_{k}=(-1,1)^{n}\cap\\{|u-w|>k\\}.$

We first recall that for $1<p<2$,

$|\nabla u(x)-\nabla w(x)|\leq Cg(x)^{1/p}+Cg(x)^{1/2}|\nabla
u(x)|^{\frac{2-p}{2}},$ (2.6)

where

$g(x):=\frac{|\nabla u(x)-\nabla w(x)|^{2}}{(|\nabla u(x)|+|\nabla
w(x)|)^{2-p}}.$

Let $T_{k}$, $k>0,$ be the truncation operator defined in (1.13). Then it
follows from (2.6) that

$|\partial_{x_{n}}(T_{2k}-T_{k})(u-w)(x^{\prime},x_{n})|\lesssim\left(\mathbf{1}_{E_{k}}g(x^{\prime},x_{n})^{\frac{1}{p}}+\mathbf{1}_{E_{k}}g(x^{\prime},x_{n})^{\frac{1}{2}}|\nabla
u(x^{\prime},x_{n})|^{\frac{2-p}{2}}\right),$

where $\mathbf{1}_{E_{k}}$ is the characteristic function of the set $E_{k}$.
Thus, for any $0<\gamma\leq 1$, by Hölder’s inequality we find

$\displaystyle\int_{(-1,1)^{n}}|(T_{2k}-T_{k})(u-w)(x)|^{\gamma}dx$
$\displaystyle\leq
2\int_{(-1,1)^{n-1}}\left(\int_{-1}^{1}|\partial_{x_{n}}(T_{2k}-T_{k})(u-w)(x^{\prime},x_{n})|dx_{n}\right)^{\gamma}dx^{\prime}$
$\displaystyle\lesssim\left(\int_{(-1,1)^{n}}\mathbf{1}_{E_{k}}g(x)^{\frac{1}{p}}dx\right)^{\gamma}$
$\displaystyle\quad+\int_{(-1,1)^{n-1}}\left(\int_{-1}^{1}\mathbf{1}_{E_{k}}g(x^{\prime},x_{n})dx_{n}\right)^{\frac{\gamma}{2}}\left(\int_{-1}^{1}|\nabla
u(x^{\prime},x_{n})|^{2-p}dx_{n}\right)^{\frac{\gamma}{2}}dx^{\prime}$
$\displaystyle\lesssim\left(\int_{E_{k}}g(x)dx\right)^{\frac{\gamma}{p}}|E_{k}|^{\frac{\gamma(p-1)}{p}}$
$\displaystyle\quad+\left(\int_{E_{k}}g(x)dx\right)^{\frac{\gamma}{2}}\left(\int_{(-1,1)^{n-1}}\left(\int_{-1}^{1}|\nabla
u(x^{\prime},x_{n})|^{2-p}dx_{n}\right)^{\frac{\gamma}{2-\gamma}}dx^{\prime}\right)^{\frac{2-\gamma}{2}}.$

On the other hand, using $T_{2k}(u-w)$ as a test function for (1.1) we have

$\int_{E_{k}}g(x)dx\leq\int_{(-1,1)^{n}\cap\\{|u-w|<2k\\}}g(x)dx\lesssim
k|\mu|((-1,1)^{n})\lesssim kA^{p-1}.$ (2.7)

Thus we get

$\displaystyle\int_{(-1,1)^{n}}|(T_{2k}-T_{k})(u-w)(x)|^{\gamma}dx\lesssim
k^{\frac{\gamma}{p}}A^{\frac{\gamma(p-1)}{p}}|E_{k}|^{\frac{\gamma(p-1)}{p}}$
$\displaystyle+k^{\frac{\gamma}{2}}|\mu|((-1,1)^{n})^{\frac{\gamma}{2}}\left(\int_{(-1,1)^{n-1}}\left(\int_{-1}^{1}|\nabla
u(x^{\prime},x_{n})|^{2-p}dx_{n}\right)^{\frac{\gamma}{2-\gamma}}dx^{\prime}\right)^{\frac{2-\gamma}{2}}.$

We now set

$\gamma=p-1.$

Observe that $E_{k}\subset F_{k}$ and on $F_{2k}$ we have
$|(T_{2k}-T_{k})(u-w)|=k$. Then in view of (2.5), this gives

$k^{\gamma}|F_{2k}|\lesssim
k^{\frac{\gamma}{p}}A^{\frac{\gamma(p-1)}{p}}|F_{k}|^{\frac{\gamma(p-1)}{p}}+k^{\frac{\gamma}{2}}A^{\frac{p-1}{2}}.$

As $|F_{2k}|\leq|F_{k}|$ and $|F_{2k}|\leq 2^{n}$, we can write for any
$\nu>0$,

$k^{\frac{\gamma}{2}}|F_{2k}|^{1+\nu}\lesssim
k^{\frac{\gamma}{p}-\frac{\gamma}{2}}A^{\frac{\gamma(p-1)}{p}}|F_{k}|^{\frac{\gamma(p-1)}{p}+\nu}+A^{\frac{p-1}{2}}.$

Then taking the supremum over $k>0$ we obtain

$||w-u||_{L^{\frac{\gamma}{2(1+\nu)},\infty}}^{\frac{\gamma}{2}}\lesssim||w-u||_{L^{\frac{\frac{\gamma}{p}-\frac{\gamma}{2}}{\frac{\gamma(p-1)}{p}+\nu},\infty}}^{\frac{\gamma}{p}-\frac{\gamma}{2}}A^{\frac{\gamma(p-1)}{p}}+A^{\frac{p-1}{2}}.$

We now choose $\nu\geq 0$ such that

$\frac{\gamma}{2(1+\nu)}=\frac{\frac{\gamma}{p}-\frac{\gamma}{2}}{\frac{\gamma(p-1)}{p}+\nu}\Longleftrightarrow\nu=\frac{2-p-\gamma(p-1)}{2(p-1)}=\frac{2-p-(p-1)^{2}}{2(p-1)}>0.$

Then using Young’s inequality, we obtain

$\|w-u\|_{L^{\frac{\gamma(p-1)}{p-\gamma(p-1)},\infty}}\lesssim A\lesssim
1,\qquad\gamma=p-1.$

Note that by (2.7) and Chebyshev’s inequality,

$\displaystyle|\\{g>\lambda\\}|$
$\displaystyle=|\\{g>\lambda\\}\cap\\{|u-w|\geq
k\\}|+|\\{g>\lambda\\}\cap\\{|u-w|<k\\}|$ $\displaystyle\lesssim
k^{-\frac{\gamma(p-1)}{p-\gamma(p-1)}}||u-w||_{L_{\omega}^{\frac{\gamma(p-1)}{p-\gamma(p-1)},\infty}}^{\frac{\gamma(p-1)}{p-\gamma(p-1)}}+\frac{1}{\lambda}\int_{(-1,1)^{n}\cap\\{|u-w|<k\\}}gdx$
$\displaystyle\lesssim
k^{-\frac{\gamma(p-1)}{p-\gamma(p-1)}}+\frac{k}{\lambda}|\mu|((-1,1)^{n}).$

Thus choosing
$k=\lambda^{1-\frac{\gamma(p-1)}{p}}|\mu|((-1,1)^{n})^{-1+\frac{\gamma(p-1)}{p}}$,
one has

$|\\{g>\lambda\\}|\lesssim\lambda^{-\frac{\gamma(p-1)}{p}}|\mu|((-1,1)^{n})^{\frac{\gamma(p-1)}{p}}\lesssim\lambda^{-\frac{(p-1)^{2}}{p}}|\mu|((-1,1)^{n})^{\frac{(p-1)^{2}}{p}}.$

It follows that for any $\beta<\frac{(p-1)^{2}}{p}$,

$\|g\|_{L^{\beta}}\lesssim|\mu|((-1,1)^{n}),$

which in view of (2.6) and (2.5) yields

$\|\nabla(u-w)\|_{L^{p\beta}}\lesssim\|g\|_{L^{\beta}}^{\frac{1}{p}}+\|g\|_{L^{\beta}}^{\frac{1}{2}}\|\nabla
u\|_{L^{p\beta}}^{\frac{2-p}{2}}\lesssim
A^{\frac{p-1}{p}}+|\mu|((-1,1)^{n})^{\frac{1}{2}}\|\nabla
u\|_{L^{p\beta}}^{\frac{2-p}{2}}.$

Next, we choose $\beta=\frac{(p-1)^{2}}{2p},$ and employ Holder’s inequality
to deduce

$\|\nabla u\|_{L^{p\beta}}=\|\nabla
u\|_{L^{\frac{(p-1)^{2}}{2}}}\lesssim\|\nabla
u\|_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{1}(0))}.$

Here we used the fact that $\frac{(p-1)^{2}}{2}<\frac{(p-1)(2-p)}{3-p}$
provided $1<p<3/2$.
These inequalities now give

$||\nabla(u-w)||_{L^{\frac{(p-1)^{2}}{2}}}\lesssim
A^{\frac{p-1}{p}}+|\mu|((-1,1)^{n})^{\frac{1}{2}}\|\nabla
u\|_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{1}(0))}^{\frac{2-p}{2}}\lesssim
1,$

which completes the proof of the lemma.

Using Lemma 2.5, we next prove a reverse Hölder type inequality for the
gradient.

###### Lemma 2.6

Suppose that $u$ is a $W^{1,p}_{\rm loc}(\Omega)$ solution of (1.1). Let
$Q_{r}(x_{0})\subset\Omega$, $n\geq 2$, and $\kappa=(p-1)^{2}/2$, where
$1<p<3/2$. Let

$\theta\in\left(0,\frac{2\kappa(p-1)}{(2-p)(p-\kappa)}\right)$

and define $s_{1},s_{2}$ by the equations

$\frac{1}{2-p}=\frac{\theta}{\kappa}+\frac{1-\theta}{s_{1}},\quad\frac{3-p}{(p-1)(2-p)}=\frac{\theta}{\kappa}+\frac{1-\theta}{s_{2}}.$
(2.8)

Then,

$2-p<s_{1}<p,\qquad s_{1}>s_{2}>\frac{(p-1)(2-p)}{3-p},$ (2.9)
$s_{2}<\frac{s_{1}(n-1)(p-1)}{s_{1}(n-1)-p+1},$ (2.10)

and moreover,

$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{\sigma
r}(x_{0}))}\lesssim\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+\left(\fint_{Q_{r}(x_{0})}|\nabla
u|^{\kappa}dx\right)^{\frac{1}{\kappa}},$ (2.11)

provided $\sigma\in(0,1)$.

###### Remark 2.7

By Hölder’s inequality, (2.11) also holds with

$s_{1}=s_{2}=\frac{(p-1)(2-p)}{3-p}>\kappa$

and then a covering/iteration argument as in [10, Remark 6.12] implies that we
can replace $\kappa$ with any $\epsilon>0$ in (2.11).

Proof. The proof of (2.9) and (2.10) is obvious. In order to show (2.11), we
first show that

$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{r/2}(x_{0}))}\lesssim\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{r}(x_{0}))}$
(2.12)

for any $Q_{r}(x_{0})\Subset\Omega$. Moreover, by scaling, to prove (2.12) we
may assume that $Q_{r}(x_{0})=Q_{1}(0)=(-1,1)^{n}$,

$|\mu|((-1,1)^{n})^{\frac{1}{p-1}}+\|\nabla
u\|_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}((-1,1)^{n})}\lesssim
1,$ (2.13)

and prove that

$\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}((-1/2,1/2)^{n})}\lesssim 1.$
(2.14)

Let $w$ be as in (1.4) with $Q_{r}(x_{0})=(-1,1)^{n}$. Then, by Lemma 2.5 and
(2.13),

$\left(\int_{Q_{1}(0)}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}\lesssim
1,\qquad\kappa=\frac{(p-1)^{2}}{2}.$

Thus for $\lambda=\fint_{Q_{7/8}(_{0})}wdx$, by Poincaré inequality we have

$\left(\int_{Q_{7/8}(0)}|u-\lambda|^{\kappa}\right)^{\frac{1}{\kappa}}\lesssim\left(\int_{Q_{7/8}(0)}|w-\lambda|^{\kappa}\right)^{\frac{1}{\kappa}}+1\lesssim\int_{Q_{7/8}(0)}|\nabla
w|dx+1.$

Then by the reverse Hölder property of $\nabla w$ and Lemma 2.5 we can now
deduce

$\displaystyle\left(\int_{Q_{7/8}(0)}|u-\lambda|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\lesssim\left(\int_{Q_{1}(0)}|\nabla
w|^{\kappa}dx\right)^{\frac{1}{\kappa}}+1\lesssim\left(\int_{Q_{1}(0)}|\nabla
u|^{\kappa}dx\right)^{\frac{1}{\kappa}}+1$ $\displaystyle\lesssim\|\nabla
u\|_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{1}(0))}+1\lesssim
1.$

On the other hand, it follows from Corollary 2.4 that

$\|\nabla u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{1/2}(0))}\lesssim
1+\left(\int_{Q_{7/8}(0)}|u-\lambda|^{\kappa}dx\right)^{\frac{1}{\kappa}}$

for any $\lambda\in\mathbb{R}$. Thus we obtain (2.14) as desired.

Next, for any cube $Q_{r}(x_{0})\subset\Omega$ we consider the cubes
$Q_{t}(x_{0})\subset Q_{s}(x_{0})$ where $0<t<s<r$. Recall that
$Q_{t}(x_{0})=\\{x_{0}^{\prime}+(-t,t)^{n-1}\\}\times\\{x_{0n}+(-t,t)\\}$. We
can cover the interval $I_{t}(x_{0n})=x_{0n}+(-t,t)$ by a sequence of
intervals $I_{i}=y_{in}+(-(s-t)/2,(s-t)/2)$, $y_{in}\in I_{t}(x_{0n})$, in
such a way that any point $y\in\mathbb{R}$ belongs to almost 3 intervals of
the collection $\\{2I_{i}\\}=\\{y_{in}+(-(s-t),(s-t))\\}$. Note that we have
$1\leq i\leq M$, where $M\lesssim t/(s-t)$.

Likewise, we can cover the $(n-1)$-dimensional cube
$J_{t}(x_{0}^{\prime})=x_{0}^{\prime}+(-t,t)^{n-1}$ by a sequence of
$(n-1)$-dimensional cubes $J_{j}=y_{j}^{\prime}+(-(s-t)/2,(s-t)/2)^{n-1}$,
$y_{j}^{\prime}\in J_{t}(x_{0}^{\prime})$, in such a way that any point
$y\in\mathbb{R}^{n-1}$ belongs to almost $N(n)$ cubes of the collection
$\\{2J_{j}\\}=\\{y_{j}^{\prime}+(-(s-t),(s-t))^{n-1}\\}$. Also, $1\leq j\leq
M^{\prime}$, where $M^{\prime}\lesssim[t/(s-t)]^{n-1}$.
Note that we have $2J_{j}\times 2I_{i}\subset Q_{s}(x_{0})$ for any $i,j$.
Then applying (2.12), we get

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(J_{j}\times
I_{i})}\lesssim\left(\frac{|\mu|(Q_{r}(x_{0}))}{(s-t)^{n-1}}\right)^{\frac{1}{p-1}}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L^{\frac{(p-1)(2-p)}{3-p}}_{x^{\prime}}L_{x_{n}}^{2-p}(2J_{j}\times
2I_{i})}.$ (2.15)

We now observe that, as $s_{2}<s_{1}$,

$\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{t}(x_{0}))}^{s_{2}}\leq\sum_{i,j}\int_{J_{j}}\left(\int_{I_{i}}|\nabla
u|^{s_{1}}dx_{n}\right)^{\frac{s_{2}}{s_{1}}}dx^{\prime}=\sum_{i,j}\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(J_{j}\times I_{i})}^{s_{2}}.$

Thus in view of (2.15) we find

$\displaystyle\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{t}(x_{0}))}^{s_{2}}$
$\displaystyle\lesssim
r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\sum_{i,j}\left(\frac{|\mu|(Q_{r}(x_{0}))}{(s-t)^{n-1}}\right)^{\frac{s_{2}}{p-1}}$
$\displaystyle+r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}(s-t)^{-s_{2}\Big{[}\frac{1}{2-p}+\frac{(n-1)(3-p)}{(2-p)(p-1)}\Big{]}}\sum_{i,j}\|\nabla
u\|^{s_{2}}_{L^{\frac{(p-1)(2-p)}{3-p}}_{x^{\prime}}L_{x_{n}}^{2-p}(2J_{j}\times
2I_{i})}.$

Since $s_{2}>(p-1)(2-p)/(3-p)$ and $(p-1)/(3-p)<1$, we then have

$\displaystyle\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{t}(x_{0}))}^{s_{2}}\lesssim\left(\frac{r}{s-t}\right)^{n}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\left(\frac{|\mu|(Q_{r}(x_{0}))}{(s-t)^{n-1}}\right)^{\frac{s_{2}}{p-1}}$
$\displaystyle\qquad+\left(\frac{r}{s-t}\right)^{\frac{2s_{2}}{p-1}}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}(s-t)^{-s_{2}\Big{[}\frac{1}{2-p}+\frac{(n-1)(3-p)}{(2-p)(p-1)}\Big{]}}\times$
$\displaystyle\qquad\qquad\times\left(\sum_{j}\int_{2J_{j}}\left(\sum_{i}\int_{2I_{i}}|\nabla
u|^{2-p}dx_{n}\right)^{\frac{p-1}{3-p}}dx^{\prime}\right)^{\frac{s_{2}(3-p)}{(p-1)(2-p)}}.$

This gives

$\displaystyle\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{t}(x_{0}))}^{s_{2}}\lesssim\left(\frac{r}{s-t}\right)^{n}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\left(\frac{|\mu|(Q_{r}(x_{0}))}{(s-t)^{n-1}}\right)^{\frac{s_{2}}{p-1}}$
$\displaystyle\quad\quad+\left(\frac{r}{s-t}\right)^{\frac{2s_{2}}{p-1}}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}(s-t)^{-s_{2}\Big{[}\frac{1}{2-p}+\frac{(n-1)(3-p)}{(2-p)(p-1)}\Big{]}}\|\nabla
u\|^{s_{2}}_{L^{\frac{(p-1)(2-p)}{3-p}}_{x^{\prime}}L_{x_{n}}^{2-p}(Q_{s}(x_{0}))},$

and thus by Hölder’s inequality,

$\displaystyle\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{t}(x_{0}))}^{s_{2}}\lesssim\left(\frac{r}{s-t}\right)^{n}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\left(\frac{|\mu|(Q_{r}(x_{0}))}{(s-t)^{n-1}}\right)^{\frac{s_{2}}{p-1}}$
$\displaystyle\qquad+\left(\frac{r}{s-t}\right)^{\frac{2s_{2}}{p-1}}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}(s-t)^{-s_{2}\Big{[}\frac{1}{2-p}+\frac{(n-1)(3-p)}{(2-p)(p-1)}\Big{]}}\|\nabla
u\|^{\theta s_{2}}_{L^{\kappa}}\|\nabla
u\|^{(1-\theta)s_{2}}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{s}(x_{0}))}.$

Then by Young’s inequality, we get

$\displaystyle\|\nabla
u\|_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{t}(x_{0}))}^{s_{2}}\leq
C\left(\frac{r}{s-t}\right)^{n}r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\left(\frac{|\mu|(Q_{r}(x_{0}))}{(s-t)^{n-1}}\right)^{\frac{s_{2}}{p-1}}+\frac{1}{2}\|\nabla
u\|^{s_{2}}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{s}(x_{0}))}$
$\displaystyle\qquad+C\left(\frac{r}{s-t}\right)^{\frac{2s_{2}}{(p-1)\theta}}r^{\frac{s_{2}}{\theta}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}(s-t)^{-\frac{s_{2}}{\theta}\Big{[}\frac{1}{2-p}+\frac{(n-1)(3-p)}{(2-p)(p-1)}\Big{]}}\|\nabla
u\|^{s_{2}}_{L^{\kappa}(Q_{s}(x_{0}))}.$

This enables us to apply [10, Lemma 6.1] and use (2.8) to obtain

$\displaystyle\|\nabla
u\|^{s_{2}}_{L^{s_{2}}_{x^{\prime}}L_{x_{n}}^{s_{1}}(Q_{\sigma
r}(x_{0}))}\lesssim
r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{s_{2}}{p-1}}$
$\displaystyle\qquad+r^{\frac{s_{2}}{\theta}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}r^{-\frac{s_{2}}{\theta}\Big{[}\frac{1}{2-p}+\frac{(n-1)(3-p)}{(2-p)(p-1)}\Big{]}}\|\nabla
u\|^{s_{2}}_{L^{\kappa}(Q_{r}(x_{0}))}$ $\displaystyle\qquad\lesssim
r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-1}}\right)^{\frac{s_{2}}{p-1}}+r^{s_{2}[\frac{1}{s_{1}}+\frac{n-1}{s_{2}}]}r^{\frac{-s_{2}n}{\kappa}}\|\nabla
u\|^{s_{2}}_{L^{\kappa}(Q_{r}(x_{0}))}.$

This completes the proof of the lemma.

We mention again that Theorem 1.2 follows from Lemmas 2.5, 2.6, and Corollary
2.4. We now describe the boundary version of Theorem 1.2. Let $u$ be a
$W^{1,p}_{0}(\Omega)$ solution of (1.1), and let $x_{0}\in\partial\Omega$,
$r<{\rm diam}(\Omega)/10$. We then extend both $u$ and $\mu$ by zero outside
$\Omega$ and consider the the unique solution $w\in W^{1,p}(Q_{r}(x_{0}))+u$
to

$\left\\{\begin{array}[]{rcl}-\operatorname{div}\left({A(x,\nabla
w)}\right)&=&0\quad\text{in}\quad Q_{r}(x_{0})\cap\Omega,\\\
w&=&u\quad\text{on}\quad\partial\left(Q_{r}(x_{0})\cap\Omega\right).\end{array}\right.$
(2.16)

Then we have the following boundary counterpart of Theorem 1.2.

###### Theorem 2.8

Let $x_{0}\in\partial\Omega$, $u\in W^{1,p}_{0}(\Omega)$ and $w$ be as in
(2.16). Then with $\kappa=(p-1)^{2}/2$ and $1<p<3/2$, for any $\Sigma\in(1,2]$
we have

$\displaystyle\left(\fint_{Q_{r}(x_{0})}|\nabla(u-w)|^{\kappa}\right)^{\frac{1}{\kappa}}+\frac{1}{r}\left(\fint_{Q_{r}(x_{0})}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\qquad\lesssim\left(\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-1}}\right)^{\frac{1}{p-1}}+\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-1}}\left(\fint_{Q_{\Sigma r}(x_{0})}|\nabla
u|^{\kappa}\right)^{\frac{2-p}{\kappa}},$

and

$\displaystyle\left(\fint_{Q_{r}(x_{0})}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\lesssim\left(\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-p}}\right)^{\frac{1}{p-1}}+\frac{|\mu|(Q_{\Sigma
r}(x_{0}))}{r^{n-p}}\left(\fint_{Q_{\Sigma
r}(x_{0})}|u|^{\kappa}\right)^{\frac{2-p}{\kappa}}.$

Note that if $u\in W^{1,p}_{0}(\Omega)$ then in Lemma 2.1 we may take $\phi\in
C_{0}^{\infty}(\mathbb{R}^{n})$ (i.e., $\phi$ does not need to have compact
support in $\Omega$). With this observation, we see that Theorem 2.8 can be
proved in the same manner as Theorem 1.2.
We now devote the rest of this section to the proof of Corollary 1.3.

Proof of Corollary 1.3. By Remark 2.7, it is enough to consider the case
$\epsilon=\kappa$. Let $w$ be as in (1.4) with $Q_{\Sigma_{0}r}(x_{0})$ in
place of $Q_{r}(x_{0})$, where we choose $\Sigma_{0}$ so that
$1<\Sigma_{0}<\Sigma\leq 2$. By $L^{1}$ Poincaré inequality, Lemma 2.5, and
Young’s inequality we find

$\displaystyle\left(\inf_{q\in\mathbb{R}}\fint_{Q_{r}(x_{0})}|u-q|^{\kappa}\right)^{\frac{1}{\kappa}}\lesssim\left(\inf_{q\in\mathbb{R}}\fint_{Q_{r}(x_{0})}|w-q|^{\kappa}\right)^{\frac{1}{\kappa}}+\left(\fint_{Q_{r}(x_{0})}|u-w|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\qquad\lesssim r\fint_{Q_{r}(x_{0})}|\nabla
w|dx+\left(\frac{|\mu|(B_{\Sigma_{0}r})(x_{0})}{r^{n-p}}\right)^{\frac{1}{p-1}}+r{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{\Sigma_{0}r}(x_{0}))}.$

We now use the reverse Hölder property of $\nabla w$ to obtain from the above
inequality that

$\displaystyle\left(\inf_{q\in\mathbb{R}}\fint_{Q_{r}(x_{0})}|u-q|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle\lesssim r\left(\fint_{Q_{\Sigma_{0}r}(x_{0})}|\nabla u-\nabla
w|^{\kappa}dx\right)^{\frac{1}{\kappa}}+r\left(\fint_{Q_{\Sigma_{0}r}(x_{0})}|\nabla
u|^{\kappa}\right)^{\frac{1}{\kappa}}$
$\displaystyle+\left(\frac{|\mu|(B_{\Sigma_{0}r})(x_{0})}{r^{n-p}}\right)^{\frac{1}{p-1}}+r{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\nabla
u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{L_{x^{\prime}}^{\frac{(p-1)(2-p)}{3-p}}L^{2-p}_{x_{n}}(Q_{\Sigma_{0}r}(x_{0}))}.$

At this point, we use Lemmas 2.5 and 2.6 to conclude the proof.

##  3 Proof of Theorem 1.4

To prove Theorem 1.4, we shall need the following sharp quantitative
regularity/decay estimates for homogeneous equations.

###### Lemma 3.1

Under (1.2)–(1.3), let $w\in W^{1,p}(\Omega)$ be a solution of
$\operatorname{div}\left({A(x,\nabla w)}\right)=0$ in $\Omega$. Then there
exists $\alpha_{0}\in(0,1]$ such that for any $Q_{\rho}(x_{0})\subset
Q_{R}(x_{0})\subset\Omega$, and $\epsilon\in(0,1)$, we have

$\fint_{Q_{\rho}(x_{0})}|w-(w)_{Q_{\rho}(x_{0})}|^{p}dx\lesssim\left(\frac{\rho}{R}\right)^{\alpha_{0}p}\fint_{Q_{R}(x_{0})}|w-(w)_{Q_{R}(x_{0})}|^{p}dx,$
(3.1)

and

$\inf_{q\in\mathbb{R}}\fint_{Q_{\rho}(x_{0})}|w-q|^{\epsilon}dx\lesssim\left(\frac{\rho}{R}\right)^{\alpha_{0}\epsilon}\inf_{q\in\mathbb{R}}\fint_{Q_{R}(x_{0})}|w-q|^{\epsilon}dx.$
(3.2)

Proof. The proof of (3.1) follows from [10, Chapter 7], whereas the proof of
(3.2) follows from (3.1) and the reverse Hölder property of $w$.

We are now ready for the proof of Theorem 1.4.

Proof of (1.4). We first observe that for each cube
$Q_{\rho}(x_{0})\Subset\Omega$ and $f\in L^{\kappa}_{\rm loc}(\Omega)$, there
exists $q_{\rho,x_{0}}=q_{\rho,x_{0}}(f)\in\mathbb{R}$ such that

$\inf_{q\in\mathbb{R}}\left(\fint_{Q_{\rho}(x_{0})}|f-q|^{\kappa}dx\right)^{\frac{1}{\kappa}}=\left(\fint_{Q_{\rho}(x_{0})}|f-q_{\rho,x_{0}}|^{\kappa}dx\right)^{\frac{1}{\kappa}}.$

Then it is known that (see, e.g., [6, Lemma 4.1])

$\lim_{\rho\rightarrow 0}q_{\rho,x_{0}}(f)=f(x_{0})\quad\text{ a.e. with
respect to }x_{0}\in\mathbb{R}^{n}.$ (3.3)

For a cube $Q_{\rho}=Q_{\rho}(x_{0})\subset\Omega$, we now define

$\mathbf{J}(\rho,x_{0})=\mathbf{J}(\rho,x_{0},u):=\inf_{q\in\mathbb{R}}\left(\fint_{Q_{\rho}(x_{0})}|u-q|^{\kappa}dx\right)^{\frac{1}{\kappa}}.$

This kind of quantity is a good substitution for the mean oscillation for
functions that may not belong to $L^{1}_{\rm loc}$ (see, e.g., [6, Section
4]).
Next, for any $\varepsilon\in(0,1)$ and $Q_{r}(x_{0})\Subset\Omega$ by Lemma
3.1 and quasi-triangle inequality, we can find $\alpha_{0}\in(0,1]$ such that

$\displaystyle\mathbf{J}(\varepsilon r/2,x_{0})$
$\displaystyle\leq\left(\fint_{Q_{\varepsilon r/2}(x_{0})}|u-q_{\varepsilon
r/2,x_{0}}(w)|^{\kappa}dx\right)^{\frac{1}{\kappa}}$ $\displaystyle\leq
C\left(\fint_{Q_{\varepsilon r/2}(x_{0})}|w-q_{\varepsilon
r/2,x_{0}}(w)|^{\kappa}dx\right)^{\frac{1}{\kappa}}+C\left(\fint_{Q_{\varepsilon
r/2}(x_{0})}|u-w|^{\kappa}dx\right)^{\frac{1}{\kappa}}$ $\displaystyle\leq
C\varepsilon^{\alpha_{0}}\left(\fint_{Q_{r/2}(x_{0})}|w-q_{r,x_{0}}(w)|^{\kappa}dx\right)^{\frac{1}{\kappa}}+C\varepsilon^{-n/\kappa}\left(\fint_{Q_{r/2}(x_{0})}|u-w|^{\kappa}dx\right)^{\frac{1}{\kappa}}.$

Here we choose $w$ as in (1.4) with $Q_{r/2}(x_{0})$ in place of
$Q_{r}(x_{0})$.
Thus after similar manipulations we get

$\displaystyle\mathbf{J}(\varepsilon r/2,x_{0})\leq
C\varepsilon^{\alpha_{0}}\mathbf{J}(r/2,x_{0})+C_{\varepsilon}\left(\fint_{Q_{r/2}(x_{0})}|u-w|^{\kappa}dx\right)^{\frac{1}{\kappa}}.$

We now apply Theorem 1.2 to bound the second term on the right-hand side of
the above inequality. This yields that

$\mathbf{J}(\varepsilon r/2,x_{0})\leq
C\varepsilon^{\alpha_{0}}\mathbf{J}(r/2,x_{0})+C_{\varepsilon}\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-p}}\right)^{\frac{1}{p-1}}+C_{\varepsilon}\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-p}}\mathbf{J}(r,x_{0})^{2-p}.$

Then by Young’s inequality we find

$\displaystyle\mathbf{J}(\varepsilon r/2,x_{0})$ $\displaystyle\leq
C\varepsilon^{\alpha_{0}}\mathbf{J}(r,x_{0})+C_{\varepsilon}\left(\frac{|\mu|(Q_{r}(x_{0}))}{r^{n-p}}\right)^{\frac{1}{p-1}}.$
(3.4)

We now choose $\varepsilon<\frac{1}{4\sqrt{n}}$ small enough so that
$C(\varepsilon)^{\alpha_{0}}\leq\frac{1}{4}$, where $C$ is the constant in
(3.4). Set $R_{j}=(\varepsilon/2)^{j}R$, $Q_{j}:=Q_{R_{j}}(x_{0})$. Applying
(3.4) yields

$\mathbf{J}(R_{j+1},x_{0})\leq\frac{1}{4}\mathbf{J}(R_{j},x_{0})+C\left(\frac{|\mu|(Q_{j})}{R_{j}^{n-p}}\right)^{\frac{1}{p-1}}.$

Summing this up over $j\in\\{2,3,...,m-1\\}$, we obtain

$\displaystyle\sum_{j=2}^{m}\mathbf{J}(R_{j},x_{0})$ $\displaystyle\leq
C\,\mathbf{J}(R_{2},x_{0})+C\sum_{j=2}^{m-1}\left(\frac{|\mu|(Q_{j})}{R_{j}^{n-p}}\right)^{\frac{1}{p-1}}.$
(3.5)

It is not hard to see that

$|q_{R_{j+1},x_{0}}(u)-q_{R_{j},x_{0}}(u)|\leq
C(\mathbf{J}(R_{j+1},x_{0})+\mathbf{J}(R_{j},x_{0}))$

for all $j\geq 1$, and then by iterating we find

$|q_{R_{m},x_{0}}(u)-q_{R_{2},x_{0}}(u)|\leq
C\sum_{j=2}^{m}\mathbf{J}(R_{j},x_{0}).$

Note also that

$\mathbf{J}(R_{2},x_{0})+|q_{R_{2},x_{0}}(u)|\leq
C\mathbf{J}(R_{2},x_{0})+C\left(\fint_{Q_{2}}|u|^{\kappa}dx\right)^{\frac{1}{\kappa}}\leq
C\left(\fint_{B_{R}(x_{0})}|u|^{\kappa}dx\right)^{\frac{1}{\kappa}}.$

Thus using these in (3.5) we get

$\displaystyle|q_{R_{m},x_{0}}(u)|$ $\displaystyle\leq
C\,\mathbf{J}(R_{2},x_{0})+C|q_{R_{2},x_{0}}(u)|+C\sum_{j=2}^{m-1}\left(\frac{|\mu|(Q_{j})}{R_{j}^{n-p}}\right)^{\frac{1}{p-1}}$
$\displaystyle\leq
C\left(\fint_{B_{R}(x_{0})}|u|^{\kappa}dx\right)^{\frac{1}{\kappa}}+C\sum_{j=2}^{m-1}\left(\frac{|\mu|(Q_{j})}{R_{j}^{n-p}}\right)^{\frac{1}{p-1}}.$

Note that

$\sum_{j=2}^{m-1}\left(\frac{|\mu|(Q_{j})}{R_{j}^{n-p}}\right)^{\frac{1}{p-1}}\leq\sum_{j=2}^{m-1}\left(\frac{|\mu|(B_{\sqrt{n}R_{j}}(x_{0}))}{R_{j}^{n-p}}\right)^{\frac{1}{p-1}}\leq
C\,{\bf W}^{R}_{1,p}(|\mu|)(x_{0}).$

Thus in view of (3.3) we obtain (1.8) as desired.

##  4 Proof of Theorem 1.10

In this section, we provide a proof of Theorem 1.10.

Step 1: In this step we assume that $\mu\in L^{\infty}(\Omega)$. Set

$E(\mu):=\left\\{u\in W^{1,p}_{0}(\Omega):|\nabla u(x)|\leq
2C_{0}\mathbf{P}[\mu](x){\rm~{}a.e.}\right\\},$

where $C_{0}$ is the constant in (1.14), and

$\mathbf{P}[\mu](x):=\Big{[}{\bf I}_{1}^{2{\rm
diam}(\Omega)}(|\mu|)(x)\Big{]}^{\frac{1}{p-1}},\qquad x\in\mathbb{R}^{n}.$

Here we extend $u$ by zero outside $\Omega$. Note that by inequality (2.10) of
[24] and (1.19), one has

$\mathbf{P}[\mathbf{P}[\mu]^{q}](x)\leq
C_{1}c_{0}^{\frac{q-p+1}{(p-1)^{2}}}\mathbf{P}[\mu](x)\qquad\forall
x\in\mathbb{R}^{n}.$ (4.1)

Now let $S:E(\mu)\rightarrow W_{0}^{1,p}(\Omega)$ be defined by $S(v)=u$ where
$u\in W_{0}^{1,p}(\Omega)$ is the unique renormalized solution of

$\displaystyle\left\\{\begin{array}[]{rcl}-{\rm div}(A(x,\nabla u))&=&|\nabla
v|^{q}+\mu\quad{\rm in}~{}\Omega,\\\
u&=&0\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
on~{}}\partial\Omega.\end{array}\right.$

By (1.14) and (4.1), one obtains

$\displaystyle|\nabla u(x)|$ $\displaystyle\leq C_{0}\mathbf{P}[|\nabla
u|^{q}+|\mu|](x)\leq
C_{0}\left((2C_{0})^{q}\mathbf{P}[\mathbf{P}[\mu]^{q}](x)+\mathbf{P}[|\mu|](x)\right)$
$\displaystyle\leq
C_{0}\left((2C_{0})^{q}C_{1}c_{0}^{\frac{q-p+1}{(p-1)^{2}}}+1\right)\mathbf{P}[|\mu|](x)\leq
2C_{0}\mathbf{P}[|\mu|](x)$

provided $c_{0}>0$ is small enough. This means that $S(E(\mu))\subset E(\mu)$.
Then, similar to [20, Proof of Theorem 1.9], we obtain that $S$ has a fixed
point in $E(\mu)$. Thus, there exists a solution $u\in E(\mu)$ to

$\displaystyle\left\\{\begin{array}[]{rcl}-{\rm div}(A(x,\nabla u))&=&|\nabla
u|^{q}+\mu\quad{\rm in}~{}\Omega,\\\
u&=&0\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
on~{}}\partial\Omega.\end{array}\right.$ (4.5)

Step 2: We now extend $\mu$ be zero out side $\Omega$ and let
$\mu_{k}=\rho_{k}*\mu$, where ${\\{\rho_{k}\\}}_{k>0}$ is a standard sequence
of mollifiers: $\rho_{k}(\cdot):=k^{n}\rho(k\cdot)$ for a nonnegative and
radial function $\rho\in C_{0}^{\infty}(B_{1}(0))$ such that
$\int_{\mathbb{R}^{n}}\rho dx=1$. It is not hard to see that there exists a
constant $B>0$ such that

$|\mu_{k}|(K)\leq Bc_{0}\,{\rm Cap}_{1,\,\frac{q}{q-p+1}}(K)$

for any compact set $K\subset\Omega$; see [25, Lemma 5.7]. Thus, by Step 1,
for each $k>0$ there exists a renormalized solution $u_{k}\in E(\mu_{k})$ to
(4.5) with datum $\mu=\mu_{k}$. Also, by (5.1) in [20] we have

${\bf I}_{1}^{2{\rm diam}(\Omega)}(|\mu_{k}|)\leq\rho_{k}*{\bf I}_{1}^{2{\rm
diam}(\Omega)}(|\mu|)\leq M\,{\bf I}_{1}^{2{\rm diam}(\Omega)}(|\mu|).$

Thus by the stability result for renormalized solutions of [5], we can find a
subsequence of $\\{u_{k}\\}_{k}$ that converges to a renormalized solution $u$
to the equation (4.5) such that

$|\nabla u(x)|\leq 2C_{0}M^{\frac{1}{p-1}}\mathbf{P}[\mu](x){\rm~{}a.e.}$

This completes the proof of the theorem.

###### Remark 4.1

Finally, we remark that due to a comparison estimate obtained earlier in [20]
for the case $\frac{3n-2}{2n-1}<p\leq 2-1/n$, all results in this paper also
hold in this case. Moreover, local and global estimates ‘below the duality
exponent’ in the spirit of [19, 26, 27] for $1<p\leq 2-1/n$ can also be
deduced from the comparison estimates of Theorems 1.2 and 2.8 and [20].

Acknowledgments: Q.H.N. is supported by the Academy of Mathematics and Systems
Science, Chinese Academy of Sciences startup fund, and the National Natural
Science Foundation of China (12050410257). N.C.P. is supported in part by
Simons Foundation (award number 426071). The authors would like to thank
Hongjie Dong for helpful comments.

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