# Existence and uniqueness of solutions to some anisotropic elliptic equations
with a singular convection term

G. di Blasio – F. Feo – G. Zecca Dipartimento di Matematica e Fisica,
Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln, 5 -
81100 Caserta, Italy. E–mail<EMAIL_ADDRESS>di Ingegneria, Università degli Studi di Napoli “Pathenope”, Centro
Direzionale Isola C4 80143 Napoli, Italy. E–mail:
<EMAIL_ADDRESS>Dipartimento di Matematica e Applicazioni “R.
Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso
Universitario Monte S. Angelo, Via Cintia 26, Napoli, Italy. E–mail:
<EMAIL_ADDRESS>
###### Abstract

We prove the existence and uniqueness of weak solutions to a class of
anisotropic elliptic equations with coefficients of convection term belonging
to some suitable Marcinkiewicz spaces. Some useful a priori estimates and
regularity results are also derived.

00footnotetext: Mathematics Subject Classifications: 35J25, 35B45 Key words:
Noncoercive Anisotropic nonlinear equations, Lower order terms, Existence,
Uniqueness, Regularity

##  1 Introduction

In this paper we obtain existence and uniqueness results for the weak
solutions of the following class of Dirichlet problems

$\left\\{\begin{array}[c]{ll}-\sum_{i=1}^{N}\partial_{x_{i}}\left[\mathcal{A}_{i}(x,\nabla
u)+\mathcal{B}_{i}(x,u)\right]+\mathcal{G}(x,u)=\mathcal{F}&\hbox{ in
}\Omega,\\\ &\\\ u=0&\hbox{ on }\partial\Omega,\end{array}\right.$ (1.1)

where $\Omega$ is a bounded domain of ${\mathbb{R}}^{N}$ with Lipschitz
boundary, $N>2$, $p_{i}>1$ for every $i=1,...,N$ with $\bar{p}<N$, denoting by
$\overline{p}$ the harmonic mean of $\vec{p}=(p_{1},\cdots,p_{N})$, i.e.

$\frac{1}{\overline{p}}=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{p_{i}}.$ (1.2)

Throughout this paper, we make the following assumptions for any $i=1,...,N,$

$(\mathcal{H}1)\quad\mathcal{A}_{i}:\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R}$
is a Carathéodory function that satisfies

$\displaystyle|\mathcal{A}_{i}(x,\xi)|\leqslant\beta_{i}|\xi_{i}|^{p_{i}-1}$
(1.3)
$\displaystyle\alpha|\xi_{i}|^{p_{i}}\leqslant\mathcal{A}_{i}(x,\xi)\,\xi_{i}$
(1.4) $\displaystyle
0<(\mathcal{A}_{i}(x,\xi)-\mathcal{A}_{i}(x,\eta))\,(\xi_{i}-\eta_{i})\quad\,\,\xi\neq\eta$
(1.5)

for a.e. $x\in\Omega$ and for any vector $\xi,\eta$ in $\mathbb{R}^{N}$, where
$0<\alpha\leqslant\beta_{i}$ are constants.

$(\mathcal{H}2)\quad\mathcal{B}_{i}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Carathéodory function such that

$|\mathcal{B}_{i}(x,s)|\leqslant
b_{i}(x)|s|^{\frac{\bar{p}}{p_{i}^{\prime}}},$ (1.6)

for a.e. $x\in\Omega$ and for every $s\in\mathbb{R}$, with
$b_{i}:\,\Omega\to[0,+\infty)$ measurable function such that

$b_{i}\in L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega).$ (1.7)

$(\mathcal{H}3)$ $\mathcal{G}:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is
a Carathéodory function such that

$|\mathcal{G}(x,s)|\leqslant\tilde{\mu}|s|^{\gamma}$ (1.8)

with $1\leq\gamma<p_{\infty}-1$ and

$\mathcal{G}(x,s)\,s\geq 0$ (1.9)

for a.e. $x\in\Omega$ and for every $s\in\mathbb{R}$, where $\tilde{\mu}$ is a
non negative constant and $p_{\infty}=\max\\{\overline{p}^{*},p_{\max}\\}$,
with $p_{\max}=\max_{i}p_{i}$.

$(\mathcal{H}4)$ $\mathcal{F}$ belongs to the dual space
$(W_{0}^{1,\vec{p}}(\Omega))^{*}$, where $W_{0}^{1,\vec{p}}(\Omega)$ is the
anisotropic Sobolev space defined in Section 2.

We observe that in our assumptions, the following definition of weak solution
is well posed.

###### Definition 1.1

For any $\mathcal{F}\in(W_{0}^{1,\vec{p}}(\Omega))^{*}$ we say that $u\in
W_{0}^{1,\vec{p}}(\Omega)$ is a weak solution to (1.1) provided

$\int_{\Omega}\left[\sum_{i=1}^{N}\left[\mathcal{A}_{i}(x,\nabla
u)+\mathcal{B}_{i}(x,u)\right]\partial_{x_{i}}\varphi+\mathcal{G}(x,u)\varphi\right]\,dx=\langle\mathcal{F}\,,\varphi\rangle$
(1.10)

$\forall\varphi\in C^{\infty}_{0}(\Omega)$, where $\langle\cdot,\cdot\rangle$
denotes the duality product of $(W_{0}^{1,\vec{p}}(\Omega))^{*}$ and
$W_{0}^{1,\vec{p}}(\Omega)$.

In the anisotropic framework the natural space where looking for weak
solutions of Dirichlet problem (1.1) is the anisotropic Sobolev space
$W_{0}^{1,\overrightarrow{p}}(\Omega)$ (see $\S$ 2 for definition). When the
harmonic mean $\bar{p}$, defined in (1.2), is less then the dimension $N$, it
is well-known (see $\S$ 2) that $W_{0}^{1,\overrightarrow{p}}(\Omega)$ is
continuously embedded in the Lorentz space $L^{\bar{p}^{*},\bar{p}}(\Omega)$.
On the other hand, by Poincaré inequality (see (2.7) below) the space
$W_{0}^{1,\overrightarrow{p}}(\Omega)$ is embedded in the Lebesgue space
$L^{p_{\max}}(\Omega)$. This suggests us to link, as in our assumption (1.8),
the growths of zero order term with $p_{\infty}$, which takes over how the
$p_{i}$ are spread out.

The prototype equation of our class of problems (1.1) is

$-\sum_{i=1}^{N}\partial_{x_{i}}\left[|\partial_{x_{i}}u|^{p_{i}-2}\partial_{x_{i}}u+\beta_{i}(x)|u|^{\frac{\bar{p}}{p_{i}^{\prime}}-1}u\right]+\tilde{\mu}|u|^{\gamma-1}u=\mathcal{F}\qquad\mbox{
in }\Omega,$ (1.11)

where $p_{i}\geq 1,\bar{p}<N,\tilde{\mu}\geq
0,1\leq\gamma<p_{\infty}-1,\beta_{i}\in
L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega)$ and $\mathcal{F}$
belongs to the dual space. We stress that when $\bar{p}^{*}\geq p_{\max}$
using the Sobolev embedding in the Lorentz space
$L^{\bar{p}^{*},\bar{p}}(\Omega)$ the summability assumption on $\beta_{i}$ is
optimal to assure $|\beta_{i}|^{p^{\prime}_{i}}|u|^{\bar{p}}\in
L^{1}(\Omega)$. Moreover we observe that when $p_{i}=p$, $\beta_{i}=0$ and
$\widetilde{\mu}=0$, the principal part in equation (1.11) becomes the so-
called pseudo-Laplacian operator (see [27] pp. 106 and 155) or orthotropic
$p$-Laplacian operator (see [11]), extensively studied in the literature.

Let us point out that term anisotropy is used in various scientific
disciplines and could have a different meaning when it is related to equations
as well. The interest in anisotropic problems has deeply increased in the last
years since their many applications in mathematical modelling for natural
phenomena in biology and fluid mechanics. For example, they are related to the
mathematical description of the dynamics of fluids in anisotropic media when
the conductivities of the media are different in different directions (see
e.g. [3]) and they also appear in biology as a model for the propagation of
epidemic diseases in heterogeneous domains (see e.g. [6]). On anisotropic
problems many results in different directions have been obtained, here we
quote a list of references that is obviously not exhaustive and we refer the
reader to references therein to extend it: [1, 4, 5, 9, 11, 12, 13, 16, 19,
22, 17, 20, 23, 26, 31].

A goal of this paper is to analyze the existence of weak solutions for this
class of problems (see [21] for the isotropic case). Since under our
assumptions the coercivity for the involved operator in problem (1.1) is not
guaranteed, we will proceed as usual by approximations. The existence of weak
solutions can be expected when the datum and the coefficients are smooth
enough. Then we first consider problem with $b_{i}\in L^{\infty}(\Omega)$ and
then we reduce to the general case $b_{i}\in
L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega)$ assuming a control on
suitable distance of $b_{i}$ from $L^{\infty}(\Omega)$ (see assumption (3.2)
in Theorem 5.1). This strategy allows us to overcome that the norms in the
Marcinkiewicz $L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}$ are not
absolutely continuous and that $L^{\infty}(\Omega)$ is not dense in
Marcinkiewicz spaces. Finally when we pass to the limit in the approximated
problems we have to deal with an extra difficulty due to the fact that in our
assumption the operator $u\in
W_{0}^{1,\overrightarrow{p}}(\Omega)\rightarrow{b_{i}}(x)|u|^{\frac{\bar{p}}{p^{\prime}_{i}}-1}u\in
L^{p^{\prime}_{i}}(\Omega)$ is not compact in general. We emphasize that our
assumption on the distance (3.2), firstly considered in [24] in the isotropic
case (see also [25], [21]), is weaker than asking the smallness of the norms
of $b_{i}$ which is the standard approach to treat the presence of the first
order term.

We also analyze the regularity for weak solutions of problem (1.1) and we
obtain Stampacchia type regularity, extending [10, 31, 20] where the case
$\mathcal{B}\equiv 0$ is studied. Regularity results for local solutions of
problem (1.1) has been recently obtained in [18].

For what concerns the uniqueness (see Theorems 7.1, 7.3 and 7.5) we emphasize
that our proofs strongly uses the presence of the zero order term and his
monotonicity assumption (7.1) below. In order to describe our result, let us
reduce ourselves to the model case (1.11) for simplicity. The main difficulty
is due to the presence of the lower order terms that are only Hölder
continuous, but not Lipschitz continuous with respect to solution $u$ when
$p_{i}<2$ for all $i$. We recall [7] for equations with a Hölder continuous
dependence on the solution of the coefficient in the main part of the
operator.

If one wants to neglect the term $\mathcal{G}$, Corollary 6.2 gives a partial
uniqueness result when the datum is zero (see [8] for isotropic case).
Otherwise also in the isotropic case the uniqueness is proved in [8] requiring
a control of the partial derivative of first order term $B(x,u)$ with respect
to $u$, which is not satisfied in the simple case
$B(x,u)=\textbf{b}(x)|u|^{p-2}u$ with
$\textbf{b}\in\left(L^{\frac{N}{p-1}}(\Omega)\right)^{N}$ .

The paper is organized as follows. Section 2 contains some preliminaries and
in Section 3 we prove some useful a priori estimates. In our strategy we need
some $L^{\infty}$ estimate of $u$ as well. For completeness we write Section 4
to analyze the regularity of the solutions. The existence theorem is stated
and proved in Section 5. The last section is devoted to the uniqueness
results.

##  2 Preliminaries

In the present section we recall some known function spaces, useful in the
sequel.

We start by recalling definitions of Lorentz spaces and their properties (see
[29] for more details).

Here we assume that $\Omega\subset{\mathbb{R}}^{N}$, $N>2$ is an open set.
Given $1<p,q<+\infty$, the Lorentz space $L^{p,q}(\Omega)$ consists of all
measurable functions $f$ defined on $\Omega$ for which the quantity

$\|f\|_{p,q}^{q}=p\int_{0}^{+\infty}|\Omega_{t}|^{\frac{q}{p}}t^{q-1}dt$ (2.1)

is finite, where $\Omega_{t}=\left\\{x\in\Omega:|f(x)|>t\right\\}$ and
$|\Omega_{t}|$ is the Lebesgue measure of $\Omega_{t}$, that is,
$\mu_{f}(t)=|\Omega_{t}|$ is the distribution function of $f$. Note that
$\|\cdot\|_{p,q}$ is equivalent to a norm and $L^{p,q}(\Omega)$ becomes a
Banach space when endowed with it. For $p=q$, the Lorentz space
$L^{p,p}(\Omega)$ reduces to the Lebesgue space $L^{p}(\Omega)$. For
$q=\infty$, the class $L^{p,\infty}(\Omega)$ consists of all measurable
functions $f$ defined on $\Omega$ such that

$\|f\|^{p}_{p,\infty}=\sup_{t>0}t^{p}\mu_{f}(t)<+\infty$ (2.2)

and it coincides with the Marcinkiewicz class or the so-called
weak-$L^{p}(\Omega)$.

It well-known that if $\Omega$ is bounded, the following inclusions hold

$L^{r}(\Omega)\subset L^{p,q}(\Omega)\subset L^{p,r}(\Omega)\subset
L^{p,\infty}(\Omega)\subset L^{q}(\Omega),$ (2.3)

whenever $1\leqslant q<p<r\leqslant\infty.$ Moreover, for $1<p<\infty$,
$1\leqslant q\leqslant\infty$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$,
$\frac{1}{q}+\frac{1}{q^{\prime}}=1$, if $f\in L^{p,q}(\Omega)$, $g\in
L^{p^{\prime},q^{\prime}}(\Omega)$ we have the Hölder–type inequality

$\int_{\Omega}|f(x)g(x)|dx\leqslant\|f\|_{p,q}\|g\|_{p^{\prime},q^{\prime}}.$

We remark that $L^{\infty}(\Omega)$ is not dense in $L^{p,\infty}(\Omega)$ for
$p\in{}]1,+\infty[$. Therefore it is possible to define the distance of a
given function $f\in L^{p,\infty}(\Omega)$ to $L^{\infty}(\Omega)$ as

${\rm{dist}}_{L^{p,\infty}(\Omega)}(f,L^{\infty}(\Omega))=\inf_{g\in
L^{\infty}(\Omega)}\|f-g\|_{L^{p,\infty}(\Omega)}.$ (2.4)

Note that, since $\|~{}\|_{p,\infty}$ is not a norm,
${\rm{dist}}_{L^{p,\infty}(\Omega)}$ is just equivalent to a metric. In [15]
is proved that

${\rm{dist}}_{L^{p,\infty}(\Omega)}(f,L^{\infty}(\Omega))=\lim_{M\to+\infty}\|f-T_{M}f\|_{L^{p,\infty}(\Omega)},$
(2.5)

where the truncation at level $M>0$ is defined as

$T_{M}\,y=\frac{y}{|y|}\min\\{|y|,M\\}.$ (2.6)

Now, let $\vec{p}=(p_{1},p_{2},...,p_{N})$ with $p_{i}>1$ for $i=1,...,N$, and
let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{N}$. As usual the
anisotropic Sobolev space is the Banach space defined as

$W^{1,\vec{p}}(\Omega)=\\{u\in W^{1,1}(\Omega):\partial_{x_{i}}u\in
L^{p_{i}}(\Omega),i=1,...,N\\}$

equipped with

$\|u\|_{W^{1,\vec{p}}(\Omega)}=\|u\|_{L^{1}(\Omega)}+\sum_{i=1}^{N}\|\partial_{x_{i}}u\|_{L^{p_{i}}(\Omega)}.$

It is well-known that in the anisotropic setting a Poincarè type inequality
holds true (see [23]). Indeed for every $u\in C_{0}^{\infty}(\Omega)$ with
$\Omega$ a bounded open set with Lipschitz boundary we have

$\|u\|_{L^{p_{i}}(\Omega)}\leq
C_{P}\|\partial_{x_{i}}u\|_{L^{p_{i}}(\Omega)},\qquad\qquad i=1,...,N$ (2.7)

for a constant $C_{P}$ proportional to the width of $\Omega$ in the direction
of $e_{i}$ and then to the diameter of $\Omega$. Moreover, for $u\in
C_{0}^{\infty}(\mathbb{R^{N}})$ the following anisotropic Sobolev inequality
holds true (see [32])

$\|u\|_{L^{p,q}(\mathbb{R^{N}})}\leq
S_{N}\prod_{i=1}^{N}\|\partial_{x_{i}}u\|_{L^{p_{i}}(\mathbb{R^{N}})}^{\frac{1}{N}},$
(2.8)

where $S_{N}$ is an universal constant and $p=\bar{p}^{*}$ and $q=\bar{p}$
whenever $\bar{p}<N$, where $\bar{p}$ is defined in (1.2) and

$\bar{p}^{*}=\frac{N\bar{p}}{N-\bar{p}}.$ (2.9)

Using the inequality between geometric and arithmetic mean we can replace the
right-hand-side of (2.8) with
$\sum_{i=1}^{N}\|\partial_{x_{i}}u\|_{L^{p_{i}}}$. In [18] is proved a
generalization of (2.8) to product of function. We recall an its simpler form
that we will need in the following

$\left\|u\right\|_{L^{\bar{p}^{*},\bar{p}}(\Omega)}\leq\widetilde{S}_{N}\left\|\left(\prod_{i=1}^{N}\left|\partial_{x_{i}}u\right|\right)^{1/N}\right\|_{L^{\overline{p}}(\Omega)},$
(2.10)

for a suitable universal constant $\widetilde{S}_{N}$.

When $\overline{p}<N$ and $\Omega$ is a bounded open set with Lipschitz
boundary, the space
$W_{0}^{1,\overrightarrow{p}}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}^{\sum_{i=1}^{N}\|\partial_{x_{i}}u\|_{L^{p_{i}}}}$
is continued embedding into $L^{q}(\Omega)$ for $q\in[1,p_{\infty}]$, with
$p_{\infty}:=\max\\{\overline{p}^{\ast},\max_{i}p_{i}\\}$ as a consequence of
(2.8) and (2.7).

We recall the following useful lemma.

###### Lemma 2.1

_(_ see [28, page 43]_)_ Let $X$ be a rearrangement invariant space and let
$0\leq\theta_{i}\leq 1$ for $i=1,...,M,$ such that
$\sum_{i=1}^{M}\theta_{i}=1$, then

$\left\|\prod_{i=1}^{M}|f_{i}|^{\theta_{i}}\right\|_{X}\leq\prod_{i=1}^{M}\|f_{i}\|_{X}^{\theta_{i}}\quad\forall
f_{i}\in X.$

##  3 An useful a priori estimates

In this section we suppose that a weak solution $u$ of the problem (1.1)
exists and we prove the following a priori estimate under suitable assumptions
on
${\rm{dist}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}(b_{i},L^{\infty}(\Omega))$,
defined in (2.4), for $i=1,\cdots,N$.

Since $\mathcal{F}$ belongs to $(W_{0}^{1,\vec{p}}(\Omega))^{*}$, in what
follows we can write it as

$\mathcal{F}=-\sum_{i=1}^{N}(f_{i})_{x_{i}},\mbox{ with }f_{i}\in
L^{p_{i}^{\prime}}(\Omega)\,\,\forall i=1,...,N.$ (3.1)

###### Lemma 3.1

Let us assume that $\Omega$ is a bounded Lipschitz domain, $p_{i}>1$ for
$i=1,..,N$, $\bar{p}<N$, $(\mathcal{H}1)-(\mathcal{H}4)$ are in force and let
$u\in W_{0}^{1,\vec{p}}(\Omega)$ be a weak solution of (1.1). Then there
exists a positive constant $d=d(N,\alpha,\vec{p})$ such that whenever

$\max_{i}\left\\{{\emph{dist}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}(b_{i},L^{\infty}(\Omega))\right\\}<d,$
(3.2)

the following uniform estimate holds

$\sum_{i=1}^{N}\int_{\Omega}|u_{x_{i}}|^{p_{i}}dx\leq C,$ (3.3)

where
$C=C(\alpha,N,\overrightarrow{p},d,\|\mathcal{F}\|_{(W_{0}^{1,\vec{p}}(\Omega))^{*}}).$

Proof. Using as test function $T_{k}u$ in (1.10), by (1.4) and (1.6) we get

$\displaystyle\alpha\sum_{i=1}^{N}\int_{\Omega_{k}}|u_{x_{i}}|^{p_{i}}dx+\int_{\Omega}\mathcal{G}(x,u)T_{k}u\,dx$
$\displaystyle\leq\sum_{i=1}^{N}\int_{\Omega_{k}}|b_{i}(x)||u|^{\frac{\bar{p}}{p_{i}^{\prime}}}|u_{x_{i}}|dx+\sum_{i=1}^{N}\int_{\Omega_{k}}|f_{i}||u_{x_{i}}|dx,$

where $\Omega_{k}=\\{x\in\Omega:|u(x)|\normalcolor<k\\}$. For all $M>0$, using
(1.9), Young and Hölder inequality we have

$\displaystyle\sum_{i=1}^{N}\int_{\Omega_{k}}|u_{x_{i}}|^{p_{i}}dx\leq$
$\displaystyle
C\left(\sum_{i=1}^{N}\int_{\Omega_{k}}|u|^{\bar{p}}dx+\sum_{i=1}^{N}\|b_{i}(x)-T_{M}b_{i}\|^{p^{\prime}_{i}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}\|T_{k}u\|_{L^{\bar{p}^{*},\bar{p}}(\Omega)}^{\bar{p}}\right.$
$\displaystyle+\left.\sum_{i=1}^{N}\int_{\Omega_{k}}|f_{i}|^{p^{\prime}_{i}}dx\right),$

where $C$ is a suitable positive constant which can vary from line to line.

Denoting $B=\prod_{i=1}^{N}\|(T_{k}u)_{x_{i}}\|_{L^{p_{i}}(\Omega)}$, by
Sobolev inequality (2.8) it follows

$\displaystyle\sum_{i=1}^{N}\int_{\Omega_{k}}|u_{x_{i}}|^{p_{i}}dx\leq$
$\displaystyle
C\left(\int_{\Omega_{k}}|u|^{\bar{p}}dx+\sum_{i=1}^{N}\|b_{i}(x)-T_{M}b_{i}\|^{p^{\prime}_{i}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}B^{\frac{\bar{p}}{N}}\right.$
(3.4)
$\displaystyle+\left.\sum_{i=1}^{N}\int_{\Omega_{k}}|f_{i}|^{p^{\prime}_{i}}dx\right).$

Previous inequality gives us an estimate of the $j^{th}$ addendum of the sum
at the left-hand side of (3.4) as well. Then, elevating to the power
$\frac{1}{Np_{j}}$, making the product on the left and right sides of (3.4) we
get

$\displaystyle B^{\frac{1}{N}}\leq$ $\displaystyle
C\left[\left(\int_{\Omega_{k}}|u|^{\bar{p}}dx\right)^{\frac{1}{\bar{p}}}+\sum_{i=1}^{N}\|b_{i}(x)-T_{M}b_{i}\|^{\frac{p^{\prime}_{i}}{\bar{p}}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}B^{\frac{1}{N}}+\left(\sum_{i=1}^{N}\int_{\Omega_{k}}|f_{i}|^{p^{\prime}_{i}}dx\right)^{\frac{1}{\bar{p}}}\right].$

At this point, assuming
$d<\left(\frac{1}{CN}\right)^{\frac{\bar{p}}{p^{\prime}_{i}}}$ in (3.2) and
using (2.5), we can fix now $M$ large enough in order to have

$\displaystyle B^{\frac{1}{N}}\leq$ $\displaystyle
C\left[\left(\int_{\Omega_{k}}|u|^{\bar{p}}dx\right)^{\frac{1}{\bar{p}}}+\left(\sum_{i=1}^{N}\int_{\Omega_{k}}|f_{i}|^{p^{\prime}_{i}}dx\right)^{\frac{1}{\bar{p}}}\right].$
(3.5)

Combining (3.4) and (3.5) we obtain

$\displaystyle\sum_{i=1}^{N}\int_{\Omega_{k}}$
$\displaystyle|u_{x_{i}}|^{p_{i}}dx\leq
C\left(\int_{\Omega_{k}}|u|^{\bar{p}}dx+\sum_{i=1}^{N}\int_{\Omega_{k}}|f_{i}|^{p^{\prime}_{i}}dx\right),$
(3.6)

for every $k>0$ and with $C$ positive costant independent of $u$ and $k.$ We
prove now that previous estimate gives (3.3). To this aim we follow the idea
of [21, Lemma 2]. We argue by contradiction and assume that there exists a
sequence of functions $\\{u_{n}\\}_{n}\subseteq W^{1,\vec{p}}_{0}(\Omega)$
satisfying (3.6) and such that

$\|u_{n}\|:=\|u_{n}\|_{W^{1,\overrightarrow{p}}_{0}(\Omega)}\rightarrow\infty$

as $n\rightarrow\infty$. For every $n\in{\mathbb{N}}$ and $\varepsilon>0$, we
set $k_{n}=\varepsilon\|u_{n}\|$ so that by (3.6), for $i=1,...,N,$

$\begin{split}\left(\int_{\Omega}|\partial_{x_{i}}(T_{k_{n}}u_{n})|^{p_{i}}\,\mathrm{d}x\right)^{\frac{1}{p_{i}}}&\leqslant
C^{\frac{1}{p_{i}}}\left(1+\int_{\Omega}|u_{n}|^{\bar{p}}\chi_{\\{|u_{n}|<k_{n}\\}}\,\mathrm{d}x\right)^{\frac{1}{p_{i}}},\\\
\end{split}$

where $T_{k_{n}}$ is defined in (2.6). Since $\bar{p}<\bar{p}^{*}$, using
Lemma 2.1 with $X=L^{\bar{p}}(\Omega)$ and
$\theta_{i}=\frac{\bar{p}}{p_{i}N}$, we get from the previous inequality

$\begin{split}\left(\int_{\Omega}\prod^{N}_{i=1}|\partial_{x_{i}}(T_{k_{n}}u_{n})|^{\frac{\bar{p}}{N}}\,\mathrm{d}x\right)^{\frac{1}{\bar{p}}}&\leqslant\prod^{N}_{i=1}\left(\int_{\Omega}|\partial_{x_{i}}(T_{k_{n}}u_{n})|^{p_{i}}\,\mathrm{d}x\right)^{\frac{1}{Np_{i}}}\\\
&\leqslant
C^{\frac{1}{\bar{p}}}\left(1+\int_{\Omega}|u_{n}|^{\bar{p}}\chi_{\\{|u_{n}|<k_{n}\\}}\,\mathrm{d}x\right)^{\frac{1}{\bar{p}}}.\\\
\end{split}$ (3.7)

We set

$w_{n}=\frac{u_{n}}{\|u_{n}\|}\,.$

so that, up to subsequence not relabeled, there exists $\bar{w}\in
W^{1,\overrightarrow{p}}_{0}(\Omega)$ such that $w_{n}\rightharpoonup\bar{w}$
weakly in $W^{1,\overrightarrow{p}}_{0}(\Omega)$, $w_{n}\to\bar{w}$ strongly
in $L^{\bar{p}}(\Omega)$ and $w_{n}\to\bar{w}$ for a.e. in $\Omega$. Dividing
both sides of (3.7) by $\|u_{n}\|^{\bar{p}}$ we have

$\int_{\Omega}\prod^{N}_{i=1}|\partial_{x_{i}}(T_{\varepsilon}w_{n})|^{\frac{\bar{p}}{N}}\,\mathrm{d}x=\int_{\Omega}\frac{\prod^{N}_{i=1}|\partial_{x_{i}}(T_{k_{n}}u_{n})|^{\frac{\bar{p}}{N}}}{\|u_{n}\|^{\bar{p}}}\,\mathrm{d}x\leqslant
C\left(\frac{1}{\|u_{n}\|^{\bar{p}}}+\int_{\Omega}|w_{n}|^{\bar{p}}\chi_{\\{|w_{n}|<\varepsilon\\}}\,\mathrm{d}x\right).$
(3.8)

Assume now that

$|\\{x\in\Omega:|\bar{w}(x)|=\varepsilon\\}|=0\,.$ (3.9)

In this case we have
$\chi_{\\{|w_{n}|<\varepsilon\\}}\to\chi_{\\{|\bar{w}|<\varepsilon\\}}$ a.e.
in $\Omega$ and hence
$w_{n}\,\chi_{\\{|w_{n}|<\varepsilon\\}}\to\bar{w}\,\chi_{\\{|\bar{w}|<\varepsilon\\}}$
strongly in $L^{\bar{p}}(\Omega)$. So, since
$T_{\varepsilon}w_{n}\rightharpoonup T_{\varepsilon}\bar{w}$ weakly in
$W^{1,\overrightarrow{p}}_{0}(\Omega)$ and $T_{\varepsilon}w_{n}\to
T_{\varepsilon}\bar{w}$ strongly in $L^{\bar{p}}(\Omega)$, letting
$n\to+\infty$ in (3.8), using the semicontinuity of the norm with respect to
weak convergence in $W^{1,\overrightarrow{p}}_{0}(\Omega)$, we arrive to the
following estimate

$\int_{\Omega}{\prod^{N}_{i=1}|\partial_{x_{i}}(T_{\varepsilon}\bar{w})|^{\frac{\bar{p}}{N}}}\,\mathrm{d}x\leqslant
C\int_{\Omega}|\bar{w}|^{\bar{p}}\chi_{\\{|\bar{w}|<\varepsilon\\}}\,\mathrm{d}x.$
(3.10)

Using Sobolev inequality (2.10) and Hölder inequality by (3.10) we get

$\varepsilon^{\bar{p}}\,|\\{x\in\Omega:|\bar{w}|\geqslant\varepsilon\\}|^{\frac{\bar{p}}{\bar{p}*}}\leqslant
C\,\varepsilon^{\bar{p}}\,|\\{x\in\Omega:0<|\bar{w}|<\varepsilon\\}|\,.$

Passing to the limit as $\varepsilon\downarrow 0$, we deduce

$|\\{x\in\Omega:|\bar{w}|>0\\}|=0\,,$

that is, $\bar{w}(x)=0$ a.e. Note that previous equality has been obtained
assuming (3.9). Nevertheless, the set of values $\varepsilon>0$ for which
(3.9) fails is at most countable. This means $w_{n}\rightharpoonup 0$ weakly
in $W^{1,\overrightarrow{p}}_{0}(\Omega)$. On the other hand, at this point we
can use again the same argument above to obtain that $w_{n}\to 0$ strongly in
$W^{1,\overrightarrow{p}}_{0}(\Omega)$, and this gives the contradiction,
since by definition $\|w_{n}\|=1$ for every $n\in{\mathbb{N}}$.

##  4 Regularity results

In this section we study the regularity of weak solutions of problem (1.1).
Here and in the following we shall assume notation (3.1) being in force. When
the datum $f_{i}\in L^{p_{i}}(\Omega)$ for $i=1,..,N$ with $\bar{p}<N$, as
follows by the anisotropic Sobolev embedding, a solution $u$ belongs to
$L^{\overline{p}^{*}}(\Omega)$, where $\bar{p}^{*}$ is defined in (2.9).
Otherwise if ${f}_{i}\in L^{s_{i}}(\Omega)$ with $s_{i}>p_{i}$ for $i=1,..,N$,
the summability of $u$ improves. In order to study the higher summability of
$u$, the following minimum

$\mu=\min_{i}\left\\{\frac{s_{i}}{p_{i}^{\prime}}\right\\}$ (4.1)

first introduced in [10], plays a crucial role. In [18, Theorem 2.1 and Remark
5.3] the following Stampacchia type regularity result is proved (see [30] as
classical reference).

###### Theorem 4.1

Let $\Omega$ be a bounded Lipschitz domain, $p_{i}>1$ for $i=1,..,N$,
$\bar{p}<N$ and let $s_{1},\cdots,s_{N}$ be such that

$1<\mu<\frac{N}{\bar{p}},$

where $\mu$ is defined in (4.1). Assume that $(\mathcal{H}1)-(\mathcal{H}3)$
are fulfilled and $f_{i}\in L^{s_{i}}(\Omega)$ for $i=1,..,N$. There exists a
positive constant $d=d(\vec{r},N,\alpha,\vec{p})$ such that if

$\max_{i}\left\\{{\emph{dist}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}(b_{i},L^{\infty}(\Omega))\right\\}<d$

and $u\in W^{1,\vec{p}}_{0}(\Omega)$ is a weak solution to (1.1), then

$u\in L^{s}(\Omega)\quad\text{with }s=\max\\{(\mu\bar{p})^{*},\mu
p_{\max}\\}.$

where

$(\mu\bar{p})^{*}=\frac{N\mu\bar{p}}{N-\mu\bar{p}}.$

Without lower order terms in [10] the authors have proved that the boundedness
of a weak solution of Dirichlet problems is guaranteed under the assumption
$\mu>\frac{N}{\bar{p}},$ where $\mu$ is defined in (4.1). However if
$\mathcal{B}_{i}\not\equiv 0$ for $i=1,\cdots,N$ the boundedness is not
assured assuming that (3.2) is in force as showed in Example 4.8 of [24] (when
$p_{i}=2$ for $i=1,\cdots,N$). The smallness of
$\|b_{i}\|_{L^{\frac{Np_{i}^{\prime}}{\bar{p}},\infty}}$ for $i=1,\cdots,N$
neither is sufficient to get boundedness, as showed in Example 2.3 of [18].

In order to get the boundedness of solutions, we need to improve the
summability of data and of the coefficients $b_{i}$.

###### Lemma 4.2

Let $\Omega$ be a bounded Lipschitz domain, $p_{i}>1$ for $i=1,..,N$,
$\bar{p}<N$, and let $s_{1},\cdots,s_{N}$ be such that

$\mu>\frac{N}{\bar{p}}$ (4.2)

where $\mu$ is defined in (4.1). Assume that $(\mathcal{H}1)-(\mathcal{H}3)$
are fulfilled and $b_{i},f_{i}\in L^{s_{i}}(\Omega)$ for $i=1,..,N$. Then
every weak solution $u$ of problem (1.1) is bounded and there exists a
positive constant
$C=C(\alpha,\Omega,N,\overrightarrow{p},\|{b}_{i}\|_{L^{s_{i}}(\Omega)},\|{f}_{i}\|_{L^{s_{i}}(\Omega)})$
such that

$\|u\|_{\infty}\leq C.$

Proof. The proof is quite standard, and for the convenience of the reader we
give some details. So, let $u$ be a weak solution of problem (1.1). For $k>0$,
we use as a test function $G_{k}u:=u-T_{k}u$ in (1.10) and by (1.3), (1.6) we
obtain

$\displaystyle\alpha\sum_{i=1}^{N}\int_{\Omega}|(G_{k}u)_{x_{i}}|^{p_{i}}dx+\int_{\Omega}\mathcal{G}(x,u)G_{k}udx$
$\displaystyle\leq\sum_{i=1}^{N}\int_{\Omega}|b_{i}(x)||u|^{\frac{\bar{p}}{p_{i}^{\prime}}}|(G_{k}u)_{x_{i}}|dx+\sum_{i=1}^{N}\int_{\Omega}|f_{i}||(G_{k}u)_{x_{i}}|dx.$

Denoting $A_{k}=\\{x\in\Omega:|u(x)|>k\\}$, we note that by Lemma 3.1 for
every $k\geq 0$ we have:

$|A_{k}|\leq\frac{\|u\|_{L^{1}(\Omega)}}{k}\leq\frac{C_{0}}{k},$ (4.3)

where $C_{0}$ is a positive constant indepedent on $u$. We stress that (3.2)
is obviously satisfied when $b_{i}\in L^{s_{i}}(\Omega)$.

Using (1.9) and Young inequality we have

$\displaystyle\alpha\sum_{i=1}^{N}\int_{\Omega}|(G_{k}u)_{x_{i}}|^{p_{i}}dx\leq
C\sum_{i=1}^{N}\int_{A_{k}}|b_{i}(x)|^{p_{i}^{\prime}}|G_{k}u|^{\bar{p}}dx+C\sum_{i=1}^{N}\int_{A_{k}}|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}dx$
$\displaystyle+\varepsilon\sum_{i=1}^{N}\int_{\Omega}|(G_{k}u)_{x_{i}}|^{p_{i}}dx+C\sum_{i=1}^{N}\int_{A_{k}}|f_{i}|^{p_{i}^{\prime}}dx+\varepsilon\sum_{i=1}^{N}\int_{A_{k}}|(G_{k}u)_{x_{i}}|^{p_{i}}dx,$

for every $\varepsilon>0$ and a suitable $C>0$. Choosing
$\varepsilon=\varepsilon(\alpha,\vec{p})$ small enough we have

$\displaystyle\sum_{i=1}^{N}\int_{\Omega}|(G_{k}u)_{x_{i}}|^{p_{i}}dx\leq
C\left(\sum_{i=1}^{N}\int_{A_{k}}|b_{i}|^{p_{i}^{\prime}}|G_{k}u|^{\bar{p}}dx+\sum_{i=1}^{N}\int_{A_{k}}|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}dx+\sum_{i=1}^{N}\int_{A_{k}}|f_{i}|^{p_{i}^{\prime}}dx\right)$
$\displaystyle\leq
C\left[\sum_{i=1}^{N}\|b_{i}\|^{p^{\prime}_{i}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}}}(A_{k})}\|G_{k}u\|_{L^{\bar{p}^{*}}(\Omega)}^{\bar{p}}+\sum_{i=1}^{N}\int_{A_{k}}\left(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}}\right)dx\right],$

where, here and below, $C$ is a suitable positive constant which can vary from
line to line.

As before, denoting
$B=\prod_{i=1}^{N}\|(G_{k}u)_{x_{i}}\|_{L^{p_{i}}(A_{k})}$, by Sobolev
inequality (2.8) it follows

$\displaystyle\sum_{i=1}^{N}\int_{\Omega}|(G_{k}u)_{x_{i}}|^{p_{i}}dx\leq$
$\displaystyle
C\left[\sum_{i=1}^{N}\|b_{i}\|^{p^{\prime}_{i}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}}}(A_{k})}B^{\frac{\bar{p}}{N}}+\sum_{i=1}^{N}\int_{A_{k}}\left(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}}\right)dx\right].$
(4.4)

Previous inequality gives us an estimate of the $j^{th}$ addendum of the sum
at the left-hand side of (4.4) as well. Then, elevating to the power
$\frac{1}{Np_{j}}$, making the product on the left and right sides of (4.4) we
get

$\displaystyle B^{\frac{1}{N}}\leq$ $\displaystyle
C\left[\sum_{i=1}^{N}\|b_{i}(x)\|^{\frac{p^{\prime}_{i}}{\bar{p}}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}}}(A_{k})}B^{\frac{1}{N}}+\left(\sum_{i=1}^{N}\int_{A_{k}}(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}})dx\right)^{\frac{1}{\bar{p}}}\right].$

At this point, we can choose $k=k(N,\vec{p},b_{i},\alpha,C)>1$ large enough in
order to have $|A_{k}|$ small and so by absolutely continuity of the Lebesgue
norm we have

$\displaystyle B^{\frac{1}{N}}\leq$ $\displaystyle
C_{1}\left[\left(\sum_{i=1}^{N}\int_{A_{k}}(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}})dx\right)^{\frac{1}{\bar{p}}}\right]$
(4.5)

for suitable $C_{1}>0$. Combining (4.4) and (4.5) we obtain

$\sum_{i=1}^{N}\int_{\Omega}|(G_{k}u)_{x_{i}}|^{p_{i}}dx\leq
C_{2}\left(\sum_{i=1}^{N}\int_{A_{k}}(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}})dx\right).$
(4.6)

As before, we note that previous inequality gives us an estimate of the
$j^{th}$ addendum of the sum at the left-hand side of (4.6) as well. Then,
elevating to the power $\frac{1}{Np_{j}}$, making the product on the left and
right sides of (4.6) and using Sobolev inequality, we get

$\displaystyle\|G_{k}u\|_{L^{\bar{p}^{*}}(\Omega)}$ $\displaystyle\leq
C_{3}\prod_{j=1}^{N}\left(\sum_{i=1}^{N}\int_{A_{k}}(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}})dx\right)^{\frac{1}{Np_{j}}}$
$\displaystyle\leq
C_{3}\left(\sum_{i=1}^{N}\int_{A_{k}}(|b_{i}(x)|^{p_{i}^{\prime}}k^{\bar{p}}+|f_{i}|^{p_{i}^{\prime}})dx\right)^{\frac{1}{\bar{p}}}$
$\displaystyle\leq
C_{3}k\left(\sum_{i=1}^{N}(\|b_{i}\|^{p_{i}^{\prime}}_{s_{i}}+\|f_{i}\|_{s_{i}}^{p_{i}^{\prime}})\,\,|A_{k}|^{1-\frac{p^{\prime}_{i}}{s_{i}}}\right)^{\frac{1}{\bar{p}}}.$

We fix now $k_{0}>1$ such that $|A_{k_{0}}|<1$. Note that in view of (4.3) we
can fix such $k_{0}$ independent on $u$.

Applying Hölder inequality at the left hand side of previous inequality, for
every $k\geq k_{0}$ we get

$\int_{\Omega}|G_{k}u|\,dx\leq
C\,k|A_{k}|^{\frac{\eta}{\bar{p}}+1-\frac{1}{\bar{p}^{*}}}\qquad\mbox{ where
}\quad\eta=\min_{i}\\{1-\frac{p^{\prime}_{i}}{s_{i}}\\}.$ (4.7)

Observe now that the function

$g(k):=\int_{\Omega}|G_{k}u|\,dx$

is a non-negative and decreasing function such that $g^{\prime}(k)=-|A_{k}|$.
Hence we can rewrite (4.7) as

$\left(\frac{g(k)}{k}\right)^{a}\leq-Cg^{\prime}(k)\qquad\mbox{ where }\quad
a:=\frac{\bar{p}\bar{p}^{*}}{\eta\bar{p}^{*}+\bar{p}\bar{p}^{*}-\bar{p}}$

and by (4.2), it holds $(1-a)>0$. Let us consider now every value of $k>k_{0}$
such that $g(k)\not=0.$ By previous inequality we get

${k}^{-a}\leq-Cg^{\prime}(k)g(k)^{-a}=-C(g(k)^{(1-a)})^{\prime}.$

Integrating previous inequality with respect to $k$ from $k_{0}$ to $k$ we get

${k}^{1-a}-{k_{0}}^{1-a}\leq-C(g(k)^{1-a}-g(k_{0})^{(1-a)}).$

This implies

$g(k)^{1-a}\leq Cg(k_{0})^{(1-a)}-{k}^{1-a}+{k_{0}}^{1-a}.$

It is obvious at this point that, since $g(k_{0})\leq\|u\|_{L^{1}(\Omega)}$,
there exists a value $\bar{k}$ (independent in $u$) such that $g(\bar{k})=0$
and the thesis follows.

##  5 Existence result

In this section we analyze the existence of weak solutions of problem (1.1).
It is well known that, if the operator in problem (1.1) is coercive, then a
solution exists in $W_{0}^{1,\vec{p}}(\Omega)$. Unfortunately, under our
assumptions the coercivity for the involved operator in problem (1.1) is not
guaranteed. Another difficulty is due to the singularity of coefficients
$b_{i}$ in the lower order term. Indeed in the Marcinkiewicz space
$L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega)$, which is slightly
larger than Lebesgue space $L^{\frac{N\,p^{\prime}_{i}}{\bar{p}}}(\Omega)$,
the bounded functions are not dense and the norm is not absolutely continuous
(i.e. a function can have large norm even if restricted to a set with small
measure).

###### Theorem 5.1

Let us assume that $\Omega$ is a bounded Lipschitz domain, $p_{i}>1$ for
$i=1,..,N$, $\bar{p}<N$ and ($\mathcal{H}1$)-($\mathcal{H}4$) are in force.
There exists a positive constant $d=d(N,\alpha,\vec{p})$ such that if (3.2)
holds, then there exists at least a weak solution to problem (1.1).

For example in our prototype (1.11) we can consider as coefficient of the
lower order term the function

$\beta_{i}(x)=\frac{\kappa_{i}}{|x|^{\gamma_{i}}}+\beta_{i}^{0}(x)$

with $\kappa_{i}$ suitable small constant, $\beta_{i}^{0}\in
L^{\infty}(\Omega)$ and $\gamma_{i}=\frac{\bar{p}}{p_{i}^{\prime}}$ for
$i=1,\cdots,N$. Indeed, in this case, an easy computation shows that
${\rm{dist}}_{L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}(\beta_{i},L^{\infty}(\Omega))=|\kappa_{i}|\omega_{N}^{\frac{\gamma_{i}}{N}}$.

Our proof is detailed in the next subsection.

###  5.1 Proof of Theorem 5.1

We split the proof in three steps.

Step 1: Existence when $b_{i}\in L^{\infty}(\Omega)$ and $f_{i}\in
C_{c}^{\infty}(\Omega)$ for $i=1,\cdots,N$.

We stress that also in this case the operator
$\sum_{i=1}^{N}\left[-\partial_{x_{i}}(\mathcal{A}_{i}(x,\nabla
u)+\mathcal{B}_{i}(x,u))\right]+\mathcal{G}(x,u)$ can be not coercive. To
overcome this difficulty we use the boundedness result proved in Lemma 4.2 and
so we suppose that a solution $u$ there exists. Using Lemma 4.2, it follows
that

$\|u\|_{\infty}\leq M,$ (5.1)

where $M$ is a constant depending only on the data of the equation.

Now we define

$\widetilde{\mathcal{B}}(x,s)=\left\\{\begin{array}[c]{ll}\mathcal{B}(x,s)&\hbox{
if }|s|\leq M,\\\ \mathcal{B}(x,M)&\hbox{ if }|s|>M.\end{array}\right.$

Problem (1.1) with $\mathcal{B}(x,s)=\widetilde{\mathcal{B}}(x,s)$, $b_{i}\in
L^{\infty}(\Omega)$ and smooth data has a solution $u$, since the operator is
coercive. Moreover this solution $u$ verifies estimate (5.1), because
$\mathcal{B}(x,s)$ verifies (1.6). It follows that
$\widetilde{\mathcal{B}}(x,s)=\mathcal{B}(x,s)$ and we have proved the
existence of a solution to problem (1.1) when $b_{i}\in L^{\infty}(\Omega)$
and data are smooth.

Step 2: Existence when $b_{i}\in L^{\infty}(\Omega)$ and
$\mathcal{F}\in(W_{0}^{1,\overrightarrow{p}}(\Omega))^{*}$.

We consider the following sequence of approximate problems

$\left\\{\begin{array}[c]{ll}-\sum_{i=1}^{N}\partial_{x_{i}}\left[\mathcal{A}_{i}(x,\nabla
u_{n})+\mathcal{B}_{i}(x,u_{n})\right]+\mathcal{G}(x,u_{n})=-\sum_{i=1}^{N}(f^{n}_{i})_{x_{i}}&\hbox{
in }\Omega,\\\ &\\\ u_{n}=0&\hbox{ on }\partial\Omega,\end{array}\right.$
(5.2)

where $f^{n}_{i}\in C_{c}^{\infty}(\Omega)$ such that

$f^{n}_{i}\rightarrow f_{i}\quad\text{ in
}L^{p_{i}^{\prime}}(\Omega),\quad\|f^{n}_{i}\|_{L^{p_{i}^{\prime}}(\Omega)}\leq\|f_{i}\|_{L^{p_{i}^{\prime}}(\Omega)}\text{
for }i=1,\cdots,N.$ (5.3)

Step 1 assure the existence of a solution $u_{n}$ of problem (5.2). We stress
that estimate (3.3) holds for problem (5.2) when $b_{i}\in L^{\infty}(\Omega)$
for $i=1,\cdots,N$.

We get that the sequence $\\{u_{n}\\}_{n}$ is bounded in
$W_{0}^{1,\overrightarrow{p}}(\Omega)$ by a constant independent of $n$. Then
there exists a function $u\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ such that
(up to a subsequence denoted again by $u_{n}$)

$u_{n}\rightharpoonup u\text{ weakly in
}W_{0}^{1,\overrightarrow{p}}(\Omega),$ (5.4) $u_{n}\to u\text{ in
}L^{q}(\Omega)\text{ for }q<p_{\infty}$ (5.5)

and

$u_{n}\to u\text{ a.e. }\Omega.$ (5.6)

To take into account the terms $\mathcal{A}_{i}(x,\nabla u_{n})$ we need the
following lemma, which is the anisotropic version of Lemma 2.2 of [27]. Note
that if $A_{i}$ is strongly monotone, the claim is obvious.

###### Lemma 5.2

Assume $(\mathcal{H}1)$ be in force. Let us suppose $u_{n},u\in
W_{0}^{1,\overrightarrow{p}}(\Omega)$, $u_{n}\rightharpoonup u$ weakly in
$W_{0}^{1,\overrightarrow{p}}(\Omega)$ and

$\int_{\Omega}\left(\mathcal{A}_{i}(x,\nabla u_{n})-\mathcal{A}_{i}(x,\nabla
u)\right)\left[\partial_{x_{i}}u_{n}-\partial_{x_{i}}u\right]\,dx\rightarrow
0$ (5.7)

for $i=1,\cdots,N$. Then for $i=1,\cdots,N$

$\partial_{x_{i}}u_{n}\rightarrow\partial_{x_{i}}u\quad\text{ a.e. in
}\Omega.$ (5.8)

Proof. The proof is standard and follows as in Lemma 2.2 of [27]. For the
convenience of the reader we give some details. Let us denote for
$i=1,\cdots,N$

$Q_{n}^{i}(x):=\left(\mathcal{A}_{i}(x,\nabla
u_{n}(x))-\mathcal{A}_{i}(x,\nabla
u(x))\right)\left[\partial_{x_{i}}u_{n}(x)-\partial_{x_{i}}u(x)\right].$

Since (1.5) and (5.7) we get (up to a subsequence) that

$Q_{n}^{i}(x)\rightarrow 0\text{ for every }x\in\Omega\setminus\Omega_{0}$

with $|\Omega_{0}|=0$. Moreover by our assumptions we get (up to a
subsequence) $u_{n}(x)\rightarrow u(x)$ for $x\in\Omega\setminus\Omega_{0}$
(using that $W_{0}^{1,\overrightarrow{p}}(\Omega)$ is compactly embedded in
$L^{q}(\Omega)$ for $q<p_{\infty}$). Let us suppose
$x\in\Omega\setminus\Omega_{0}$ and let us denote
$\partial_{x_{i}}u_{n}(x)=\xi_{n}^{i}$ and $\partial_{x_{i}}u(x)=\xi^{i}$ and
by $\bar{\xi}^{i}$ one of the limit of $\xi_{n}^{i}$ for $i=1,\cdots,N$. By
(1.4) and (1.3) we have

$Q_{n}^{i}(x)\geq\alpha|\xi_{n}^{i}|^{p_{i}}-\beta_{i}|\xi_{n}^{i}|^{p_{i}-1}|\xi^{i}|-\beta_{i}|\xi_{n}^{i}||\xi^{i}|^{p_{i}-1}.$
(5.9)

If we assume $|\bar{\xi}^{i}|=+\infty$ then by (5.9) we obtain
$Q_{n}^{i}(x)\rightarrow+\infty$, which is a contradiction (see (5.7)). Then
$|\bar{\xi}^{i}|<+\infty\quad\forall i=1,\cdots,N$. Using the continuity of
$\mathcal{A}_{i}$ with respect to $\xi$ we conclude that

$\left(\mathcal{A}_{i}(x,\bar{\xi})-\mathcal{A}_{i}(x,\xi)\right)\left[\bar{\xi}^{i}-\xi^{i}\right]=0\quad\forall
i=1,\cdots,N.$

So assumption (1.5) yields then $\bar{\xi}=\xi$, i.e.
$\partial_{x_{i}}u_{n}(x)\rightarrow\partial_{x_{i}}u(x)\quad\forall
i=1,\cdots,N$ for every $x\in\Omega\setminus\Omega_{0}$.

Now we prove that (5.7) is in force. We use $\varphi=u_{n}-u$ as test function
in the variational formulation of the approximating problems (5.2):

$\sum_{i=1}^{N}\int_{\Omega}\left[\mathcal{A}_{i}(x,\nabla
u_{n})+\mathcal{B}_{i}(x,u_{n})\right]\partial_{x_{i}}(u_{n}-u)\,dx+\int_{\Omega}\mathcal{G}(x,u_{n})(u_{n}-u)\,dx=\sum_{i=1}^{N}\int_{\Omega}f_{i}^{n}\partial_{x_{i}}(u_{n}-u)\,dx.$

Adding and subtracting in the previous equality the term
$\sum_{i=1}^{N}\int_{\Omega}\mathcal{A}_{i}(x,\nabla
u)\partial_{x_{i}}(u_{n}-u)$, by (1.3), (5.4) and (5.3) we easily obtain that

$\sum_{i=1}^{N}\int_{\Omega}\mathcal{A}_{i}(x,\nabla
u)\partial_{x_{i}}(u_{n}-u)\rightarrow 0\quad\text{ and
}\quad\sum_{i=1}^{N}\int_{\Omega}f_{i}^{n}\partial_{x_{i}}(u_{n}-u)\,dx\rightarrow
0.$

Then recalling again (5.4) in order to prove (5.7) it is enough to show that

$\mathcal{B}_{i}(x,u_{n})\rightarrow\mathcal{B}_{i}(x,u)\text{ in
}L^{p^{\prime}_{i}}(\Omega)\quad\forall i=1,\cdots,N,$ (5.10)
$\mathcal{G}(x,u_{n})\rightarrow\mathcal{G}(x,u)\text{ in
}L^{p^{\prime}_{\infty}}(\Omega).$ (5.11)

To this aim we observe that by continuity of $\mathcal{B}_{i}$ with respect to
$s$ and by (5.6) we get
$\mathcal{B}_{i}(x,u_{n})\rightarrow\mathcal{B}_{i}(x,u)$ a.e. in $\Omega$.
Now (1.6), Hölder inequality, Sobolev inequality (2.8) and (3.3) yields that
for every measurable set $\Omega^{\prime}\subset\Omega$

$\begin{split}\int_{\Omega^{\prime}}|\mathcal{B}_{i}(x,u_{n})|^{p^{\prime}_{i}}\,dx&\leq\|b_{i}\|^{p^{\prime}_{i}}_{\infty}\int_{\Omega^{\prime}}|u_{n}|^{\bar{p}}\,dx\leq\|b_{i}\|^{p^{\prime}_{i}}_{\infty}\left(\int_{\Omega^{\prime}}|u_{n}|^{\bar{p}^{*}}dx\right)^{\frac{\bar{p}}{\bar{p}^{*}}}|\Omega^{\prime}|^{1-\frac{\bar{p}}{\bar{p}^{*}}}\\\
&\leq
C\|b_{i}\|^{p^{\prime}_{i}}_{\infty}|\Omega^{\prime}|^{1-\frac{\bar{p}}{\bar{p}^{*}}},\end{split}$

where $C$ is independent of $n$. By Vitali convergence Theorem we conclude
that (5.10) holds.

On the other hand by continuity of $\mathcal{G}$ with respect to $s$ we get
$\mathcal{G}(x,u_{n})\rightarrow\mathcal{G}(x,u)$ a.e. in $\Omega$, by (5.6).
Now (1.8), Hölder inequality, Sobolev inequality (2.8) and (3.3) yields that
for every measurable set $\Omega^{\prime}\subset\Omega$

$\int_{\Omega^{\prime}}|\mathcal{G}(x,u_{n})|^{p^{\prime}_{\infty}}\,dx\leq\widetilde{\mu}\int_{\Omega^{\prime}}|u_{n}|^{\gamma
p^{\prime}_{\infty}}\,dx\leq\mu\left(\int_{\Omega^{\prime}}|u_{n}|^{p_{\infty}}\,dx\right)^{\frac{\gamma}{p_{\infty}-1}}|\Omega^{\prime}|^{1-\frac{\gamma}{p_{\infty}-1}}\leq
C|\Omega^{\prime}|^{1-\frac{\gamma}{p_{\infty}-1}},$

where $C$ is independent of $n$. Also in this case we conclude that (5.11)
holds by Vitali convergence Theorem.

Then (5.7) is in force, we can apply Lemma 5.2 and convergence (5.8) follows
for $i=1,\cdots,N$. Then

$\mathcal{A}_{i}(x,\nabla u_{n})\rightarrow\mathcal{A}_{i}(x,\nabla
u)\quad\text{ a.e. }\Omega,$

because of continuity of $\mathcal{A}_{i}$ with respect to $\xi$. Moreover by
(1.3) and (3.3) we get that $\mathcal{A}_{i}(x,\nabla u_{n})$ is bounded in
$L^{p^{\prime}_{i}}(\Omega)$ and then

$\mathcal{A}_{i}(x,\nabla u_{n})\rightharpoonup\mathcal{A}_{i}(x,\nabla
u)\text{ weakly in }L^{p^{\prime}_{i}}(\Omega).$ (5.12)

Taking $\phi\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ as test function in
(5.2) and using (5.12), (5.10),(5.11) and (5.3) we can pass to the limit
obtaining that $u$ is a solution of problem (1.1) when $b_{i}\in
L^{\infty}(\Omega)$ and
$\mathcal{F}\in(W_{0}^{1,\overrightarrow{p}}(\Omega))^{*}$.

Step 3 : Existence dropping assumptions that $b_{i}\in
L^{\infty}(\Omega)\quad\forall i=1,\cdots,N$.

In order to remove the assumptions on $b_{i}\in L^{\infty}(\Omega)$, we set
for each $n\in\mathbb{N}$ and $i=1,...,N$ for almost every $x\in\Omega$,

$\vartheta^{i}_{n}(x)=\left\\{\begin{array}[]{ll}\frac{T_{n}b_{i}(x)}{b_{i}(x)}&\hbox{
if }b_{i}(x)\neq 0\\\ 1&\hbox{ if }b_{i}(x)=0\\\ \end{array}\right.$

and we consider the following sequence of approximating problems:

$\left\\{\begin{array}[c]{ll}-\sum_{i=1}^{N}\partial_{x_{i}}\left[\mathcal{A}_{i}(x,\nabla
u_{n})+\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\right]+\mathcal{G}(x,u_{n})=\mathcal{F}&\hbox{
in }\Omega,\\\ &\\\ u_{n}=0&\hbox{ on }\partial\Omega.\end{array}\right.$
(5.13)

Applying the previous Step 2 with $\vartheta^{i}_{n}(x)b_{i}(x)\in
L^{\infty}(\Omega)$ in place of $b_{i}$, for every $n\in{\mathbb{N}}$, we find
a solution $u_{n}\in W^{1,\overrightarrow{p}}_{0}(\Omega)$ to problem (5.13).
Moreover, by Lemma 3.1, using estimate (3.3) we get that the sequence
$\\{u_{n}\\}_{n}$ is bounded in $W^{1,\overrightarrow{p}}_{0}(\Omega)$ by a
constant independent of $n$. Then, unless to pass to subsequences not
relabeled there exists $u\in W^{1,\overrightarrow{p}}_{0}(\Omega)$ such that
(5.4), (5.5) and (5.6) hold.

We shall conclude our proof showing that such function $u$ solves problem
(1.1). We emphasize that in our assumptions the compactness of the sequence
$\mathcal{B}_{i}(x,u_{n})$ could fail (see Remark 5.4).

To this end we use an idea contained in [21]. In the rest of our proof we let
for simplicity $\eta(t):=\arctan t$. Obviously, $\eta\in C^{1}(\mathbb{R})$,
$|\eta(t)|\leq|t|$ and $0\leq\eta^{\prime}(t)\leq 1$ for all $t\in\mathbb{R}$.
In particular, $\eta$ is Lipschitz continuous in the whole of $\mathbb{R}$ and
therefore

$u_{n},u\in
W^{1,\overrightarrow{p}}_{0}(\Omega)\quad\Longrightarrow\quad\eta(u_{n}-u)\in
W^{1,\overrightarrow{p}}_{0}(\Omega)\,.$

Moreover, since $\eta(0)=0$ we have

$\eta(u_{n}-u)\rightharpoonup 0\qquad\text{in
$W^{1,\overrightarrow{p}}_{0}(\Omega)$ weakly}\,.$ (5.14)

Testing equation in (5.13) by the function $\varphi_{n}=\eta(u_{n}-u)$, we get

$\sum_{i=1}^{N}\int_{\Omega}\left[\mathcal{A}_{i}(x,\nabla
u_{n})+\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\right]\partial_{x_{i}}\varphi_{n}\,dx+\int_{\Omega}\mathcal{G}(x,u_{n})\varphi_{n}\,dx=\langle\mathcal{F}\,,\varphi_{n}\rangle$
(5.15)

As before, we add and subtract the term
$\sum_{i=1}^{N}\int_{\Omega}\mathcal{A}_{i}(x,\nabla
u)\partial_{x_{i}}\varphi_{n}$ in previous equality.

In view of (1.3) and (1.8), by (5.14) we obviously have

$\lim_{n\rightarrow\infty}\sum_{i=1}^{N}\int_{\Omega}\mathcal{A}_{i}(x,\nabla
u)\partial_{x_{i}}\varphi_{n}\rightarrow
0,\quad\qquad\lim_{n\rightarrow\infty}\langle\mathcal{F}\,,\varphi_{n}\rangle=0.$

Moreover also in this case we get (5.11) and so by (5.14) we obtain

$\int_{\Omega}\mathcal{G}(x,u_{n})\varphi_{n}\,dx\rightarrow 0.$

At this point it suffices to show that (up a subsequence)

$\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\rightarrow\mathcal{B}_{i}(x,u)\,\text{
strongly in }L^{p^{\prime}_{i}}(\Omega)\quad\forall i=1,\cdots,N.$ (5.16)

We preliminary observe that combining (5.6) with the property that
$\vartheta^{i}_{n}\rightarrow 1$ as $n\rightarrow\infty$, we have

$\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\eta^{\prime}(u_{n}-u)=\frac{\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})}{1+|u_{n}-u|^{2}}\rightarrow\mathcal{B}_{i}(x,u)\qquad\text{a.e.\
in }\Omega\,.$ (5.17)

Moreover, using (1.6) we note that for every measurable set
$\Omega^{\prime}\subset\Omega$ it holds

$\int_{\Omega^{\prime}}\left[\frac{\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})}{1+|u_{n}-u|^{2}}\right]^{p^{\prime}_{i}}dx\leq
C\int_{\Omega^{\prime}}\left[b_{i}(x)^{p^{\prime}_{i}}|u|^{\bar{p}}+\frac{b_{i}(x)^{p^{\prime}_{i}}|u_{n}-u|^{\bar{p}}}{1+|u_{n}-u|^{2}}\right]dx,$
(5.18)

for some positive constant $C=C(\bar{p})$. Hence, if $1<\bar{p}\leq 2$, (5.16)
immediately follows by Vitali convergence Theorem combining (5.17) and (5.18).
For $\bar{p}>2$ we choose $s$ satisfying

$\frac{\bar{p}^{\ast}}{\bar{p}}<s<\frac{\bar{p}^{\ast}}{\bar{p}-2}\,,$

so that $s^{\prime}<\frac{N}{\bar{p}}$ and $s(\bar{p}-2)<\bar{p}^{\ast}.$
Then, using Hölder inequality and (5.5) it follows that

$\int_{\Omega^{\prime}}\frac{b_{i}(x)^{p^{\prime}_{i}}|u_{n}-u|^{\bar{p}}}{1+|u_{n}-u|^{2}}\leq\int_{\Omega^{\prime}}b_{i}(x)^{p^{\prime}_{i}}|u_{n}-u|^{(\bar{p}-2)}dx\leq\|b_{i}\|^{p^{\prime}_{i}}_{L^{s^{\prime}\,p^{\prime}_{i}}(\Omega^{\prime})}\|u_{n}-u\|^{(\bar{p}-2)}_{s(\bar{p}-2)}\leq
C\|b_{i}\|^{p^{\prime}_{i}}_{L^{s^{\prime}\,p^{\prime}_{i}}(\Omega^{\prime})},$

where $C$ is a constant independent of $n$. So, also in this case (5.16)
follows by Vitali convergence theorem. So we obtain that for $i=1,\cdots,N$,

$\int_{\Omega}\left(\mathcal{A}_{i}(x,\nabla u_{n})-\mathcal{A}_{i}(x,\nabla
u)\right)\left[\partial_{x_{i}}\eta(u_{n}-u)\right]\,dx\rightarrow 0.$ (5.19)

Now, since $\eta^{\prime}(u_{n}-u)\rightarrow 1$ a.e. in $\Omega$, arguing as
in Lemma 5.2 we have that (5.19) implies (5.8) for $i=1,\cdots,N$. The end of
proof runs as in the previous step, obtaining that $u$ is a solution of
problem (1.1)

###### Remark 5.3

We observe that in case $p_{\max}:=\max_{i}p_{i}>\bar{p}^{*}$, the compactness
of sequence $\mathcal{B}_{i}(x,u_{n})$ in $L^{p^{\prime}_{i}}(\Omega)$ holds
and then Step 3 in the proof of previous Theorem 5.1 can be simplified. For
completeness we gives some details. In order to obtain that
$\partial_{x_{i}}u_{n}\to\partial_{x_{i}}u$ a.e. in $\Omega$ for every
$i=1,...,N$, we can choose as a test function $\varphi_{n}=(u_{n}-u)$ instead
of $\varphi_{n}=\eta(u_{n}-u)$ in (5.15) and prove that, unless to pass to a
subsequence:

$\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\rightarrow\mathcal{B}_{i}(x,u)\,\text{
strongly in }L^{p^{\prime}_{i}}(\Omega)\quad\forall i=1,\cdots,N\,.$ (5.20)

To prove (5.20), we note that

$\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\rightarrow\mathcal{B}_{i}(x,u)\qquad\text{a.e.\
in }\Omega\,.$

and that, using (1.6) for every measurable set $\Omega^{\prime}\subset\Omega$
it holds

$\int_{\Omega^{\prime}}\left[\mathcal{\vartheta}^{i}_{n}(x)\mathcal{B}_{i}(x,u_{n})\right]^{p^{\prime}_{i}}dx\leq
C\int_{\Omega^{\prime}}\left[b_{i}(x)^{p^{\prime}_{i}}|u|^{\bar{p}}+b_{i}(x)^{p^{\prime}_{i}}|u_{n}-u|^{\bar{p}}\right]dx$

for some positive constant $C=C(\bar{p})$. Hence, since $p_{\max}>\bar{p}^{*}$
and $1<\bar{p}<N$, we can choose $s$ satisfying

$\frac{N}{N-\bar{p}}<s\leq\frac{p_{\max}}{\bar{p}}\,,$

so that $s^{\prime}<\frac{N}{\bar{p}}$ and $s\bar{p}\leq p_{\max}.$ Then,
using again Hölder inequality, by Poincaré inequality it follows that

$\
\int_{\Omega^{\prime}}b_{i}(x)^{p^{\prime}_{i}}|u_{n}-u|^{\bar{p}}dx\leq\|b_{i}\|^{p^{\prime}_{i}}_{L^{s^{\prime}\,p^{\prime}_{i}}(\Omega^{\prime})}\|u_{n}-u\|^{\bar{p}}_{s\bar{p}}\leq
C\|b_{i}\|^{p^{\prime}_{i}}_{L^{s^{\prime}\,p^{\prime}_{i}}(\Omega^{\prime})},$

where $C$ is a constant independent of $n$. Hence, by Vitali convergence
theorem (5.20) is proved.

###### Remark 5.4

When $p_{i}=p<N$ for all $i$ and $N\geq 2$, the compactness of operator

$T:u\in W_{0}^{1,\overrightarrow{p}}(\Omega)\rightarrow b|u|^{p-2}u\in
L^{p^{\prime}}(\Omega)$

could fail. In this case the usual norm in $W_{0}^{1,p}(\Omega)$ gives an
equivalent norm in $W_{0}^{1,\overrightarrow{p}}(\Omega)$, then Example 3 in
[21] allows us to built a sequence of functions $\\{u_{n}\\}_{n\in\mathbb{N}}$
in $W_{0}^{1,\overrightarrow{p}}(\Omega)$ and a function $b\in
L^{\frac{N}{p-1},\infty}(\Omega)$ such that $\\{\nabla
u_{n}\\}_{n\in\mathbb{N}}$ is bounded in $L^{p}(\Omega,\mathbb{R}^{N})$, but
it is not possible to extract from $\\{b|u_{n}|^{p-1}\\}_{n\in\mathbb{N}}$ any
sequence strongly converging in $L^{p^{\prime}}(\Omega)$.

##  6 Positivity of solutions

In the isotropic case it is well known (see [8]) that when the data is non-
negative and the coefficient $b(x)\in L^{N/p-1}(\Omega)$, then the solution is
non-negative. In this section we prove a similar result for the anisotropic
problem (1.1). In order to give a precise statement we introduce the following
notation: $\mathcal{F}\geq 0$ means $<\mathcal{F},\phi>\geq 0$ for every
$\phi\geq 0$ when $\mathcal{F}\in(W_{0}^{1,\overrightarrow{p}}(\Omega))^{*}$.
We emphasize that the following result holds assuming $\mathcal{G}\equiv 0$ as
well.

###### Proposition 6.1

Let $u\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ be a weak solution to problem
(1.1) under assumptions ($\mathcal{H}1$)-($\mathcal{H}4$). If $\mathcal{F}\geq
0$, then we have $u\geq 0$.

Proof. Let us denote $u^{-}(x):=\min\\{0,u(x)\\}$. Taking $T_{h}(u^{-})$ as
test function with $h>0$ in (1.10), by (1.4), (1.6), (1.9) and assumption on
the data we get

$\displaystyle\alpha\sum_{i=1}^{N}\int_{\\{-h<u<0\\}}|\partial_{x_{i}}T_{h}(u^{-})|^{p_{i}}dx$
$\displaystyle\leq\sum_{i=1}^{N}\int_{\\{-h<u<0\\}}|b_{i}(x)||u|^{\frac{\bar{p}}{p^{\prime}_{i}}}|\partial_{x_{i}}T_{h}(u^{-})|dx.$

By Young inequality it follows that

$\sum_{i=1}^{N}\int_{\\{-h<u<0\\}}|\partial_{x_{i}}T_{h}(u^{-})|^{p_{i}}dx\leq
C\sum_{i=1}^{N}\int_{\\{-h<u<0\\}}|b_{i}(x)|^{p^{\prime}_{i}}|u|^{\bar{p}}dx,$
(6.1)

where $C$ is a suitable positive constant. Using the definition (2.2) in (6.1)
we get for every $j$

$\displaystyle\int_{\\{-h<u<0\\}}|\partial_{x_{j}}T_{h}(u^{-})|^{p_{j}}$
$\displaystyle\leq\sum_{i=1}^{N}\int_{\\{-h<u<0\\}}|\partial_{x_{i}}T_{h}(u^{-})|^{p_{i}}$
$\displaystyle\leq
Ch^{\bar{p}}\sum_{i=1}^{N}\|b_{i}\|^{p_{i}^{\prime}}_{L^{\frac{Np_{i}^{\prime}}{\bar{p}},\infty}(\Omega)}\int_{0}^{|\\{-h<u<0\\}|}s^{-\bar{p}/N}\,ds.$

By Sobolev inequality (2.8), the previous inequality becomes

$\displaystyle\left(\int_{\Omega}|T_{h}(u^{-})|^{\bar{p}^{*}}\,dx\right)^{\frac{1}{\bar{p}^{*}}}\leq
C\prod_{j=1}^{N}\left(\int_{\\{-h<u<0\\}}|\partial_{x_{j}}T_{h}(u^{-})|^{p_{j}}\,dx\right)^{\frac{1}{Np_{j}}}$
(6.2) $\displaystyle\leq
h\left(\int_{0}^{|\\{-h<u<0\\}|}s^{-\bar{p}/N}\,ds\right)^{1/\bar{p}}\prod_{j=1}^{N}\left(\sum_{i=1}^{N}\|b_{i}\|^{p_{i}^{\prime}}_{L^{\frac{Np_{i}^{\prime}}{\bar{p}},\infty}(\Omega)}\right)^{\frac{1}{Np_{j}}}.$

Now we observe that for $\delta>h$,

$h|\\{u<-\delta\\}|^{1/\bar{p}^{*}}=\left(\int_{\\{u<-\delta\\}}|T_{h}(u^{-})|^{\bar{p}^{*}}\,dx\right)^{\frac{1}{\bar{p}^{*}}}.$
(6.3)

Combining (6.2) and (6.3), it follows that

$|\\{u<-\delta\\}|^{1/\bar{p}^{*}}\leq
C\left(\int_{0}^{|\\{-h<u<0\\}|}s^{-\bar{p}/N}\,ds\right)^{1/\bar{p}}\prod_{j=1}^{N}\left(\sum_{i=1}^{N}\|b_{i}\|^{p_{i}^{\prime}}_{L^{\frac{Np_{i}^{\prime}}{\bar{p}},\infty}(\Omega)}\right)^{\frac{1}{Np_{j}}}.$

Observing that

$\int_{0}^{|\\{-h<u<0\\}|}s^{-\bar{p}/N}\,ds\rightarrow 0\text{ as
}h\rightarrow 0^{+},$

we conclude that the measure of the set $\\{u<-\delta\\}$ is zero for every
$\delta>0$.

As a consequence of Proposition 6.1 we get the following partial uniqueness
result when $\mathcal{F}\equiv 0$.

###### Corollary 6.2

Let $u\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ be a weak solution to problem
(1.1) under assumptions ($\mathcal{H}1$)-($\mathcal{H}4$). If
$\mathcal{F}\equiv 0$, then we have $u\equiv 0$.

Proof. The thesis follows arguing as in the previous proposition taking
$T_{h}(u^{-})$ and $T_{h}(u^{+})$ as test function.

We stress that a similar argument can be used to prove the uniqueness when
$p_{i}=2$ for all $i$ and the coefficient $b_{i}(x)=b(x)\in
L^{N,\infty}(\Omega)$ (see [8] when $b_{i}(x)=b(x)\in L^{N}(\Omega)$).

##  7 Uniqueness results

First of all we observe that prototype (1.11) verifies a strongly monotone
condition, but not the assumption of Lipschitz continuity in $u$ when $p_{i}$
are not all equals and $p_{i}\leq 2$ for $i=1,\cdots,N$ as required in the
classical uniqueness result. Then the novelty of our result consists in the
possibility to deal with cases when $\mathcal{B}(x,u)$ is only Hölder
continuous with respect to $u$ (see (7.3) below). In general the presence of a
zero order term could help in order to obtain uniqueness result, even in the
case $p_{i}\geq 2$ for $i=1,\cdots,N$. In what follows we assume that

$\mathcal{G}\left(x,s\right)\text{ is strictly monotone increasing function in
$s$.}$ (7.1)

Note that whenever $\mathcal{G}$ satisfies (7.1) and $\mathcal{G}(x,0)\equiv
0$ for all $x\in\Omega,$ then (1.9) obviously holds. For example
$\mathcal{G}\left(x,s\right)=\widetilde{\mu}|s|^{\gamma-1}s$ with
$\widetilde{\mu}>0,0<\gamma<p_{\infty}-1$ verifies (7.1) and (1.8) and (1.9).
We observe that we can relax condition (7.1) assuming that

$\left(\mathcal{G}\left(x,s\right)-\mathcal{G}\left(x,s^{\prime}\right)\right)(s-s^{\prime})>0\text{
for }s>s^{\prime}.$

As usual when we deal with uniqueness results for equation in which we have
the dependence on the unknown $u$ we consider two cases: first when $p_{i}\leq
2$ for every $i=1,\cdots,N$, then when $p_{i}\geq 2$ for every $i=1,\cdots,N$
and finally the mixed case.

###  7.1 First case: $p_{i}\leq 2$ for every $i=1,\cdots,N$

In this subsection we prove the uniqueness of weak solutions of problem (1.1)
when all $p_{i}\leq 2$. For example in model cases (1.11) the main difficulty
is due to the terms
$B_{i}(x,u)=\beta_{i}(x)|u|^{\frac{\bar{p}}{p^{\prime}_{i}}-1}u$ with
$\beta_{i}\in L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)$, who are
Hölder continuous but not Lipschitz continuous with respect to solution $u$.
To prove uniqueness we will strongly use the presence of zero order term. We
assume that

$\left(\mathcal{\mathcal{A}}_{i}\left(x,\xi\right)-\mathcal{A}_{i}\left(x,\xi^{\prime}\right)\right)\left(\xi_{i}-\xi_{i}^{\prime}\right)\geq\widetilde{\alpha}\left(\left|\xi_{i}\right|+\left|\xi_{i}^{\prime}\right|\right)^{p_{i}-2}\left|\xi_{i}-\xi_{i}^{\prime}\right|^{2}$
(7.2)

with $\widetilde{\alpha}>0$ and

$\left|\mathcal{B}_{i}\left(x,s\right)-\mathcal{B}_{i}\left(x,s^{\prime}\right)\right|\leq\widetilde{b}_{i}(x)\left|s-s^{\prime}\right|^{\frac{\bar{p}}{p_{i}^{\prime}}}$
(7.3)

with $\widetilde{b}_{i}:\,\Omega\to[0,+\infty)$ measurable function such that

$\widetilde{b}_{i}\in L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega),$
(7.4)

for $i=1,..,N$. We stress that
$\mathcal{A}_{i}\left(x,\xi\right)=\alpha|\xi_{i}|^{p_{i}-2}\xi_{i}$ verifies
(7.2) and
$\mathcal{B}_{i}\left(x,s\right)=\beta_{i}(x)|s|^{\frac{\bar{p}}{p_{i}^{\prime}}-1}s$
verifies (7.3), recalling that $\frac{\bar{p}}{p_{i}^{\prime}}\leq 1$, because
$\bar{p}\leq 2\leq p_{i}^{\prime}$ .

###### Theorem 7.1

Let $u\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ be a weak solution to problem
(1.1) under assumptions ($\mathcal{H}1$)-($\mathcal{H}4$) with $p_{i}\leq 2$
for $i=1,..,N$, $\bar{p}<N$ and $\mathcal{G}\not\equiv 0$. If
$2\min_{i}\\{\frac{\bar{p}}{p_{i}^{\prime}}\\}\geq 1$, (7.1),(7.2) and (7.3)
are in force, then $u$ is the unique weak solution to problem (1.1).

Proof. We generalize the ideas contained in [14] where a linear isotropic
operator is considered.

Let $u$ and $v$ be two weak solutions to problem (1.1). Let us denote
$w=\left(u-v\right)^{+}$ and $D=\left\\{x\in\Omega:w>0\right\\}$. Denoting
$r_{\min}=\min_{i}\\{\frac{\bar{p}}{p_{i}^{\prime}}\\}$ and recalling that
$2r_{\min}\geq 1$, for every $\varepsilon>0$ there exists
$\delta(\varepsilon)$ such that

$\int_{\delta(\varepsilon)}^{\varepsilon}\frac{1}{s^{2r_{\min}}}ds=1.$

Let us define

$\Psi^{\sigma}_{\varepsilon}(t)=\left\\{\begin{array}[c]{ll}0&\hbox{ se
}t\leq\delta(\varepsilon),\\\
\int_{\delta(\varepsilon)}^{t}\frac{1}{s^{\sigma}}ds&\hbox{ se
}\delta(\varepsilon)<t<\varepsilon\\\ 1&\hbox{ se
}t\geq\varepsilon.\end{array}\right.$ (7.5)

We take $\Psi_{\varepsilon}(t)=\Psi^{\sigma}_{\varepsilon}$ with
$\sigma=2r_{\min}$. We stress that $\Psi_{\varepsilon}(t)$ is Lipschitz
continuous, $\Psi^{\prime}_{\varepsilon}(t)\geq 0$ and
$\varphi\Psi_{\varepsilon}(w)\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ taking
$\varphi\geq 0$ and $\varphi\in W^{1,\overrightarrow{p}}(\Omega)\cap
L^{\infty}(\Omega)$.

Supposing that $D$ has positive measure, we use $\varphi\Psi_{\varepsilon}(w)$
as test function in the difference of the equations. It follows

$\displaystyle I_{\varepsilon}:=$
$\displaystyle\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}\varphi\Psi_{\varepsilon}(w)\,dx$
$\displaystyle+\int_{\Omega}\left[\mathcal{G}(x,u)-\mathcal{G}(x,v)\right]\varphi\Psi_{\varepsilon}(w)\,dx=$
$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx.$

By (7.2) and (7.3) we get

$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}$
$\displaystyle\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx$
$\displaystyle\leq-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\widetilde{\alpha}\frac{|\partial_{x_{i}}w|^{2}}{(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{2-p_{i}}}\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx+\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\widetilde{b}_{i}(x)w^{r_{i}}|\partial_{x_{i}}w|\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx,$

where $r_{i}=\frac{\bar{p}}{p^{\prime}_{i}}$. Using Young inequality with
$\theta<\widetilde{\alpha}$ we obtain

$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}$
$\displaystyle\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx$
$\displaystyle\leq$
$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}(\widetilde{\alpha}-\theta)\frac{|\partial_{x_{i}}w|^{2}}{(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{2-p_{i}}}\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx+$
$\displaystyle+C(\theta)\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\widetilde{b}^{2}_{i}(x)w^{2r_{i}}(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{2-p_{i}}\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx$
$\displaystyle\leq
C(\theta)\|\varphi\|_{\infty}\overset{N}{\underset{i=1}{\sum}}\varepsilon^{2(r_{i}-r_{\min})}\int_{\\{\delta(\varepsilon)<w<\varepsilon\\}}\widetilde{b}^{2}_{i}(x)(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{2-p_{i}}\,dx=J_{\varepsilon}.$

We underline that
$\widetilde{b}^{2}_{i}(x)(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{2-p_{i}}\in
L^{1}(\Omega)$. Indeed we have

$\begin{split}\int_{\Omega}\widetilde{b}^{2}_{i}(x)(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{2-p_{i}}\,dx&\leq\left(\int_{\Omega}\widetilde{b}_{i}^{p^{\prime}_{i}}(x)\,dx\right)^{\frac{2}{p^{\prime}_{i}}}\left(\int_{\Omega}(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{p_{i}}\,dx\right)^{\frac{2-p_{i}}{p_{i}}}\\\
&\leq
C(\Omega,p_{i})\|\widetilde{b}_{i}\|^{p^{\prime}_{i}}_{L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}\left(\int_{\Omega}(|\partial_{x_{i}}u|+|\partial_{x_{i}}v|)^{p_{i}}\,dx\right)^{\frac{2-p_{i}}{p_{i}}}.\end{split}$

Since $(r_{i}-r_{\min})\geq 0$, letting $\varepsilon\rightarrow 0$ we get
$J_{\varepsilon}\rightarrow 0$. Moreover
$\Psi_{\varepsilon}(w)\rightarrow\chi_{\\{u-v>0\\}}$, where
$\chi_{\\{u-v>0\\}}$ denotes the characteristic function of the set
$\\{u-v>0\\}$. By the Lebesgue dominated convergence Theorem we obtain

$\overset{N}{\underset{i=1}{\sum}}\int_{\\{u-v>0\\}}\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}\varphi+\left[\mathcal{G}(x,u)-\mathcal{G}(x,v)\right]\varphi\,dx\leq
0.$ (7.6)

Taking $\varphi=1$ in (7.6) it follows

$\int_{\\{u-v>0\\}}\left[\mathcal{G}(x,u)-\mathcal{G}(x,v)\right]\,dx\leq 0.$

Assumption (7.1) allows us to conclude that ${\\{u-v>0\\}}$ ha zero measure.
Exchanging $u$ for $v$ we conclude.

###### Remark 7.2

  * i)

If $3/2\leq p_{i}\leq 2$ for all $i$ then
$2\min_{i}\\{\frac{\bar{p}}{p^{\prime}_{i}}\\}\geq 1$.

  * ii)

Obviously when all $p_{i}=2$ the uniqueness results holds even
$\mathcal{G}\equiv 0$, because we have Lipschitz dependence on $u$.

  * iii)

Theorem 7.1 holds replacing the condition $p_{i}\leq 2$ for every
$i=1,\cdots,N$ with $\frac{\bar{p}}{p^{\prime}_{i}}\leq 1$ for every
$i=1,\cdots,N$.

###  7.2 Second case: $p_{i}\geq 2$ for every $i=1,\cdots,N$

In this subsection we prove the uniqueness of weak solutions of problem (1.1)
when all $p_{i}\geq 2$. In the model cases (1.11) the term
$B_{i}(x,u)=\beta_{i}(x)|u|^{\frac{\bar{p}}{p^{\prime}_{i}}-1}u$ with
$|\beta_{i}|\in L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)$ are
locally Lipschitz continuous with respect to solution $u$, but it is well-
known that even in the isotropic case when $p>2$ the uniqueness can fail (see
for example in [2] at the end of Section 2). Again to prove uniqueness we will
strongly use the presence of zero order term. We assume

$\left(\mathcal{\mathcal{A}}_{i}\left(x,\xi\right)-\mathcal{A}_{i}\left(x,\xi^{\prime}\right)\right)\left(\xi_{i}-\xi_{i}^{\prime}\right)\geq\widehat{\alpha}\left|\xi_{i}-\xi_{i}^{\prime}\right|^{p_{i}}$
(7.7)

with $\widehat{\alpha}>0$ and

$\left|\mathcal{B}_{i}\left(x,s\right)-\mathcal{B}_{i}\left(x,s^{\prime}\right)\right|\leq\widehat{b}_{i}(x)\left|s-s^{\prime}\right|\left(|s|+|s^{\prime}|+\zeta\right)^{\frac{\bar{p}}{p^{\prime}_{i}}-1}$
(7.8)

with $\widehat{b}_{i}(x)$ defined as in (7.4) and $\zeta\geq 0$. We stress
that $\frac{\bar{p}}{p^{\prime}_{i}}\geq 1$ under our assumptions. Moreover
$\mathcal{A}_{i}\left(x,\xi\right)=\alpha|\xi_{i}|^{p_{i}-2}\xi_{i}$ verifies
(7.7) and
$\mathcal{B}_{i}\left(x,s\right)=\beta_{i}(x)|s|^{\frac{\bar{p}}{p_{i}^{\prime}}-1}s$
verifies (7.3), recalling that $p_{i}^{\prime}\leq 2\leq\bar{p}$.

###### Theorem 7.3

Let $u\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ be a weak solution to problem
(1.1) under assumptions ($\mathcal{H}1$)-($\mathcal{H}4$) with $p_{i}\geq 2$
for $i=1,..,N$, $\bar{p}<N$ and $\mathcal{G}\not\equiv 0$. If (7.1), (7.7) and
(7.8) are in force, then $u$ is the unique weak solution to problem (1.1).

Proof. The idea of the proof follows the previous theorem. Let $u$ and $v$ be
two weak solutions to problem (1.1). Let us denote $w=\left(u-v\right)^{+}$
and $D=\left\\{x\in\Omega:w>0\right\\}$.

Let $\Psi_{\varepsilon}(t)=\Psi^{\sigma}_{\varepsilon}(t)$ with
$\sigma=\min_{i}{p^{\prime}_{i}}$ defined in (7.5). Supposing that $D$ has
positive measure, we use $\varphi\Psi_{\varepsilon}(w)$ as test function in
the difference of the equations. By (7.7) and (7.8) we get

$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}$
$\displaystyle\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx$
$\displaystyle\leq-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\widehat{\alpha}|\partial_{x_{i}}w|^{p_{i}}\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx+\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\widehat{b}_{i}w\left(|u|+|v|+\zeta\right)^{r_{i}-1}|\partial_{x_{i}}w|\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx,$

where we put $r_{i}=\frac{\bar{p}}{p^{\prime}_{i}}$. Using Young inequality
with $\theta<\widehat{\alpha}$ we obtain

$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}$
$\displaystyle\left\\{\left[\mathcal{A}_{i}\left(x,\nabla
u\right)-\mathcal{A}_{i}\left(x,\nabla
v\right)\right]+\left[\mathcal{B}_{i}\left(x,u\right)-\mathcal{B}_{i}\left(x,v\right)\right]\right\\}\partial_{x_{i}}w\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx$
$\displaystyle\leq$
$\displaystyle-\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}(\widehat{\alpha}-\theta)|\partial_{x_{i}}w|^{p_{i}}\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx+$
$\displaystyle+C(\theta)\overset{N}{\underset{i=1}{\sum}}\int_{\Omega}\widehat{b}_{i}^{p^{\prime}_{i}}w^{p^{\prime}_{i}}\left(|u|+|v|+\zeta\right)^{(r_{i}-1)p^{\prime}_{i}}\Psi^{\prime}_{\varepsilon}(w)\varphi\,dx$
$\displaystyle\leq
C(\theta)\|\varphi\|_{\infty}\overset{N}{\underset{i=1}{\sum}}\varepsilon^{(p^{\prime}_{i}-\min_{i}p^{\prime}_{i})}\int_{\\{\delta(\varepsilon)<w<\varepsilon\\}}\widehat{b}_{i}^{p^{\prime}_{i}}\left(|u|+|v|+\zeta\right)^{(\bar{p}-p^{\prime}_{i})}\,dx=J_{\varepsilon}.$

We underline that
$\widehat{b}^{p^{\prime}_{i}}_{i}\left(|u|+|v|+\zeta\right)^{(\bar{p}-p^{\prime}_{i})}\in
L^{1}(\Omega)$. Indeed we have

$\begin{split}\int_{\Omega}\widehat{b}_{i}^{p^{\prime}_{i}}\left(|u|+|v|+\zeta\right)^{(r_{i}-1)p^{\prime}_{i}}\,dx&\leq
C\left(\|\widehat{b}_{i}\|^{{}^{p^{\prime}_{i}}}_{L^{\frac{N\,p^{\prime}_{i}}{\bar{p}},\infty}(\Omega)}\||u|+|v|\|^{\bar{p}-p^{\prime}_{i}}_{L^{\frac{N(\bar{p}-p^{\prime}_{i})}{N-\bar{p}},\bar{p}-p^{\prime}_{i}}(\Omega)}+\|\widehat{b}_{i}\|^{p^{\prime}_{i}}_{L^{p^{\prime}_{i}}(\Omega)}\right).\end{split}$

The right-hand side is finite, since $u,v\in
L^{\frac{N(\bar{p}-p^{\prime}_{i})}{N-\bar{p}},\bar{p}-p^{\prime}_{i}}(\Omega)$
and $L^{\bar{p}^{*},\bar{p}}(\Omega)\subset
L^{\frac{N(\bar{p}-p^{\prime}_{i})}{N-\bar{p}},\bar{p}-p^{\prime}_{i}}(\Omega)$.
We conclude as in Theorem 7.1 letting $\varepsilon\rightarrow 0$.

###### Remark 7.4

Theorem 7.1 holds replacing the condition $p_{i}\geq 2$ for every
$i=1,\cdots,N$ with $\frac{\bar{p}}{p^{\prime}_{i}}\geq 1$ for every
$i=1,\cdots,N$.

###  7.3 The mixed case: when $\min_{i}p_{i}\leq 2$ and $\max_{i}p_{i}>2$.

We end the section studying the mixed case when $\min_{i}p_{i}\leq 2$ and
$\max_{i}p_{i}>2$. For simplicity of presentation we take into account the
model problem (1.11) when $\min_{i}p_{i}\leq 2$ and $\max_{i}p_{i}\geq 2$.

###### Theorem 7.5

Let $u\in W_{0}^{1,\overrightarrow{p}}(\Omega)$ be a weak solution to problem
(1.11) with $p_{i}>1$ for $i=1,..,N$, $\min_{i}p_{i}\leq 2$ and
$\max_{i}p_{i}>2$,$\bar{p}<N$, $\widetilde{\mu}>0$ and $\beta_{i}\in
L^{\frac{Np^{\prime}_{i}}{\bar{p}},\infty}(\Omega)$. If
$2\min_{i}\\{\frac{\bar{p}}{p^{\prime}_{i}}\\}\geq 1$, then $u$ is the unique
weak solution to problem (1.11).

We stress that problem (1.11) verifies assumptions (7.2) and (7.3) for
$p_{i}\leq 2$. Otherwise for $p_{i}>2$ conditions (7.7) and (7.8) are
fulfilled. Then we can blend the proof of Theorem 7.1 and Theorem 7.3 to prove
Theorem 7.5. In particular
$\Psi_{\varepsilon}(t)=\Psi^{\sigma}_{\varepsilon}(t)$ with
$\sigma=\min_{i}\\{p^{\prime}_{i},2\frac{\bar{p}}{p^{\prime}_{i}}\\}$ defined
in (7.5).

## Acknowledgments

The authors are partially supported by GNAMPA of the Italian INdAM (National
Institute of High Mathematics).

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