# Positive solutions of quasilinear elliptic equations with Fuchsian
potentials in
Wolff class

Ratan Kr. Giri  Ratan Kr. Giri

Department of Mathematics, Technion - Israel Institute of Technology, 3200
Haifa, Israel
Department of Mathematics, The LNM Institute of Information Technology, Jaipur
- 302031, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Yehuda
Pinchover  Yehuda Pinchover

Department of Mathematics, Technion - Israel Institute of Technology
3200 Haifa, Israel<EMAIL_ADDRESS>
###### Abstract.

Using Harnack’s inequality and a scaling argument we study Liouville-type
theorems and the asymptotic behaviour of positive solutions near an isolated
singular point $\zeta\in\partial\Omega\cup\\{\infty\\}$ for the quasilinear
elliptic equation

$-\text{div}(|\nabla u|_{A}^{p-2}A\nabla u)+V|u|^{p-2}u=0\quad\text{ in
}\Omega,$

where $\Omega$ is a domain in $\mathbb{R}^{d}$, $d\geq 2$, $1<p<d$, and
$A=(a_{ij})\in L_{\rm loc}^{\infty}(\Omega;\mathbb{R}^{d\times d})$ is a
symmetric and locally uniformly positive definite matrix. It is assumed that
the potential $V$ belongs to a certain Wolff class and has a generalized
Fuchsian-type singularity at an isolated point
$\zeta\in\partial\Omega\cup\\{\infty\\}$.

###### Key words and phrases:

Fuchsian singularity, Morrey spaces, Kato Class, Wolff class, Liouville
theorem, Quasilinear elliptic equation, $(p,A)$-Laplacian

###### 2010 Mathematics Subject Classification:

Primary 35B53; Secondary 35B09, 35J62, 35B40

##  1\. Introduction

This paper studies positive Liouville-type theorems and removable singularity
theorems for the $(p,A)$-Laplacian type elliptic equation

$Q(u)=Q_{p,A,V}(u):=-\Delta_{p,A}(u)+V|u|^{p-2}u=0\qquad\text{in }\Omega$
(1.1)

near an isolated singular point $\zeta\in\partial\Omega\cup\\{\infty\\}$. Here
$\Omega$ is a domain in $\mathbb{R}^{d}$ ($d\geq 2)$ with boundary
$\partial\Omega$, $1<p<d$, $V$ is a real valued potential belonging to a
certain Wolff space $\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ (see
Definition 2.8), and

$\Delta_{p,A}(u):=\text{div}(|\nabla u|_{A}^{p-2}A\nabla u)$

is the $(p,A)$-Laplacian, where $A=(a_{ij})\in L_{\rm
loc}^{\infty}(\Omega;\mathbb{R}^{d\times d})$ is a symmetric and locally
uniformly positive definite matrix valued function, and

$|\xi|_{A}^{2}=|\xi|_{A(x)}^{2}:=A(x)\xi\cdot\xi=\\!\\!\\!\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\;\;\;\;\;x\in\Omega,\;\xi=(\xi_{1},...,\xi_{d})\\!\in\\!\mathbb{R}^{d}.$

Liouville theorems and asymptotic behaviours of positive solutions of second-
order quasilinear elliptic operators near an isolated singular point
$\zeta\in\partial\Omega\cup\\{\infty\\}$ was studied extensively in the past
few decades, see [8, 9, 12, 15, 16, 17, 25, 28] and references therein. In
particular, we mention the work of Frass and Pinchover[8], where Liouville
theorems and removable singularity theorems for positive solutions of (1.1)
have been obtained for the entire range $1\\!<\\!p\\!<\\!\infty$ under the
assumptions that $A$ is the identity matrix, and
$V\\!\in\\!L^{\infty}_{\operatorname{loc}}(\Omega)$ has a pointwise Fuchsian-
type singularity at $\zeta\\!\in\\!\\{0,\infty\\}$, namely,

$|V(x)|\leq\frac{C}{|x|^{p}}\qquad\mbox{near }\zeta.$

Furthermore, in the same paper and in [9], the asymptotic behaviour of the
quotient of two positive solutions near the singular point $\zeta$ has been
established. The outcomes in [8, 9] extend to the quasilinear case results
obtained in [19, and references therein] for positive classical solutions of a
second-order linear elliptic operators in a non-divergence form.

Recently, based on criticality theory for $Q_{p,A,V}$ with $V$ in a local
Morrey space $M^{q}_{\operatorname{loc}}(p;\Omega)$ (see Definition 2.2)
established in [20], the aforementioned results were further extended in [11]
to the case where $A=(a_{ij})\in L_{\rm
loc}^{\infty}(\Omega;\mathbb{R}^{d\times d})$ is a symmetric and locally
uniformly positive definite matrix valued function, and $V$ belongs to
$M^{q}_{\operatorname{loc}}(p;\Omega)$ which has a generalized Fuchsian-type
singularity at $\zeta$. More precisely, let
$\mathcal{A}_{R}:=\\{x\in\mathbb{R}^{d}:R/2\leq|x|<3R/2\\}$. $V\in
M^{q}_{\operatorname{loc}}(p;\Omega)$ is said to have a generalized Fuchsian-
type singularity at $\zeta$ if for every relative punctured neighbourhood
$\Omega^{\prime}\subset\Omega$ of $\zeta$, there exists a positive constant
$C$ and $R_{0}>0$ such that

$\displaystyle\begin{cases}\||x|^{p-{d}/{q}}\,V\|_{M^{q}(p;\mathcal{A}_{R}\cap\Omega^{\prime})}\leq
C&\text{if }p\neq d,\\\\[5.69054pt]
\|V\|_{M^{q}(d;\mathcal{A}_{R}\cap\Omega^{\prime})}\leq C&\text{if
}p=d,\end{cases}$ (1.2)

for all $0<R<R_{0}$ if $\zeta=0$, and $R>R_{0}$ if $\zeta=\infty$.

The results in [11] covers the full range of $1<p<\infty$ as in [8, 9]. Note
that for $p>d$,
$M^{q}_{\operatorname{loc}}(p;\Omega)=L^{1}_{\operatorname{loc}}(p;\Omega)$,
which is mainly the weakest assumption for the well definiteness of weak
solutions, and for ensuring the local Harnack inequality and the Hölder
continuity of weak solutions of the equation $Q(u)=0$.

On the other and, by Lemma 2.4 and remarks 2.6 and 2.9, the local Morrey space
$M^{q}_{\operatorname{loc}}(p;\Omega)$ for $1<p<d$ is a proper subset of the
local Stummel-Kato class $K^{p}_{\operatorname{loc}}(\Omega)$ (see Definition
2.3), which in turn is a proper subset of the local Wolff class
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ (see Definition 2.8). It is
important to remark here that when $1<p<d$, the assumption that $V\in
M^{q}_{\operatorname{loc}}(p;\Omega)$ is the weakest to ensure the local
Hölder continuity of weak solutions of (1.1) [20]. That is, if $V\in
K^{p}_{\operatorname{loc}}(\Omega)$ or
$V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, then weak solutions of
(1.1) may not be Hölder continuous, see [5], [7] and [24] for counter
examples.

The purpose of this paper is to extend the results obtained in [8, 9, 11] for
$1<p<d$. We study the asymptotic behaviour of positive weak solutions near a
Fuchsian-type singular point $\zeta$, and prove removable singularity theorems
for positive solutions of (1.1) when the potential $V$ lies in a local Wolff
space and having a Fuchsian-type singularity at $\zeta$ (see Definition 4.1).
More precisely, after establishing the needful parts of criticality theory for
$Q_{p,A,V}$ with $V\\!\in\\!\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, we
study the uniqueness (up to a multiplicative constant) of certain positive
weak solutions and the asymptotic behaviour of the quotients of two positive
weak solutions of the quasilinear elliptic equation (1.1) near the singular
point $\zeta$. Our main tool is a uniform Harnack inequality and a scaling
argument using the quasi-invariance of (1.1) under the map $x\mapsto Rx$,
where $x\in\Omega$ and $R>0$. The novelty of this work lies in the fact that
$V$ is assumed to be in the Wolff class
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ which implies that solutions
of (1.1) are not necessarily Hölder continuous. Another difficulty arises
since $\mathfrak{W}^{p}(\omega)$ is not a Banach space for $p>2$ for
subdomains $\omega\Subset\Omega$.

The paper is organized as follows. In Section 2, we recall various function
spaces such as Morrey, Kato and Wolff spaces, and provide basic results
related to these spaces. Section 3 is devoted to criticality theory for the
operator $Q_{p,A,V}$ with $V$ in the local Wolff class
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$. In particular, we prove the
corresponding Harnack convergence principle, weak maximum, and weak comparison
principles. Finally, in Section 4, we introduce the notion of a Fuchsian-type
singularity, discuss some related notions, and prove the main results of the
paper. In particular, we present a uniform Harnack inequality and a ratio
limit theorem for positive solutions defined near the singular point $\zeta$.
Finally, we obtain a positive Liouville-type theorem under the further
assumption that $Q$ has a weak Fuchsian-type singularity at $\zeta$.

##  2\. Morrey, Kato and Wolff spaces

We begin with some standard notation. Let $\Omega$ be a domain (i.e., a
nonempty open connected set) in $\mathbb{R}^{d}$, $d\geq 2$. We denote the
ball in $\mathbb{R}^{d}$ of radius $r>0$ centered at $x$ by $B_{r}(x)$.
Similarly, $S_{r}(x):=\partial B_{r}(x)$ denotes the sphere of radius $r>0$
centered at $x$, and we set $B_{r}:=B_{r}(0)$, $S_{r}:=S_{r}(0)$. Further,
denote by $B_{r}^{*}:=\mathbb{R}^{d}\setminus\overline{B_{r}}$ and
$(\mathbb{R}^{d})^{*}:=\mathbb{R}^{d}\setminus\\{0\\}$ the corresponding
exterior domains.

Let $f,g\in C(\Omega)$ be two positive functions. By $f\asymp g$ in $\Omega$,
we mean that there exists positive constant $C$ such that

$C^{-1}g(x)\leq f(x)\leq Cg(x)\quad\text{ for all }\,x\in\Omega.$

We write $\Omega_{1}\Subset\Omega_{2}$ if $\Omega_{2}$ is open and
$\overline{\Omega}_{1}$ is compact (proper) subset of $\Omega_{2}$. By a
compact exhaustion of a domain $\Omega$, we mean a sequence
$\\{\Omega_{i}\\}_{i=1}^{\infty}$ of smooth, relatively compact domains in
$\Omega$ such that $\Omega_{1}\neq\emptyset$, $\Omega_{i}\Subset\Omega_{i+1}$,
and $\cup_{i=1}^{\infty}\Omega_{i}=\Omega$. Finally, throughout the paper, $C$
refers to a positive constant which may vary from line to line.

In this section we introduce various function spaces such as Morrey, Kato and
Wolff spaces, and discuss the relationships between them.

###  2.1. Morrey spaces

We recall certain classes of Morrey spaces.

######  Definition 2.1 (Morrey spaces).

Let $f\in L^{1}_{\operatorname{loc}}(\Omega)$, for $q\in[1,\infty]$, we say
that $f$ belongs to the local Morrey space
$M^{q}_{\operatorname{loc}}(\Omega)$ if for any $\omega\Subset\Omega$

$\|f\|_{M^{q}(\omega)}:=\sup_{\begin{subarray}{c}y\in\omega\\\
0<r<\text{diam}(\omega)\end{subarray}}\frac{1}{r^{d/q^{\prime}}}\int_{\omega\cap
B_{r}(y)}|f(x)|\,\mathrm{d}x<\infty,$

where $q^{\prime}\\!=\\!q/(q\\!-\\!1)$ is the Hölder conjugate exponent of
$q$. An application of Hölder inequality implies that
$L^{q}_{\operatorname{loc}}(\Omega)\\!\subsetneq\\!M^{q}_{\operatorname{loc}}(\Omega)\\!\subsetneq\\!L^{1}_{\operatorname{loc}}(\Omega)$
for any $q\\!\in\\!(1,\infty)$.

Next we recall a special local Morrey space
$M^{q}_{\operatorname{loc}}(p;\Omega)$ which depends on the underlying
exponent $1<p<\infty$.

######  Definition 2.2 (Local Morrey spaces).

Let

$M^{q}_{\operatorname{loc}}(p;\Omega):=\begin{cases}M^{q}_{\operatorname{loc}}(\Omega)\mbox{
with }q>d/p&\mbox{ if }p<d,\\\ L^{1}_{\operatorname{loc}}(\Omega)&\mbox{ if
}p>d,\end{cases}$

while for $p=d$, a function $f\in L^{1}_{\operatorname{loc}}(\Omega)$ is said
to belong to the Morrey space $M^{q}_{\operatorname{loc}}(d;\Omega)$ if for
some $q>d$ and any $\omega\Subset\Omega$

$\|f\|_{M^{q}(d;\omega)}:=\sup_{\begin{subarray}{c}y\in\omega\\\
0<r<\text{diam}(\omega)\end{subarray}}\varphi_{q}(r)\int_{\omega\cap
B_{r}(y)}|f|\,\mathrm{d}x<\infty,$

where
$\varphi_{q}(r)\\!:=\\!\log^{q/d^{\prime}}\\!\\!\left(\\!\\!\frac{\text{diam}(\omega)}{r}\\!\\!\right)\\!$
(see [18, Theorem 1.94], and references therein).

###  2.2.  Stummel-Kato class

Next, we introduce the Stummel-Kato class of locally integrable functions for
$1<p<d$.

######  Definition 2.3 (Stummel-Kato class).

Let $1<p<d$. We say that $V\in L^{1}_{\operatorname{loc}}(\Omega)$ belongs to
the class $\bar{K}^{p}_{\operatorname{loc}}(\Omega)$ if for any
$\omega\Subset\Omega$

$\eta_{V,\omega}(r):=\sup_{x\in\omega}\int_{\omega\cap
B_{r}(x)}\frac{|V(y)|}{|x-y|^{d-p}}\>\\!\mathrm{d}y<\infty\qquad\forall\,0<r<\operatorname{diam}(\omega).$

A function $V\in\bar{K}^{p}_{\operatorname{loc}}(\Omega)$ is said to belong to
the local Stummel-Kato class $K^{p}_{\operatorname{loc}}(\Omega)$ if for every
$\omega\Subset\Omega$ we have $\displaystyle{\lim_{r\rightarrow
0}\eta_{V,\omega}(r)=0}$. (see for example, [7, Definition 1.1] for $p=2$, and
[24, Definition 2.1] for $1<p<d$). If $\Omega=\mathbb{R}^{d}$ we write
$\eta_{V}(r):=\eta_{V,\mathbb{R}^{d}}(r)$.

We call $\eta_{V,\omega}(r)$ the Stummel-Kato modulus of $V$ in $\omega$. It
is known that $\eta_{V,\omega}(r)$ is a nondecreasing continuous function of
$r$ with the doubling property [24], i.e., there exists constant
$\mathcal{K}\in(0,1)$ such that

$\mathcal{K}\eta_{V,\omega}(2r)\leq\eta_{V,\omega}(r)\qquad\forall\,r>0.$

The constant $\mathcal{K}$ is called the doubling constant of
$\eta_{V,\omega}$. The Stummel-Kato class $K^{p}_{\operatorname{loc}}(\Omega)$
is equipped with the seminorms given by

$\|V\|_{K^{p}(\omega)}:=\eta_{V,\omega}(\operatorname{diam}(\omega)),$

for every $\omega\Subset\Omega$, and therefore,
$K^{p}_{\operatorname{loc}}(\Omega)$ is a Fréchet space. In fact, for any
$\omega\Subset\Omega$ and a fixed $0<r<\operatorname{diam}(\omega)$,
$\eta_{V,\omega}(r)$ defines a semi-norm in
$K^{p}_{\operatorname{loc}}(\Omega)$ and
$\eta_{V,\omega}(r)\leq\|V\|_{K^{p}(\omega)}$.

When $p=2$, $K^{2}_{\operatorname{loc}}(\Omega)$ is the classical Kato class
which was introduced by Kato [13] to study the self-adjointness of Schrödinger
operators. Aizenman and Simon [1] for $A=I$, and later Chiarenza et al. [5]
established for $p=2$ a local Harnack inequality and the continuity of
solutions for the Schrödinger-type equation $-\mathrm{div}(A\nabla u)+Vu=0$,
where $A$ is a symmetric matrix such that $A\in
L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{d})$ is locally uniformly
elliptic, and $V$ belongs to the Stummel-Kato class. For $1<p<d$, the higher
order Stummel-Kato class potentials can be traced back to Davies and Hinz work
[6]. In general, the study of higher-order Schrödinger operators with
potentials in Stummel-Kato class $K^{p}_{\operatorname{loc}}(\Omega)$ turns
out to be more difficult than in the case $p=2$. More details about the
Stummel-Kato class and related theory in PDEs can be found for example in
Rasgusa and Zamboni [24], Zamboni [29], Zheng and Yao [30], and in references
therein.

Lemma 2.4 below asserts that for $1<p<d$, the local Morrey space
$M^{q}_{\operatorname{loc}}(p;\Omega)$ is a (proper) subset of
$K^{p}_{\operatorname{loc}}(\Omega)$. For a proof see [18, Lemma 1.29].

######  Lemma 2.4.

Let $1<p<d$ and $f\in M^{q}_{\operatorname{loc}}(p;\Omega)$ with $q>d/p$. Then
there exists a constant $C>0$ such that for $0<r<\operatorname{diam}(\omega)$

$\int_{\omega\cap B_{r}(x)}\frac{|f(y)|}{|x-y|^{d-p}}\>\\!\mathrm{d}y\leq
Cr^{p-d/q}\|f\|_{M^{q}(\omega)}.$

We next give a subclass of $K^{p}_{\operatorname{loc}}(\Omega)$ in which each
Stummel-Kato modulus $\eta_{V,\omega}(r)$ of $V\in
K^{p}_{\operatorname{loc}}(\Omega)$ satisfies a particular integral condition
near zero.

######  Definition 2.5 (See [24, 29]).

Let $1<p<d$, $\sigma\in(0,1)$, and let $\theta:={p}/{(\sigma p^{\prime}+p)}$,
so, $0<1-\theta=\frac{\sigma}{\sigma+p-1}<1$. A function $V\in
K^{p}_{\operatorname{loc}}(\Omega)$ is said to belong to the class
$\tilde{K}^{p}_{\operatorname{loc}}(\Omega)$ if there exists $\delta>0$ such
that

$\int_{0}^{\delta}\frac{1}{t}\left(\int_{0}^{t}\frac{\eta_{V}^{1-\theta}(s)}{s}\,\mathrm{d}s\right)^{\frac{1}{p}}\>\\!\mathrm{d}t<\infty.$

######  Remark 2.6.

Clearly, $\tilde{K}^{p}_{\operatorname{loc}}(\Omega)\subset
K^{p}_{\operatorname{loc}}(\Omega)\subset\bar{K}^{p}_{\operatorname{loc}}(\Omega)$.
Also, it can be seen that for $1<p<d$ the local Morrey space
$M^{q}_{\operatorname{loc}}(p;\Omega)$ is in fact, a proper subspace of the
class $\tilde{K}^{p}_{\operatorname{loc}}(\Omega)$. Indeed, if $f\in
M^{q}(p;\omega)$, then by Lemma 2.4, we have $\eta_{V}^{1-\theta}(s)\leq
Cs^{(p-d/q)(1-\theta)}\|f\|^{1-\theta}_{M^{q}(\omega)}$. Hence for $\delta>0$,

$\displaystyle\int_{0}^{\delta}\frac{1}{t}$
$\displaystyle\left(\int_{0}^{t}\frac{\eta_{V}^{1-\theta}(s)}{s}\,\mathrm{d}s\right)^{\frac{1}{p}}\\!\\!\,\mathrm{d}t\leq
C\|f\|^{(1-\theta)/p}_{M^{q}(\omega)}\int_{0}^{\delta}\frac{1}{t}\left(\int_{0}^{t}s^{(p-d/q)(1-\theta)-1}ds\right)^{1/p}\\!\\!\\!\\!\,\mathrm{d}t$
$\displaystyle=C\|f\|^{(1-\theta)/p}_{M^{q}(\omega)}\int_{0}^{\delta}t^{\frac{(p-d/q)(1-\theta)}{p}-1}\>\\!\mathrm{d}t=C\|f\|^{(1-\theta)/p}_{M^{q}(\omega)}\delta^{\frac{(p-d/q)(1-\theta)}{p}}<\infty.$

The next result is an uncertainty-type inequality due to Ragusa and Zamboni
[24, Corollaries 2.6, 2.7] (see also [29, Theorem 2.1, Corollary 2.2]).

######  Theorem 2.7.

Let $1<p<d$, $0<\theta<1$ as above, and $\omega\Subset\mathbb{R}^{d}$. Suppose
that a function $V\in K^{p}(\omega)$ satisfies

$\int_{0}^{\varrho}\frac{\eta_{V}^{1-\theta}(s)}{s}\,\mathrm{d}s<\infty\qquad\mbox{for
some $\varrho>0$},$

and let

$\Phi(r):=C(d)\int_{0}^{r}\frac{\eta_{V}^{1-\theta}(t)}{t}\>\\!\mathrm{d}t.$

Then

  * (i).

There exists a constant $C(d,p)$ such that

$\int_{\omega\cap B_{r}}|V(x)||u(x)|^{p}\,\mathrm{d}x\leq
C(d,p)\Phi(r)\int_{\omega\cap B_{r}}|\nabla u(x)|^{p}\,\mathrm{d}x$

for every $u\in C_{c}^{\infty}(w)$ with $\mbox{\rm
supp}\,(u)\subseteq\omega\cap B_{r}$.

  * (ii).

For any $\delta>0$ the following inequality holds true for any $u\in
C_{c}^{\infty}(\omega)$

$\int_{\omega}|V(x)||u(x)|^{p}\,\mathrm{d}x\leq\delta\int_{\omega}|\nabla
u(x)|^{p}\,\mathrm{d}x+\frac{C_{1}(\operatorname{diam}(\omega))\delta}{\left[\Phi^{-1}\left(\frac{\delta}{C(d,p,\eta_{V})}\right)\right]^{d+p}}\int_{\omega}|u(x)|^{p}\,\mathrm{d}x.$

Using a standard density argument one can replace in Theorem 2.7 the space
$C_{c}^{\infty}(\omega)$ with $W_{0}^{1,p}(\omega)$. Theorem 2.7 is an
extension of the Morrey-Adams theorem to the Kato class (see for example [20,
11] and references therein, where the potential $V$ is assumed to be in the
local Morrey space $M^{q}_{\operatorname{loc}}(p;\Omega)$). Using Theorem 2.7
and the Moser iteration technique, Ragusa and Zamboni [24] proved the local
boundedness of weak solutions and the validity of Harnack inequality for
nonnegative weak solutions of the equation $-\Delta_{p}u+V|u|^{p-2}u=0$ for
$V\in\tilde{K}^{p}_{\operatorname{loc}}(\Omega)$. Hence, as a consequence of
the Harnack inequality, the local continuity of weak solutions is obtained
(see [24, Theorem 6.1]). Note that if the potential $V$ lies in the Morrey
space $M^{q}_{\operatorname{loc}}(p;\Omega)$ with $1<p<d$, then weak solutions
are in fact Hölder continuous (see Malý and Ziemer [18, Theorem 4.11] and also
Di Fazio[7]).

###  2.3. Wolff class

In this subsection, we introduce another space of locally integrable functions
for $1<p<d$ which is known as the Wolff class.

######  Definition 2.8 (Wolff class).

Let $1<p<d$. We say that $f\in L^{1}_{\operatorname{loc}}(\Omega)$ belongs to
the Wolff class $\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ if for every
$\omega\Subset\Omega$

$\lim_{r\rightarrow 0}W_{f}(r):=\lim_{r\rightarrow
0}\left(\sup_{x\in\omega}\int_{0}^{r}\left[\frac{1}{s^{d-p}}\int_{\omega\cap
B_{s}(x)}|f(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\\!\frac{\,\mathrm{d}s}{s}\right)=0.$

We call $W_{f}(r)$ the Wolff modulus of $f$. Note that we should consider
$d\geq 2$ when $1<p<2$, and $d\geq 3$ when $p=2$. For $p=2$, by [18, Lemma
1.27] we have

$\int_{0}^{r}\frac{1}{s^{d-1}}\left(\int_{\omega\cap
B_{s}(x)}|f(y)|\,\mathrm{d}y\right)\,\mathrm{d}s=\frac{1}{(d-2)}\int_{\omega\cap
B_{r}(x)}\frac{|f(y)|}{|x-y|^{d-2}}\,\mathrm{d}y.$

Thus,
$\mathfrak{W}^{2}_{\operatorname{loc}}(\Omega)=K^{2}_{\operatorname{loc}}(\Omega)$,
i.e., for $p=2$ the Wolff space
$\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ is in fact the standard Kato
class. The following remark gives a relationship between the Kato class
$K^{p}_{\operatorname{loc}}(\Omega)$ and the Wolff class
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ for $p\in(1,d)$.

######  Remark 2.9.

Let $1<p<d$ and $r>0$ be sufficiently small. Then

$\displaystyle W_{f}(r)$
$\displaystyle\\!=\\!\sup_{x\in\omega}\int_{0}^{r}\left[\frac{1}{s^{d-p}}\int_{\omega\cap
B_{s}(x)}|f(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\frac{\>\\!\mathrm{d}s}{s}$
$\displaystyle\\!\leq\\!\sup_{x\in\omega}\\!\int_{0}^{r}\\!\\!\left[\\!\int_{\omega\cap
B_{s}(x)}\\!\\!\frac{|f(y)|}{|x\\!-\\!y|^{d-p}}\\!\,\mathrm{d}y\\!\right]^{\frac{1}{p-1}}\\!\\!\\!\frac{\,\mathrm{d}s}{s}\\!\leq\\!\int_{0}^{r}\\!\frac{\eta_{f,\omega}^{\frac{1}{p-1}}(s)}{s}\\!\,\mathrm{d}s\\!\leq\\!\int_{0}^{r}\\!\frac{\eta_{f,\omega}^{1-\theta}(s)}{s}\\!\,\mathrm{d}s,$

since

$1-\theta=\frac{\sigma}{\sigma+p-1}<\frac{1}{p-1}\,\,\,\,\,\mbox{with
}0<\sigma<1.$

This implies that if $f\in K_{\operatorname{loc}}^{p}(\Omega)$ and for every
$\omega\Subset\Omega$ there exists $\delta>0$ such that
$\int_{0}^{\delta}\frac{\eta_{f,\omega}^{1-\theta}(s)}{s}\>\\!\mathrm{d}s<\infty$,
then $f$ is in the Wolff class
$\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$.

######  Lemma 2.10.

For $1<p\leq 2$ and $0<r\leq\operatorname{diam}(\omega)$, the expression

$\|f\|_{\mathfrak{W}^{p}(\omega)}:=W_{f}^{p-1}(r)=\left[\sup_{x\in\omega}\int_{0}^{r}\left[\frac{1}{s^{d-p}}\int_{\omega\cap
B_{s}(x)}|f(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\frac{\>\\!\mathrm{d}s}{s}\right]^{p-1}$

defines a norm on the Wolff class $\mathfrak{W}^{p}(\omega)$. Moreover,
$(\mathfrak{W}^{p}(\omega),\|\cdot\|_{\mathfrak{W}^{p}(\omega)})$ is a Banach
space.

###### Proof.

We will show only the triangle inequality. Since $1<p\leq 2$ implies
$\frac{1}{p-1}\geq 1$, the Minkowski inequality implies that

$\displaystyle\|f_{1}+f_{2}\|_{\mathfrak{W}^{p}(\omega)}=\left[\sup_{x\in\omega}\int_{0}^{r}\left[\frac{1}{s^{d-p}}\int_{\omega\cap
B_{s}(x)}|f_{1}(y)+f_{2}(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\frac{\>\\!\mathrm{d}s}{s}\right]^{p-1}$
$\displaystyle\leq\sup_{x\in\omega}\left[\int_{0}^{r}\left[\frac{1}{s^{d-1}}\int_{\omega\cap
B_{s}(x)}|f_{1}(y)|\>\\!\mathrm{d}y+\frac{1}{s^{d-1}}\int_{\omega\cap
B_{s}(x)}|f_{2}(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\>\\!\mathrm{d}s\right]^{p-1}$
$\displaystyle\leq\sup_{x\in\omega}\left[\int_{0}^{r}\left[\frac{1}{s^{d-1}}\int_{\omega\cap
B_{s}(x)}|f_{1}(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\\!\\!\,\mathrm{d}s\right]^{p-1}$
$\displaystyle+\sup_{x\in\omega}\left[\int_{0}^{r}\left[\frac{1}{s^{d-1}}\int_{\omega\cap
B_{s}(x)}|f_{2}(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\\!\\!\\!\>\\!\mathrm{d}s\right]^{p-1}=\|f_{1}\|_{\mathfrak{W}^{p}(\omega)}+\|f_{2}\|_{\mathfrak{W}^{p}(\omega)}.$

Following [3, Lemma 2.7], we infer that the Wolff space
$\mathfrak{W}^{p}(\omega)$ with the norm
$\|\cdot\|_{\mathfrak{W}^{p}(\omega)}$ is a Banach space for $1<p\leq 2$. ∎

######  Lemma 2.11.

For $p>2$, the expression

$\|f\|_{\mathfrak{W}^{p}(\omega)}:=W_{f}^{p-1}(r)=\left[\sup_{x\in\omega}\int_{0}^{r}\left[\frac{1}{s^{d-p}}\int_{\omega\cap
B_{s}(x)}|f(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\frac{\>\\!\mathrm{d}s}{s}\right]^{p-1}$

defines a quasinorm on $\mathfrak{W}^{p}(\omega)$ for every fixed
$0<r\leq\operatorname{diam}(\omega)$.

###### Proof.

If $p>2$, then $0<\frac{1}{p-1}<1$. Using the same argument as in the proof of
Lemma 2.10, we obtain

$\|f_{1}+f_{2}\|_{\mathfrak{W}^{p}(\omega)}\leq
2^{p-2}\bigg{(}\|f_{1}\|_{\mathfrak{W}_{p}(\omega)}+\|f_{2}\|_{\mathfrak{W}^{p}(\omega)}\bigg{)},$
(2.1)

where we use the following two inequalities with $a,b>0$: $(a+b)^{\lambda}\leq
a^{\lambda}+b^{\lambda}\,\,\,\mbox{for }\,0<\lambda<1$ and
$(a+b)^{\lambda}\leq 2^{\lambda-1}(a^{\lambda}+b^{\lambda})\,\,\,\mbox{for
}\,\lambda>1$. ∎

The following crucial Morrey-Adams type estimates for potentials in the Wolff
class $\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ extend Theorem 2.7, and
will be used in our study.

######  Theorem 2.12.

Let $1<p<d$, $\omega\Subset\Omega$, and $V\in\mathfrak{W}^{p}(\omega)$.

  * (i).

For any $\varepsilon>0$ there exist $0<r_{0}<1$ and $\tau_{\varepsilon}>0$
(depending on $\varepsilon,d,p$) such that if $B_{4r_{0}}(x_{0})\subset\omega$
and $W_{V}(r_{0})<\tau_{\varepsilon}$, then

$\int_{B_{r}(x_{0})}|V||u|^{p}\,\mathrm{d}x\leq\varepsilon\int_{B_{r}(x_{0})}|\nabla
u|^{p}\,\mathrm{d}x\qquad\forall r\leq r_{0}\mbox{ and }u\in
W_{0}^{1,p}(B_{r}(x_{0})).$

  * (ii).

Let $r>0$ and $B_{2r}(x_{0})\subset\omega$. Then there exists a constant $C>0$
depending on $d,p$ such that

$\int_{B_{r}(x_{0})}|V||u|^{p}\,\mathrm{d}x\leq
CW^{p-1}_{V}(2r)\int_{B_{r}(x_{0})}|\nabla u|^{p}\,\mathrm{d}x\qquad\forall
u\in W_{0}^{1,p}(B_{r}(x_{0})).$

###### Proof.

See [16, Lemma 2.1] and [27, Lemma 2.1]. ∎

As a consequence of Theorem 2.12 (ii), we obtain the following two Morrey-
Adams type theorems in $\omega\Subset\Omega$.

######  Theorem 2.13.

Let $1<p<d$ and $V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$. Then for
any $\omega^{\prime}\Subset\omega\Subset\Omega$ and $\delta\\!>\\!0$, there
exists a positive constant
$C_{\delta}\\!=\\!C(d,p,\delta,\omega^{\prime},\omega,W_{V}^{p}(\operatorname{diam}(\omega)))$
such that

$\int_{\omega^{\prime}}|V||u|^{p}\,\mathrm{d}x\leq\delta\|\nabla
u\|^{p}_{L^{p}(\omega^{\prime},\mathbb{R}^{d})}+C_{\delta}\|u\|^{p}_{L^{p}(\omega^{\prime})}\qquad\forall
u\in W_{0}^{1,p}(\omega^{\prime}).$

###### Proof.

The proof follows straight forwardly from Theorem 2.12 by using a partition of
unity of $\omega^{\prime}$ (see [24, Corollary 2.7.] for a similar proof). ∎

######  Theorem 2.14 (Morrey-Adams type theorem).

Let $1<p<d$ and $V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$. Then for
any
$\omega^{\prime}\\!\Subset\\!\omega\\!\Subset\\!\tilde{\omega}\Subset\Omega$
with $\partial\omega^{\prime}$ being Lipschitz and $\delta\\!>\\!0$, there
exists
$C_{\delta}\\!=\\!C(d,p,\delta,\omega^{\prime},\omega,\tilde{\omega},W_{V}(\operatorname{diam}(\tilde{\omega})))>0$
such that

$\int_{\omega^{\prime}}|V||u|^{p}\,\mathrm{d}x\leq\delta\|\nabla
u\|^{p}_{L^{p}(\omega^{\prime},\mathbb{R}^{d})}+C_{\delta}\|u\|^{p}_{L^{p}(\omega^{\prime})}\qquad\forall
u\in W^{1,p}(\omega^{\prime})$ (2.2)

###### Proof.

To prove (2.2), we use an extension operator
$E:W^{1,p}(\omega^{\prime})\rightarrow W_{0}^{1,p}(\omega)$ such that for any
$u\in W^{1,p}(\omega^{\prime})$

$\begin{cases}E(u)=u\quad\mbox{in }\omega^{\prime},\\\
\|E(u)\|_{L^{p}(\omega)}\leq
C(d,p,\omega^{\prime},\omega)\|u\|_{L^{p}(\omega^{\prime})},\\\ \|\nabla
E(u)\|_{L^{p}(\omega,\mathbb{R}^{d})}\leq
C(d,p,\omega^{\prime},\omega)\|\nabla
u\|_{L^{p}(\omega^{\prime},\mathbb{R}^{d})}.\end{cases}$ (2.3)

Then applying Theorem 2.13 in $\omega$, we have

$\displaystyle\int_{\omega^{\prime}}|V||u|^{p}\,\mathrm{d}x\leq\int_{\omega}|V||E(u)|^{p}\,\mathrm{d}x$
$\displaystyle\leq\delta\|\nabla
E(u)\|^{p}_{L^{p}(\omega,\mathbb{R}^{d})}+C(d,p,\delta,\omega,\tilde{\omega},W_{V}(\operatorname{diam}(\tilde{\omega})))\|E(u)\|^{p}_{L^{p}(\omega)}.$

Therefore, by using (2.3) in the latter term, (2.2) follows. ∎

We conclude the present section by the following corollary which demonstrates
that the Wolff class $\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ is
continuously embedded in $L^{1}_{\operatorname{loc}}(\Omega)$.

######  Corollary 2.15.

Let $1<p<d$ and $V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$. Suppose
$x_{0}\in\Omega$ and $B_{2r}(x_{0})\Subset\Omega$. Then there exists
$C=(C(d,p))$ such that

$\int_{B_{r/2}(x_{0})}|V|\,\mathrm{d}x\leq Cr^{d-p}W_{V}^{p-1}(2r).$

###### Proof.

The proof is immediate by choosing $u\in C_{c}^{\infty}(B_{2r}(x_{0}))$ with
$\mbox{\rm supp}\,{u}\subset B_{r}(x_{0})$, $0\leq u\leq 1$ in $B_{r}(x_{0})$,
$u=1$ in $B_{r/2}(x_{0})$ and $|\nabla u|<C/r$, in part (ii) of Theorem 2.12.
∎

##  3\. Criticality theory for $Q_{p,A,V}$ with potential in Wolff class

The present section is devoted to criticality theory for weak positive
solutions of the equation (1.1) with potentials in the local Wolff class
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$.

###  3.1. The setting

Let $\Omega\subseteq\mathbb{R}^{d}$ be a domain, and fix $1<p<d$. Throughout
the paper we assume that the operator $Q=Q_{p,A,V}$ satisfies the following
local regularity assumptions.

######  Assumptions 3.1.

  *   * •

$A\\!=\\!(a_{ij})_{i,j=1}^{d}\\!\in\\!L_{\rm
loc}^{\infty}(\Omega;\mathbb{R}^{d\times d})$ is a symmetric matrix valued
function.

  * •

$A$ is locally uniformly elliptic in $\Omega$, that is, for any
$\omega\Subset\Omega$ there exists $\Theta_{\omega}>0$ such that for all
$\xi\in\mathbb{R}^{d}$ and $x\in\omega$

$\displaystyle\hskip
42.67912pt(\Theta_{\omega})^{-1}\sum_{i=1}^{d}\xi_{i}^{2}\leq|\xi|_{A(x)}^{2}:=\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\leq\Theta_{\omega}\sum_{i=1}^{d}\xi_{i}^{2}.$

  * •

$V\in\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ is a real valued function.

By Assumptions 3.1, the function $\xi\mapsto|\xi|_{A(x)}^{p}$ is strictly
convex a.e. $x\in\Omega$. At some points, we assume that $|\xi|_{A}^{p}$ is
locally strong convexity.

######  Assumption 3.2.

We say that $|\xi|_{A}^{p}$ is a locally strong convex function with respect
to $\xi\in\mathbb{R}^{d}$ if there exists $\bar{p}\geq p$ such that for every
$\omega\Subset\Omega$ there exists a positive constant $C_{\omega}(\bar{p})$
such that

$|\xi|_{A}^{p}-|\eta|_{A}^{p}-p|\eta|_{A}^{p-2}A(x)\eta\cdot(\xi-\eta)\geq
C_{\omega}(\bar{p})|\xi-\eta|_{A}^{\bar{p}}\quad\forall\,\xi,\eta\in\mathbb{R}^{d}\mbox{
and a.e. }x\in\omega.$

In fact, by [21, Lemma 3.4] and [22, Lemma 2.2], there exists a constant
$C(p)>0$ such that for all $\xi,\eta\in\mathbb{R}^{n}$ ($\eta\neq 0$ if $p<2$)
and a.e. $x\in\omega$,

$|\xi|^{p}_{A}-|\eta|^{p}_{A}-p|\eta|^{p-2}_{A}A(x)\eta\cdot(\xi-\eta)\geq
C(p)[\xi,\eta]_{p,A},$ (3.1)

where

$[\xi,\eta]_{p,A}\triangleq\begin{cases}|\xi-\eta|_{A}^{p}&\text{if~{}$p\geq
2$},\\\
|\xi-\eta|_{A}^{2}(|\xi|_{A}+|\eta|_{A})^{p-2}&\text{if~{}$1<p<2$}.\end{cases}$

######  Definition 3.3.

A function $v$ is said to be a (weak) solution of the equation
$Q_{p,A,V}(u)=0$ in $\Omega$ if $v\in W_{\operatorname{loc}}^{1,p}(\Omega)$
and $v$ satisfies

$\int_{\Omega}(|\nabla v|^{p-2}_{A}A\nabla
v\cdot\nabla\varphi+V|v|^{p-2}v\varphi)\,\mathrm{d}x=0\qquad\forall\varphi\in
C_{c}^{\infty}(\Omega).$ (3.2)

We say the $v\in W_{\operatorname{loc}}^{1,p}(\Omega)$ is a supersolution of
(1.1) in $\Omega$, if the integral in (3.2) is nonnegative for every
nonnegative test function $\varphi\in\mathrm{C}_{c}^{\infty}(\Omega)$. A
function $v$ is a subsolution of (1.1) if $-v$ is supersolution of (1.1).

Theorem 2.12 implies that the integral in (3.2) is well defined. Note that
(1.1) is the Euler-Lagrange equation associated with the energy functional

$\mathcal{Q}_{p,A,V}(\varphi)=\mathcal{Q}_{p,A,V}(\varphi;\Omega):=\int_{\Omega}(|\nabla\varphi|^{p}_{A}+V|\varphi|^{p})\,\mathrm{d}x\qquad\varphi\in
C_{c}^{\infty}(\Omega).$

Throughout the paper we write $\mathcal{Q}_{p,A,V}\geq 0$ in $\Omega$ if
$\mathcal{Q}_{p,A,V}(\varphi)\geq 0$ for all $\varphi\in
C_{c}^{\infty}(\Omega)$.

###  3.2. Harnack convergence principle

In this subsection we prove the Harnack convergence principle. First we recall
the local Harnack inequality for nonnegative weak solutions of the equation
$Q_{p,A,V}(u)=0$ for $1<p<d$ proved in[2, Theorem 2.2]. We also refer to [26,
Theorem 2.2], where the local Harnack inequality of nonnegative solutions is
proved for certain quasilinear equations with coefficients in Wolff class. The
Harnack inequality of positive solutions is further extended in [17, Theorem
1.2] for a certain class of divergence-type elliptic equations with
nonstandard growth conditions and coefficients from appropriate Wolff classes.

######  Theorem 3.4 (Local Harnack inequality).

Let $1<p<d$ and $\omega^{\prime}\Subset\omega\Subset\Omega$. Then under
Assumptions 3.1, any nonnegative solution $v$ of the equation $Q_{p,A,V}(u)=0$
in $\omega$ satisfies

$\sup_{\omega^{\prime}}v\leq C\inf_{\omega^{\prime}}v,$

where $C$ is a positive constant depending only $d,p,\Theta_{\omega}$,
$W_{V}(\operatorname{diam}(\omega))$ and $\mbox{\rm
dist}\,(\omega^{\prime},\omega)$, but not on $v$.

We recall the weak Harnack inequality which is valid for nonnegative
supersolutions of $Q_{p,A,V}(u)=0$. For the proof see [26, Theorem 2.1].

######  Theorem 3.5 (Weak Harnack inequality).

Let $1<p<d$ and $\omega^{\prime}\Subset\omega\Subset\Omega$. Then under
Assumptions 3.1, for every $0<q<\frac{d(p-1)}{d-p}$ any nonnegative
supersolution $v$ of the equation of $Q_{p,A,V}(u)=0$ in $\omega$ satisfies

$\|v\|_{L^{q}(\omega^{\prime})}\leq C\inf_{\omega^{\prime}}v,$

where the constant $C>0$ depends only on $d$, $p$, $q$, $\Theta_{\omega}$,
$\mathcal{L}^{d}(\omega^{\prime})$, $\mbox{\rm
dist}\,(\omega^{\prime},\omega)$ and $W_{V}(\operatorname{diam}(\omega))$.
Here $\mathcal{L}^{d}(\omega^{\prime})$ is the Lebesgue measure of
$\omega^{\prime}$.

As a consequence of Theorem 3.4, we have the following result.

######  Corollary 3.6 (Local continuity).

Let $1<p<d$. Then under Assumptions 3.1, any solution $v$ of (1.1) is
continuous and has the following bound on its modulus of continuity: if
$B_{r_{0}}:=B_{r_{0}}(x_{0})\Subset\Omega$ and $\beta\in(0,1)$, then there
exist $\alpha\in(0,1)$ depending on $\beta$, and positive constants
$C=C\big{(}d,p,\Theta_{B_{r_{0}}},W_{V}(r_{0})\big{)}$,
$\gamma=\gamma\big{(}\>\\!\mathrm{d},p,\Theta_{B_{r_{0}}},W_{V}(r_{0})\big{)}$,
such that

$\underset{B_{r}}{\operatorname*{osc}}\,v\leq
C\left[\left(\frac{r}{r_{0}}\right)^{\alpha}\underset{B_{r_{0}}}{\operatorname*{osc}}\,v+\gamma
W_{V}(r_{0}^{1-\beta}r^{\beta})\right]\quad\forall\,0<r<r_{0}.$

###### Proof.

Since the solution $v$ of (1.1) is locally bounded [2, Theorem 3.5], hence
$\|v\|_{L^{\infty}(B_{r_{0}})}<\infty$. For $0<r<r_{0}$, define

$M(r):=\sup_{B_{r}}v,\quad
m(r):=\inf_{B_{r}}v,\quad\tau(r):=\underset{B_{r}}{\operatorname*{osc}}\,v=M(r)-m(r).$

Following the arguments in [18, Theorem 4.11] or in[10, Theorem 8.22], one
obtains that there exists a positive constant
$\vartheta=\vartheta(d,p,\Theta_{B_{r_{0}}},W_{V}(r_{0}))\in(0,1)$, such that

$\tau\bigg{(}\frac{r}{2}\bigg{)}\leq\vartheta\tau(r)+\gamma
W_{V}(r)\qquad\forall\,0<r<r_{0},$

where $\gamma>0$ is a constant depending on
$\>\\!\mathrm{d},p,\Theta_{B_{r_{0}}},W_{V}(r_{0})$ (see [17, Corollary 1.5.]
for $p(x)\equiv p$).

Then by applying [18, Lemma 4.12], or [10, Lemma 8.23], it follows that for
any $\beta\in(0,1)$, there exist a positive constant $C=C(\vartheta)>0$ and
$\alpha\in(0,1)$ such that

$\tau(r)\leq C\bigg{[}\left(\frac{r}{r_{0}}\right)^{\alpha}\tau(r_{0})+\gamma
W_{V}(r_{0}^{1-\beta}r^{\beta})\bigg{]}\qquad\forall\,0<r<r_{0}.$

This implies the continuity of solutions since $W_{V}(r)\rightarrow 0$ as
$r\rightarrow 0$. ∎

######  Remark 3.7.

When the potential $V$ lies in the Wolff class
$\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ or in the Stummel-Kato class
$\tilde{K}^{p}_{\operatorname{loc}}(\Omega)$, one should not expect Hölder
continuity of solutions of (1.1) (see for example, [5, 24]).

The following version of the Harnack convergence principle will be used
several times throughout the paper.

######  Theorem 3.8 (Harnack convergence principle).

Let $\\{\Omega_{i}\\}$ be a compact exhaustion of the domain $\Omega$. Suppose
that $\\{A_{i}\\}\subset L^{\infty}(\Omega_{i};\mathbb{R}^{d\times d})$ is a
sequence of symmetric and locally uniformly positive definite matrices such
that their local ellipticity constants are uniformly bounded away from $0$,
and $\\{A_{i}\\}$ converges weakly in
$L^{\infty}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d\times d})$ to
$\mathbb{A}\in L^{\infty}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d\times d})$
satisfying Assumptions 3.1. Furthermore, assume that

  * (i)

for $1<p\leq 2$, $\\{V_{i}\\}\in\mathfrak{W}^{p}(\Omega_{i})$ converges weakly
in the Wolff class $\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ to
$\mathbb{V}\in\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$,

  * (ii)

for $p>2$, $\\{V_{i}\\}\in\mathfrak{W}^{p}(\Omega_{i})$ converges with respect
to the quasinorm of the Wolff class
$\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ to
$\mathbb{V}\in\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$.

For each $i\in\mathbb{N}$, let $v_{i}$ be a positive (continuous) solution of
$Q_{p,A_{i},V_{i}}(v)=0$ in $\Omega_{i}$ with $v_{i}(x_{0})=1$, where
$x_{0}\in\Omega_{1}$ is a fixed reference point.

Then, up to a subsequence, $\\{v_{i}\\}$ converges weakly in
$W^{1,p}_{\operatorname{loc}}(\Omega)$ and also locally uniformly to a
positive solution $v$ of the equation $Q_{p,\mathbb{A},\mathbb{V}}(v)=0$ in
$\Omega$.

###### Proof.

(i) Assume first that $1<p\leq 2$. So,
$\\{V_{i}\\}\in\mathfrak{W}^{p}(\Omega_{i})$ converges weakly in the Wolff
class $\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ to
$\mathbb{V}\in\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$. The sequence
$\\{A_{i}\\}$ of matrices is locally uniformly elliptic and converges in the
weak topology of $L^{\infty}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d\times
d})$, so $\|A_{i}\|_{L^{\infty}(\tilde{\omega};\mathbb{R}^{d\times d})}\leq C$
for every $\tilde{\omega}\Subset\Omega$. Therefore, the matrices $A_{i}$ are
locally uniformly positive definite matrices which are uniformly bounded in
every $\tilde{\omega}\Subset\Omega$ except for a set of measure zero.

Since $v_{i}$ is a positive weak solution of the equation
$Q_{p,A_{i},V_{i}}(v)=0$ in $\Omega_{i}$, we have

$\int_{\Omega_{i}}|\nabla v_{i}|_{A_{i}}^{p-2}A_{i}\nabla v_{i}\cdot\nabla
u\,\mathrm{d}x+\int_{\Omega_{i}}V_{i}v_{i}^{p-1}u\,\mathrm{d}x=0\quad\text{for
all}\,\,u\in W_{0}^{1,p}(\Omega_{i}).$

Fixing a natural number $k$ and $u\in C_{c}^{\infty}(\Omega_{k})$, we may plug
$v_{i}|u|^{p}\in W_{0}^{1,p}(\Omega_{k})$, $i\geq k$, as a test function in
the above equality to obtain

$\|\,|\nabla v_{i}|_{A_{i}}u\|^{p}_{L^{p}(\Omega_{k})}\leq
p\int_{\Omega_{k}}|\nabla v_{i}|_{A_{i}}^{p-1}|u|^{p-1}v_{i}|\nabla
u|_{A_{i}}\,\mathrm{d}x+\int_{\Omega_{k}}|V_{i}|v_{i}^{p}|u|^{p}\,\mathrm{d}x.$

Applying Young’s inequality $pab\\!\leq\\!\varepsilon
a^{p^{\prime}}\\!+\\![(p\\!-\\!1)/\varepsilon]^{p-1}b^{p}$,
$\varepsilon\\!\in\\!(0,1)$, with $a\\!=\\!|\nabla
v_{i}|_{A_{i}}^{p-1}|u|^{p-1}$ and $b\\!=\\!v_{i}|\nabla u|_{A_{i}}$ on the
first term, and Morrey-Adams theorem (Theorem 2.13) for the second term of the
above inequality, we get

$\displaystyle(1-\varepsilon)\||\nabla
v_{i}|_{A_{i}}u\|^{p}_{L^{p}(\Omega_{k})}$
$\displaystyle\leq\left(\frac{p-1}{\varepsilon}\right)^{p-1}\\!\\!\|v_{i}|\nabla
u|_{A_{i}}\|^{p}_{L^{p}(\Omega_{k})}+\delta\|\nabla(v_{i}u)\|^{p}_{L^{p}(\Omega_{k};\mathbb{R}^{d})}+C_{\delta}\|v_{i}u\|^{p}_{L^{p}(\Omega_{k})},$

where
$C_{\delta}=C\big{(}d,p,\delta,W_{V_{i}}(\operatorname{diam}(\Omega_{k+1}))\big{)}$.
Since the sequence $\\{A_{i}\\}$ is locally uniformly elliptic and bounded
a.e., therefore, using the inequality

$\|\nabla(v_{i}u)\|^{p}_{L^{p}(\Omega_{k};\mathbb{R}^{d})}\leq
2^{p-1}\left(\|v_{i}\nabla
u\|^{p}_{L^{p}(\Omega_{k};\mathbb{R}^{d})}+\|u\nabla
v_{i}\|^{p}_{L^{p}(\Omega_{k};\mathbb{R}^{d})}\right),$

we obtain the following estimates for every $i\geq k$ and any $u\in
C_{c}^{\infty}(\Omega_{k})$

$\displaystyle\left((1\\!-\\!\varepsilon)C_{\Omega_{k}}^{p}\\!-\\!2^{p-1}\delta
C_{\Omega_{k}}^{-p}\right)\\!\||\nabla v_{i}|u\|^{p}_{L^{p}(\Omega_{k})}$
$\displaystyle\leq\left(\\!\\!\left(\\!\frac{p\\!-\\!1}{\varepsilon}\\!\right)^{p\\!-\\!1}C^{-p}_{\Omega_{k}}\\!+\\!2^{p-1}\delta\\!\right)\\!\|v_{i}|\nabla
u|\|^{p}_{L^{p}(\Omega_{k})}\\!\\!+\\!C_{\delta}\|v_{i}u\|^{p}_{L^{p}(\Omega_{k})}.$
(3.3)

Pick an arbitrary subdomain $\omega\Subset\Omega$. Without loss of generality,
we may assume that $x_{0}\in\omega$. Now for a subdomain
$\omega^{\prime}\Subset\Omega$ with $\omega\Subset\omega^{\prime}$, there
exists $k\geq 1$ such that $\omega^{\prime}\Subset\Omega_{k}$. Let
$0<\delta<(1-\varepsilon)2^{1-p}C^{2p}_{\Omega_{k}}$ and specialize $u\in
C_{c}^{\infty}(\Omega_{k})$ such that

$\text{supp}\\{u\\}\subset\omega^{\prime},\,0\leq u\leq
1,\,\,u=1\,\text{in}\,\omega,\text{ and }|\nabla
u|\leq\frac{1}{\text{dist}(\omega,\partial\omega^{\prime})}\,\,\text{in}\,\,\omega^{\prime}.$
(3.4)

Our assumption on the weak convergence of $\\{V_{i}\\}$ in
$\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ implies that
$\\{W_{V_{i}}(\operatorname{diam}(\omega^{\prime}))\\}$ is bounded in
$\mathbb{R}$. Hence, by the local Harnack inequality, the sequence of
solutions $\\{v_{i}\\}$ is bounded in $L^{\infty}(\omega)$. Also, by Corollary
3.6, the solutions $\\{v_{i}\\}$ have a common modulus of continuity in
$\omega$. Thus, Arzelà-Ascoli theorem implies that up to a subsequence,
$\\{v_{i}\\}$ converges uniformly in $\omega$. Moreover, using $u$ as in (3.4)
to the estimates (3.3), we get

$\|\nabla
v_{i}\|^{p}_{L^{p}(\omega;\mathbb{R}^{d})}+\|v_{i}\|^{p}_{L^{p}(\omega)}\leq
C\big{(}d,p,\varepsilon,\delta,\text{dist}(\omega,\partial\omega^{\prime}),C_{\Omega_{k}},W_{V_{i}}(\operatorname{diam}(\Omega_{k+1}))\big{)},$

for every $i\geq k$. This shows that the sequence $\\{v_{i}\\}$ is bounded in
$W^{1,p}(\omega)$. Hence, up to a subsequence, $\\{v_{i}\\}$ converges weakly
to a nonnegative function $v\in W^{1,p}(\omega)\cap C(\omega)$ with
$v(x_{0})=1$. Since up to subsequence, $\\{v_{i}\\}$ converges uniformly in
$\omega$, it follows that

$\displaystyle v_{i}\rightarrow v\quad\text{ uniformly in }\omega,\quad\text{
and }\nabla v_{i}\rightharpoonup\nabla
v\,\,\text{in}\,L^{p}(\omega;\mathbb{R}^{d}).$

Claim: $v$ is a weak solution of $Q_{\mathbb{A},p,\mathbb{V}}(u)=0$ in
$\tilde{\omega}\Subset\omega$. Indeed, using the uniform convergence of
$\\{v_{i}\\}$ to $v$ in $\omega$, we get

$\displaystyle\left|\int_{\tilde{\omega}}(V_{i}v_{i}^{p-1}\varphi-\mathbb{V}v^{p-1}\varphi)\,\mathrm{d}x\right|$
$\displaystyle\leq\int_{\tilde{\omega}}|V_{i}|\,|v_{i}^{p-1}-v^{p-1}|\,|\varphi|\,\mathrm{d}x+\left|\int_{\tilde{\omega}}(V_{i}-\mathbb{V})v^{p-1}\varphi\,\mathrm{d}x\right|$
$\displaystyle\leq
C\|v_{i}-v\|_{L^{\infty}(\tilde{\omega})}\int_{\tilde{\omega}}|V_{i}|\,\mathrm{d}x+\left|\int_{\tilde{\omega}}(V_{i}-\mathbb{V})v^{p-1}\varphi\,\mathrm{d}x\right|\quad\forall\varphi\in
C_{c}^{\infty}(\tilde{\omega}).$ (3.5)

Since $\\{W_{V_{i}}(\operatorname{diam}(\omega^{\prime}))\\}$ is bounded in
$\mathbb{R}$, the sequence $\\{V_{i}\\}$ is bounded in
$L_{\operatorname{loc}}^{1}(\Omega)$ (see Corollary 2.15). Thus, the first
term of the above inequality converges to zero, while the second term
converges to zero by our assumption on the weak convergence of $\\{V_{i}\\}$
to $\mathbb{V}$. Therefore, from (3.5), we get

$\int_{\tilde{\omega}}V_{i}v_{i}^{p-1}\varphi\,\mathrm{d}x\rightarrow\int_{\tilde{\omega}}\mathbb{V}v^{p-1}\varphi\,\mathrm{d}x\qquad\text{for
every }\varphi\in C_{c}^{\infty}(\tilde{\omega}).$

On the other hand, the sequence $\\{A_{i}\\}$ of matrices converges weakly in
$L^{\infty}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d\times d})$ to
$\mathbb{A}\in L^{\infty}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d\times d})$
and $\nabla v_{i}\rightharpoonup\nabla
v\,\,\text{in}\,L^{p}(\omega;\mathbb{R}^{d})$, consequently, by [11,
Proposition 2.7] it follows that

$|\nabla v_{i}|^{p-2}_{A_{i}}A_{i}\nabla
v_{i}\rightharpoonup_{i\rightarrow\infty}|\nabla
v|^{p-2}_{\mathbb{A}}\mathbb{A}\nabla
v\,\,\,\text{in}\,L^{p^{\prime}}(\tilde{\omega};\mathbb{R}^{d}).$

Thus, the claim is proved.
(ii) Assume now that $p>2$. In this case, we only need to show that

$\int_{\tilde{\omega}}V_{i}v_{i}^{p-1}\varphi\,\mathrm{d}x\rightarrow\int_{\tilde{\omega}}\mathbb{V}v^{p-1}\varphi\,\mathrm{d}x\qquad\text{for
every }\varphi\in C_{c}^{\infty}(\tilde{\omega}),$ (3.6)

since the rest of the proof follows as in (i). Obviously,

$\displaystyle\left|\int_{\tilde{\omega}}\\!\\!(V_{i}v_{i}^{p-1}u\\!-\\!\mathbb{V}v^{p-1}u)\\!\,\mathrm{d}x\right|\\!\leq\\!\int_{\tilde{\omega}}\\!\\!|V_{i}\\!-\\!\mathbb{V}||v_{i}|^{p-1}\\!|u|\\!\,\mathrm{d}x\\!+\\!\int_{\tilde{\omega}}\\!\\!|\mathbb{V}||v_{i}^{p-1}\\!-\\!v^{p-1}||u|\\!\,\mathrm{d}x$
$\displaystyle\leq
C\\!\int_{\tilde{\omega}}\\!\\!|V_{i}-\mathbb{V}||u|\,\mathrm{d}x+\\!\int_{\tilde{\omega}}\\!|\mathbb{V}||v_{i}^{p-1}-v^{p-1}||u|\,\mathrm{d}x.$
(3.7)

Clearly, $\mathbb{V}v_{i}^{p-1}\rightarrow\mathbb{V}v^{p-1}$ pointwise in
$\tilde{\omega}$, and by the uniform bound of $\\{v_{i}\\}$ in
$L^{\infty}(\tilde{\omega})$, $|\mathbb{V}v_{i}^{p-1}|\leq C|\mathbb{V}|$,
where $C$ is independent of $i$. Since
$|\mathbb{V}|\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)\subset
L^{1}_{\operatorname{loc}}(\Omega)$, the dominated convergence theorem implies
that

$\int_{\omega}|\mathbb{V}|\,|v_{i}^{p-1}-v^{p-1}|\,|u|\,\mathrm{d}x\rightarrow
0\qquad\forall u\in C_{c}^{\infty}(\tilde{\omega}).$

Moreover, $V_{i}\rightarrow\mathbb{V}$ in the quasinorm of
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, and
$\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ is continuously embedded in
$L^{1}_{\operatorname{loc}}(\Omega)$ (Corollary 2.15), it follows that the
first integral of the right-hand side of (3.7) converges to $0$ as
$i\rightarrow\infty$. Thus, (3.6) follows. ∎

###  3.3. The Dirichlet problem

The present subsection is devoted to the existence of a solution of the
nonhomogeneous Dirichlet problem (3.11) in a bounded Lipschitz subdomain. For
a uniqueness result, see Theorem 3.26. We begin with a lemma concerning the
weakly lower semicontinuity and coerciveness of a functional $J$ associated
with this Dirichlet problem.

######  Lemma 3.9.

Let $1<p<d$, and
$\omega\Subset\tilde{\omega}\Subset\omega^{\prime}\Subset\mathbb{R}^{d}$,
where $\omega$ is a Lipschitz domain. Assume that $A$ and $V$ satisfy
Assumptions 3.1, and let $\mathcal{G}\in\mathfrak{W}^{p}(\omega^{\prime})$.
Define a functional $J:W^{1,p}(\omega)\rightarrow\mathbb{R}\cup\\{\infty\\}$
by

$J[u]:=\mathcal{Q}_{p,A,V}(u;\omega)-\int_{\omega}\mathcal{G}|u|\,\mathrm{d}x=\int_{\omega}|\nabla
u|_{A}^{p}\,\mathrm{d}x+\int_{\omega}V|u|^{p}\,\mathrm{d}x-\int_{\omega}\mathcal{G}|u|\,\mathrm{d}x.$

Then:

  * (i)

The functional $J$ is weakly lower semicontinuous in $W^{1,p}(\omega)$.

  * (ii)

Moreover, if $V\geq 0$, then for any $g\in W^{1,p}(\omega)$ the functional $J$
is coercive in

$S_{g}:=\\{u\in W^{1,p}(\omega)\mid u=g\,\,\,\mbox{on }\partial\omega\mbox{ in
the trace sense}\\}.$

###### Proof.

(i). Let $\\{u_{n}\\}$ be a sequence in $W^{1,p}(\omega)$ such that
$u_{n}\rightharpoonup u$ in $W^{1,p}(\omega)$. Consequently,
$\\{\|u_{n}\|_{W^{1,p}(\omega)}\\}$ is bounded. Since $W^{1,p}(\omega)$ is
compactly embedded in $L^{p}(\omega)$, it follows that, up to a subsequence,
$u_{n}\rightarrow u$ in $L^{p}(\omega)$ and $u_{n}\rightarrow u$ a.e. in
$\omega$.
Let $\delta>0$. Using Minkowski’s inequality and Theorem 2.14, we get

$\displaystyle\left(\int_{\omega}V^{\pm}|u_{n}|^{p}\,\mathrm{d}x\right)^{1/p}-\left(\int_{\omega}V^{\pm}|u|^{p}\,\mathrm{d}x\right)^{1/p}\leq\left(\int_{\omega}V^{\pm}|u_{n}-u|^{p}\,\mathrm{d}x\right)^{1/p}$
$\displaystyle\leq\left(\delta\|\nabla(u_{n}-u)\|^{p}_{L^{p}(\omega,\mathbb{R}^{d})}+C(d,p,\delta,\|V^{\pm}\|^{1/(p-1)}_{\mathfrak{W}^{p}(\omega^{\prime})})\|u_{n}-u\|^{p}_{L^{p}(\omega)}\right)^{1/p}$
$\displaystyle\leq\delta^{1/p}\left(C+\|\nabla
u\|_{L^{p}(\omega,\mathbb{R}^{d})}\right)+C(d,p,\delta,\|V^{\pm}\|^{1/(p-1)}_{\mathfrak{W}^{p}(\omega^{\prime})})\|u_{n}-u\|_{L^{p}(\omega)},$

where $C=\underset{n\in\mathbb{N}}{\sup}\|u_{n}\|_{W^{1,p}(\omega)}$. This
implies that
$\underset{n\rightarrow\infty}{\lim\sup}\int_{\omega}V^{\pm}|u_{n}|^{p}\,\mathrm{d}x\leq\int_{\omega}V^{\pm}|u|^{p}\,\mathrm{d}x$.
On the other hand, Fatou’s lemma implies
$\int_{\omega}V^{\pm}|u|^{p}\,\mathrm{d}x\leq\underset{n\rightarrow\infty}{\lim\inf}\int_{\omega}V^{\pm}|u_{n}|^{p}\,\mathrm{d}x$.
Therefore,

$\underset{n\rightarrow\infty}{\lim}\int_{\omega}V|u_{n}|^{p}\,\mathrm{d}x=\int_{\omega}V|u|^{p}\,\mathrm{d}x.$

In addition, the weak lower semicontinuity of the gradient term is immediate
since the mapping $\xi\mapsto|\xi|^{p}_{A(x)}$ is convex. Thus,

$\mathcal{Q}_{p,A,V}(u)\leq\underset{n\rightarrow\infty}{\liminf}\mathcal{Q}_{p,A,V}(u_{n}).$

Similarly, for the last term of the functional $J$, we have

$\displaystyle\int_{\omega}\mathcal{G}^{\pm}|u_{n}|\,\mathrm{d}x-\int_{\omega}\mathcal{G}^{\pm}|u|\,\mathrm{d}x\leq\|\mathcal{G}^{\pm}\|^{1/p^{\prime}}_{L^{1}(\omega)}\left(\int_{\omega}\mathcal{G}^{\pm}|u_{n}-u|^{p}\,\mathrm{d}x\right)^{1/p}$
$\displaystyle\leq\|\mathcal{G}^{\pm}\|^{1/p^{\prime}}_{L^{1}(\omega)}\\!\\!\left[\delta^{1/p}\left(\\!C\\!\\!+\\!\|\nabla
u\|_{L^{p}(\omega,\mathbb{R}^{d})}\\!\right)\\!\\!+\\!C(d,p,\delta,\|\mathcal{G}^{\pm}\|^{1/(p-1)}_{\mathfrak{W}^{p}(\omega^{\prime})})\|u_{n}-u\|_{L^{p}(\omega)}\\!\right]\\!.$

This shows that
$\underset{n\rightarrow\infty}{\lim\sup}\int_{\omega}\mathcal{G}^{\pm}|u_{n}|\,\mathrm{d}x\leq\int_{\omega}\mathcal{G}^{\pm}|u|\,\mathrm{d}x.$
On the other hand, by Fatou’s lemma we have
$\int_{\omega}\mathcal{G}^{\pm}|u|\,\mathrm{d}x\leq\underset{n\rightarrow\infty}{\lim\inf}\int_{\omega}\mathcal{G}^{\pm}|u_{n}|\,\mathrm{d}x.$
Thus,

$\underset{n\rightarrow\infty}{\lim}\int_{\omega}\mathcal{G}|u_{n}|\,\mathrm{d}x=\int_{\omega}\mathcal{G}|u|\,\mathrm{d}x,$

and part (i) is proved.

(ii) Fix a number $t\in\mathbb{R}$, and suppose $u\in S_{g}$ such that
$J[u]\leq t$. In order to prove that the functional $J$ is coercive in
$S_{g}$, it is enough to show that $\|u\|_{W^{1,p}(\omega)}\leq C$, where the
positive constant $C$ is independent of $u$.
Since $V\geq 0$ in $\omega$ and $J[u]\leq t$, it follows that

$\displaystyle\int_{\omega}|\nabla u|_{A}^{p}\,\mathrm{d}x\leq
t+\int_{\omega}\mathcal{G}|u|\,\mathrm{d}x$ $\displaystyle\leq
t+\|\mathcal{G}\|^{1/p^{\prime}}_{L^{1}(\omega)}\left(\int_{\omega}|\mathcal{G}||u|^{p}\,\mathrm{d}x\right)^{1/p}\leq
t+C\|u\|_{W^{1,p}(\omega)},$ (3.8)

where the last inequality follows from the Morrey-Adams theorem (Theorem
2.14), and the constant $C$ depends only on
$d,p,\delta,\omega,\omega^{\prime},\|\mathcal{G}\|^{1/(p-1)}_{\mathfrak{W}^{p}(\omega^{\prime})}$
and $\|\mathcal{G}\|_{L^{1}(\omega)}$. Since $A$ is locally uniformly
elliptic, (3.8) implies

$\|\nabla u\|^{p}_{L^{p}(\omega,\mathbb{R}^{d})}\leq
C_{1}+C_{2}\|u\|_{W^{1,p}(\omega)},$ (3.9)

where the positive constants $C_{1}$ and $C_{2}$ are independent of $u$. Since
$u=g$ on $\partial\omega$, it follows $u-g\in W_{0}^{1,p}(\omega)$.
Furthermore, by the Poincaré inequality in $W_{0}^{1,p}(\omega)$, the uniform
ellipticity of $A$ and inequality (3.8), we have

$\displaystyle\|u\|_{L^{p}(\omega)}$
$\displaystyle\leq\|u-g\|_{L^{p}(\omega)}+\|g\|_{L^{p}(\omega)}\leq
C_{p}\|\nabla(u-g)\|_{L^{p}(\omega,\mathbb{R}^{d})}+\|g\|_{L^{p}(\omega)}$
$\displaystyle\leq C_{p}(\|\nabla u\|_{L^{p}(\omega,\mathbb{R}^{d})})+\|\nabla
g\|_{L^{p}(\omega,\mathbb{R}^{d})})+\|g\|_{L^{p}(\omega)}$
$\displaystyle\leq\frac{C_{p}}{\Theta_{\omega}}\left(\Big{(}\int_{\omega}|\nabla
u|_{A}^{p}\,\mathrm{d}x\Big{)}^{1/p}+\|\nabla
g\|_{L^{p}(\omega,\mathbb{R}^{d})}\right)+\|g\|_{L^{p}(\omega)}$
$\displaystyle\leq\frac{C_{p}}{\Theta_{\omega}}\left(\left(t+C\|u\|_{W^{1,p}(\omega)}\right)^{1/p}+\|\nabla
g\|_{L^{p}(\omega,\mathbb{R}^{d})}\right)+\|g\|_{L^{p}(\omega)},$

where the constant $C_{p}$ depends only on $d,\omega$ and $C$ as in (3.8).
Therefore,

$\|u\|_{L^{p}(\omega)}^{p}\leq C_{3}+C_{4}\|u\|_{W^{1,p}(\omega)},$ (3.10)

where $C_{3}$, $C_{4}$ are positive numbers independent of $u$. Thus, from
(3.9) and (3.10) we get

$\|u\|_{W^{1,p}(\omega)}^{p}\leq C_{5}+C_{6}\|u\|_{W^{1,p}(\omega)},$

where the positive constants $C_{5}$ and $C_{6}$ are independent of $u$. Since
$p>1$ the latter inequality implies that there exists $C>0$ (independent of
$u$) such that $\|u\|_{W^{1,p}(\omega)}\leq C$. ∎

Consequently, we have:

######  Theorem 3.10.

Assume $1<p<d$ and let $\omega\Subset\Omega$ be any Lipschitz subdomain. Let
$A$ satisfy Assumptions 3.1, and let
$V,\mathcal{G}\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, where
$V,\mathcal{G}\geq 0$ a.e. in $\omega$, and let $g\in W^{1,p}(\omega)\cap
C(\bar{\omega})$ be nonnegative. Then there exists a nonnegative solution
$v\in W^{1,p}(\omega)\cap C(\bar{\omega})$ to the Dirichlet problem

$\displaystyle\begin{cases}Q_{p,A,V}(u)=\mathcal{G}&\mbox{ in }\omega,\\\
u=g&\mbox{ on }\partial\omega.\end{cases}$ (3.11)

###### Proof.

Consider the functional
$I:W^{1,p}(\omega)\rightarrow\mathbb{R}\cup\\{\infty\\}$ defined by

$I[u]:=\mathcal{Q}_{p,A,V}(u;\omega)-\int_{\omega}\mathcal{G}u\,\mathrm{d}x=\int_{\omega}|\nabla
u|_{A}^{p}\,\mathrm{d}x+\int_{\omega}V|u|^{p}\,\mathrm{d}x-\int_{\omega}\mathcal{G}u\,\mathrm{d}x.$

Let $\\{u_{n}\\}\subset S_{g}=\\{u\in W^{1,p}(\omega)\mid u\\!=\\!g\mbox{ on
}\partial\omega\\}$ be such that $I[u_{n}]\searrow m:=\underset{u\in
S_{g}}{\inf}I[u]$. Since $g\geq 0$, we have $\\{|u_{n}|\\}\subset S_{g}$. Now,
consider the functional
$J:W^{1,p}(\omega)\rightarrow\mathbb{R}\cup\\{\infty\\}$ defined as in Lemma
3.9. Since $\mathcal{G}\geq 0$, it follows that $J[u]\leq I(u)$ and $m\leq
I[|u_{n}|]=J[u_{n}]\leq I[u_{n}]$. This implies that $J[u_{n}]\rightarrow m$
as $n\rightarrow\infty$ and $m=\underset{u\in S_{g}}{\inf}J[u]$.
By Lemma 3.9, the functional $J$ is weakly lower semicontinuous and also
coercive in $S_{g}$. So, in light of Mazur’s theorem, and since $J$ is even,
there exists a nonnegative $u\in S_{g}$ such that $J[u]=m$. Also we have
$J[u]=I[u]=m$. Thus, $u$ is a minimizer of the functional $I$ in $S_{g}$, and
hence, a solution of problem (3.11). ∎

###  3.4. Principal eigenvalue and the maximum principle

In this subsection we recall the definition of a principal eigenvalue of the
operator $Q_{p,A,V}$ satisfying Assumptions 3.1, and discuss its properties.

######  Definition 3.11 (Eigenvalue).

Let $\omega\Subset\Omega$ be a bounded domain.

1\. A real number $\Lambda$ is said to be an eigenvalue with an eigenfunction
$v$ of the Dirichlet eigenvalue problem

$\displaystyle\begin{cases}Q_{p,A,V}(u)=\Lambda|u|^{p-2}u&\mbox{ in
}\omega,\\\ u=0&\mbox{ on }\partial\omega,\end{cases}$ (3.12)

if $v\in W_{0}^{1,p}(\omega)\setminus\\{0\\}$ and $v$ satisfies

$\int_{\omega}|\nabla v|_{A}^{p-2}A\nabla
v\cdot\nabla\varphi\,\mathrm{d}x+\int_{\omega}V|v|^{p-2}v\varphi\,\mathrm{d}x=\Lambda\int_{\omega}|v|^{p-2}v\varphi\,\mathrm{d}x\quad\forall\varphi\in
C_{c}^{\infty}(\omega).$

2\. A principal eigenvalue of the Dirichlet eigenvalue problem (3.12) is an
eigenvalue of (3.12) with a nonnegative eigenfunction.

The following theorem on the existence of principal eigenvalue extends the
result in [20, Thorem 3.9], where the potential $V$ is assumed to be in the
Morrey space $M^{q}(p;\omega)$. The proof follows the same arguments as in
[20, Thorem 3.9] (using the Morrey-Adams theorem for
$V\in\mathfrak{W}^{p}(\omega)$), and therefore it is omitted.

######  Theorem 3.12.

Assume that $A$ and $V$ satisfy Assumptions 3.1. Then $Q_{p,A,V}$ on a
subdomain $\omega\Subset\Omega$ admits a unique principal eigenvalue, denoted
by $\Lambda_{1}$. Moreover, $\Lambda_{1}$ is given by the Rayleigh-Ritz
variational formula

$\Lambda_{1}:=\Lambda_{1}(Q_{p,A,V};\omega)=\Lambda_{1}(\omega)=\inf_{u\in
W_{0}^{1,p}(\omega)\setminus\\{0\\}}\frac{\mathcal{Q}_{p,A,V}(u;\omega)}{\|u\|^{p}_{L^{p}(\omega)}}.$

Furthermore, the principal eigenvalue $\Lambda_{1}$ is simple and isolated.

We recall a generalized Picone identity which implies the strict monotonicity
of $\Lambda_{1}(Q_{p,A,V};\omega)$ as as a function of subdomains
$\omega\Subset\Omega$. For the proof of Picone identity see [21, Lemma 3.1]
and Remark 3.2 therein.

######  Lemma 3.13 (Picone identity).

Let $v>0,u\geq 0$ be in $W_{\operatorname{loc}}^{1,p}(\Omega)$ such that
$uv^{-1}\in L^{\infty}_{\operatorname{loc}}(\Omega)$, and assume that $A$ and
$V$ satisfy Assumptions 3.1. For $x\in\Omega$ define

$L_{A}(u,v)(x):=|\nabla u|_{A}^{p}+(p-1)\frac{u^{p}}{v^{p}}|\nabla
v|_{A}^{p}-p\frac{u^{p-1}}{v^{p-1}}\nabla u\cdot A(x)\nabla v|\nabla
v|_{A}^{p-2},$ (3.13)

and

$R_{A}(u,v)(x):=|\nabla
u|_{A}^{p}-\nabla\left(\frac{u^{p}}{v^{p-1}}\right)\cdot A(x)\nabla v|\nabla
v|_{A}^{p-2}.$ (3.14)

Then

$L_{A}(u,v)(x)=R_{A}(u,v)(x)\geq 0\quad\mbox{a.e. }\,x\in\Omega.$

Moreover, $L_{A}(u,v)=0$ a.e. in $\Omega$ if and only if $u=cv$ in $\Omega$
for some constant $c\geq 0$.

Using this Picone identity we have the following theorem on the positivity of
principal eigenvalue which implies that if $\mathcal{Q}_{p,A,V}(\varphi)\geq
0$ for all $\varphi\in C_{c}^{\infty}(\Omega)$, then $\Lambda_{1}(\omega)>0$
for any bounded subdomain $\omega\Subset\Omega$. This extends the result
obtained in [21, Lemma 5.1] for potential $V\in
L^{\infty}_{\operatorname{loc}}(\Omega)$.

######  Theorem 3.14.

Let $\omega_{1}\Subset\omega_{2}\Subset\Omega$ be Lipschitz subdomains.
Suppose that $A$ and $V$ satisfy Assumptions 3.1, and $\mathcal{Q}_{p,A,V}\geq
0$ in $\omega_{2}$. Then $\Lambda_{1}(\omega_{1})>\Lambda_{1}(\omega_{2})\geq
0$.

###### Proof.

From the Rayleigh-Ritz variational formula, we get
$\Lambda_{1}(\omega_{2})\geq 0$. Let $\phi_{i}\in W^{1,p}_{0}(\omega_{i})$ be
the normalized principal eigenfunctions corresponding to the principal
eigenvalues $\Lambda_{1}(\omega_{i})$, $i=1,2$. By the Harnack inequality,
$\phi_{i}$ does not vanish in $\omega_{i}$ for $i=1,2$, and hence
$\frac{\phi_{1}}{\phi_{2}}\in L^{\infty}_{\operatorname{loc}}(\omega_{1})$.
Now consider a nonnegative minimizing sequence $\\{\varphi_{n}\\}\subset
C_{c}^{\infty}(\omega_{1})$ that converges to $\phi_{1}$ in
$W^{1,p}_{0}(\omega_{1})$. Then by using the Picone identity (Lemma 3.13) we
have

$\displaystyle 0\leq$
$\displaystyle\int_{\omega_{2}}L_{A}(\varphi_{n},\phi_{2})\>\\!\mathrm{d}x=\int_{\omega_{1}}L_{A}(\varphi_{n},\phi_{2})\>\\!\mathrm{d}x=\int_{\omega_{1}}R_{A}(\varphi_{n},\phi_{2})\>\\!\mathrm{d}x$
$\displaystyle=\int_{\omega_{1}}|\nabla\varphi_{n}|_{A}^{p}\>\\!\mathrm{d}x-\int_{\omega_{1}}\nabla\left(\frac{\varphi_{n}^{p}}{\phi_{2}^{p-1}}\right)\cdot
A(x)\nabla\phi_{2}|\nabla\phi_{2}|_{A}^{p-2}\>\\!\mathrm{d}x$
$\displaystyle=\int_{\omega_{1}}|\nabla\varphi_{n}|_{A}^{p}\>\\!\mathrm{d}x+\int_{\omega_{1}}V\varphi_{n}^{p}\>\\!\mathrm{d}x-\Lambda_{1}(\omega_{2})\int_{\omega_{1}}\varphi_{n}^{p}\>\\!\mathrm{d}x,$

where we have used $\frac{\varphi_{n}^{p}}{\phi_{2}}\in
W_{0}^{1,p}(\omega_{2})$ as a test function in weak formulation of
$\eqref{evp}$ in $\omega_{2}$. Letting $n\rightarrow\infty$ and using Fatou’s
lemma, we obtain

$\displaystyle 0\leq\int_{\omega_{1}}L_{A}(\phi_{1},\phi_{2})\>\\!\mathrm{d}x$
$\displaystyle\leq\lim_{n\rightarrow\infty}\int_{\omega_{1}}L_{A}(\varphi_{n},\phi_{2})\>\\!\mathrm{d}x$
$\displaystyle=\left(\Lambda_{1}(\omega_{1})-\Lambda_{1}(\omega_{2})\right)\int_{\omega_{1}}\varphi_{1}^{p}\>\\!\mathrm{d}x.$

This implies that $\Lambda_{1}(\omega_{1})\geq\Lambda_{1}(\omega_{2})$. If
$\Lambda_{1}(\omega_{1})=\Lambda_{1}(\omega_{2})$, then
$L_{A}(\phi_{1},\phi_{2})=0$ a.e. in $\omega_{1}$. Therefore, Lemma 3.13,
implies $\phi_{2}|_{\omega_{1}}=c\phi_{1}$ for some $c>0$. But by the Harnack
inequality $\phi_{2}>0$ on $\partial\omega_{1}$ which contradicts the fact
that $\phi_{1}=0$ on $\partial\omega_{1}$ in the trace sense. Thus,
$\Lambda_{1}(\omega_{1})>\Lambda_{1}(\omega_{2})$. ∎

The positivity of the principal eigenvalue in a bounded subdomain implies the
generalized maximum principle for $\mathcal{Q}_{p,A,V}\geq 0$.

######  Theorem 3.15 (Maximum principle).

Suppose that $A$ and $V$ satisfy Assumptions 3.1, and assume that
$\mathcal{Q}\geq 0$ in $\Omega$. Let $\omega\Subset\Omega$ be a subdomain.
Then the following assertions are equivalent:

  * (i)

The principal eigenvalue $\Lambda_{1}$ of $Q_{p,A,V}$ in $\omega$ is positive.

  * (ii)

The weak maximum principle for $Q_{p,A,V}$ holds true in $\omega$: If $v\in
W^{1,p}(\omega)$ is a solution of

$\displaystyle\begin{split}Q_{p,A,V}(u)&=\mathcal{G}\quad\mbox{in }\omega,\\\
u&\geq 0\quad\mbox{on }\partial\omega,\end{split}$ (3.15)

where $\mathcal{G}\in L^{1}_{\operatorname{loc}}(\omega)$ is nonnegative, then
$v\geq 0$ in $\omega$.

  * (iii)

The strong maximum principle for $Q_{p,A,V}$ holds true in $\omega$: a
solution $v\in W^{1,p}(\omega)$ of (3.15) is either $v=0$ or $v>0$ in
$\omega$.

###### Proof.

$\mbox{(i)}\Rightarrow\mbox{(ii)}$: Let $v\in W^{1,p}(\omega)$ be a solution
of (3.15). Then choosing $v^{-}\in W_{0}^{1,p}(\omega)$ as a test function in
the weak formulation of (3.15) we have

$\mathcal{Q}_{p,A,V}(v^{-};\omega)=\int_{\\{x\in\omega:v(x)<0\\}}\mathcal{G}v\,\mathrm{d}x.$

Since $\mathcal{G}\geq 0$ it follows that
$\mathcal{Q}_{p,A,V}(v^{-};\omega)\leq 0$. If $v^{-}\neq 0$, then
$\Lambda_{1}\leq 0$, which contradicts (i). Thus, $v^{-}=0$ a.e. in $\omega$
and, hence $v\geq 0$ a.e. in $\omega$.

$\mbox{(ii)}\Rightarrow\mbox{(iii)}$: Since $\mathcal{G}\geq 0$, it follows
that $v$ is a nonnegative supersolution of $Q_{p,A,V}(u)=0$ in $\omega$. The
weak Harnack inequality (Theorem 3.5) implies that $v>0$ in $\omega$.

$\mbox{(iii)}\Rightarrow\mbox{(i)}$: Suppose that the principal eigenvalue
$\Lambda_{1}\leq 0$ and let $v\in W_{0}^{1,p}(\omega)$ be its principal
eigenfunction. Then $u:=-v$ is a supersolution of $Q_{p,A,V}(u)=0$ in $\omega$
with $u=0$ on $\partial\omega$. Since $u\neq 0$, the strong maximum principle
implies that $u>0$ which is a contradiction. Thus, $\Lambda_{1}>0$. ∎

Next we prove the existence and uniqueness for the Dirichlet problem.

######  Theorem 3.16.

Assume that $A$ and $V$ satisfy Assumptions 3.1, and let
$\omega\Subset\omega^{\prime}\Subset\Omega$ be subdomains. Suppose that
$\Lambda_{1}$ of $Q_{p,A,V}$ in $\omega$ is positive. Then for $0\leq
g\\!\in\\!L^{p^{\prime}}(\omega)$ there exists a unique positive solution in
$W_{0}^{1,p}(\omega)$ of

$\displaystyle\begin{cases}Q_{p,A,V}(u)=g&\mbox{ in }\omega,\\\ u=0&\mbox{ on
}\partial\omega.\end{cases}$ (3.16)

###### Proof.

Consider the functional

$J[u]:=\mathcal{Q}_{p,A,V}(u;\omega)-\int_{\omega}gu\,\mathrm{d}x\qquad u\in
W_{0}^{1,p}(\omega).$

Using Theorem 2.13, and following the same steps as in Lemma 3.9 (i), we
obtain that the functional $\mathcal{Q}_{p,A,V}$ is weakly lower
semicontinuous in $W_{0}^{1,p}(\omega)$. On the other hand, since $g\in
L^{p^{\prime}}(\omega)$, the functional $I(u):=\int_{\omega}gu\,\mathrm{d}x$
is a bounded linear functional in $W_{0}^{1,p}(\omega)$. Therefore, the
functional $J[u]$ is weakly lower semicontinuous in $W_{0}^{1,p}(\omega)$.

We claim that the functional $J[u]$ is coercive in $W_{0}^{1,p}(\omega)$.
Assume that $J[u]\leq t$. Since $\Lambda_{1}>0$ hence we have

$\Lambda_{1}\|u\|^{p}_{L^{p}(\omega)}\leq\mathcal{Q}_{p,A,V}(u;\omega)\quad\forall
u\in W_{0}^{1,p}(\omega).$

Then by applying Hölder inequality, we get

$\Lambda_{1}\|u\|^{p}_{L^{p}(\omega)}\leq
J[u]+\int_{\omega}gu\,\mathrm{d}x\leq
t+\|g\|_{L^{p^{\prime}}(\omega)}\|u\|_{L^{p}(\omega)}.$

This inequality implies that

$\|u\|_{L^{p}(\omega)}\leq
m:=\max\left\\{1,\left(\frac{t+\|g\|_{L^{p^{\prime}}(\omega)}}{\Lambda_{1}}\right)^{1/{(p-1)}}\right\\}.$
(3.17)

On the other hand, applying Hölder’s inequality and Morrey-Adams theorem
(Theorem 2.13) in the inequality $J[u]\leq t$ we obtain

$\displaystyle\int_{\omega}|\nabla u|_{A}^{p}\,\mathrm{d}x$ $\displaystyle\leq
t+\int_{\omega}gu\,\mathrm{d}x+\int_{\omega}|V||u|^{p}\,\mathrm{d}x$
$\displaystyle\leq
t+\|g\|_{L^{p^{\prime}}(\omega)}\|u\|_{L^{p}(\omega)}+\delta\|\nabla
u\|^{p}_{L^{p}(\omega;\mathbb{R}^{d})}+C_{\delta}\|u\|^{p}_{L^{p}(\omega)},$
(3.18)

where
$C_{\delta}=C(d,p,\delta,\omega,\omega^{\prime},\|V\|_{\mathfrak{W}^{p}(\omega^{\prime})})$.
Then, (3.17), (3.18) and Assumptions 3.1 imply

$\|\nabla
u\|^{p}_{L^{p}(\omega;\mathbb{R}^{d})}\leq(\Theta_{\omega}^{-p}-\delta)^{-1}(t+\|g\|_{L^{p^{\prime}}(\omega)}m+C_{\delta}m^{p}),$

for $\delta<\Theta_{\omega}$. This together with (3.17) implies that
$\|u\|_{W^{1,p}(\omega)}\leq C$, where the constant $C>0$ is independent of
$u$.

Thus, the functional $J[u]$ is weakly lower semicontinuous and coercive in
$W_{0}^{1,p}(\omega)$. Consequently, the functional has a minimizer in
$W_{0}^{1,p}(\omega)$ and the corresponding Dirichlet problem has a
nonnegative solution $v_{1}\in W_{0}^{1,p}(\omega)$. Since $\Lambda_{1}>0$ by
the strong maximum principle (Theorem 3.15), the solution $v_{1}$ is either
$0$ or positive in $\omega$.

Uniqueness: Let $v_{i}\in W_{0}^{1,p}(\omega)$, $i=1,2$, be two solutions of
(3.16). If $v_{1}=0$, then $g=0$ in $\omega$, and then $v_{2}$ is a principal
eigenvalue with eigenvalue $0$, but in light of the uniqueness of the
principal eigenvalue, this contradicts the assumption that $\Lambda_{1}>0$.

Thus, $v_{i}>0$ in $\omega$ for $i=1,2$. Consider $v_{i,h}:=v_{i}+h$, where
$h$ is a positive constant and $i=1,2$. Taking
$\varphi_{1,h}:=\frac{v_{1,h}^{p}-v_{2,h}^{p}}{v_{1,h}^{p-1}}\in
W_{0}^{1,p}(\omega)$ as a test function in the definition of $v_{1}$ being a
solution of the equation (3.16), we have

$\displaystyle\int_{\omega}\\!(v_{1,h}^{p}\\!-\\!v_{2,h}^{p})|\nabla\log
v_{1,h}|^{p}_{A}\\!\,\mathrm{d}x\\!$
$\displaystyle-\\!p\\!\int_{\omega}\\!\\!v_{2,h}^{p}|\nabla\log
v_{1,h}|^{p-2}_{A}\\!A\nabla\\!\log
v_{1,h}\\!\cdot\\!\nabla\\!\log(\frac{v_{2,h}}{v_{1,h}})\\!\,\mathrm{d}x$
$\displaystyle=\int_{\omega}\frac{(g-Vv_{1}^{p-1})(v_{1,h}^{p}-v_{2,h}^{p})}{v_{1,h}^{p-1}}.$

Similarly, using
$\varphi_{2,h}=\frac{v_{2,h}^{p}-v_{1,h}^{p}}{v_{2,h}^{p-1}}\in
W_{0}^{1,p}(\omega)$ as a test function in the definition of $v_{2}$ being a
solution of the Dirichlet problem (3.16), we get

$\displaystyle\int_{\omega}\\!(v_{2,h}^{p}\\!-\\!v_{1,h}^{p})|\nabla\\!\log
v_{2,h}|^{p}_{A}\\!\,\mathrm{d}x$
$\displaystyle\\!-\\!p\\!\int_{\omega}\\!\\!v_{1,h}^{p}|\nabla\log
v_{2,h}|^{p-2}_{A}\\!A\nabla\\!\log
v_{2,h}\\!\cdot\\!\nabla\\!\log(\frac{v_{1,h}}{v_{2,h}})\\!\,\mathrm{d}x$
$\displaystyle=\int_{\omega}\frac{(g-Vv_{2}^{p-1})(v_{2,h}^{p}-v_{1,h}^{p})}{v_{2,h}^{p-1}}.$

Adding the above two equations, we reach at

$\displaystyle\int_{\omega}\\!v_{1,h}^{p}\\!\left(\\!|\nabla\\!\log\\!v_{1,h}|^{p}_{A}\\!-\\!|\nabla\\!\log\\!v_{2,h}|^{p}_{A}\\!-\\!p|\nabla\log
v_{2,h}|^{p-2}_{A}\\!A\nabla\\!\log
v_{2,h}\\!\cdot\\!\nabla\\!\log(\frac{v_{1,h}}{v_{2,h}})\\!\right)\\!\\!\,\mathrm{d}x$
$\displaystyle+\\!\\!\int_{\omega}\\!\\!v_{2,h}^{p}\\!\\!\left(\\!|\nabla\\!\log\\!v_{2,h}|^{p}_{A}\\!-\\!|\nabla\\!\log\\!v_{1,h}|^{p}_{A}\\!-\\!p\nabla\\!\log
v_{1,h}|^{p-2}_{A}\\!A\nabla\\!\log\\!v_{1,h}\\!\cdot\\!\nabla\\!\log(\frac{v_{2,h}}{v_{1,h}})\\!\right)\\!\\!\,\mathrm{d}x$
$\displaystyle\,\,\,\,\,=\int_{\omega}\left[g\left(\frac{1}{v_{1,h}^{p-1}}-\frac{1}{v_{2,h}^{p-1}}\right)-V\left(\frac{v_{1}^{p-1}}{v_{1,h}^{p-1}}-\frac{v_{2}^{p-1}}{v_{2,h}^{p-1}}\right)\right](v_{1,h}^{p}-v_{2,h}^{p})\,\mathrm{d}x$

Using the convexity inequality (3.1) we arrive at

$\displaystyle\int_{\omega}\left[g\left(\frac{1}{v_{1,h}^{p-1}}-\frac{1}{v_{2,h}^{p-1}}\right)-V\left(\frac{v_{1}^{p-1}}{v_{1,h}^{p-1}}-\frac{v_{2}^{p-1}}{v_{2,h}^{p-1}}\right)\right](v_{1,h}^{p}-v_{2,h}^{p})\,\mathrm{d}x\geq
C(p)\,\times$
$\displaystyle\int_{\omega}\\!\\!\\!\,\mathrm{d}x\\!\begin{cases}\\!\\!(v_{1,h}^{p}\\!+\\!v_{2,h}^{p})|\nabla\log(\frac{v_{1,h}}{v_{2,h}})|_{A}^{p}\qquad\qquad
p\in[2,\infty),\\\\[5.69054pt]
\\!(v_{1,h}^{p}\\!+\\!v_{2,h}^{p})|\nabla\\!\log(\frac{v_{1,h}}{v_{2,h}})|_{A}^{2}\\!\left(|\nabla\\!\log\\!v_{1,h}|_{A}\\!+\\!|\nabla\\!\log\\!v_{2,h}|_{A}\\!\right)^{p-2}\,\
p\\!\in\\!(1,2).\end{cases}$

This implies that

$0\\!\geq\\!\int_{\omega}\\!\\!g\left(\frac{1}{v_{1,h}^{p-1}}-\frac{1}{v_{2,h}^{p-1}}\right)(v_{1,h}^{p}-v_{2,h}^{p})\\!\,\mathrm{d}x\geq\\!\\!\int_{\omega}\\!V\\!\\!\left(\frac{v_{1}^{p-1}}{v_{1,h}^{p-1}}-\frac{v_{2}^{p-1}}{v_{2,h}^{p-1}}\right)(v_{1,h}^{p}-v_{2,h}^{p})\\!\,\mathrm{d}x.$

Since $v_{i,h}\rightarrow v_{i}$ as $h\rightarrow 0$ for $i=1,2$, it follows
that the integrand of the integral of the right hand side above converges to
$0$ a.e. in $\omega$. Also,

$\left|V\\!\\!\left(\frac{v_{1}^{p-1}}{v_{1,h}^{p-1}}-\frac{v_{2}^{p-1}}{v_{2,h}^{p-1}}\right)(v_{1,h}^{p}-v_{2,h}^{p})\right|\leq
2|V|(\left(v_{1}+1)^{p}+(v_{2}+1)^{p}\right)\in L^{1}(\omega).$

Therefore, by the dominated converges theorem, it follows that

$\lim_{h\rightarrow
0}\int_{\omega}g\left(\frac{1}{v_{1,h}^{p-1}}-\frac{1}{v_{2,h}^{p-1}}\right)(v_{1,h}^{p}-v_{2,h}^{p})\,\mathrm{d}x=0.$

This together with the Fatou lemma implies that

$0=\int_{\omega}\\!\\!\\!\,\mathrm{d}x\\!\begin{cases}\\!\\!(v_{1}^{p}\\!+\\!v_{2}^{p})|\nabla\log(\frac{v_{1}}{v_{2}})|_{A}^{p}&p\in[2,\infty),\\\\[5.69054pt]
\\!(v_{1}^{p}\\!+\\!v_{2}^{p})|\nabla\\!\log(\frac{v_{1}}{v_{2}})|_{A}^{2}\\!\left(|\nabla\\!\log\\!v_{1}|_{A}\\!+\\!|\nabla\\!\log\\!v_{2}|_{A}\\!\right)^{p-2}&\
p\\!\in\\!(1,2).\end{cases}$

Thus, $v_{1}=v_{2}$ a.e. in $\omega$. ∎

###  3.5. Agmon-Allegretto-Piepenbrink-type theorem

The next result, known as Agmon-Allegretto-Piepenbrink (AAP)-type theorem,
asserts that the nonnegativity of the functional $\mathcal{Q}_{p,A,V}$ on
$C_{c}^{\infty}(\Omega)$ is equivalent to the existence of a weak positive
solution or supersolution of $Q_{p,A,V}(u)=0$ in $\Omega$. This result
generalizes the AAP Theorem $4.3$ obtained in [20] for a potential $V\in
M^{q}_{\operatorname{loc}}(p;\Omega)$ (see [20] for the history of AAP
theorem).

######  Theorem 3.17 (AAP-type theorem).

Suppose that $A$ and $V$ satisfy Assumptions 3.1. Then the following
assertions are equivalent:

  * (i)

$\mathcal{Q}_{p,A,V}\geq 0$ in $\Omega$;

  * (ii)

the equation $Q_{p,A,V}(u)=0$ in $\Omega$ admits a positive solution;

  * (iii)

the equation $Q_{p,A,V}(u)=0$ in $\Omega$ admits a positive supersolution.

###### Proof.

$\mbox{(i)}\Rightarrow\mbox{(ii)}$: Fix a point $x_{0}\in\Omega$ and consider
a sequence $\\{\omega_{i}\\}_{i\in\mathbb{N}}$ of Lipschitz compact exhaustion
$\cup_{i\in\mathbb{N}}\omega_{i}=\Omega$ such that $x_{0}\in\omega_{1}$, and
consider the following Dirichlet problem for $i\geq 2$

$\displaystyle\begin{split}Q_{p,A,V+1/i}(u)&=f_{i}\quad\mbox{ in
}\omega_{i},\\\ u&=0\quad\mbox{ on }\partial\omega_{i},\end{split}$ (3.19)

where $f_{i}\in
C_{c}^{\infty}(\omega_{i}\setminus\bar{\omega}_{i-1})\setminus\\{0\\}$ and
$f_{i}\geq 0$. By (i), we have

$\Lambda_{1}=\inf_{u\in
W_{0}^{1,p}(\omega_{i})\setminus\\{0\\}}\frac{\mathcal{Q}_{p,A,V+1/i}(u;\omega_{i})}{\|u\|^{p}_{L^{p}(\omega_{i})}}\geq\frac{1}{i}\quad\mbox{for
all }i\geq 1.$

This together with Theorem 3.16 implies that the Dirichlet problem (3.19)
admits a positive solution $v_{i}\in W_{0}^{1,p}(\omega_{i})$. Set
$\omega^{\prime}_{i}=\omega_{i-1}$. As $\mbox{\rm
supp}\,(f_{i})\subset\omega_{i}\setminus\bar{\omega^{\prime}}_{i}$, we get for
$i\geq 2$

$\int_{\omega^{\prime}_{i}}|\nabla v_{i}|_{A}A\nabla v_{i}\cdot\nabla
u\,\mathrm{d}x+\int_{\omega^{\prime}_{i}}\left(V+\frac{1}{i}\right)v_{i}^{p-1}u\,\mathrm{d}x=0\quad\forall
u\in W_{0}^{1,p}(\omega^{\prime}_{i}).$

By Theorem 3.6, $v_{i}\in C(\omega^{\prime}_{i})$ for all $i\geq 2$. Normalize
$f_{i}$ to get $v_{i}(x_{0})=1$ for all $i\geq 2$. The Harnack convergence
principle (Theorem 3.8) with $A_{i}=A$ and $V_{i}=V+{1}/{i}$, implies that, up
to a subsequence, $v_{i}\to v$, where $v$ is a positive solution of
$Q_{p,A,V}(u)=0$ in $\Omega$.

$\mbox{(ii)}\Rightarrow\mbox{(iii)}$ is trivial.

$\mbox{(iii)}\Rightarrow\mbox{(i)}$: Let $\tilde{v}\in
W_{\operatorname{loc}}^{1,p}(\Omega)$ be a positive supersolution of the
equation $Q_{p,A,V}(u)=0$ in $\Omega$. The weak Harnack inequality (Theorem
3.5) implies that $1/\tilde{v}\in L^{\infty}_{\operatorname{loc}}(\Omega)$.
For any $\varphi\in C_{c}^{\infty}(\Omega)$, choose
$|\varphi|^{p}\tilde{v}^{1-p}\in W_{0}^{1,p}(\Omega)$ as a test function and
denote $T:=-|\nabla\log\tilde{v}|_{A}^{p-2}\nabla\log\tilde{v}$, then

$(p-1)\int_{\Omega}|T|_{A}^{p^{\prime}}|\varphi|^{p}\,\mathrm{d}x\leq
p\int_{\Omega}|T|_{A}|\varphi|^{p-1}|\nabla\varphi|_{A}\,\mathrm{d}x+\int_{\Omega}V|\varphi|^{p}\,\mathrm{d}x.$

Using the Young’s inequality: $pab\leq(p-1)a^{p^{\prime}}+b^{p}$ with
$a=|T|_{A}|\varphi|^{p-1}$ and $b=|\nabla\varphi|_{A}$, we obtain that
$\mathcal{Q}_{p,A,V}(\varphi)\geq 0$ for all $\varphi\in
C_{c}^{\infty}(\Omega)$. ∎

###  3.6. Null-sequence and ground state

In this subsection we recall the notion of a null-sequence and a ground state
corresponding to the functional $\mathcal{Q}_{p,A,V}$. We show that if
$\mathcal{Q}_{p,A,V}$ admits a ground state $\Phi$ in $\Omega$, then $\Phi$ is
the unique (up to a multiplicative constant) positive supersolution of
${Q}_{p,A,V}$ in $\Omega$. Moreover, in a bounded domain, the positive
principal eigenfunction is in fact a ground sate if $A$ and $V$ satisfy
assumptions 3.1, and 3.2.

######  Definition 3.18 (Null-sequence).

Assume that $\mathcal{Q}_{p,A,V}\geq 0$ in $\Omega$. A sequence
$\\{u_{n}\\}\subset W_{c}^{1,p}(\Omega)$ is said to be a null-sequence of
$\mathcal{Q}_{p,A,V}$ in $\Omega$ if

  * (a)

$u_{n}\geq 0$ for all $n\in\mathbb{N}$;

  * (b)

there exists an open set $K\\!\Subset\\!\Omega$ such that
$\|u_{n}\|_{L^{p}(K)}\\!=\\!1$ for all $n\\!\in\\!\mathbb{N}$;

  * (c)

$\displaystyle{\lim_{n\to\infty}\mathcal{Q}_{p,A,V}(u_{n})=0}$.

######  Definition 3.19 (Ground state).

A positive $\Phi\in W^{1,p}_{\operatorname{loc}}(\Omega)$ is an (Agmon) ground
state of $\mathcal{Q}_{p,A,V}$ in $\Omega$ if $\Phi$ is an
$L^{p}_{\operatorname{loc}}(\Omega)$ limit of a null-sequence.

######  Lemma 3.20.

Assume that $\mathcal{Q}_{p,A,V}\\!\geq\\!0$ in $\Omega$, where $A$ and $V$
satisfy Assumptions 3.1. If $p<2$ assume further that Assumption 3.2 is
satisfied. Suppose that $\mathcal{Q}_{p,A,V}$ admits a null-sequence
$\\{u_{n}\\}\subset W_{c}^{1,p}(\Omega)$ in $\Omega$. Let $v$ be a positive
supersolution of $Q_{p,A,V}(u)\\!=\\!0$ in $\Omega$, and consider the sequence
$\\{w_{n}\\!:=\\!u_{n}/v\\}$. Then $\\{w_{n}\\}$ is bounded in
$W_{\operatorname{loc}}^{1,p}(\Omega)$, and $\nabla w_{n}\rightarrow 0$ in
$L^{p}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d})$.

###### Proof.

Fix $K\Subset\Omega$ such that $\|u_{n}\|_{L^{p}(K)}=1$ for all
$n\in\mathbb{N}$, and let $K\Subset\omega\Subset\Omega$ be a Lipschitz
subdomain. By the Minkowski inequality we have

$\|w_{n}\|_{L^{p}(\omega)}\leq\|w_{n}-\langle
w_{n}\rangle_{K}\|_{L^{p}(\omega)}+\langle
w_{n}\rangle_{K}(\mathcal{L}^{d}(\omega))^{1/p},$

where $\langle w_{n}\rangle_{K}=\frac{1}{|K|}\int_{K}w_{n}(x)\,\mathrm{d}x$.
Applying the Poincaré inequality to the first term, and the weak Harnack
inequality (Theorem 3.5) to the second term of the right hand side of the
above inequality, we have

$\displaystyle\|w_{n}\|_{L^{p}(\omega)}$ $\displaystyle\leq
C(d,p,K,\omega)\|\nabla
w_{n}\|_{L^{p}(\omega;\mathbb{R}^{d})}+\frac{1}{\inf_{K}v}\langle
u_{n}\rangle_{K}(\mathcal{L}^{d}(\omega))^{1/p}$ $\displaystyle\leq
C(d,p,K,\omega)\|\nabla
w_{n}\|_{L^{p}(\omega;\mathbb{R}^{d})}+\frac{1}{\inf_{K}v}\left(\frac{\mathcal{L}^{d}(\omega)}{\mathcal{L}^{d}(K)}\right)^{1/p},$
(3.20)

where in the second term of the latter inequality, we used the Hölder
inequality, and $\|u_{n}\|_{L^{p}(K)}=1$. Let

$I(v,w_{k}):=\left\\{\begin{array}[]{ll}C(\bar{p})\displaystyle{\int_{\Omega}v^{\bar{p}}|\nabla
w_{k}|_{A}^{\bar{p}}\,\mathrm{d}x}&1\leq p<2,\\\\[8.53581pt]
C(p)\displaystyle{\int_{\Omega}v^{p}|\nabla w_{k}|_{A}^{p}\,\mathrm{d}x}&p\geq
2,\end{array}\right.$

where $\bar{p}\geq p$, and $C(p)$ and $C(\bar{p})$ are the constants in (3.2)
and (3.1), respectively. Using (3.2) and (3.1) to obtain

$\displaystyle I(v,w_{k})$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}|\nabla
u_{k}|_{A}^{p}\,\mathrm{d}x-\int_{\Omega}w_{k}^{p}|\nabla
v|_{A}^{p}\,\mathrm{d}x-\int_{\Omega}v|\nabla v|_{A}^{p-2}A\nabla
v\\!\cdot\\!\nabla(w_{k}^{p})\,\mathrm{d}x$ (3.21) $\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla
u_{k}|_{A}^{p}\,\mathrm{d}x-\int_{\Omega}|\nabla v|_{A}^{p-2}A\nabla
v\\!\cdot\\!\nabla(w_{k}^{p}v)\,\mathrm{d}x.$

Since $v$ is a positive supersolution, we get

$I(v,w_{k})\leq\int_{\Omega}|\nabla
u_{k}|_{A}^{p}\,\mathrm{d}x+\int_{\Omega}Vu_{k}^{p}\,\mathrm{d}x=Q_{A,p,V}[u_{k}].$
(3.22)

Using the definition of $I$, the weak Harnack inequality, (and for $p\leq 2$
use also Hölder inequality), we obtain from (3.22) that

$c_{\omega}\\!\int_{\omega}\\!|\nabla
w_{k}|^{p}\\!\,\mathrm{d}x\\!\leq\\!C(p)\int_{\Omega}v^{p}|\nabla
w_{k}|^{p}_{A}\\!\,\mathrm{d}x\\!\leq\\!\mathcal{Q}_{A,p,V}[u_{k}]\rightarrow
0\mbox{ as }k\rightarrow\infty,$ (3.23)

where $c_{\omega}>0$ is a positive constant. By the weak compactness of
$W^{1,p}(\omega)$, we infer that (up to a subsequence)

$\nabla w_{k}\rightarrow 0\qquad\mbox{in }L^{p}_{\rm
loc}(\Omega;\mathbb{R}^{d}).$ (3.24)

By (3.20) and (3.23), we have that $w_{k}$ is bounded in $W_{\rm
loc}^{1,p}(\omega)$. ∎

######  Remark 3.21.

In Lemma 3.20, for $p<2$ we may replace Assumption 3.2 with the assumption
that the gradient of the positive supersolution $v$ is locally bounded (see
the proof of Proposition 4.11 in [20]).

As a consequence of this lemma, we have the following uniqueness result:

######  Theorem 3.22.

Assume that $\mathcal{Q}_{p,A,V}\\!\geq\\!0$ in $\Omega$, where $A$ and $V$
satisfy Assumptions 3.1, and if $p<2$ assume further that Assumption 3.2 is
satisfied. Then any null-sequence of the functional $\mathcal{Q}_{p,A,V}$
converges, both in $L^{p}_{\operatorname{loc}}(\Omega)$ and a.e. in $\Omega$
to a unique (up to a multiplicative constant) positive supersolution (which is
in fact a solution) of the equation $Q_{p,A,V}(u)\\!=\\!0$ in $\Omega$.

###### Proof.

By the AAP theorem (Theorem 3.17), the equation $Q_{p,A,V}(u)=0$ in $\Omega$
admits a positive solution $v\in W_{\operatorname{loc}}^{1,p}(\Omega)$ and a
positive supersolution $\tilde{v}\in W_{\operatorname{loc}}^{1,p}(\Omega)$.
Suppose that $\\{u_{n}\\}$ is a null-sequence of the functional
$\mathcal{Q}_{p,A,V}$ in $\Omega$. Consider $w_{n}:=u_{n}/\tilde{v}$. Lemma
3.20 implies that $\nabla w_{n}\rightarrow 0$ in
$L^{p}_{\operatorname{loc}}(\Omega;\mathbb{R}^{d})$. Then the Rellich-
Kondrachov theorem implies that up to a subsequence (still denoted by
$\\{w_{n}\\}$), $w_{n}\rightarrow c$ in $W^{1,p}_{\operatorname{loc}}(\Omega)$
for a constant $c\geq 0$. Since $\tilde{v}$ is locally bounded and bounded
away from zero, $u_{n}\rightarrow c\tilde{v}$ a.e. in $\Omega$ and also in
$L^{p}_{\operatorname{loc}}(\Omega)$ (up to a subsequence). In particular,
$c=\|\tilde{v}\|_{L^{p}(K)}^{-1}>0$. It follows that any null-sequence
$\\{u_{n}\\}$ converges (up to a positive multiplicative constant) to the same
positive supersolution $\tilde{v}$. Since the positive solution $v$ is also a
positive supersolution, we conclude that $\tilde{v}=Cv$ for some $C>0$, and
thus $\tilde{v}$ is also the unique positive solution of the equation
$Q_{p,A,V}(u)=0$ in $\Omega$. ∎

######  Lemma 3.23.

Suppose that $A$ and $V$ satisfy Assumptions 3.1. Let $\omega\Subset\Omega$ be
a subdomain. Then, up to a multiplicative constant, the positive principal
eigenfunction of $\mathcal{Q}_{p,A,V}$ in $\omega$ is a ground state of
$\mathcal{Q}_{p,A,V-\Lambda_{1}}$ in $\omega$.

###### Proof.

Let $v_{1}$ be the positive principal eigenfunction of $\mathcal{Q}_{p,A,V}$
in $\omega$ with principal eigenvalue $\Lambda_{1}$. Fix $K\Subset\omega$ such
that $C_{K}:=\|v_{1}\|_{L^{p}(K)}>0$. Then the sequence
$\\{C^{-1}_{K}v_{1}\\}$ is a null-sequence and, $C^{-1}_{K}v_{1}$ is a ground
state of the functional $\mathcal{Q}_{p,A,V-\Lambda_{1}}$ in $\omega$. ∎

As a corollary of Theorem 3.22, we have:

######  Theorem 3.24.

Let $\omega\Subset\Omega$ be a subdomain. Assume that $A$ and $V$ satisfy
Assumptions 3.1, and if $p<2$ assume further that Assumption 3.2 is satisfied.
Then the following assertions are equivalent:

  * (i)

the principal eigenvalue $\Lambda_{1}>0$;

  * (ii)

the equation $Q_{p,A,V}(u)=0$ in $\omega$ admits a strict positive
supersolution in $W^{1,p}_{0}(\omega)$;

  * (iii)

the equation $Q_{p,A,V}(u)=0$ in $\omega$ admits a strict positive
supersolution in $W^{1,p}(\omega)$.

###### Proof.

$\text{ (i) }\Rightarrow\text{ (ii) }$: Assume that the principal eigenvalue
$\Lambda_{1}>0$ with a principal eigenvalue $v\in W^{1,p}_{0}(\omega)$. Then
by the strong maximum principle, $v$ is a strict positive supersolution of
$Q_{p,A,V}(u)=0$ in $\omega$.
$\text{ (ii) }\Rightarrow\text{ (iii) }$: This is immediate.
$\text{ (iii) }\Rightarrow\text{ (i) }$: Suppose that the equation
$Q_{p,A,V}(u)=0$ in $\omega$ admits a strict positive supersolution in
$W^{1,p}(\omega)$. By the AAP theorem (Theorem 3.17),
$\mathcal{Q}_{p,A,V}(\varphi)\geq 0$ for all $\varphi\in
C_{c}^{\infty}(\omega)$. This implies that the principal eigenvalue
$\Lambda_{1}\geq 0$. Now if $\Lambda_{1}=0$, then in light of Theorem 3.22 and
Lemma 3.23, the associated principal eigenfunction is the unique (up to a
multiplicative constant) positive supersolution of the equation
$Q_{p,A,V}(u)=0$ in $\omega$. This contradicts our assumption that the
equation $Q_{p,A,V}(u)=0$ in $\omega$ admits a strict positive supersolution
in $W^{1,p}(\omega)$. Therefore, $\Lambda_{1}>0$. ∎

###  3.7. Weak comparison principle

The first result in this subsection is a simple version of the weak comparison
principle between sub and supersolutions of the operator $Q_{p,A,V}$ when the
potential $V\in\mathfrak{W}_{\operatorname{loc}}^{p}(\Omega)$ is nonnegative.
We omit the proof since it is quite standard and follows exactly as in [20,
Lemma 5.1], where the potential is assumed to be in $M^{q}(p;\omega)$.

######  Lemma 3.25.

Let $1<p<d$, and $\omega\Subset\Omega$ be a Lipschitz subdomain. Suppose that
$A$ and $V$satisfy Assumptions 3.1,
$\mathcal{G}\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, and $V\geq 0$
in $\omega$. Let $v_{1}$, $v_{2}$ be respectively a subsolution and a
supersolution of the problem

$Q_{p,A,V}(u)=\mathcal{G}\quad\mbox{in }\omega.$

If $v_{1}\leq v_{2}$ a.e. on $\partial\omega$ in the trace sense, then
$v_{1}\leq v_{2}$ a.e. in $\omega$.

######  Theorem 3.26.

Let $1<p<d$ and $\omega\Subset\Omega$ be a Lipschitz subdomain. Suppose that
$A$ and $V$satisfy Assumptions 3.1 and
$g\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, where $g\geq 0$. Let
$f,u_{1},u_{2}\in W^{1,p}(\omega)\cap C(\bar{\omega})$, where $f\geq 0$ a.e.
in $\omega$ and

$\displaystyle Q_{p,A,V}(u_{1})$ $\displaystyle\leq g\leq
Q_{p,A,V}(u_{2})\quad\mbox{in }\omega,$ $\displaystyle u_{1}$
$\displaystyle\leq f\leq u_{2}\quad\mbox{on }\partial\omega,$ $\displaystyle
0$ $\displaystyle\leq u_{1}\leq u_{2}\quad\mbox{in }\omega.$

Then there exists a nonnegative solution $u\in W^{1,p}(\omega)\cap
C(\bar{\omega})$ of the Dirichlet problem

$\displaystyle\begin{cases}Q_{p,A,V}(u)=g\quad&\mbox{ in }\omega,\\\
u=f\quad&\mbox{ on }\partial\omega,\end{cases}$ (3.25)

such that $u_{1}\leq u\leq u_{2}$ a.e. in $\omega$. Moreover, if $f>0$ a.e. on
$\partial\omega$, then the solution $u$ of the Dirichlet problem (3.25) is
unique.

###### Proof.

Existence: Define the set

$S:=\\{v\in W^{1,p}(\omega)\cap C(\bar{\omega}):0\leq u_{1}\leq v\leq
u_{2}\,\,\mbox{in }\omega\\}.$

For $v\in S$, consider the function
$\mathcal{G}(x,v):=g(x)+2V^{-}(x)v^{p-1}(x)$, $x\in\omega$. Since $0\leq v\in
C(\bar{\omega})$, it follows that $\mathcal{G}\in\mathfrak{W}^{p}(\omega)$ and
also $\mathcal{G}\geq 0$ a.e. in $\omega$. For $v\in S$ consider the following
Dirichlet problem

$\displaystyle\begin{cases}Q_{p,A,|V|}(u)=\mathcal{G}(x,v)\quad&\mbox{ in
}\omega,\\\ u=f\qquad&\mbox{ on }\partial\omega.\end{cases}$ (3.26)

By Theorem 3.10, the Dirichlet problem (3.26) admits a solution
$u\\!\in\\!W^{1,p}(\omega)$. Moreover, the solution is unique. Indeed, if $u$
and $\tilde{u}$ are solutions of the above problem, then we have

$\displaystyle Q_{p,A,|V|}(u)=\mathcal{G}(x,v)=Q_{p,A,|V|}(\tilde{u})$
$\displaystyle\mbox{in }\omega,$ $\displaystyle u=f=\tilde{u}$
$\displaystyle\mbox{on }\partial\omega.$

By applying Lemma 3.25 with $u$ as a subsolution and $\tilde{u}$ as a
supersolution, we get $u\leq\tilde{u}$ in $\omega$. By interchanging the roles
of $u$ and $\tilde{u}$, it follows that $\tilde{u}\leq u$ in $\omega$. Hence,
$u=\tilde{u}$ in $\omega$.

Define a map $\psi:S\rightarrow W^{1,p}(\omega)$ by $\psi(v)=u$, where $u$ is
a solution of the problem (3.26). The mapping $\psi$ is well-defined in the
set $S$ and by using Lemma 3.25, it can be seen that the map $\psi$ is
monotone. Let $v\in W^{1,p}(\omega)\cap C(\bar{\omega})$ be a subsolution of
the Dirichlet problem (3.25). Then

$Q_{p,A,|V|}(v)=Q_{p,A,V}(v)+\mathcal{G}(x,v)-g(x)\leq\mathcal{G}(x,v)$

in the weak sense, and hence $v$ is a subsolution of the problem (3.26). Also,
by the definition of $\psi$, $\psi(v)$ is a solution of (3.26). By Lemma 3.25,
we have $v\leq\psi(v)$ a.e. in $\omega$. Moreover,

$\displaystyle Q_{p,A,V}(\psi(v))$
$\displaystyle=Q_{p,A,|V|}(\psi(v))-\mathcal{G}(x,\psi(v))+g(x)$
$\displaystyle=2V^{-}(|v|^{p-2}v-|\psi(v)|^{p-2}\psi(v))+g(x)\leq
g(x)\quad\mbox{in }\,\omega,$

in the weak sense. This shows that if $v$ is a subsolution of the Dirichlet
(3.25), then $\psi(v)$ is also a subsolution of (3.25) with $v\leq\psi(v)$
a.e. in $\omega$. Similarly, it can be seen that if $v\in W^{1,p}(\omega)\cap
C(\bar{\omega})$ be a supersolution of (3.25), then $\psi(v)$ also a
supersolution of (3.25) with $\psi(v)\leq v$ a.e. in $\omega$.
Now define the following two sequences

$\underaccent{\bar}{v}_{0}:=u_{1},\quad\underaccent{\bar}{v}_{n}:=\psi(\underaccent{\bar}{v}_{n-1})\quad\mbox{
and }\quad\bar{v}_{0}:=u_{2},\quad\bar{v}_{n}:=\psi(\bar{v}_{n-1}).$

From the above arguments, the sequences
$\underaccent{\bar}{v}_{n}\nearrow\underaccent{\bar}{v}$ and
$\bar{v}_{n}\searrow\bar{v}$ for every $x\in\omega$. Also due to Brezis-Lieb
Lemma[14, Theorem 1.9], both the sequences converges in $L^{p}(\omega)$. By
using a similar argument as in Theorem 3.8, it can be seen that
$\underaccent{\bar}{v}$ and $\bar{v}$ are fixed point of $\psi$ and both are
solutions of the Dirichlet problem (3.25) with
$u_{1}\leq\underaccent{\bar}{v}\leq\bar{v}\leq u_{2}$ in $\omega$.

Uniqueness: Let $v_{1},v_{2}\in W^{1,p}(\omega)\cap C(\bar{\omega})$ be
positive solutions of (3.25) with $f>0$ a.e. on $\partial\omega$. Then using
$\varphi_{1}=\frac{v_{1}^{p}-v_{2}^{p}}{v_{1}^{p-1}}\in W_{0}^{1,p}(\omega)$
as a test function in the definition of $v_{1}$ being a solution of (3.25), we
have

$\displaystyle\int_{\omega}(v_{1}^{p}-v_{2}^{p})|\nabla\log
v_{1}|^{p}_{A}\,\mathrm{d}x$ $\displaystyle-p\int_{\omega}v_{2}^{p}|\nabla\log
v_{1}|^{p-2}_{A}A\nabla\log
v_{1}\cdot\nabla\log(\frac{v_{2}}{v_{1}})\,\mathrm{d}x$
$\displaystyle=\int_{\omega}\frac{(g-Vv_{1}^{p-1})(v_{1}^{p}-v_{2}^{p})}{v_{1}^{p-1}}.$

Similarly, using $\varphi_{2}=\frac{v_{2}^{p}-v_{1}^{p}}{v_{2}^{p-1}}\in
W_{0}^{1,p}(\omega)$ as a test function in the definition of $v_{2}$ being a
solution of the Dirichlet (3.25), we get

$\displaystyle\int_{\omega}(v_{2}^{p}-v_{1}^{p})|\nabla\log
v_{2}|^{p}_{A}\,\mathrm{d}x$ $\displaystyle-p\int_{\omega}v_{1}^{p}|\nabla\log
v_{2}|^{p-2}_{A}A\nabla\log
v_{2}\cdot\nabla\log(\frac{v_{1}}{v_{2}})\,\mathrm{d}x$
$\displaystyle=\int_{\omega}\frac{(g-Vv_{2}^{p-1})(v_{2}^{p}-v_{1}^{p})}{v_{2}^{p-1}}.$

Adding the above two equations, we reach at

$\displaystyle\int_{\omega}v_{1}^{p}\left(|\nabla\log
v_{1}|^{p}_{A}-|\nabla\log v_{2}|^{p}_{A}-p|\nabla\log
v_{2}|^{p-2}_{A}A\nabla\log
v_{2}\cdot\nabla\log(\frac{v_{1}}{v_{2}})\right)\,\mathrm{d}x$
$\displaystyle+\int_{\omega}v_{2}^{p}\left(|\nabla\log
v_{2}|^{p}_{A}-|\nabla\log v_{1}|^{p}_{A}-p\nabla\log
v_{1}|^{p-2}_{A}A\nabla\log
v_{1}\cdot\nabla\log(\frac{v_{2}}{v_{1}})\right)\,\mathrm{d}x$
$\displaystyle\,\,\,\,\,=\int_{\omega}g\left(\frac{1}{v_{1}^{p-1}}-\frac{1}{v_{2}^{p-1}}\right)(v_{1}^{p}-v_{2}^{p})\,\mathrm{d}x\leq
0$

Now,(3.1) implies that

$\displaystyle
0\\!\geq\\!C(p)\\!\\!\\!\int_{\omega}\\!\\!\\!\,\mathrm{d}x\\!\begin{cases*}(v_{1}^{p}+v_{2}^{p})|\nabla\log(\frac{v_{1}}{v_{2}})|_{A}^{p}\qquad\mbox{
if }p\geq 2,\\\\[5.69054pt]
\\!(v_{1}^{p}\\!+\\!v_{2}^{p})|\nabla\log(\frac{v_{1}}{v_{2}})|_{A}^{2}\left(|\nabla\log
v_{1}|_{A}\\!+\\!|\nabla\log v_{2}|_{A}\right)^{p-2}\mbox{ if
}1\\!<p\\!<\\!2.\end{cases*}$

Consequently, $v_{2}\nabla v_{1}=v_{1}\nabla v_{2}$ a.e. in $\omega$. Hence,
there is a constant $c$ such that $v_{1}=cv_{2}$ a.e. in $\omega$. Since
$v_{1}=v_{2}=f$ on $\partial\omega$, we infer that $v_{1}=v_{2}$ in
$\bar{\omega}$. Therefore, the Dirichlet problem (3.25) admits a unique
solution. ∎

######  Theorem 3.27 (Weak Comparison Principle).

Let $\omega\Subset\Omega$. Assume that $A$ and $V$ satisfy Assumptions 3.1,
and $\mathcal{Q}_{p,A,V}\geq 0$ in $\Omega$. Suppose that $v_{1},v_{2}\in
W^{1,p}(\omega)\cap C(\bar{\omega})$ satisfy the following inequalities

$\begin{cases*}Q_{p,A,V}(v_{2})=g\,\,\text{in}\,\,\omega,\\\
v_{2}>0\,\,\,\,\,\text{on}\,\,\partial\omega,\end{cases*}\quad\mbox{and}\quad\begin{cases*}Q_{p,A,V}(v_{1})\leq
Q_{p,A,V}(v_{2})\,\,\text{in}\,\,\omega,\\\ v_{1}\leq
v_{2}\,\,\,\,\,\text{on}\,\,\partial\omega.\end{cases*}$

where $g\in\mathfrak{W}^{p}(\omega)$ with $g\geq 0$ a.e. in $\omega$. Then
$v_{1}\leq v_{2}$ a.e. in $\omega$.

###### Proof.

By the assumption on $v_{2}$, we have that $v_{2}$ is a supersolution of
equation (1.1) in $\omega$ and $v_{2}>0$ on $\partial\omega$. Hence, by the
strong maximum principle we have $v_{2}>0$ in $\bar{\omega}$. Define
$c:=\max\\{1,\frac{\max_{\bar{\omega}}v_{1}}{\min_{\bar{\omega}}v_{2}}\\}$.
Then $v_{1}\leq cv_{2}$ in $\bar{\omega}$. Now consider the following boundary
value problem

$\begin{cases*}Q_{p,A,V}(u)=g\,\,\text{in}\,\,\omega,\\\
u=v_{2}\,\,\,\,\,\text{on}\,\,\partial\omega.\end{cases*}$ (3.27)

By our assumption and the choice of the constant $c$, it follows that $cv_{2}$
is a supersolution of the problem (3.27) with $v_{1}\leq v_{2}\leq cv_{2}$ on
$\partial\omega$. Also note that $v_{1}$ is a subsolution of the problem
(3.27). By applying Theorem 3.26 with $u_{1}=v_{1}$ and $u_{2}=cv_{2}$, we
have a solution $v$ of (3.27) such that $v_{1}\leq v\leq cv_{2}$ in $\omega$
with $v=v_{2}$ on $\partial\omega$. Again by the strong maximum principle, we
have $v>0$ in $\bar{\omega}$. Now, the uniqueness of the boundary value
problem (3.27) (Theorem 3.26) implies that $v=v_{2}$. Therefore, $v_{1}\leq
v_{2}$ a.e. in $\omega$. ∎

##  4\. Fuchsian-type singularity and positive Liouville-type theorem

The present section is devoted to the study of positive Liouville-type
theorems, Picard-type principles, and removable singularity theorems for the
equation

$Q_{p,A,V}(u)=-\Delta_{p,A}(u)+V|u|^{p-2}u=0\quad\mbox{in }\Omega,$ (4.1)

where the potential $V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ has an
isolated Fuchsian-type singularity at $\zeta$ which belongs to the (ideal)
boundary of $\Omega$. Since we permit the domain $\Omega$ to be unbounded and
the singular point $\zeta=\infty$, it is worthy to consider the one-point
compactification $\hat{\mathbb{R}}^{d}:=\mathbb{R}^{d}\cup\\{\infty\\}$ of
$\mathbb{R}^{d}$. We denote by $\hat{\Omega}$ the closure of $\Omega$ in
$\hat{\mathbb{R}}^{d}$.

Throughout this section, we assume that $\mathcal{Q}_{p,A,V}\geq 0$ in
$\Omega$. Note that under this assumption the weak comparison principle in
bounded subdomains is valid when $A$ and $V$ satisfy Assumptions 3.1.

For $R>0$, we denote by $\mathcal{A}_{R}$ the annuls given by

$\mathcal{A}_{R}:=\\{x\in\mathbb{R}^{d}\mid{R}/{2}\leq|x|<{3R}/{2}\\}.$

The dilated domain $\Omega/R$ of $\Omega$ is defined by

$\Omega/R:=\\{x\in\mathbb{R}^{d}\,|\,x=R^{-1}y,\,\text{where}\,y\in\Omega\\}.$

We assume that the singular point $\zeta\in\partial\hat{\Omega}$ is an
isolated component of $\partial\hat{\Omega}$ and $\zeta$ is either equal to
$0$ or $\infty$. We are interested in the behavior of positive solutions of
(4.1) near $\zeta$. In fact, we may assume that $\Omega$ is one of the
followings domains: $B_{r}\setminus\\{0\\}$, $\mathbb{R}^{d}\setminus B_{r}$
for some $r>0$, $\mathbb{R}^{d}$ and $\mathbb{R}^{d}\setminus\\{0\\}$.

With some abuse of notation, we write

$x\to\zeta\;\;\mbox{if}\;\;\begin{cases}x\to 0&\mbox{in }\mathbb{R}\,\mbox{
and }\zeta=0,\\\ x\to\infty&\mbox{in }\mathbb{R}\,\mbox{ and
}\zeta=\infty.\end{cases}$

######  Definition 4.1 (Fuchsian-type singularity).

Let $1<p<d$, and $A$ and $V$ satisfy Assumptions 3.1. Let
$\zeta\in\\{0,\infty\\}$ be an isolated point of $\partial\hat{\Omega}$. Then
we say that the operator $Q_{p,A,V}$ has a Fuchsian-type singularity at
$\zeta$ if there exists a punctured neighbourhood
$\Omega^{\prime}\subset\Omega$ of $\zeta$ such that

  * (i)

The matrix $A$ is bounded and uniformly elliptic in $\Omega^{\prime}$, i.e.,
there exists a constant $C>0$ such that

$C^{-1}|\xi|^{2}\leq|\xi|_{A(x)}^{2}\leq C|\xi|^{2}\qquad\forall
x\in\Omega^{\prime}\mbox{ and }\xi\in\mathbb{R}^{d}.$ (4.2)

  * (ii)

There exists a positive constant $C$ and $R_{0}>0$ such that

$\|V\|_{\mathfrak{W}^{p}(\mathcal{A}_{R}\cap\Omega^{\prime})}\leq C$ (4.3)

for all $0<R<R_{0}$ if $\zeta=0$, and $R>R_{0}$ if $\zeta=\infty$.

A set $\mathcal{A}\subset\Omega$ is said to be an essential set with respect
to the singular point $\zeta$ if there exist $a\in(0,1)$, $b\in(1,\infty)$,
and a sequence $\\{R_{n}\\}$ of positive numbers converging to $\zeta$ such
that

$\mathcal{A}=\bigcup_{n=1}^{\infty}\mathcal{A}_{n},\quad\text{where}\,\,\mathcal{A}_{n}=\\{x\in\Omega:aR_{n}<|x|<bR_{n}\\}.$

Similar to the linear case as in [19] or for the quasi-linear cases as in [8,
11], it is enough to assume that inequalities (4.2) and (4.3) hold only on an
essential set $\mathcal{A}$ with respect to the singular point $\zeta$, i.e.,
there exists a constant $C>0$ (independent of $n$) such that

$\mathrm{C}^{-1}|\xi|^{2}\leq|\xi|_{A(x)}^{2}\leq C|\xi|^{2}\quad\forall
x\in\mathcal{A}\mbox{ and }\xi\in\mathbb{R}^{d},\mbox{ and
}\quad\|V\|_{\mathfrak{W}^{p}(\mathcal{A}_{n})}\leq C,$

for some essential set $\mathcal{A}=\cup_{n=1}^{\infty}\mathcal{A}_{n}$ of
$\zeta$.

######  Example 4.2.

Let $1<p<d$, and $\Omega=\mathbb{R}^{d}\setminus\\{0\\}$. Consider the
$(p,A)$-Laplacian with the Hardy potential $V(x)=\lambda|x|^{-p}$, and the
equation

$-\Delta_{p,A}(u)-\frac{\lambda}{|x|^{p}}|v|^{p-2}v=0\quad\mbox{in }\Omega,$
(4.4)

where the matrix $A$ satisfies (4.2) outside a compact set in $\Omega$. We
assert that equation (4.4) has Fuchsian-type singularity both at $\zeta=0$ and
$\zeta=\infty$.
Let us check that (4.3) is satisfied near $\zeta=0$. It is enough to show that
for sufficiently small $R>0$, (4.3) is fulfilled over the annular set
$\mathcal{A}_{R}=\\{x\in\Omega:R/2\leq|x|<3R/2\\}$. Indeed,

$\displaystyle\|V\|^{\frac{1}{p-1}}_{\mathfrak{W}^{p}(\mathcal{A}_{R})}$
$\displaystyle=\sup_{x\in\mathcal{A}_{R}}\int_{0}^{\operatorname{diam}(\mathcal{A}_{R})}\frac{1}{r^{\frac{d-1}{p-1}}}\left[\int_{\mathcal{A}_{R}\cap
B_{r}(x)}|y|^{-p}\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\>\\!\mathrm{d}r$
$\displaystyle\leq
C\sup_{x\in\mathcal{A}_{R}}\frac{1}{R^{\frac{p}{p-1}}}\int_{0}^{3R}\frac{1}{r^{\frac{d-1}{p-1}}}r^{\frac{d}{p-1}}\>\\!\mathrm{d}r\leq
C\frac{1}{R^{\frac{p}{p-1}}}R^{\frac{p}{p-1}}=C<\infty.$

Note that the constant $C>0$ is independent of $R$. Hence, (4.4) has a
Fuchsian-type singularity at $\zeta=0$. The case $\zeta=\infty$ follows
similarly for large enough $R>0$.

Next, we present a dilation process using the quasi-invariance of (4.1) under
the scaling $x\mapsto Rx$, where $R>0$. For $R>0$, let $A_{R}$ and $V_{R}$ be
the scaled matrix and the scaled potential defined by

$A_{R}(x):=A(Rx),\quad V_{R}(x):=R^{p}V(Rx)\qquad x\in\Omega/R.$

If $u$ is a solution of (1.1), then for any $R>0$, the function
$u_{R}(x):=u(Rx)$ is a solution of the equation

$-\Delta_{p,A_{R}}(u_{R})+V_{R}(x)|u_{R}|^{p-2}u_{R}=0\quad\mbox{in
}\Omega/R.$

Now take an annular subset
$\mathcal{A}_{R}:=(B_{3R/2}\setminus\bar{B}_{R/2})\cap\Omega$. Since $\zeta$
is an isolated singular point of $\partial\hat{\Omega}$, it follows that for
$R$ ‘near’ $\zeta$, the set $\mathcal{A}_{R}/R$ is a fixed annular set
$\tilde{\mathcal{A}}$. Moreover, for such $R$ we have

$\displaystyle\|V_{R}\|_{\mathfrak{W}^{p}(\mathcal{A}_{R}/R)}^{\frac{1}{p-1}}=\sup_{x\in\mathcal{A}_{R}/R}\int_{0}^{\operatorname{diam}(\mathcal{A}_{R}/R)}\frac{1}{r^{\frac{d-1}{p-1}}}\left[\int_{\mathcal{A}_{R}/R\cap
B_{r}(x)}\\!\\!\\!R^{p}|V(Ry)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\\!\\!\\!\\!\\!\>\\!\mathrm{d}r$
$\displaystyle=\sup_{x\in\mathcal{A}_{R}/R}\int_{0}^{\operatorname{diam}(\mathcal{A}_{R}/R)}\frac{R^{\frac{p-d}{p-1}}}{r^{\frac{d-1}{p-1}}}\left[\int_{\mathcal{A}_{R}\cap
B_{Rr}(Rx)}|V(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\\!\\!\\!\\!\>\\!\mathrm{d}r$
$\displaystyle=\sup_{Rx\in\mathcal{A}_{R}}\int_{0}^{\operatorname{diam}(\mathcal{A}_{R})}\frac{R^{\frac{p-d}{p-1}}\cdot
R^{\frac{d-1}{p-1}}}{(Rr)^{\frac{d-1}{p-1}}}\left[\int_{\mathcal{A}_{R}\cap
B_{Rr}(Rx)}|V(y)|\>\\!\mathrm{d}y\right]^{\frac{1}{p-1}}\\!\\!\\!\\!\>\\!\mathrm{d}r$
$\displaystyle=\\!\\!\sup_{Rx\in\mathcal{A}_{R}}\int_{0}^{\operatorname{diam}(\mathcal{A}_{R})}\\!\\!\\!\\!\\!\frac{1}{(Rr)^{\frac{d-1}{p-1}}}\\!\left[\\!\int_{\mathcal{A}_{R}\cap
B_{Rr}(Rx)}\\!\\!\\!|\\!V(y)|\>\\!\mathrm{d}y\\!\right]^{\frac{1}{p-1}}\\!\\!\\!\\!\>\\!\mathrm{d}(Rr)\\!=\\!\|V\|_{\mathfrak{W}^{p}(\mathcal{A}_{R})}^{\frac{1}{p-1}}.$

Thus, if $Q_{p,A,V}$ has a Fuchsian-type singularity at $\zeta$, then

$\|V_{R}\|_{\mathfrak{W}^{p}(\tilde{\mathcal{A}})}=\|V_{R}\|_{\mathfrak{W}^{p}(\mathcal{A}_{R}/R)}=\|V\|_{\mathfrak{W}^{p}(\mathcal{A}_{R})}\leq
C,$ (4.5)

where $C>0$ is a positive constant independent of $R$.

The limiting dilation process is defined as follows. Let $\\{R_{n}\\}$ be a
sequence of positive numbers satisfying $R_{n}\rightarrow\zeta$, and set
$Y:=\lim_{n\to\infty}\Omega/{R_{n}}$. Clearly,
$Y=(\mathbb{R}^{d})^{*}=\mathbb{R}^{d}\setminus\\{0\\}$. Let $\zeta=0$ or
$\zeta=\infty$, and assume that

$\displaystyle A_{R_{n}}\longrightarrow\mathbb{A}\quad\mbox{in the weak
topology of }L^{\infty}_{\operatorname{loc}}(Y,\mathbb{R}^{d\times
d}),\mbox{and }$ (4.6)
$\displaystyle\begin{cases}V_{R_{n}}\longrightarrow\mathbb{V}\quad\mbox{in the
weak toplogy of }\mathfrak{W}^{p}_{\operatorname{loc}}(Y)\mbox{ if }1<p\leq
2,\\\ V_{R_{n}}\longrightarrow\mathbb{V}\quad\mbox{with resect to the
quasinorm of }\mathfrak{W}^{p}_{\operatorname{loc}}(Y)\mbox{ if
}p>2,\end{cases}$ (4.7)

as $n\rightarrow\infty$. Define the limiting dilated equation with respect to
(4.1) and the sequence $\\{R_{n}\\}$ (which satisfies the conditions (4.6) and
(4.7)) by

$\displaystyle\mathcal{D}^{\\{R_{n}\\}}(Q)(w)=\mathcal{D}^{\\{R_{n}\\}}(Q_{p,A,V})(w):=Q_{p,\mathbb{A},\mathbb{V}}(w)$
$\displaystyle=-\Delta_{p,\mathbb{A}}(w)+\mathbb{V}|w|^{p-2}w=0\quad\mbox{on
}Y.$ (4.8)

Next, we show that the property of having a Fuchsian-type singularity at
$\zeta$ is preserved under the limiting dilation process
$Q_{p,A,V}\mapsto\mathcal{D}^{\\{R_{n}\\}}(Q)$.

######  Lemma 4.3.

Let $1<p<d$ and assume that $A$ and $V$ satisfy Assumptions 3.1. Suppose that
(4.1) has a Fuchsian-type singularity at
$\zeta\in\\{0,\infty\\}\subset\partial\hat{\Omega}$. Then for any sequence
$R_{n}\rightarrow\zeta$ which satisfies conditions (4.6) and (4.7), the
corresponding limiting dilated equation $\mathcal{D}^{\\{R_{n}\\}}(Q)(w)=0$ in
$Y$ has Fuchsian-type singularity at $\zeta$.

###### Proof.

By our assumption, there exists an essential set
$\mathcal{A}=\cup_{n=1}^{\infty}\mathcal{A}_{n}$, where
$\mathcal{A}_{n}=\\{x\in\Omega:aR_{n}<|x|<bR_{n}\\}$ and a constant $C>0$ such
that

$\|V\|_{\mathfrak{W}^{p}(\mathcal{A}_{n})}\leq
C,\quad\forall\,n\in\mathbb{N}.$

We claim that

$\|\mathbb{V}\|_{\mathfrak{W}^{p}(\mathcal{A}_{n})}\leq
C,\quad\forall\,n\in\mathbb{N}.$

Since for each $n$, $\mathcal{A}_{n}/R_{n}$ is a fixed annular set
$\tilde{\mathcal{A}}=\\{x\in\Omega:a<|x|<b\\}$, we have for any fixed $k$
sufficiently large:

$\displaystyle\\!\|\mathbb{V}\|_{\mathfrak{W}^{p}(\mathcal{A}/R_{k})}=\|\mathbb{V}\|_{\mathfrak{W}^{p}(\tilde{\mathcal{A}})}\\!\leq\underset{n\rightarrow\infty}{\lim\inf}\|V_{R_{n}}\|_{\mathfrak{W}^{p}(\mathcal{A}_{n}/R_{n})}\\!\\!=\\!\underset{n\rightarrow\infty}{\lim\inf}\|V\|_{\mathfrak{W}^{p}(\mathcal{A}_{n})}\\!\\!\leq\\!\\!C.$

where $C>0$ is independent on $k$, and the weak convergence of
$\\{V_{R_{n}}\\}$ is used for $1<p\leq 2$, while for $p>2$, a Fatou type lemma
on quasinormed space follows from [4, Lemma 3.3 and Lemma 3.5]. Similarly,
$\mathrm{C}^{-1}|\xi|^{2}\leq|\xi|_{\mathbb{A}(x)}^{2}\leq C|\xi|^{2}$ for all
$x\in\mathcal{A}/R_{k}$, $k\geq k_{0}$. Therefore, the lemma follows. ∎

Let us now recall the notion of a regular point (see [8, 9, 11]) of the
operator $Q_{p,A,V}$ which turns out to be closely related to positive
Liouville-type theorems for (4.1).

######  Definition 4.4.

Let $\mathcal{G}_{\zeta}$ be the germs of all positive solutions $u$ of the
equation $Q_{p,A,V}(w)=0$ in relative punctured neighbourhoods of
$\zeta\in\\{0,\infty\\}$. The singular point $\zeta$ is said to be a regular
point of the equation $Q_{p,A,V}(w)=0$ if for any two positive solutions
$u,v\in\mathcal{G}_{\zeta}$

$\underset{x\in\Omega^{\prime}}{\lim_{x\rightarrow\zeta}}\frac{u(x)}{v(x)}\,\,\,\mbox{
exists in the generalized sense},$

where $\Omega^{\prime}$ is a relative punctured neighbourhood of $\zeta$.

The following result states that for any $1<p<\infty$ and $d\geq 2$, the
singular point $\zeta$ is a regular point of the equation
$-\Delta_{p,\mathbb{A}}(w)=0$ in $\mathbb{R}^{d}\setminus\\{0\\}$, where
$\mathbb{A}\in\mathbb{R}^{d\times d}$ is a fixed symmetric and positive
definite matrix and $\zeta$ is either $0$ or $\infty$. For a proof see [11,
Theorem 4.1].

######  Theorem 4.5.

Assume that $1<p<\infty$ and $d\geq 2$. Let $\mathbb{A}\in\mathbb{R}^{d\times
d}$ be a fixed symmetric and positive definite matrix. Then

  * (i)

for $p\leq d$, $\zeta=0$ is a regular point of $-\Delta_{p,\mathbb{A}}(w)=0$
in $\mathbb{R}^{d}\setminus{\\{0\\}}$.

  * (ii)

for $p\geq d$, $\zeta=\infty$ is a regular point of
$-\Delta_{p,\mathbb{A}}(w)=0$ in $\mathbb{R}^{d}$.

Next we recall the notion of positive solutions of minimal growth [8, 21, 23,
11].

######  Definition 4.6.

(1) Let $K\Subset\Omega$. A positive solution $u$ of (1.1) in $\Omega\setminus
K$ is said to be a positive solution of minimal growth in a neighbourhood of
infinity in $\Omega$ if for any $K^{\prime}\Subset\Omega$ with smooth boundary
such that $K\Subset\mbox{int}\,K^{\prime}$, and any positive supersolution
$v\in C(\Omega\setminus\mbox{int}\,K^{\prime})$ of (1.1) in $\Omega\setminus
K^{\prime}$, we have

$u\leq v\quad\mbox{ on }\partial K^{\prime}\quad\Rightarrow\quad u\leq
v\quad\mbox{ in }\Omega\setminus K^{\prime}.$

(2) A positive solution of (1.1) in $\Omega$ which has minimal growth in a
neighbourhood of infinity in $\Omega$ is called a minimal positive solution of
(1.1) in $\Omega$.

(3) Let $\zeta\in\partial\hat{\Omega}$ and let $u$ be a positive solution of
(1.1) in $\Omega$. Then $u$ is said to be a positive solution of minimal
growth in a neighbourhood of $\partial\hat{\Omega}\setminus\\{\zeta\\}$ if for
any relative punctured neighbourhood $K\Subset\hat{\Omega}$ of $\zeta$ such
that $\Gamma:=\partial K\cap\Omega$ is smooth, and any positive supersolution
$v\in C((\Omega\setminus K)\cup\Gamma)$ of (1.1) in $\Omega\setminus K$, we
have

$u\leq v\quad\mbox{ on }\Gamma\quad\Rightarrow\quad u\leq v\quad\mbox{ in
}\Omega\setminus K.$

Next we discuss the existence of a positive solution of minimal growth of
$Q(u)=0$ in a neighbourhood of $\partial\hat{\Omega}\setminus\\{\zeta\\}$,
where $\zeta\in\\{0,\infty\\}$ is an isolated singular point.

######  Theorem 4.7.

Let $1<p<d$. Suppose that $\mathcal{Q}_{p,A,V}\geq$ in $\Omega$, where $A,V$
satisfy Assumptions 3.1. Assume that the operator $Q_{p,A,V}$ has an isolated
Fuchsian-type singularity at $\zeta\in\partial\hat{\Omega}$, where
$\zeta\in\\{0,\infty\\}$. Then the equation $Q_{p,A,V}(u)=0$ in $\Omega$
admits a positive solution of minimal growth in a neighbourhood of
$\partial\hat{\Omega}\setminus\\{\zeta\\}$.

###### Proof.

The proof follows the same steps as in [20, Theorem 5.7] and [11, Proposition
3.9], in which $V$ is assumed to be in $M^{q}_{\operatorname{loc}}(p;\Omega)$.
∎

######  Remark 4.8.

It turns out that $\Phi$ is a ground state of (1.1) in $\Omega$ if and only if
$\Phi$ is a positive minimal positive solution (see, [20, Theorem 5.9]).

The next lemma concerns a monotonicity property for the quotient of two
positive solutions near the isolated singular point $\zeta\in\\{0,\infty\\}$.

######  Lemma 4.9.

Let $1<p<d$, and $A$, $V$ satisfy Assumptions 3.1. Assume that
$u,v\in\mathcal{G}_{\zeta}$ are defined in a punctured neighbourhood
$\Omega^{\prime}$ of $\zeta\in\\{0,\infty\\}$. For $r>0$, denote $\partial
E_{A}(r):=\\{x\in\mathbb{R}^{d}:|x|_{A^{-1}}=r\\}$ where
$A\in\mathbb{R}^{d\times d}$ is a symmetric, positive definite matrix (in
particular if $A=I$ then $\partial E_{A}(r)=S_{r}$). Define

$m_{r}:=\underset{\partial
E_{A}(r)\cap\Omega^{\prime}}{\inf}\,\frac{u(x)}{v(x)}\,,\hskip
14.22636ptM_{r}:=\underset{\partial
E_{A}(r)\cap\Omega^{\prime}}{\sup}\,\frac{u(x)}{v(x)}\,.$ (4.9)

The following assertions hold:

  * (i)

The functions $m_{r}$ and $M_{r}$ are finally monotone as $r\rightarrow\zeta$.
In particular, there are numbers $0\leq m\leq M\leq\infty$ such that

$m:=\underset{r\rightarrow\zeta}{\lim}\,m_{r},\quad\mbox{and}\quad
M:=\underset{r\rightarrow\zeta}{\lim}\,M_{r}.$ (4.10)

  * (ii)

Suppose further that $u$ and $v$ are both positive solutions of (4.1) in
$\Omega$ of minimal growth in $\partial\hat{\Omega}\setminus\\{\zeta\\}$, then
$0<m\leq M<\infty$ and $m_{r}\searrow m$ and $M_{r}\nearrow M$ when
$r\rightarrow\zeta$.

The proof of Lemma 4.9 is similar to [8, Lemma 4.2] for the case when $A=I$
and $V\in L^{\infty}_{\operatorname{loc}}(\Omega)$. We also refer to [11,
Lemma 3.14] in which $V\in M^{q}_{\operatorname{loc}}(p,\Omega)$. Part (ii) of
Lemma 4.9 implies the following result.

######  Corollary 4.10.

Suppose that operator $Q_{p,A,V}$ has a Fuchsian-type isolated singularity at
$\zeta\in\\{0,\infty\\}$. Let $u,v$ be two positive solutions of (4.1) of
minimal growth in a neighbourhood of
$\partial\hat{\Omega}\setminus\\{\zeta\\}$. Then

$mv(x)\leq u(x)\leq Mv(x)\quad x\in\Omega,$

where $0<m\leq M<\infty$ are defined in (4.10).

As in [8, 11], where $V\in L_{\operatorname{loc}}^{\infty}(\Omega)$ or
$V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$, the regularity at
$\zeta\in\\{0,\infty\\}$ imply a Picard-type principle and a positive
Liouville-type theorem, respectively.

######  Proposition 4.11.

Suppose that the operator $Q_{p,A,V}$ has and an isolated Fuchsian-type
singularity at $\zeta\in\\{0,\infty\\}$ which is regular. Then (4.1) admits a
unique (up to a multiplicative constant) positive solution in $\Omega$ of
minimal growth in a neighbourhood of
$\partial\hat{\Omega}\setminus\\{\zeta\\}$.

###### Proof.

Existence: Follows from Theorem 4.7.

Uniqueness: Let $u$ and $v$ be two solutions of (4.1) of minimal growth in a
neighbourhood of $\partial\hat{\Omega}\setminus\\{\zeta\\}$. Then Corollary
4.10 implies

$mv(x)\leq u(x)\leq Mv(x)\quad x\in\Omega,$

where $0<m\leq M<\infty$ are defined in (4.9) and (4.10). In addition, since
$\zeta$ is a regular point, it follows that

$\underset{x\in\Omega}{\displaystyle{\lim_{x\rightarrow\zeta}}}\,\frac{u(x)}{v(x)}\quad\mbox{
exists and is positive}.$

Therefore, we get $m=M$ and hence $u(x)=Mv(x)$. ∎

Let $\mathcal{A}_{r}:=(B_{3r/2}\setminus\bar{B}_{r/2})\cap\Omega^{\prime}$ be
an annular set, where $\Omega^{\prime}$ is a punctured neighbourhood of
$\zeta$. For $u,v\in\mathcal{G}_{\zeta}$, denote

$\mathbf{a}_{r}:=\underset{x\in\mathcal{A}_{r}}{\inf}\frac{u(x)}{v(x)}\,,\hskip
28.45274pt\mathbf{A}_{r}:=\underset{x\in\mathcal{A}_{r}}{\sup}\frac{u(x)}{v(x)}\,.$

The local Harnack inequality (Theorem 3.4) implies that there exists $r_{0}>0$
such that $0<\mathbf{A}_{r}\leq C_{r}\mathbf{a}_{r}<\infty$ for every
$0<r<r_{0}$. For a Fuchsian-type singularity of (4.1) at
$\zeta\in\\{0,\infty\\}$ we have the following uniform Harnack inequality in
annular sets $\mathcal{A}_{r}$ for $r$ near $\zeta$.

######  Theorem 4.12 (Uniform Harnack inequality).

Let $1<p<d$. Suppose that $A$ and $V$ satisfy Assumptions 3.1. Further assume
that $Q_{p,A,V}$ has an isolated Fuchsian-type singularity at
$\zeta\in\\{0,\infty\\}$. Let $u,v\in\mathcal{G}_{\zeta}$. Then there exists
positive constant $C$ independent of $r$, $u$ and $v$ such that

$\mathbf{A}_{r}\leq C\mathbf{a}_{r}\qquad\forall\,r\mbox{ near }\zeta.$

###### Proof.

Let $\Omega^{\prime}\subset\Omega$ be a fixed punctured neighbourhood of
$\zeta$ and $u,v\in\mathcal{G}_{\zeta}$ be two positive solutions in
$\Omega^{\prime}$. For $r>0$, consider the annular set
$\tilde{\mathcal{A}}_{r}:=(B_{2r}\setminus\bar{B}_{r/4})\cap\Omega^{\prime}$.
By our assumption, $\zeta=0$ (or $\zeta=\infty$) is an isolated singular
point. Therefore, for $r<r_{0}$ (respectively, $r>r_{0}$)
$\mathcal{A}:=\mathcal{A}_{r}/r$ and
$\tilde{\mathcal{A}}:=\tilde{\mathcal{A}}_{r}/r$ are fixed annulus with
$\mathcal{A}\Subset\tilde{\mathcal{A}}$.

For such $r$, Let $u_{r}(x):=u(rx)$, $v_{r}(x):=v(rx)$, where
$x\in\Omega^{\prime}/r$. Then the functions $u_{r}$ and $v_{r}$ are positive
solution of the equation

$Q_{r}[u_{r}]:=-\Delta_{p,A_{r}}(u_{r})+V_{r}(x)|u_{r}|^{p-2}u_{r}=0\,\,\text{in}\,\,\tilde{\mathcal{A}},$

where $A_{r}(x):=A(rx)$ and $V_{r}=r^{p}V(rx)$. By estimate (4.5), the norms
$\|V_{r}\|_{\mathfrak{W}^{p}(\tilde{\mathcal{A}})}$ of the scaled potentials
are uniformly bounded $\tilde{\mathcal{A}}$. Also, by (4.2), the scaled
matrices $A_{r}(x)$ are uniformly elliptic and bounded in
$\tilde{\mathcal{A}}$. Thus, by applying the local Harnack inequality (Theorem
3.4) in the annular domain $\tilde{\mathcal{A}}$ we get

$\mathbf{A}_{r}=\underset{x\in\mathcal{A}_{r}}{\sup}\frac{u(x)}{v(x)}=\underset{x\in\mathcal{A}}{\sup}\frac{u_{r}(x)}{v_{r}(x)}\leq
C\underset{x\in\mathcal{A}}{\inf}\frac{u_{r}(x)}{v_{r}(x)}=C\underset{x\in\mathcal{A}_{r}}{\inf}\frac{u(x)}{v(x)}=C\mathbf{a}_{r},$

where the constant $C>0$ is independent of $u$ and $v$ for $r$ near $\zeta$. ∎

The next result is an extension to the case
$V\in\mathfrak{W}^{p}_{\operatorname{loc}}(\Omega)$ of Proposition 3.18 in
[11], where the potential $V$ is assumed to be in
$M^{q}_{\operatorname{loc}}(p;\Omega)$.

######  Theorem 4.13.

Let $1<p<d$, and $A,V,$ satisfy Assumptions 3.1. Assume that the operator
$Q_{p,A,V}$ has an isolated Fuchsian-type singularity at
$\zeta\in\partial\hat{\Omega}$, and there is a sequence
$R_{n}\rightarrow\zeta$, such that either $0$ or $\infty$ is a regular point
of a limiting dilated equation $\mathcal{D}^{\\{R_{n}\\}}(Q)(w)=0$ in $Y$.
Then $\zeta$ is a regular point of the equation $Q_{p,A,V}(u)=0$ in $\Omega$.

###### Proof.

The proof follows the same steps as in [11, Proposition 3.18.]. For the
completeness, we provide the proof. Let $u,v\in\mathcal{G}_{\zeta}$. Define

$m_{r}:=\underset{S_{r}\cap\Omega^{\prime}}{\inf}\frac{u(x)}{v(x)},\quad
M_{r}:=\underset{S_{r}\cap\Omega^{\prime}}{\sup}\frac{u(x)}{v(x)},$

where $\Omega^{\prime}$ is a punctured neighbourhood of $\zeta$. By Lemma 4.9,
$M\\!:=\\!\lim_{r\rightarrow\zeta}{M_{r}}$ and
$m:=\lim_{r\rightarrow\zeta}{m_{r}}$ exist in the generalized sense. Thus, we
only need to show that $M=m$.

If $M:=\lim_{r\rightarrow\zeta}{M_{r}}=\infty$ (respectively,
$m:=\lim_{r\rightarrow\zeta}{m_{r}}=0$), then by the uniform Harnack
inequality, Lemma 4.12, we get $m=\infty$ (respectively, $M=0$), and hence the
limit

$\underset{\begin{subarray}{c}x\rightarrow\zeta\\\
x\in\Omega^{\prime}\end{subarray}}{\lim}\frac{u(x)}{v(x)}\quad\text{exists in
the generalized sense.}$

So, it can be assumed that $u\asymp v$ in some neighbourhood
$\Omega^{\prime}\subset\Omega$ of $\zeta$. Let us now fix
$x_{0}\in\mathbb{R}^{d}$ such that $R_{n}x_{0}\in\Omega$ for all
$n\in\mathbb{N}$ and consider

$u_{n}(x):=\frac{u(R_{n}x)}{u(R_{n}x_{0})}\,,\qquad
v_{n}(x):=\frac{v(R_{n}x)}{u(R_{n}x_{0})}\,.$

Then by the definition of $\mathcal{G}_{\zeta}$, $u_{n}$ and $v_{n}$ are
positive solutions of the equation

$-\Delta_{p,A_{n}}(w)+V_{n}(x)|w|^{p-2}w=0\quad\mbox{in
}\Omega^{\prime}/{R_{n}},$

where $A_{n}(x):=A_{R_{n}}(x)$ and $V_{n}(x):=V_{R_{n}}(x)$, are the
associated scaled matrix and potential, respectively. Since $u_{n}(x_{0})=1$
and $v_{n}(x_{0})\asymp 1$, the Harnack convergence principle (Proposition
3.8) implies that the sequence $\\{R_{n}\\}$ admits a subsequence (still
denoted by $\\{R_{n}\\}$) such that

$\lim_{n\rightarrow\infty}u_{n}(x):=u_{\infty}(x),\mbox{ and
}\lim_{n\rightarrow\infty}v_{n}(x):=v_{\infty}(x)$

locally uniformly in
$Y=\displaystyle{\lim_{n\rightarrow\infty}\Omega^{\prime}/R_{n}}$, and
$u_{\infty}$ and $v_{\infty}$ are positive solutions of the equation

$\mathcal{D}^{\\{R_{n}\\}}(Q)(w)=-\Delta_{p,\mathbb{A}}(w)+\mathbb{V}|w|^{p-2}w=0\qquad\text{in
}Y.$ (4.11)

So, for any fixed $R>0$ we get

$\displaystyle\underset{x\in S_{R}}{\sup}\frac{u_{\infty}(x)}{v_{\infty}(x)}$
$\displaystyle\\!=\\!\underset{x\in
S_{R}}{\sup}\lim_{n\rightarrow\infty}\frac{u_{n}(x)}{v_{n}(x)}=\lim_{n\rightarrow\infty}\underset{x\in
S_{R}}{\sup}\frac{u_{n}(x)}{v_{n}(x)}$
$\displaystyle\\!=\\!\lim_{n\rightarrow\infty}\underset{x\in
S_{R}}{\sup}\frac{u(R_{n}x)}{v(R_{n}x)}\\!=\\!\lim_{n\rightarrow\infty}\underset{R_{n}x\in
S_{RR_{n}}}{\sup}\frac{u(R_{n}x)}{v(R_{n}x)}\\!=\\!\lim_{n\rightarrow\infty}M_{RR_{n}}\\!=\\!M,$

where we have used the existence of $\lim_{r\rightarrow\zeta}{M_{r}}=M$, and
the local uniform convergence of the sequence $\\{u_{n}/v_{n}\\}$ in $Y$.
Similarly, we get

$\underset{x\in S_{R}}{\inf}\frac{u_{\infty}(x)}{v_{\infty}(x)}=m.$

By our assumption, either $\zeta_{1}=0$ or $\zeta_{1}=\infty$ is a regular
point of the dilated equation (4.11), so the limit

$\underset{\begin{subarray}{c}x\rightarrow\zeta_{1}\\\ x\in
Y\end{subarray}}{\lim}\frac{u_{\infty}(x)}{v_{\infty}(x)}\quad\text{exists.}$

Hence, $m=M$, this shows that

$\underset{\begin{subarray}{c}x\rightarrow\zeta\\\
x\in\Omega^{\prime}\end{subarray}}{\lim}\frac{u(x)}{v(x)}\quad\text{exists.}$

Therefore, $\zeta$ is a regular point of the equation $Q_{p,A,V}(u)=0$ in
$\Omega$. ∎

We now introduce the notion of a weak Fuchsian-type singularity, and prove a
Liouville-type theorem for $Q_{p,A,V}$ having a such singularity at $\zeta$.

######  Definition 4.14.

Let $1<p<d$, and $A,V$ satisfy Assumptions 3.1. Assume that the operator
$Q_{p,A,V}$ has an isolated Fuchsian-type singularity
$\zeta\in\\{0,\infty\\}$. The operator $Q_{p,A,V}$ is said to have a weak
Fuchsian-type singularity at $\zeta$ if there exist $m$ sequences
$\\{R_{n}^{(j)}\\}_{n=1}^{\infty}\subset\mathbb{R}_{+}$, $1\leq j\leq m$,
satisfying $R_{n}^{(j)}\rightarrow\zeta^{j}$, where $\zeta^{(1)}=\zeta$, and
$\zeta^{(j)}=0$ or $\zeta^{(j)}=\infty$ for $2\leq j\leq m$, such that

$\mathcal{D}^{\\{R_{n}^{(m)}\\}}\circ\cdots\circ\mathcal{D}^{\\{R_{n}^{(1)}\\}}(Q)(u)=-\Delta_{p,\mathbb{A}}(u)\qquad\text{on
}Y,$

where $\mathbb{A}\\!\in\\!\mathbb{R}^{d\times d}$ is a symmetric, positive
definite matrix, and
$Y\\!=\\!\\!\underset{n\rightarrow\infty}{\lim}\Omega/R_{n}^{(1)}$.

######  Theorem 4.15 (Liouville theorem).

Let $1<p<d$ and $A,V$ satisfy Assumptions 3.1. Suppose that
$\zeta\in\partial\hat{\Omega}$ is an isolated singular point, where $\zeta=0$
or $\zeta=\infty$. Further, assume that the operator $Q=Q_{p,A,V}$ has a weak
Fuchsian-type singularity at $\zeta$. Then $\zeta$ is a regular point of
(4.1).

In other words, if $u$ and $v$ are two positive solutions of the equation
$Q_{p,A,V}(w)=0$ in a punctured neighborhood of $\zeta$, then

  * (i)

$\;\;\displaystyle{\lim_{x\rightarrow\zeta}\frac{u(x)}{v(x)}}$ exists in the
generalized sense.

  * (ii)

the equation $Q_{p,A,V}(w)=0$ admits a unique positive solution in $\Omega$ of
minimal growth in a neighbourhood of
$\partial\hat{\Omega}\setminus\\{\zeta\\}$.

###### Proof.

By Proposition 4.11, we have (i) $\Rightarrow$ (ii). Thus, we only need to
show that $\underset{x\rightarrow\zeta}{\lim}\,\frac{u(x)}{v(x)}$ exists in
the generalized sense. Since the operator $Q_{p,A,V}$ has a weak Fuchsian-type
singularity at $\zeta$, we have

$\mathcal{D}^{\\{R_{n}^{(m)}\\}}\circ\cdots\circ\mathcal{D}^{\\{R_{n}^{(1)}\\}}(Q)(w)=-\Delta_{p,\mathbb{A}}(w)=0\qquad\mbox{in
}\,\mathbb{R}^{d}\setminus\\{0\\},$ (4.12)

where $\mathbb{A}\in\mathbb{R}^{d\times d}$ is a symmetric, positive definite
matrix. Recall that by Theorem 4.5 either $0$ or $\infty$ is a regular point
of $-\Delta_{p,\mathbb{A}}$. Therefore, Theorem 4.13 and a reverse induction
argument imply that $\zeta$ is a regular point of the equation
$Q_{p,A,V}(w)=0$. ∎

Acknowledgements

The authors acknowledge the support of the Israel Science Foundation (grant
637/19) founded by the Israel Academy of Sciences and Humanities.

## References

  * [1] M. Aizenman, and B. Simon, _Brownian motion and Harnack inequality for Schrödinger operators_ , Comm. Pure Appl. Math. 35 (1982), 209–273.
  * [2] M. Biroli, _Schrödinger type and relaxed Dirichlet problems for the subelliptic $p$-Laplacian_, Potential Anal. 15 (2001), 1–16.
  * [3] M. Biroli, and U. Mosco, _Kato space for Dirichlet forms_ , Potential Anal. 10 (1999), 327–345.
  * [4] A. Caetano, A. Gogatishvili, and B. Opic, _Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces_ , Proc. Roy. Soc. Edinburgh Sect. A 146A (2016), 905–927.
  * [5] F. Chiarenza, E. Fabes, and N. Garofalo, _Harnack’s inequality for Schrödinger operators and the continuity of solutions_ , Proc. Am. Math. Soc. 98 (1986), 415–425.
  * [6] E. B. Davies, and A. M. Hinz, _Kato class potentials for higher-order elliptic operators_ , J. London Math. Soc. 58 (1998), 669–678.
  * [7] G. Di. Fazio, _Hölder-continuity of solutions for some Schrödinger equations_ , Rend. Sem. Mat. Univ. Padova 79 (1988), 173–183.
  * [8] M. Frass, and Y. Pinchover, _Positive Liouville theorems and asymptotic behavior for $p$-Laplacian type elliptic equations with a Fuchsian potential_, Confluentes Mathematici 3 (2011), 291–323.
  * [9] M. Fraas, and Y. Pinchover, _Isolated singularities of positive solutions of $p$-Laplacian type equations in $\mathbb{R}^{d}$_, J. Differential Equations 254 (2013), 1097–1119.
  * [10] D. Gilbarg, and N.S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, reprint of the 1998 edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001.
  * [11] R. Kr. Giri, and Y. Pinchover, _Positive Liouville theorem and asymptotic behaviour for $(p,A)$-Laplacian type elliptic equations with Fuchsian potential in Morrey space_, in: Anal. Math. Physics 10, (2020), Paper No. 67, 34 pp..
  * [12] J. Heinonen, T. Kilpeläinen, and O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006.
  * [13] T. Kato, _Schrödinger operators with singular potentials_ , Israel. J. Math. 13 (1972), 135–148.
  * [14] E. H. Lieb, and M. Loss, “Analysis”, 2nd ed., Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001.
  * [15] V. Liskevich, S. Lyakhova and V. Moroz, _Positive solutions to nonlinear $p\,$-Laplace equations with Hardy potential in exterior domains_, J. Differential Equations 232 (2007), 212–252.
  * [16] V. Liskevich, and I. I. Skrypnik, _Isolated singularities of solutions to quasilinear elliptic equations_ , Potential Anal. 28 (2008), 1–16.
  * [17] V. Liskevich and I. I. Skrypnik, _Harnack inequality and continuity of solutions to elliptic equations with nonstandard growth conditions and lower order terms_ , Ann. Mat. Pura Appl. 189 (2010), 333–356.
  * [18] J. Malý, and W. P. Ziemer, “Fine Regularity of Solutions of Elliptic Partial Differential Equations”, Mathematical Survey and Monographs 51, American Mathematical Society, Providence, RI, 1997.
  * [19] Y. Pinchover, _On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators_ , Ann. Inst. Henri Poincaré Anal. Non Linéaire, 11 (1994), 313–341.
  * [20] Y. Pinchover, and G. Psaradakis , _On positive solutions of the $(p,A)$-Laplacian with potential in Morrey space_, Analysis & PDE, 9 (2016), 1357–1358.
  * [21] Y. Pinchover, and N. Regev, _Criticality theory of half-linear equations with the $(p,A)$-Laplacian_, Nonlinear Anal. 119 (2015), 295–314.
  * [22] Y. Pinchover, A. Tertikas, and K. Tintarev, A Liouville-type theorem for the $p$-Laplacian with potential term, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), 357–368.
  * [23] Y. Pinchover, and K. Tintarev, _Ground state alternative for $p$-Laplacian with potential term_, Calc. Var. Partial Differential Equations 28 (2007), 179–201.
  * [24] M. A. Ragusa, and P. Zamboni, _Local regularity of solutions to quasilinear elliptic equations with general structure_ , Commun. Appl. Anal. 3 (1999), 131–147.
  * [25] J. Serrin, _Isolated singularities of solutions of quasi-linear equations_ , Acta Math. 113 (1965), 219–240.
  * [26] I. I. Skrypnik, _The Harnack inequality for a nonlinear elliptic equation with coefficients from the Kato class_ , Ukr. Math. Visn. 2 (2005), 219–235.
  * [27] I. I. Skrypnik, _Harnack’s inequality for quasilinear elliptic equations with singular absorption term_ , Potential Anal. 48 (2019), 521–539.
  * [28] L. Véron, “Singularities of Solutions of Second Order Quasilinear Equations”, Pitman Research Notes in Mathematics Series, Vol 353, Longman, 1996.
  * [29] P. Zamboni, _The Harnack inequality for quasilinear elliptic equations under minimal assumptions_ , Manuscripta Math. 102 (2000), 311–323.
  * [30] Q. Zheng and Xi. Yao, _Higher-order Kato class potentials for Schrödinger operators_ , Bull. London Math. Soc. 41 (2009), 293–301.