\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 167, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/167\hfil Solvability of elliptic equations]
{Solvability of degenerate anisotropic elliptic
second-order  equations with $L^1$-data}

\author[A. A. Kovalevsky, Y. S. Gorban \hfil EJDE-2013/167\hfilneg]
{Alexander A. Kovalevsky, Yuliya S. Gorban}  % in alphabetical order

\address{Alexander A. Kovalevsky \newline
Department of Nonlinear Analysis,
Institute of Applied Mathematics and Mechanics,
National Academy of Sciences of Ukraine, Donetsk, Ukraine}
\email{alexkvl@iamm.ac.donetsk.ua}

\address{Yuliya S. Gorban \newline
Department of Differential Equations,
Donetsk National University, Donetsk, Ukraine}
\email{yuliya\_gorban@mail.ru}

\thanks{Submitted November 28, 2012. Published July 22, 2013.}
\subjclass[2000]{35J25, 35J60, 35J70, 35R05}
\keywords{Degenerate anisotropic elliptic second-order equations;
 $L^1$-data; \hfill\break\indent
Dirichlet problem; entropy solution; $T$-solution;
 $W$-solution; weighted weak solution; \hfill\break\indent
 existence of solutions}

\begin{abstract}
 In this article, we study the Dirichlet problem for degenerate anisotropic
 elliptic second-order equations  with $L^1$-right-hand sides on a
 bounded open set of $\mathbb{R}^n$  ($n\geqslant 2$).
 These equations are described with a set of exponents and of a set of
 weighted functions. The exponents characterize the rates of growth of the
 coefficients of the equations with respect to the corresponding derivatives 
 of the unknown  function, and the weighted functions characterize 
 degeneration or  singularity of the coefficients of the equations with 
 respect to the spatial variable.
 We prove theorems on the existence of entropy solutions, $T$-solutions,
 $W$-solutions, and weighted weak solutions of the problem under consideration.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction} 

In the previous twenty years, the investigations on the existence and
properties of solutions to nonlinear equations and variational
inequalities with $L^1$-data, or measures as data, have been developed
intensively.
 As is generally known, an effective approach to the
solvability of second-order equations in divergence form with
$L^1$-right-hand sides was proposed in \cite{b3}. Then closely related
research has been developed for nondegenerate isotropic nonlinear 
second-order equations with $L^1$-data, and measures as data, involving
entropy and renormalized solutions \cite{a2,b4,b5,b6,b7,b9,d1,k2,k3}.

As for the solvability of nonlinear elliptic second-order
equations with anisotropy and degeneracy (with respect to the
spatial variable), we note the following works. The existence of a
weak (distributional) solution to the Dirichlet problem for a
model nondegenerate anisotropic equation with right-hand side
measure was established in \cite{b8}. The existence of weak solutions
for a class of nondegenerate anisotropic equations with locally
integrable data in $\mathbb{R}^n$ ($n\geqslant 2$) was proved in \cite{b1}.
An analogous existence result concerning the Dirichlet problem
for a system of nondegenerate anisotropic equations with measure
data was obtained in \cite{b2}. Moreover, in \cite{l1}, the existence of weak
solutions to the Dirichlet problem for nondegenerate anisotropic
equations with right-hand sides from Lebesgues spaces close to
$L^1$ was established. Solvability of the Dirichlet problem for
degenerate isotropic equations with $L^1$-data and measures as
data was studied in \cite{a1,a3,c1,c3,l2}.
 We remark that in \cite{a1,c1},
the existence of entropy solutions to the given problem was proved
in the case of $L^1$-data. In \cite{a3}, the existence of a
renormalized solution of the problem for the same case was
established. In \cite{a3,c3,l2}, the existence of distributional
solutions of the problem was obtained in the case of right-hand
side measures.

In this article, we study the Dirichlet problem for a class of
degenerate anisotropic elliptic second-order equations
with $L^1$-right-hand sides in a bounded open set $\Omega$ of
$\mathbb{R}^n$ ($n\geqslant 2$). This class is described by 
a set of exponents $q_1,\dots,q_n$ and of a set of
weighted functions $\nu_1,\dots,\nu_n$. 
The exponents $q_i$ characterize the rates of growth of the
coefficients of the equations with respect to the corresponding derivatives of
unknown function. The functions $\nu_i$ characterize
degeneration or singularity of the coefficients of the equations
with respect to
the spatial variable. This is the most general situation in
comparison with the above-mentioned works: the nondegenerate
isotropic case means that $\nu_i\equiv 1$ and $q_i=q_1$,
$i=1,\dots,n$, the nondegenerate anisotropic case means that
$\nu_i\equiv 1$, $i=1,\dots,n$, and $q_i$, $i=1,\dots,n$, are
generally different, and the degenerate isotropic case means that
$\nu_i=\nu_1$, $i=1,\dots,n$, as in \cite{a3,c1,c3,l2} or $\nu_i$,
$i=1,\dots,n$, are generally different as in \cite{a1} but $q_i=q_1$,
$i=1,\dots,n$.

Our initial assumptions on the exponents $q_i$ and the functions
$\nu_i$ are as follows: $q_i\in (1,n)$, 
$\nu_i:\Omega\to \mathbb{R}$, $\nu_i\geqslant 0$ in $\Omega$, 
$\nu_i>0$ a.e. in $\Omega$,
$\nu_i\in L^1_{\rm loc}(\Omega)$ and $(1/\nu_i)^{1/(q_i-1)}\in
L^1(\Omega)$. Considering such kinds of solutions to the given
problem as entropy solutions, $T$-solutions, $W$-solutions and weighted weak solutions, 
we prove the corresponding existence results. In so doing, the
theorem on the existence and uniqueness of an entropy solution
does not require additional conditions on $q_i$ and $\nu_i$, while
the existence of other kinds of solutions is established under
additional conditions on the numbers $q_i$ and the exponents of
increased summability (that should be assumed) of functions
$1/\nu_i$ and $\nu_i$.

In this connection, we observe that in the nondegenerate
anisotropic case our additional conditions for the existence of
$W$-solutions are equivalent to a two-sided bound for $q_i$ which
coincides with that given in \cite{b1,b2}. Moreover, we note that,
unlike the present article, in \cite{c1},  the existence of entropy
solutions was proved under the assumption that the involved
weighted function belongs to an appropriate Muckenhoupt class. We
also remark that in the case where $q_i=q_1$ and $\nu_i=\nu_1$,
$i=1,\dots,n$, our conditions for the existence of $T$-solutions
are reduced to such requirements on the summability of the
functions $1/\nu_1$ and $\nu_1$ as in \cite{l2}. At last, we observe
that in \cite{a1}, in the case where the functions $\nu_i$,
$i=1,\dots,n$, are generally different and $q_i=q_1$,
$i=1,\dots,n$, the existence of entropy solutions was established
under some implicit hypotheses on $\nu_1,\dots,\nu_n$.

This article is organized as follows. In Section 2, we describe a
weighted anisotro\-pic Sobolev space and a set of functions which
are used in the sequel. In Section 3, we formulate the problem in
question, consider different kinds of its solutions and give the
statements of the main results. Section 4 is devoted to the proofs
of these results. Observe that the proofs are based on the use of
some results of \cite{k4,k5,k6} on the existence and properties of
solutions of second-order variational inequalities with
$L^1$-right-hand sides and sufficiently general constraints.
Finally, in Section 5, we consider particular cases concerning the
exponents $q_i$ and the weighted functions $\nu_i$, and give
examples where conditions of the main theorems are satisfied.

For completeness we note that an extensive  bibliography on
the existence and properties of solutions of second-order
variational inequalities with $L^1$-data and measure data one can
find in \cite{k6}.

As far as the solvability of nonlinear elliptic high-order
equations with anisotropy, degeneracy and $L^1$-data is concerned,
we refer the reader for instance to \cite{k7,k8,k9,k10} where classes of
elliptic equations of fourth and higher order with coefficients,
satisfying appropriate strengthened coercivity conditions, were
considered.

In \cite{c2}, a class of nondegenerate anisotropic nonlinear
elliptic equations of arbitrary even order with $L^1$-data was
considered, and the solvability of the Dirichlet problem in the
corresponding energy space was established. However, this was made
under a condition on the involved parameters which provides the
imbedding of the energy space into the space of bounded functions.


\section{Preliminaries} 

Let $n \in \mathbb{N}$, $n \geqslant 2$, $\Omega$ be a bounded open
set of $\mathbb{R}^n$, and let for every $i \in \{1,\dots,n\}$ we
have $q_i \in (1, n)$. We set $q = \{q_i : i = 1,\dots,n\}$.

For  $i\in \{1,\dots,n\}$, let $\nu_i$ be  nonnegative
functions on $\Omega$ such that $\nu_i> 0$ a.e. in $\Omega$,
\begin{equation}
\nu_i \in L^1_{\rm loc}(\Omega), \quad   \Big(\frac
1{\nu_i}\Big )^{1/(q_i - 1)}\in L^1(\Omega ). \label{e2.1}
\end{equation}
We set $\nu = \{\nu_i : i = 1,\dots,n\}$. We denote by $W^{1,q}
(\nu,\Omega)$ the set of all functions $u \in L^1(\Omega)$ such
that for every $i\in \{1,\dots,n\}$ there exists the weak
derivative $D_i u$ and $\nu_i |D_i u|^{q_i}\in L^1(\Omega)$.

Let $\| \cdot \|_{1,q,\nu}$ be the mapping from $W^{1,q}
(\nu,\Omega)$ into $\mathbb{R}$ such that for every function $u\in
W^{1,q}(\nu,\Omega )$,
$$
\| u \|_{1,q,\nu} = \int_\Omega |u|dx + \sum^n_{i = 1}
\Big(\int_\Omega \nu_i |D_i u|^{q_i}dx\Big )^{1/q_i}.
$$
The mapping $\| \cdot \|_{1,q,\nu}$ is a norm in
$W^{1,q}(\nu,\Omega)$, and, in view of the second inclusion of
\eqref{e2.1}, the set $W^{1,q}(\nu,\Omega)$ is a Banach space with
respect to the norm $\| \cdot \|_{1,q,\nu}$. Moreover, by the first inclusion of \eqref{e2.1}, we have
$C^\infty_0 (\Omega)\subset W^{1,q}(\nu,\Omega)$.

We denote by ${\mathaccent"7017 W}^{1,q}(\nu,\Omega)$ the closure of
the set $C^\infty_0 (\Omega)$ in the space $W^{1,q}(\nu,\Omega)$.
Obviously, the set ${\mathaccent"7017 W}^{1,q}(\nu,\Omega)$ is a
Banach space with respect to the norm induced by the norm 
$\|\cdot \|_{1,q,\nu}$. We observe that 
$C^1_0(\Omega)\subset {\mathaccent"7017 W}^{1,q}(\nu,\Omega)$.

Further,  for  every $k > 0$, let $T_k: \mathbb{R} \to \mathbb{R}$
be the function such that
$$
T_k (s) = \begin{cases}
s &\text{if } |s| \leqslant k, \\
k\operatorname{sign}s & \text{if } |s| > k.
\end{cases}
$$
By analogy with known results for nonweighted Sobolev spaces 
(see for instance \cite[Chapter 2]{k1}) we have: 
if $u\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)$ and $k >0$, then 
$T_k(u) \in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)$ and for every 
$i\in \{1,\dots,n\}$, $D_i T_k(u) = D_i u \cdot 1_{\{|u|<k\}}$
 a.e.\, in $\Omega$.

We denote by ${\mathaccent"7017 \mathcal{T}}^{1,q} (\nu,\Omega)$
the set of all functions $u : \Omega \to \mathbb{R}$ such that for
every $k > 0$, $T_k (u) \in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)$.
 Clearly, ${\mathaccent"7017 W}^{1,q}(\nu,\Omega) 
\subset {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$.
For every $u : \Omega \to \mathbb{R}$ and for every $x \in \Omega$
we set
$$
k(u,x) = \min\{l \in \mathbb{N} : |u (x)| \leqslant l\}.
$$

\begin{definition} \label{def2.1}\rm
 Let $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$ and 
$i \in \{1,\dots,n\}$. Then $\delta_i u:\Omega\to\mathbb{R}$ 
is the function such that for every 
$x \in \Omega$, $\delta_i u(x) = D_i T_{k (u,x)}(u)(x)$.
\end{definition}

\begin{definition} \label{def2.2} \rm
If $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$, then 
$\delta u:\Omega\to \mathbb{R}^n$ is
the mapping such that for every $x \in \Omega$ and for every 
$i \in \{1,\dots,n\}$, $(\delta u (x))_i = \delta_i u(x)$.
\end{definition}

Now we give several propositions which will be used in the next
sections.

\begin{proposition} \label{prop2.3}  
Let $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$  and
$i \in \{1,\dots,n\}$. 
Then for every $k > 0$  we have $D_i T_k (u) = \delta_i u
\cdot 1_{\{|u| < k \}}$  a.e. in $\Omega$.
\end{proposition}

The proof of this proposition is analogous to the proof of the
corresponding result given in \cite{k2} for the nonweighted case.

\begin{proposition} \label{prop2.4} 
 Let $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$  and
$w \in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty (\Omega)$.
 Then $u - w \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$, 
 and for every $i \in \{1,\dots,n\}$  and for every $k> 0$ 
we have
$$
D_i T_k (u - w) = \delta_i u - D_i w \quad   \text{a.e. in }
  \{|u - w| < k \}.
$$
\end{proposition}


\begin{proposition} \label{prop2.5} 
 Let $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$  and
$|\delta u | \in L^1 (\Omega)$.  Then 
$u \in {\mathaccent"7017 W}^{1,1}(\Omega)$
and for every $i \in \{1,\dots,n\}$  we have 
$D_i u = \delta_i u$  a.e. in $\Omega$.
\end{proposition}

The proofs of the two propositions above can be found in \cite{k4}.


\section{Statement of main results}


Let $c_1, c_2 > 0$, $g_1, g_2 \in L^1 (\Omega)$, 
$g_1, g_2 \geqslant 0$ in $\Omega$, and  for every $i\in \{1,\dots,n\}$,
let $a_i:\Omega\times \mathbb{R}^n\to \mathbb{R}$ be a Carath\'eodory
functions. We suppose that for almost every $x \in \Omega$ and for
every $\xi \in \mathbb{R}^n$,
\begin{gather}
\sum^n_{i=1}\,(1/\nu_i)^{1/(q_i-1)}(x)|a_i(x,\xi)|^{q_i/(q_i-1)}
\leqslant c_1 \sum^n_{i=1}\,\nu_i(x)|\xi_i |^{q_i} + g_1(x),
\label{e3.1} \\
\sum^n_{i=1}\,a_i(x,\xi)\xi_i \geqslant c_2 \sum^n_{i = 1}\,\nu_i
(x)|\xi_i |^{q_i}-g_2 (x). \label{e3.2}
\end{gather}
Moreover, we assume that for almost every $x\in \Omega$ and for
every $\xi,\xi' \in \mathbb{R}^n$, $\xi \neq  \xi'$,
\begin{equation}
\sum^n_{i=1}[a_i(x,\xi)-a_i(x,\xi')](\xi_i-\xi_i')> 0. \label{e3.3}
\end{equation}
Note that the following assertions hold:
if $ u,w \in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)$
and $i\in \{1,\dots,n\}$, then
\begin{equation}
 a_i(x,\nabla u)D_i w\in L^1(\Omega);
\label{e3.4}
\end{equation}
if $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$,
$w\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty(\Omega)$, $k>0$,
$l\geqslant k+\|w\|_{L^\infty(\Omega)}$ and
$i\in\{1,\dots,n\}$, then
\begin{equation} a_i(x,\delta u)D_iT_k(u-w) = a_i(x,\nabla T_l(u))D_iT_k(u-w)
\quad \text{a.e. in }\Omega; \label{e3.5}
\end{equation}
if $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$,
$w\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty(\Omega)$,
$k>0$ and $i\in\{1,\dots,n\}$, then
\begin{equation}
a_i(x,\delta u)D_iT_k(u-w)\in L^1(\Omega).
\label{e3.6}
\end{equation}


Assertion \eqref{e3.4} is established with the use of \eqref{e3.1}.
Assertion \eqref{e3.5} is proved by means of Propositions 
\ref{prop2.3} and \ref{prop2.4}. 
Assertion \eqref{e3.6} is derived from Proposition \ref{prop2.4} and assertions
\eqref{e3.4} and \eqref{e3.5}.


Let $f\in L^1(\Omega)$, and consider the Dirichlet
problem
\begin{gather}
-\sum_{i=1}^n \frac{\partial}{\partial x_i}\,a_i(x,\nabla u)=f
\quad \text{in }\Omega, \label{e3.7} \\
u=0 \quad \text{on } \partial \Omega. \label{e3.8}
\end{gather}


\begin{definition} \label{def3.1}\rm
 An entropy solution of problem \eqref{e3.7}, \eqref{e3.8}
is a function $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$ 
such that for every function $w\in {\mathaccent"7017 W}^{1,q}
(\nu,\Omega)\cap L^\infty(\Omega)$ and for every $k\geqslant 1$,
$$
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iT_k(u-w)\Big\}dx \leqslant \int_\Omega f\,T_k(u-w)dx.
$$
\end{definition}

\begin{theorem} \label{thm3.2} 
 There exists a unique entropy solution of
problem \eqref{e3.7}, \eqref{e3.8}.
\end{theorem}

\begin{definition} \label{def3.3} \rm
A $T$-solution of problem \eqref{e3.7}, \eqref{e3.8} is a
function $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$
such that:
\begin{itemize}
\item[(i)] for every $i\in\{1,\dots,n\}$, $a_i(x,\delta u)\in L^1(\Omega)$;

\item[(ii)] for every function $w\in C_0^1(\Omega)$,
$$
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\delta u)D_iw\Big\}dx =
\int_\Omega fw\,dx.
$$
\end{itemize}
\end{definition}

The next theorem shows that under additional conditions on $q$ and
$\nu$ the entropy solution of problem \eqref{e3.7}, \eqref{e3.8} is a
$T$-solution of the same problem. For the statement of this and
further results we need the following numbers depending on the set
$q$.
We define
$$
\overline q = \Big(\frac 1n\,\sum_{i=1}^n \frac1{q_i}\Big)^{-1}
$$
and for every $m \in \mathbb{R}^n$ such that $m_i
> 0$, $i=1,\dots,n$, we set
$$
p_m = n \Big(\sum^n_{i = 1} \frac{1+m_i}{m_i q_i}-1\Big)^{-1}\,.
$$
Observe that if $m\in \mathbb{R}^n$ and for every 
$i \in \{1, \dots,n\}$,  $m_i \geqslant 1/(q_i-1)$, then $p_m > 1$.
Moreover, if $m\in \mathbb{R}^n$ and for every $i\in \{1,\dots,n\}$
we have $m_i \geqslant 1/(q_i-1)$ and $1/\nu_i \in
L^{m_i}(\Omega)$, then the space 
${\mathaccent"7017 W}^{1,q}(\nu,\Omega)$ is continuously imbedded into 
the space $L^{p_m}(\Omega)$. This fact follows from \cite[Proposition 2.8]{k6}.
In turn, the mentioned proposition was established with the use of
an imbedding result for the non-weighted anisotropic case \cite{t1}.

\begin{theorem} \label{thm3.4} 
Suppose that there exist  $m,\sigma\in\mathbb{R}^n$  such that the 
following conditions are satisfied:
\begin{gather}
 m_i\geqslant 1/(q_i-1), \quad 1/\nu_i \in L^{m_i}(\Omega) 
\quad \forall i\in\{1,\dots,n\}; \label{e3.9} 
\\
 \sigma_i>0, \quad \frac 1{\sigma_i} < 1 - \frac{(q_i -1)
\overline q}{p_m(\overline q-1)}, \quad
 \nu_i\in L^{\sigma_i}(\Omega) \quad \forall i\in\{1,\dots,n\}. \label{e3.10}
\end{gather}
Let  $u$ be the entropy solution of problem \eqref{e3.7},
\eqref{e3.8}.  Then  $u$ is a  $T$-solution of problem
\eqref{e3.7}, \eqref{e3.8}.
\end{theorem}

From Theorems \ref{thm3.2} and \ref{thm3.4} we deduce the following result.

\begin{corollary} \label{coro3.5}
 Suppose that there exist $m,\sigma\in\mathbb{R}^n$  such that conditions
 \eqref{e3.9} and \eqref{e3.10}  are satisfied. 
 Then there exists a
 $T$-solution of problem \eqref{e3.7}, \eqref{e3.8}.
\end{corollary}

As we see, $T$-solutions of the given problem belong to function
set ${\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$, and in
general such solutions do not have weak derivatives. Now let us
consider a kind of solutions having weak derivatives.

\begin{definition} \label{def3.6}\rm
 A $W$-solution of problem \eqref{e3.7}, \eqref{e3.8} is a
function $u\in {\mathaccent"7017 W}^{1,1}(\Omega)$ such that:
\begin{itemize}
\item[(i)] for every $i\in\{1,\dots,n\}$,  $a_i(x,\nabla u)\in L^1(\Omega)$;

\item[(ii)] for every function $w\in C_0^1(\Omega)$,
$$
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\nabla u)D_iw\Big\}dx =
\int_\Omega fw\,dx.
$$
\end{itemize}
\end{definition}


\begin{proposition} \label{prop3.7}  
Let  $u\in  {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$. 
Then  $u$ is a  $W$-solution of problem \eqref{e3.7}, \eqref{e3.8}
 if and only if  $u$ is a  $T$-solution of problem
 \eqref{e3.7}, \eqref{e3.8}  and $|\delta u|\in L^1(\Omega)$.
\end{proposition}

For the proof of this result it suffices to use 
Propositions \ref{prop2.3} and \ref{prop2.5} along with the fact that 
$D_iT_k(w)=D_iw\cdot 1_{\{|w|<k\}}$ a.e. in $\Omega$
 if $w\in {\mathaccent"7017 W}^{1,1}(\Omega)$, $k>0$ and $i\in\{1,\dots,n\}$.

\begin{theorem} \label{thm3.8}
 Suppose that there exist  $m,\sigma\in\mathbb{R}^n$  with positive 
coordinates such that the following conditions are satisfied:
\begin{gather}
 \frac{\overline q}{p_m(\overline q-1)} < q_i-1-\frac 1{m_i}, \quad
1/\nu_i \in L^{m_i}(\Omega) \quad \forall i\in\{1,\dots,n\};  \label{e3.11}
 \\
  \frac 1{\sigma_i} < 1 - \frac{(q_i -1)\overline q}{p_m(\overline q-1)}, \quad
\nu_i\in L^{\sigma_i}(\Omega)\quad \forall i\in\{1,\dots,n\}. \label{e3.12}
\end{gather}
Let  $u$  be the entropy solution of problem \eqref{e3.7},
\eqref{e3.8}.  Then  $u$  is a  $W$-solution of problem
\eqref{e3.7}, \eqref{e3.8}.
\end{theorem}

From Theorems \ref{thm3.2} and \ref{thm3.8} we infer the following result.

\begin{corollary} \label{coro3.9} 
Suppose that there exist  $m,\sigma\in\mathbb{R}^n$  with positive coordinates 
such that conditions \eqref{e3.11}  and \eqref{e3.12}  are satisfied. 
Then there exists a  $W$-solution of problem \eqref{e3.7}, \eqref{e3.8}.
\end{corollary}

Now we consider another kind of solutions (in the sense of an
integral identity) whose existence requires less additional
conditions as compared with $W$-solutions.

We denote by ${\mathaccent"7017 V}^{1,q}(\nu,\Omega)$ the set of all
functions $w\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap
L^\infty(\Omega)$ such that for every $i\in\{1,\dots,n\}$,
 $\nu_i^{1/q_i}D_iw\in L^\infty(\Omega)$. Obviously, the set
${\mathaccent"7017 V}^{1,q}(\nu,\Omega)$ is nonempty. Moreover, if
for every $i\in\{1,\dots,n\}$ we have $\nu_i\in
L^\infty_{\rm loc}(\Omega)$, then $C^1_0(\Omega)\subset
{\mathaccent"7017 V}^{1,q}(\nu,\Omega)$.



\begin{definition} \label{def3.10} \rm
A weighted weak solution of problem \eqref{e3.7},
\eqref{e3.8} is a function $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$
 such that:
\begin{itemize}
\item[(i)] for every $i\in\{1,\dots,n\}$,
 $\nu_i^{1/q_i}\delta_i u \in L^1(\Omega)$;


\item[(ii)] for every $i\in\{1,\dots,n\}$,
 $(1/\nu_i)^{1/q_i}a_i(x,\delta u)\in L^1(\Omega)$;


\item[(iii)] for every function $w\in {\mathaccent"7017 V}^{1,q}(\nu,\Omega)$,
$$
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\delta u)D_iw\Big\}dx =
\int_\Omega fw\,dx.
$$
\end{itemize}
\end{definition}


Observe that if for every $i\in\{1,\dots,n\}$,  $1/\nu_i\in
L^\infty(\Omega)$, and $u$ is a weighted weak solution of problem
\eqref{e3.7}, \eqref{e3.8}, then $u\in {\mathaccent"7017 W}^{1,1}(\Omega)$.
Moreover, if for every $i\in\{1,\dots,n\}$,  $\nu_i\equiv 1$, and
$u$ is a weighted weak solution of problem \eqref{e3.7}, \eqref{e3.8}, then $u$
is a $W$-solution of the same problem. These facts are easily
established with the use of Proposition \ref{prop2.5}.

\begin{theorem} \label{thm3.11} 
 Suppose that there exists  $m\in \mathbb{R}^n$ such that the following 
 conditions are satisfied:
\begin{gather}
 m_i\geqslant 1/(q_i-1), \quad 1/\nu_i \in L^{m_i}(\Omega) \quad  
\forall i\in\{1,\dots,n\}; \label{e3.13} 
\\
 p_m > \frac{\overline q}{\overline q-1}\max\Big\{\frac 1{q_i-1},\; q_i-1\Big\}
\quad \forall i\in\{1,\dots,n\}.\label{e3.14}
\end{gather}
 Let  $u$  be the entropy solution of problem \eqref{e3.7},
\eqref{e3.8}.  Then  $u$  is a weighted weak solution of
problem \eqref{e3.7}, \eqref{e3.8}.
\end{theorem}

From Theorems \ref{thm3.2} and \ref{thm3.11} we obtain the following result.

\begin{corollary} \label{coro3.12}
 Suppose that there exists 
 $m\in\mathbb{R}^n$  such that conditions \eqref{e3.13}  and
\eqref{e3.14}  are satisfied.  Then there exists a weighted weak
solution of problem \eqref{e3.7}, \eqref{e3.8}.
\end{corollary}

From Theorems \ref{thm3.4}, \ref{thm3.8} and \ref{thm3.11}
 we deduce the following result.


\begin{corollary} \label{coro3.13} 
 Suppose that there exist  $m,\sigma \in\mathbb{R}^n$ 
 with positive coordinates such that conditions \eqref{e3.11}  and
\eqref{e3.12}  are satisfied.  Then the entropy solution of problem
 \eqref{e3.7}, \eqref{e3.8}  is also a
 $T$-solution, a  $W$-solution and a weighted weak
solution of the same problem.
\end{corollary}

\section{Proofs}


\subsection{Basis for the proofs}
 Here we give two results which were established in \cite{k4,k5,k6}.
They form a basis for the proof of the theorems stated in the previous section.

\begin{theorem} \label{thm4.1} 
 Let $V$  be a closed convex set in
${\mathaccent"7017 W}^{1,q}(\nu, \Omega)$  satisfying the
conditions:
\begin{gather}
V \cap L^\infty (\Omega) \neq  \emptyset, \label{e4.1} \\
\text{if $u,w \in V$  and $k >0$,  then $u - T_k (u - w) \in V$.} \label{e4.2}
\end{gather}
Then there exists a unique function  
$u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$
 such that the following assertions hold:
\begin{itemize}
\item[(i)] for every  $w \in V \cap L^\infty (\Omega)$
 and for every $k \geqslant 1$  we have  $w - T_k (w- u) \in V$;

\item[(ii)]  if  $w\in V \cap L^\infty (\Omega)$, $k \geqslant 1$
 and  $l = k + \| w \|_{L^\infty (\Omega)}$, 
then
$$
\int_\Omega\Big\{\sum_{i=1}^n\,a_i(x,\nabla T_l(u))D_iT_k(u-w)\Big\}dx
\leqslant \int_\Omega f\,T_k(u - w)dx.
$$
\end{itemize}
\end{theorem}

We note that conditions \eqref{e3.2} and \eqref{e3.3} are essential in
the proof of the given theorem.

\begin{proposition} \label{prop4.2}  
Let  $m\in \mathbb{R}^n$,  and let condition \eqref{e3.9} 
 be satisfied. Let $V$  be a closed
convex set in ${\mathaccent"7017 W}^{1,q}(\nu, \Omega)$ 
satisfying conditions \eqref{e4.1}  and \eqref{e4.2}.  Let  
$u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$, 
 and let assertions (i) and (ii)  of Theorem \ref{thm4.1}  hold.
Then for every  $i\in\{1,\dots,n\}$  and for every
$\lambda$, $0<\lambda < \frac{q_ip_m(\overline q-1)}{p_m(\overline
q-1)+\overline q}$,  we have  
$\nu_i^{1/q_i}\delta_i u\in L^\lambda(\Omega)$.
\end{proposition}


\subsection{Proof of Theorem \ref{thm3.2}}
 Applying Theorem \ref{thm4.1} for the case
where $V={\mathaccent"7017 W}^{1,q}(\nu,\Omega)$, we obtain that
there exists a unique function $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}
(\nu,\Omega)$ such that the following assertion holds:
if $w \in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap
L^\infty(\Omega)$, $k \geqslant 1$ and 
$l = k + \| w \|_{L^\infty (\Omega)}$, then
$$
\int_\Omega\Big\{\sum_{i=1}^n\,a_i(x,\nabla T_l(u))D_iT_k(u-w)\Big\}dx
\leqslant \int_\Omega f\,T_k(u - w)dx.
$$
This and assertion \eqref{e3.5} imply that $u$ is the unique entropy
solution of problem \eqref{e3.7}, \eqref{e3.8}. 
The proof is complete.


\subsection{Proof of  Theorem \ref{thm3.4}}
 First of all, taking into account Proposition \ref{prop2.4}
 and assertion \eqref{e3.5}, 
from Proposition \ref{prop4.2} we deduce the following result.


\begin{proposition} \label{prop4.3}  
Let  $m\in \mathbb{R}^n$,  and let condition \eqref{e3.9}  be satisfied. 
Let  $u$  be the entropy solution of problem \eqref{e3.7}, \eqref{e3.8}. 
Then for every  $i\in\{1,\dots,n\}$  and for every
 $\lambda$, $0<\lambda < \frac{q_ip_m(\overline q-1)}{p_m(\overline
q-1)+\overline q}$,  we have  $\nu_i^{1/q_i}\delta_iu\in
L^\lambda(\Omega)$.
\end{proposition}

Now, suppose that there exist $m,\sigma\in\mathbb{R}^n$ such that
conditions \eqref{e3.9} and \eqref{e3.10} are satisfied, and let $u$ be the
entropy solution of problem \eqref{e3.7}, \eqref{e3.8}.

Let us show that for every $i\in \{1,\dots,n\}$,  
$a_i(x,\delta u)\in L^1(\Omega)$. In fact, let $i\in \{1,\dots,n\}$. 
By \eqref{e3.1}, we have
\begin{equation}
|a_i(x,\delta u)| \leqslant (c_1+1)\sum_{j=1}^n\,\nu_i^{1/q_i}
|\nu_j^{1/q_j}\delta_ju|^{q_j(q_i-1)/q_i} +
\nu_i^{1/q_i}g_1^{(q_i-1)/q_i} \quad \text{a.e. in }
\Omega. \label{e4.3}
\end{equation}
Using Young's inequality with the exponents $q_i$ and
$q_i/(q_i-1)$, we obtain that $\nu_i^{1/q_i}g_1^{(q_i-1)/q_i} \leqslant
\nu_i + g_1$. Hence, taking into account that $g_1\in L^1(\Omega)$
and, by condition \eqref{e3.10}, $\nu_i\in L^1(\Omega)$, we infer that
\begin{equation}
\nu_i^{1/q_i}g_1^{(q_i-1)/q_i}\in L^1(\Omega). \label{e4.4}
\end{equation}
Next, we fix $j\in\{1,\dots,n\}$ and set
$$
\lambda_{ij}=\frac{\sigma_i(q_i-1)q_j}{\sigma_iq_i-1}\,.
$$
Using Young's inequality with the exponents $\sigma_iq_i$ and
$\sigma_iq_i/(\sigma_iq_i-1)$, we obtain
\begin{equation}
\nu_i^{1/q_i}|\nu_j^{1/q_j}\delta_ju|^{q_j(q_i-1)/q_i} \leqslant
\nu_i^{\sigma_i} + |\nu_j^{1/q_j}\delta_ju|^{\lambda_{ij}}\,.
\label{e4.5}
\end{equation}
Observe that, by condition \eqref{e3.10}, we have
\begin{gather}
\nu_i \in L^{\sigma_i}(\Omega), \label{e4.6}\\
\lambda_{ij} < \frac{q_jp_m(\overline q -1)}{p_m(\overline q -1) +
\overline q}\,. \nonumber
\end{gather}
Since condition \eqref{e3.9} is satisfied, from the latter inequality and
Proposition \ref{prop4.3} it follows that $\nu_j^{1/q_j}\delta_ju \in
L^{\lambda_{ij}}(\Omega)$. This inclusion along with \eqref{e4.6} and
\eqref{e4.5} implies that for every $j\in\{1,\dots,n\}$,
\begin{equation}
\nu_i^{1/q_i}|\nu_j^{1/q_j}\delta_ju|^{q_j(q_i-1)/q_i} \in
L^1(\Omega). \label{e4.7}
\end{equation}
From \eqref{e4.3}, \eqref{e4.4} and \eqref{e4.7} we deduce that for every
$i\in \{1,\dots,n\}$,  $a_i(x,\delta u)\in L^1(\Omega)$.

Further, we fix $w\in C_0^1(\Omega)$ and for every $h\in \mathbb{N}$ 
we set $w_h=T_h(u)-w$. Now let us fix $k \geqslant
\|w\|_{L^\infty(\Omega)}+1$, and let $h\in \mathbb{N}$. 
Since $u$ is the entropy solution of problem \eqref{e3.7}, \eqref{e3.8} 
and $w_h\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty(\Omega)$, 
by Definition \ref{def3.1}, we have
\begin{equation}
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iT_k(u-w_h)\Big\}dx \leqslant \int_\Omega f\,T_k(u-w_h)dx.
\label{e4.8}
\end{equation}
From Propositions \ref{prop2.3} and \ref{prop2.4}, it follows that for every $i\in
\{1,\dots,n\}$,
$$
D_iT_k(u-w_h) = (\delta_iu\cdot 1_{\{|u|\geqslant h\}} +
D_iw)\cdot 1_{\{|u-w_h|<k\}} \quad  \text{a.e. in }\Omega.
$$
Using this fact and \eqref{e3.2}, we obtain
\begin{align*}
&\int_\Omega\Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iT_k(u-w_h)\Big\}dx\\
& \geqslant \int_{\{|u-w_h|<k\}}\Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iw\Big\}dx - \int_{\{|u|\geqslant h\}}g_2dx.
\end{align*}
This and \eqref{e4.8} imply that for every $h\in \mathbb{N}$,
\begin{equation}
\int_{\{|u-w_h|<k\}}\Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iw\Big\}dx \leqslant \int_\Omega f\,T_k(u-w_h)dx +
\int_{\{|u|\geqslant h\}}g_2dx. \label{e4.9}
\end{equation}
Observe that for every $h\in\mathbb{N}$,
 $\operatorname{meas}(\Omega\setminus\{|u-w_h|<k\}) \leqslant
\operatorname{meas}\{|u|\geqslant h\}$. Then, taking into account that
$\operatorname{meas}\{|u|\geqslant h\}\to 0$ as $h\to +\infty$ and the
functions $g_2$ and $a_i(x,\delta u)D_iw$, $i=1,\dots,n$, are
summable in $\Omega$, we obtain
\begin{gather}
\int_{\{|u-w_h|<k\}}\Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iw\Big\}dx \to \int_\Omega\Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iw\Big\}dx, \label{e4.10} \\
\int_{\{|u|\geqslant h\}}g_2dx \to 0. \label{e4.11}
\end{gather}
 Finally, since $u-w_h\to w$ in $\Omega$ and $k \geqslant
\|w\|_{L^\infty(\Omega)}$, we have $T_k(u-w_h)\to w$ in $\Omega$.
Hence, applying Dominated Convergence Theorem, we obtain
\begin{equation}
\int_\Omega f\,T_k(u-w_h)dx\to \int_\Omega fw\,dx. \label{e4.12}
\end{equation}
From \eqref{e4.9}--\eqref{e4.12} we infer that
$$
\int_\Omega\Big\{\sum_{i=1}^n\,a_i(x,\delta u)D_iw\Big\}dx
\leqslant \int_\Omega fw\,dx.
$$
Therefore, for every $w\in C_0^1(\Omega)$,
$$
\int_\Omega\Big\{\sum_{i=1}^n\,a_i(x,\delta u)D_iw\Big\}dx =
\int_\Omega fw\,dx.
$$
This completes the proof of Theorem \ref{thm3.4}.


We remark that the idea of using the functions $w_h=T_h(u)-w$ in the
above proof is taken from  \cite[Corollary 4.3]{b3}.

\subsection{Proof of Theorem \ref{thm3.8}}
 Suppose that there exist
$m,\sigma\in\mathbb{R}^n$ with positive coordinates such that
conditions \eqref{e3.11} and \eqref{e3.12} are satisfied, and let $u$ be the
entropy solution of problem \eqref{e3.7}, \eqref{e3.8}.
Let us show that $|\delta u|\in L^1(\Omega)$. In fact, let
$i\in\{1,\dots,n\}$. Clearly,
\begin{equation}
|\delta_iu| = (1/\nu_i)^{1/q_i}|\nu_i^{1/q_i}\delta_iu| \quad
\text{a.e. in } \Omega. \label{e4.13}
\end{equation}
Using Young's inequality with the exponents $m_iq_i$ and
$m_iq_i/(m_iq_i-1)$, we obtain
\begin{equation}
(1/\nu_i)^{1/q_i}|\nu_i^{1/q_i}\delta_iu| \leqslant
(1/\nu_i)^{m_i} + |\nu_i^{1/q_i}\delta_iu|^{m_iq_i/(m_iq_i-1)}\,.
\label{e4.14}
\end{equation}
By condition \eqref{e3.11}, we have $1/\nu_i \in L^{m_i}(\Omega)$ and
$$
\frac{m_iq_i}{m_iq_i-1} < \frac{q_ip_m(\overline
q-1)}{p_m(\overline q-1)+\overline q}\,.
$$
This along with Proposition \ref{prop4.3} and \eqref{e4.13} and 
\eqref{e4.14} implies that
$|\delta_iu|\in L^1(\Omega)$, $i=1,\dots,n$. Hence,
$|\delta u|\in L^1(\Omega)$. Then, taking into account that conditions
\eqref{e3.9} and \eqref{e3.10} are satisfied and using
Theorem \ref{thm3.4} and Proposition \ref{prop3.7}, we obtain that $u$ is a $W$-solution
of problem \eqref{e3.7}, \eqref{e3.8}. The
proof is complete.



\subsection{An integral identity for the entropy solution}
According to Theorem \ref{thm3.4}, under conditions \eqref{e3.9} and \eqref{e3.10} the
entropy solution of problem \eqref{e3.7}, \eqref{e3.8} is a solution in the
sense of an integral identity for functions in $C_0^1(\Omega)$. In
this subsection, for every function 
$u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$
 we introduce a function set $ \mathcal{M}(u)$ and show that if $u$ 
is the entropy solution of the problem
under consideration, then $u$ satisfies the corresponding integral
identity  for functions in $\mathcal{M}(u)$. This result, having a
self-contained interest, will be used in the proof of Theorem \ref{thm3.11}.

For every function $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$ we set
$$
\mathcal{M}(u)=\{w\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap
L^\infty(\Omega): a_i(x,\delta u)D_iw \in L^1(\Omega),
\,i=1,\dots,n\}.
$$
Clearly, if $u \in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$, 
then the set $\mathcal{M}(u)$ is non-empty.


\begin{proposition} \label{prop4.4}  Let  $u$  be the entropy
solution of problem \eqref{e3.7}, \eqref{e3.8}.  Then for every 
$w\in \mathcal{M}(u)$,
$$
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\delta u)D_iw\Big\}dx =
\int_\Omega fw\,dx.
$$
\end{proposition}


\begin{proof} We fix $w\in \mathcal{M}(u)$ and for every $h\in
\mathbb{N}$ we set $w_h=T_h(u)-w$. Then we fix $k \geqslant
\|w\|_{L^\infty(\Omega)}+1$, and let $h\in \mathbb{N}$. Since $u$
is the entropy solution of problem \eqref{e3.7}, \eqref{e3.8} and $w_h\in
{\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty(\Omega)$, by
 Definition \ref{def3.1}, inequality \eqref{e4.8} holds. Then, arguing as
in the proof of Theorem \ref{thm3.4}, for every $h\in\mathbb{N}$ we obtain
inequality \eqref{e4.9}. At the same time limit relations 
\eqref{e4.10}--\eqref{e4.12} hold. We only note that now the
 convergence in \eqref{e4.10} is justified by the fact that for
 every $i\in\{1,\dots,n\}$,  $a_i(x,\delta u)D_iw\in L^1(\Omega)$,
 which holds due to the inclusion $w\in
\mathcal{M}(u)$. From \eqref{e4.9}--\eqref{e4.12} we derive the required result.
The proposition is proved.
\end{proof}

\begin{corollary} \label{coro4.5} Let  $u$  be the entropy solution
of problem \eqref{e3.7}, \eqref{e3.8}.  Then for every function
 $w\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty(\Omega)$  and
for every $k>0$,
$$
\int_\Omega \Big\{\sum_{i=1}^n\,a_i(x,\delta
u)D_iT_k(u-w)\Big\}dx = \int_\Omega f\,T_k(u-w)dx.
$$
\end{corollary}

\begin{proof} 
Let $w\in {\mathaccent"7017 W}^{1,q}(\nu,\Omega)\cap L^\infty(\Omega)$ and
 $k>0$. By Proposition \ref{prop2.4} and
assertion \eqref{e3.6}, we have $T_k(u-w)\in \mathcal{M}(u)$. Then from
Proposition \ref{prop4.4} we deduce the required equality.
\end{proof}


\subsection{Proof of Theorem \ref{thm3.11}}
 Suppose that there exists
$m\in\mathbb{R}^n$ such that conditions \eqref{e3.13} and \eqref{e3.14} are
satisfied, and let $u$ be the entropy solution of problem \eqref{e3.7},
\eqref{e3.8}.

Let $i\in \{1,\dots,n\}$. By condition \eqref{e3.14}, we have
$p_m(\overline q-1)> \overline q/(q_i-1)$ and $p_m(\overline q-1)>
\overline q(q_i-1)$. Hence,
\begin{equation}
1< \frac{q_ip_m(\overline q-1)}{p_m(\overline q-1)+\overline
q}, \quad  \frac{q_i-1}{q_i} < \frac{p_m(\overline
q-1)}{p_m(\overline q-1)+\overline q}\,. \label{e4.15}
\end{equation}
Since condition \eqref{e3.13} coincides with condition \eqref{e3.9}, in view of
Proposition \ref{prop4.3} and inequalities \eqref{e4.15}, we have
$\nu_i^{1/q_i}\delta_iu\in L^1(\Omega)$ and
$$
|\nu_j^{1/q_j}\delta_ju|^{q_j(q_i-1)/q_i}\in L^1(\Omega)\quad
\forall j\in \{1,\dots,n\}\,.
$$
Therefore, taking into account that, by  \eqref{e3.1},
$$
(1/\nu_i)^{1/q_i}|a_i(x,\delta u)| \leqslant (c_1+1)\sum_{j=1}^n
|\nu_j^{1/q_j}\delta_ju|^{q_j(q_i-1)/q_i} +
g_1^{(q_i-1)/q_i}\quad  \text{a.e. in }\Omega,
$$
we obtain the inclusion $(1/\nu_i)^{1/q_i}a_i(x,\delta u)\in
L^1(\Omega)$.

Thus, $u\in {\mathaccent"7017 \mathcal{T}}^{1,q}(\nu,\Omega)$ and
properties (i) and (ii) of Definition \ref{def3.10} hold. At the same time,
property (ii) of this definition implies that 
${\mathaccent"7017 V}^{1,q}(\nu,\Omega)\subset \mathcal{M}(u)$. Then, by
Proposition \ref{prop4.4}, property (iii) of Definition \ref{def3.10} holds. Hence,
$u$ is a weighted weak solution of problem \eqref{e3.7}, \eqref{e3.8}. This
completes the proof.


\section{Particular cases and examples}

First of all we note that Definitions \ref{def3.1}, \ref{def3.3} and 
\ref{def3.6} have the
same form with the definitions of the corresponding kinds of
solutions studied in \cite{b3,b5,b6} in the case of nondegenerate
isotropic elliptic second-order equations with $L^1$-data. It is
easy to see that in this case ($q_i=q_1$ and $\nu_i\equiv 1$ for
every $i\in\{1,\dots,n\}$) there exist $m,\sigma\in \mathbb{R}^n$,
satisfying conditions \eqref{e3.9} and \eqref{e3.10}, and the existence of
$m,\sigma\in \mathbb{R}^n$ with positive coordinates, satisfying
conditions \eqref{e3.11} and \eqref{e3.12}, is equivalent to the requirement
$q_1>2-1/n$. Thus, the results of Section 3 on entropy, $T$- and
$W$-solutions of problem \eqref{e3.7}, \eqref{e3.8} generalize the known results
concerning solutions of nondegenerate isotropic elliptic
second-order equations with $L^1$-right-hand sides.

In regard to the nondegenerate anisotropic case we state the
following proposition.

\begin{proposition} \label{prop5.1}  
Let $\nu_i \equiv 1$ for all  $i\in \{1,\dots,n\}$.
Then 
\begin{itemize}
\item[(i)] the existence of  $m,\sigma\in \mathbb{R}^n$
 satisfying conditions \eqref{e3.9} and  \eqref{e3.10} is
equivalent to the requirement
\begin{equation}
q_i<\frac{(n-1)\overline q}{n-\overline q} \quad
\forall i\in \{1,\dots, n\};\label{e5.1}
\end{equation}

\item[(ii)]   the existence of  $m,\sigma\in \mathbb{R}^n$
with positive coordinates satisfying conditions \eqref{e3.11} 
and \eqref{e3.12}  is equivalent to the requirement
\begin{equation}
 \frac{(n-1)\overline q}{n(\overline q-1)}<q_i
<\frac{(n-1)\overline q}{n-\overline q} \quad \forall i\in \{1,\dots, n\};
\label{e5.2}
\end{equation}

\item[(iii)]  the existence of  $m\in \mathbb{R}^n$  satisfying
conditions \eqref{e3.13}  and \eqref{e3.14}  is equivalent to
requirement \eqref{e5.2}.
\end{itemize}
\end{proposition}

We omit the proof of the proposition because of its simplicity.
Observe that requirement \eqref{e5.2} coincides with the condition
imposed on the corresponding exponents in \cite{b1,b2} where only the
nondegenerate case was considered.


\begin{example} \label{examp5.2} \rm
Let $n\geqslant 3$, $1<\alpha<n/2$,
$\alpha<\beta<n$, and let $q_i=\alpha$ if $i=1,\dots,n-1$, and
$q_n=\beta$. We have
$$
\alpha <\frac{\alpha (n-2)}{n-1-\alpha}< n.
$$
It is easy to verify that requirement \eqref{e5.1} is equivalent to the
condition
\begin{equation}
\beta< \frac{\alpha(n-2)}{n-1-\alpha}\,,\label{e5.3}
\end{equation}
and if $n\geqslant 4$ and $\alpha\geqslant 2-1/n$, then
requirement \eqref{e5.2} is also equivalent to condition \eqref{e5.3}.
\end{example}

As far as the degenerate isotropic case is concerned, the
following proposition holds.


\begin{proposition} \label{prop5.3} 
 For every $i\in \{1,\dots,n\}$, let
$q_i=q_1$  and  $\nu_i=\nu_1$.  Then
\begin{itemize}

\item[(i)] the existence of  $m,\sigma\in \mathbb{R}^n$
 satisfying conditions \eqref{e3.9}  and \eqref{e3.10}  is
equivalent to the existence of  $t,s\in \mathbb{R}$ such that
 $t\geqslant 1/(q_1-1)$, $t>n/q_1$, $s>nt/(tq_1-n)$, $1/\nu_1\in
L^t(\Omega)$  and  $\nu_1\in L^s(\Omega)$;

\item[(ii)] the existence of  $m,\sigma\in \mathbb{R}^n$
 with positive coordinates satisfying conditions \eqref{e3.11} 
and \eqref{e3.12}  is equivalent to the existence of
  $t,s\in \mathbb{R}$  such that  $t>n/q_1$, $1/t<q_1-2+1/n$, $s>nt/(tq_1-n)$,
$1/\nu_1\in L^t(\Omega)$  and $\nu_1\in L^s(\Omega)$;


\item[(iii)] the existence of  $m\in \mathbb{R}^n$  satisfying
conditions \eqref{e3.13}  and \eqref{e3.14}  is equivalent to the
existence of  $t\in \mathbb{R}$  such that  
$t\geqslant 1/(q_1-1)$, $t>n/q_1$, $1/t<q_1(q_1-2+1/n)$  and
 $1/\nu_1\in L^t(\Omega)$.
\end{itemize}
\end{proposition}


\begin{proof} Let $m,\sigma\in\mathbb{R}^n$, and let conditions
\eqref{e3.9} and \eqref{e3.10} be satisfied. Setting
\begin{equation}
t=\max\{m_i: i=1,\dots,n\}, \quad s=\sigma_1, \label{e5.4}
\end{equation}
by conditions \eqref{e3.9} and \eqref{e3.10}, we immediately have
$t\geqslant 1/(q_1-1)$, $1/\nu_1\in L^t(\Omega)$ and
$\nu_1\in L^s(\Omega)$. Moreover, since $\overline q=q_1$ and
$q_1/p_m\geqslant 1-q_1/n+1/t$, from condition \eqref{e3.10} we derive
that $t>n/q_1$ and $s>nt/(tq_1-n)$. Conversely,
let $t,s\in\mathbb{R}$, and let $t\geqslant 1/(q_1-1)$,
$t>n/q_1$, $s>nt/(tq_1-n)$,
$1/\nu_1\in L^t(\Omega)$ and $\nu_1\in L^s(\Omega)$. Then, taking
$m,\sigma\in\mathbb{R}^n$ such that for every $i\in\{1,\dots,n\}$,
 $m_i=t$ and $\sigma_i=s$, without any difficulties we obtain
that conditions \eqref{e3.9} and \eqref{e3.10} are satisfied. Thus, assertion
(i) is valid.

Next, let $m,\sigma\in\mathbb{R}^n$, for every $i\in\{1,\dots,n\}$,
 $m_i>0$ and $\sigma_i>0$, and let conditions \eqref{e3.11} and \eqref{e3.12}
be satisfied. Using these conditions, for $t,s\in \mathbb{R}$
defined by \eqref{e5.4} we easily establish that $t>n/q_1$,
$1/t<q_1-2+1/n$, $s>nt/(tq_1-n)$, $1/\nu_1\in L^t(\Omega)$ and
$\nu_1\in L^s(\Omega)$. Conversely, if we have $t,s\in \mathbb{R}$
with the given properties, then, taking $m,\sigma\in\mathbb{R}^n$
such that for every $i\in\{1,\dots,n\}$,  $m_i=t$ and
$\sigma_i=s$, we easily get that conditions \eqref{e3.11} and \eqref{e3.12} are
satisfied. Thus, assertion (ii) is valid.

Finally, let $m\in\mathbb{R}^n$, and let conditions \eqref{e3.13} and
\eqref{e3.14} be satisfied. Setting $t=\max\{m_i: i=1,\dots,n\}$, we have
\begin{equation}
1-\frac {q_1}n+\frac 1t \leqslant \frac {q_1}{p_m}\,. \label{e5.5}
\end{equation}
At the same time, from condition \eqref{e3.13} we infer that
$t\geqslant 1/(q_1-1)$ and $1/\nu_1\in L^t(\Omega)$, and from
condition \eqref{e3.14} we obtain that $q_1/p_m < \min\{(q_1-1)^2, \,1\}$.
This and \eqref{e5.5} imply that $t>n/q_1$ and $1/t<q_1(q_1-2+1/n)$.
Conversely, if
$t\in \mathbb{R}$, and $t\geqslant 1/(q_1-1)$, $t>n/q_1$,
$1/t<q_1(q_1-2+1/n)$ and $1/\nu_1\in L^t(\Omega)$, then, taking
$m\in\mathbb{R}^n$ such that for every $i\in\{1,\dots,n\}$,
 $m_i=t$, we easily get that conditions \eqref{e3.13} and \eqref{e3.14} are
satisfied. Thus, assertion (iii) is valid. This completes the
proof of the proposition.
\end{proof}

We remark that the conditions on $t$, $s$ and $\nu_1$ in assertion
(i) of Proposition \ref{prop5.3} are of the same kind as in \cite{l2}.
The following two examples concern the degenerate anisotropic
case.

\begin{example} \label{examp5.4} \rm
Let $n\geqslant 3$ and $1<\alpha<n-1$. We have
$\alpha<\alpha(n-2)/(n-1-\alpha)$. Let
\begin{equation}
\alpha\leqslant \beta
<\min\big\{\frac{\alpha(n-2)}{n-1-\alpha}, \,n \big\}.
\label{e5.6}
\end{equation}
Since, by \eqref{e5.6}, $\beta(n-1-\alpha)<\alpha(n-2)$, we have
$(\beta-\alpha)/(\beta-1)<\alpha/(n-1)$. Let
\begin{equation}
0<\gamma<n\min \big\{\frac{\alpha}{n-1}-\frac{\beta-\alpha}{\beta-1},
\,\alpha-1\big\}. \label{e5.7}
\end{equation}
Since, by \eqref{e5.7},
$$
\frac{\gamma}{n}+\frac{\beta-\alpha}{\beta-1}<\frac{\alpha}{n-1}\,,
$$
we have
$$
1-\frac{n-1}{\alpha}\Big(\frac{\gamma}{n}+\frac{\beta-\alpha}{\beta-1}\Big)>0.
$$
Let
\begin{equation}
0<\tau< n\min\big\{\beta
\big[1-\frac{n-1}{\alpha}\big(\frac{\gamma}{n}
+\frac{\beta-\alpha}{\beta-1}\big)\big],
\,\beta-1\big\}. \label{e5.8}
\end{equation}

Next, assume that $\Omega=\{x\in \mathbb{R}^n:|x|<1\}$. Moreover, let
$q_i=\alpha$ and for every $x\in \Omega$, $\nu_i(x)=|x|^\gamma$ if
$i=1,\dots , n-1$, and let $q_n=\beta$ and for every $x\in
\Omega$, $\nu_n(x)=|x|^\tau$.

It is easy to see that for every $i\in \{1,\dots , n\}$, 
$q_i\in (1,n)$ and $\nu_i\in L^1(\Omega)$. Besides, since in view 
of \eqref{e5.7} and \eqref{e5.8}, $\gamma<n(\alpha-1)$ and 
$\tau<n(\beta-1)$, for every
$i\in \{1,\dots , n\}$ we have $(1/\nu_i)^{1/(q_i-1)}\in
L^1(\Omega)$.

Taking into account \eqref{e5.7} and \eqref{e5.8}, we fix a positive number
$\varepsilon_1$ such that
\begin{gather}
\varepsilon_1 \leqslant \min \big\{\frac{n(\alpha-1)}{\gamma}-1,
 \frac{n(\beta-1)}{\tau}-1\big\}, \label{e5.9} \\
\frac{\varepsilon_1}{n}\big[(n-1)\frac{\gamma}{\alpha}+\frac{\tau}{\beta}\big]
< 1- \frac{\tau}{n\beta} -
\frac{n-1}{\alpha}\Big(\frac{\gamma}{n}+\frac{\beta-\alpha}{\beta-1}\Big).
\label{e5.10}
\end{gather}
Now, define $\varepsilon=1+\varepsilon_1$, and let $m\in \mathbb{R}^n$
be such that $m_i=\frac{n}{\gamma\varepsilon}$ if $i=1,\dots ,
n-1$, and $m_n=\frac{n}{\tau\varepsilon}$.

Using \eqref{e5.9} and the inequality $\varepsilon>1$, we establish that
condition \eqref{e3.9} is satisfied.  Moreover, using \eqref{e5.10}, we obtain
\begin{align*}
\frac{1}{p_m} 
&= \frac{1}{n}\Big(\sum^n_{i=1} \frac{1+m_i}{m_iq_i}-1\Big) \\
&=\frac{1}{n}\Big\{\frac{n-1}{\alpha}+\frac{1}{\beta}
+\frac{(n-1)\gamma}{n\alpha}+
\frac{\tau}{n\beta}+\frac{\varepsilon_1}{n}
\big[\frac{(n-1)\gamma}{\alpha}+\frac{\tau}{\beta}\big]-1\Big\} \\
&<\frac{1}{n}\Big\{\frac{1}{\beta}+\frac{(n-1)(\alpha-1)}{\alpha(\beta-1)}\Big\}
=\frac{\overline
q-1}{\overline q(\beta-1)}\,.
\end{align*}
Hence
\begin{equation}
1-\frac{(\beta-1)\overline q}{p_m(\overline q-1)}>0.  \label{e5.11}
\end{equation}
Then, fixing $\beta_0>0$ such that
$$
\frac{1}{\beta_0}<1-\frac{(\beta-1)\overline q}{p_m(\overline
q-1)}
$$
and taking $\sigma\in \mathbb{R}^n$ such that for every
$i\in \{1,\dots , n\}$, $\sigma_i=\beta_0$, due to the inequality
$\alpha\leqslant \beta$, we establish that condition \eqref{e3.10} is
satisfied.

Next, suppose additionally that $n>3$ and $\alpha>2$. Obviously,
$\alpha-1>1/(\beta-1)$, and from \eqref{e5.11} it follows  that condition
\eqref{e3.14} is satisfied. Moreover, if additionally we have
\begin{gather*}
\frac{\gamma}{n}<\alpha-1-\frac{1}{\beta-1}\,,\quad
\frac{\tau}{n}< \alpha-1 -\frac{1}{\beta-1}\,, \\
\frac{\gamma}{n}\, \varepsilon_1<
\alpha-1-\frac{1}{\beta-1}-\frac{\gamma}{n}\,, \quad
\frac{\tau}{n}\,\varepsilon_1<\alpha-1-\frac{1}{\beta-1}-\frac{\tau}{n}\,,
\end{gather*}
then for every $i\in \{1,\dots , n\}$,
$$
\frac{1}{\beta-1}< \alpha-1-\frac{1}{m_i}\,,
$$
and from \eqref{e5.11} it follows that condition \eqref{e3.11} is satisfied.
\end{example}


\begin{example} \label{examp5.5}\rm
 Let $n\geqslant 3$ and $(2n-3)/(n-1)<\alpha<n-1$. We have $\alpha n>2(n-1)$ and
$$
\max\Big\{\frac{\alpha}{\alpha n-2(n-1)}\,, \,\alpha\Big\}<\min
\Big\{\frac{\alpha(n-2)}{n-1-\alpha}\,, \,n\Big\}.
$$
Let
\begin{equation}
\max\Big\{\frac{\alpha}{\alpha n-2(n-1)}\,, \,\alpha\Big\}<\beta
< \min \Big\{\frac{\alpha(n-2)}{n-1-\alpha}\,, \,n\Big\}. \label{e5.12}
\end{equation}
We set
\[
r=n\Big(\frac{n-1}\alpha + \frac 1\beta\Big)^{-1}\,.
\]
Since, by \eqref{e5.12},
$$
\frac{\alpha}{\alpha n -2(n-1)}<\beta <
\frac{\alpha(n-2)}{n-1-\alpha}\,,
$$
we have
\begin{equation}
\Big(\frac 1r - \frac 1n\Big)\frac r{r-1} < \min \Big\{\frac
1{\beta-1}\,, \,\alpha-1\Big\}. \label{e5.13}
\end{equation}
Consequently, taking into account that $\alpha<\beta$,
we obtain
$$
\Big(\frac 1r - \frac 1n\Big) \frac{(\alpha-1)r}{r-1}<1.
$$
We define $\sigma_{\ast}$ by
$$
\frac 1{\sigma_{\ast}}=1-\Big(\frac 1r -\frac
1n\Big)\frac{(\alpha-1)r}{r-1}
$$
and fix $\gamma$ and $\tau$ such that
$n/\sigma_{\ast}\leqslant \gamma<n$ and $0<\tau<n$.

Next, assume that $\Omega=\{x\in \mathbb{R}^n:|x|<1\}$. Moreover,
let $q_i=\alpha$ and for every $x\in \Omega\setminus \{0\}$,
$\nu_i(x)=|x|^{-\gamma}$ if $i=1,\dots , n-1$, and let $q_n=\beta$
and for every $x\in \Omega\setminus \{0\}$,
$\nu_n(x)=|x|^{-\tau}$. It is easy to see that for every  $i\in
\{1,\dots,n\}$,  $q_i\in(1,n)$, $\nu_i\in L^1(\Omega)$ and
$(1/\nu_i)^{1/(q_i-1)}$ $\in L^1(\Omega)$. Besides, we have
\begin{equation}
\overline q=r. \label{e5.14}
\end{equation}
Taking into account \eqref{e5.13}, we fix a number $r_1$ such that
\begin{equation}
\frac 1r - \frac 1n < r_1 < \frac {r-1}r\min\Big\{\frac
1{\beta-1}\,, \,\alpha-1\Big\}, \label{e5.15}
\end{equation}
and then we fix a number $t$ such that $t\geqslant 1/(\alpha-1)$
and
\begin{equation}
\frac 1{tr} < \frac {r-1}r\min\Big\{\frac 1{\beta-1}\,,
\,\alpha-1\Big\} - r_1. \label{e5.16}
\end{equation}
Now let $b\in\mathbb{R}^n$ be such that $b_i=t$, $i=1,\dots,n$. For
every $i\in\{1,\dots,n\}$ we have $b_i\geqslant 1/(q_i-1)$ and
$1/\nu_i\in L^{b_i}(\Omega)$. Moreover,
$$
\frac 1{p_b} = \frac 1r - \frac 1n + \frac 1{tr}\,.
$$
This equality along with \eqref{e5.14}--\eqref{e5.16} implies that
$$
\frac 1{p_b} < \frac {\overline q-1}{\overline q}\min\Big\{\frac
1{\beta-1}\,, \,\alpha-1\Big\}.
$$
Hence it follows that for every $i\in\{1,\dots,n\}$,
$$
p_b > \frac {\overline q}{\overline q-1}\max\Big\{\frac
1{q_i-1}\,, \,q_i-1\Big\}.
$$

Thus, we conclude that there exists $m\in\mathbb{R}^n$ such that
conditions \eqref{e3.13} and \eqref{e3.14} are satisfied. At the same time,
since $\gamma\sigma_{\ast}\geqslant n$, we have 
$\nu_1\notin L^{\sigma_{\ast}}(\Omega)$. This and \eqref{e5.14} 
imply that there are no $m,\sigma\in \mathbb{R}^n$ such that both 
conditions \eqref{e3.9} and \eqref{e3.10} are satisfied, and there are no
 $m,\sigma\in \mathbb{R}^n$ with
positive coordinates such that both conditions \eqref{e3.11} and \eqref{e3.12}
are satisfied.
\end{example}

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\end{document}