# Left ideal LRPC codes and a ROLLO-type cryptosystem based on group algebras

Martino Borello, Paolo Santonastaso and Ferdinando Zullo

###### Abstract

In this paper we introduce left ideal low-rank parity-check codes by using
group algebras and we finally use them to extend ROLLO-I KEM.

Keywords: Group algebra, low-rank parity-check code, ideal matrix, ROLLO

##  1 Introduction

Low-rank parity-check (LRPC) codes are rank-metric codes introduced by Gaborit
et al. in [1, 5] as the rank-metric analogs of low-density parity-check codes
in the Hamming metric. These codes turned out to be very interesting for their
efficient probabilistic decoding algorithm and their applications in several
contexts, such as powerline communications [12], network coding [4] and
cryptography [9]. Regarding the last one, in ROLLO cryptosystem special types
of LRPC codes are used, known as ideal LRPC codes, which can be efficiently
stored: they can be simply represented by a vector instead of the entire
generator matrix (as in the case with circulant matrices).

Motivated by the application in cryptography, in this paper we introduce left
ideal LRPC codes arising from group algebras, deriving some interesting
properties, such as a systematic parity-check matrix. Thank to this latter
property, we are able to describe a new variant of the Key-Encapsulation
scheme ROLLO-I with the use of group algebras. A similar approach has been
recently used in [3] in the Hamming-metric context (MDPC codes), to generalize
the BIKE cryptosystems.

Throughout the paper, $q$ is a prime power, $m$ is a positive integer,
${\mathbb{F}}_{q^{m}}$ is the field with $q^{m}$ elements and $G$ is a finite
group of size $n$.

##  2 Group algebras and left ideal matrices

The group algebra ${\mathbb{F}}_{q^{m}}G$ is the set $\left\\{\sum_{g\in
G}a_{g}g\colon a_{g}\in{\mathbb{F}}_{q^{m}}\right\\}$, which is an
${\mathbb{F}}_{q^{m}}$-vector space of dimension $n$ in a natural way and it
is also an ${\mathbb{F}}_{q^{m}}$-algebra via the multiplication

$ab:=\sum_{g\in G}\left(\sum_{h\in G}a_{h}b_{h^{-1}g}\right)g,$

for $a=\sum_{g\in G}a_{g}g$ and $b=\sum_{g\in G}b_{g}g$. From now, _we fix an
ordering_ $1_{G}=g_{1},\ldots,g_{n}$ of the elements of $G$. The
${\mathbb{F}}_{q^{m}}$-vector spaces ${\mathbb{F}}_{q^{m}}^{n}$ and
${\mathbb{F}}_{q^{m}}G$ may be identified by the following
${\mathbb{F}}_{q^{m}}$-isomorphism:

$\begin{array}[]{lccl}\Psi\colon&{\mathbb{F}}_{q^{m}}^{n}&\longrightarrow&{\mathbb{F}}_{q^{m}}G\\\
&(u_{1},\ldots,u_{n})&\longmapsto&\sum_{i=1}^{n}u_{i}g_{i},\end{array}$

Via the isomorphism $\Psi$, we may endow the ${\mathbb{F}}_{q^{m}}$-vector
space ${\mathbb{F}}_{q^{m}}^{n}$ with an algebra structure, by defining the
product of two elements $u,v\in{\mathbb{F}}_{q^{m}}^{n}$ in the following way:

$uv:=\Psi^{-1}(\Psi(u)\Psi(v)).$

######  Definition 2.1.

Let $a\in{\mathbb{F}}_{q^{m}}G$. The left ideal matrix generated by $a$ is the
matrix

$\mathcal{LIM}(a):=\begin{pmatrix}\Psi^{-1}(g_{1}a)\\\ \Psi^{-1}(g_{2}a)\\\
\vdots\\\ \Psi^{-1}(g_{n}a)\end{pmatrix}\in{\mathbb{F}}_{q^{m}}^{n\times n}$

######  Remark 2.2.

The image via $\Psi$ of the vector space generated by the lines of
$\mathcal{LIM}(a)$ is the left ideal generated by $a$ in
${\mathbb{F}}_{q^{m}}G$.

Note that the product of two vectors of ${\mathbb{F}}_{q^{m}}$ can be
expressed as the usual product vector-matrix: for
$u=(u_{1},\ldots,u_{n}),v=(v_{1},\ldots,v_{n})\in{\mathbb{F}}_{q^{m}}^{n}$,

$\displaystyle uv$
$\displaystyle=\Psi^{-1}(\Psi(u)\Psi(v))=\Psi^{-1}\left(\displaystyle\sum_{i=1}^{n}u_{i}g_{i}\Psi(v)\right)=\displaystyle\sum_{i=1}^{n}u_{i}\Psi^{-1}\left(g_{i}\Psi(v)\right)$
$\displaystyle=(u_{1},\ldots,u_{n})\left(\begin{matrix}\Psi^{-1}(g_{1}\Psi(v))\\\
\Psi^{-1}(g_{2}\Psi(v))\\\ \vdots\\\
\Psi^{-1}(g_{n}\Psi(v))\end{matrix}\right)=u\mathcal{LIM}(\Psi(v)).$

Moreover, $\mathcal{LIM}(1)$ is clearly the identity. We are now interested in
invertible left ideal matrices. In this regard, since the group algebra is
finite, if an element is left invertible, it is also right invertible and
conversely. Moreover, the left inverse and the right inverses coincide.

######  Proposition 2.3 (Proposition 9 of [3]).

An element $a\in{\mathbb{F}}_{q^{m}}G$ is invertible if and only if the left
ideal matrix $\mathcal{LIM}(a)$ is invertible.

The investigations of units in finite group algebra is a subject of intense
research (see for example [11, 10, 8] and references therein).

##  3 Low-rank parity-check codes from group algebras

The LRPC codes have been introduced in [6]. This family of codes can be seen
as the equivalent of classical LDPC codes for the rank metric. There is a
natural analogy between low density matrices and matrices with low rank.

######  Definition 3.1.

Let $\lambda,K,N$ be positive integers with $0<K<N$. A low-rank parity-check
code (LRPC) with parameter $(\lambda,K,N)$ is a code with a parity-check
matrix $H$ such that $H=(h_{ij})\in{\mathbb{F}}_{q^{m}}^{(N-K)\times N}$ is a
full-rank matrix such that its coefficients generate an
${\mathbb{F}}_{q}$-subspace $\mathcal{F}$ of dimension $\lambda$, i.e.
$\dim_{{\mathbb{F}}_{q}}(\langle h_{ij}\colon
i\in\\{1,\ldots,N-K\\},j\in\\{1,\ldots,N\\}\rangle_{{\mathbb{F}}_{q}})=\lambda$.

For “small” values of $\lambda$, LRPC codes admit efficient probabilistic
decoding algorithm (RSR) to recover the support of the error vector. More
precisely, given as input an ${\mathbb{F}}_{q}$-basis of $\mathcal{F}$,
$s=(s_{1},\ldots,s_{\lambda})=eH^{\top}\in{\mathbb{F}}_{q^{m}}^{N}$ a syndrome
of an error $e\in{\mathbb{F}}_{q^{m}}^{N}$ whose components belong to an
${\mathbb{F}}_{q}$-subspace $\mathcal{E}$ of ${\mathbb{F}}_{q^{m}}$, then the
RSR algorithm outputs $\mathcal{E}$. For more details see [6].

We may describe a low-rank parity-check code via its parity-check matrix. In
order to reduce the size of a representation of this code, we can introduce
left ideal low-rank parity-check codes via left ideal matrices.

######  Definition 3.2.

Let $\mathcal{F}$ be an ${\mathbb{F}}_{q}$-subspace of dimension $\lambda$ of
${\mathbb{F}}_{q^{m}}$ and let $h_{1},h_{2}\in{\mathbb{F}}_{q^{m}}G$. Suppose
that $\langle h_{i,j}\colon 1\leq i\leq n,1\leq j\leq
2\rangle_{{\mathbb{F}}_{q}}=\mathcal{F}$ where
$\Psi^{-1}(h_{1})=(h_{1,1},\ldots,h_{n,1}),\Psi^{-1}(h_{2})=(h_{1,2},\ldots,h_{n,2})$
and either $h_{1}$ or $h_{2}$ is an invertible element of
${\mathbb{F}}_{q^{m}}G$. A LRPC code is called left ideal if it has a parity-
check matrix

$H_{h_{1},h_{2}}:=\begin{pmatrix}\mathcal{LIM}(h_{1})^{\top}\mathcal{LIM}(h_{2})^{\top}\end{pmatrix}\in{\mathbb{F}}_{q^{m}}^{n\times
2n}.$

Since either $h_{1}$ or $h_{2}$ is invertible, by Proposition 2.3 either
$\mathcal{LIM}(h_{1})$ or $\mathcal{LIM}(h_{2})$ is invertible, so that
$H_{h_{1},h_{2}}$ is a full-rank matrix.

######  Theorem 3.3.

If $\mathcal{C}$ is a left ideal LRPC code with parity-matrix
$H_{h_{1},h_{2}}$, with $h_{1},h_{2}\in\mathbb{F}_{q^{m}}G$ and either $h_{1}$
or $h_{2}$ is invertible, then the parameters of $\mathcal{C}$ are
$(\lambda,n,2n)$, where

$\lambda=\dim_{{\mathbb{F}}_{q}}(\langle h_{i,j}\colon
i\in\\{1,\ldots,n\\},j\in\\{1,2\\}\rangle_{{\mathbb{F}}_{q}}),$

where $\Psi^{-1}(h_{1})=(h_{1,1},\ldots,h_{n,1})$ and
$\Psi^{-1}(h_{2})=(h_{1,2},\ldots,h_{n,2})$. If $h_{1}$ is invertible, then
the systematic parity-check matrix of $\mathcal{C}$ is
$H_{1,h_{2}h_{1}^{-1}}=\begin{pmatrix}I_{n}\
\mathcal{LIM}(h_{2}{h_{1}}^{-1})^{\top}\end{pmatrix}$.

###### Proof.

Note that the entries of the vectors $\Psi^{-1}(g_{i}h_{j})$ are the same of
those of $h_{j}$, for any $i$ and $j$, since the map $x\in G\mapsto xg_{i}\in
G$ is a permutation of $G$. This implies that the ${\mathbb{F}}_{q}$-span of
the entries of $h_{1}$ and $h_{2}$ corresponds to $\mathcal{F}$ (that is the
${\mathbb{F}}_{q}$-span of the entries of $H$). Straightforward computations
show the second part, which we omit for brevity. ∎

A way to hide the structure of a left ideal LRPC code is to reveal only its
systematic parity-check matrix.

##  4 ROLLO-I using group algebra left ideal LRPC codes

A Key-Encapsulation scheme KEM = (KeyGen; Encap; Decap) is a triple of
probabilistic algorithms together with a key space $\mathcal{K}$. The key
generation algorithm KeyGen generates a pair of public and secret key
$(pk;sk)$. The encapsulation algorithm Encap uses the public key $pk$ to
produce an encapsulation $c$ and a key $K\in\mathcal{K}$. Finally Decap, using
the secret key $sk$ and an encapsulation $c$, recovers the key $K$ or fails
and return error.

We adapt the Key-encapsulation scheme ROLLO I by using this left ideal LRPC
codes constructed using group algebras.

KeyGen:

  * •

Pick randomly an ${\mathbb{F}}_{q}$-subspace $\mathcal{F}$ of
${\mathbb{F}}_{q^{m}}$ with dimension $\lambda$ and pick
$(x,y)\in\mathcal{F}^{n}\times\mathcal{F}^{n}$ such that the
${\mathbb{F}}_{q}$-span of the components of $x$ and $y$ is $\mathcal{F}$ and
$\Psi(x)$ is invertible (such a choice of $x$ depends on the group algebra,
but it is important to remark that a “general” element in a group algebra is
invertible and that it is easy to check if an element is invertible or not).

  * •

Compute $\Psi(h)=\Psi(y)\Psi(x)^{-1}$.

  * •

Define $pk=(h,G)$ and $sk=(x,y)$.

Encap(pk):

  * •

Pick randomly an ${\mathbb{F}}_{q}$-subspace $\mathcal{E}$ of
${\mathbb{F}}_{q^{m}}$ with dimension $r$ and pick
$(e_{1},e_{2})\in\mathcal{E}^{n}\times\mathcal{E}^{n}$ such that the
${\mathbb{F}}_{q}$-span of the components of $e_{i}$ is $\mathcal{E}$, for
$i\in\\{1,2\\}$

  * •

Compute $\Psi(c)=\Psi(e_{1})+\Psi(e_{2})\Psi(h)$.

  * •

Compute $K=Hash(\mathcal{E})$ and give as output $c$.

Dec(sk):

  * •

Compute $\Psi(c)\Psi(x)=\Psi(e_{1})\Psi(x)+\Psi(e_{2})\Psi(y)$, since
$\Psi(h)=\Psi(y)\Psi(x)^{-1}$ and recover $\mathcal{E}$ with the algorithm
RSR.

  * •

$K=Hash(\mathcal{E})$.

The above KEM scheme relies on the supposed hardness of following problem.

######  Problem 4.1.

Given an element $h\in{\mathbb{F}}_{q^{m}}G$, it is “hard” to distinguish
whether the code $\mathcal{C}$ with the parity-check matrix
$H=\begin{pmatrix}I_{n}\ \mathcal{LIM}(h)^{\top}\end{pmatrix}$ is a random
left ideal code or if it is an left ideal LRPC code with parameter
$(\lambda,n,2n)$.
In other words, it is “hard” to distinguish if $h$ was sampled uniformly at
random or as $h_{2}{h_{1}}^{-1}$ where the ${\mathbb{F}}_{q}$-span of the
components of the vectors $h_{1}$ and $h_{2}$ has dimension $\lambda$.

The large number of invertible elements in general group algebras and the fact
that in general no canonical generators of ideals are known seems to suggest
that the problem is likely to be hard. Note that if $G$ is a cyclic group our
construction of left ideal codes coincide with the the notion of quasi-cyclic
code of index $2$. In this case, there is a structural attack against this
construction (see [7]). This implies that we have to suppose that $G$ is not
cyclic and suggests also to avoid the abelian case. If $G$ is not cyclic, left
ideal codes are similar to quasi-$G$ codes of index $2$ (see [2]), even if the
transposition prevents them from being properly. It would be interesting to
analyse if a similar attack exists also in this case.

It is possible also to readapt the Cryptosystems ROLLO II and ROLLO III, using
this approach.

## Acknowledgments

The last two authors were supported by the project “VALERE: VAnviteLli pEr la
RicErca” of the University of Campania “Luigi Vanvitelli” and by the Italian
National Group for Algebraic and Geometric Structures and their Applications
(GNSAGA - INdAM).

## References

  * [1] N. Aragon, P. Gaborit, A. Hauteville, O. Ruatta, and G. Zémor.  Low rank parity check codes: New decoding algorithms and applications to cryptography.  IEEE Transactions on Information Theory, 65(12):7697–7717, 2019\.
  * [2] M. Borello and W. Willems.  On the algebraic structure of quasi group codes.  Journal of Algebra and its Applications, 2022.
  * [3] H. Chimal-Dzul, N. Gassner, J. Rosenthal, and R. Schnyder.  Efficient description of some classes of codes using group algebras.  In 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2022), 2022.
  * [4] I. El Qachchach, O. Habachi, J.-P. Cances, and V. Meghdadi.  Efficient multi-source network coding using low rank parity check code.  In 2018 IEEE Wireless Communications and Networking Conference (WCNC), pages 1–6. IEEE, 2018.
  * [5] P. Gaborit, G. Murat, O. Ruatta, and G. Zémor.  Low rank parity check codes and their application to cryptography.  In Proceedings of the Workshop on Coding and Cryptography WCC, volume 2013, 2013.
  * [6] P. Gaborit, G. Murat, O. Ruatta, and G. Zemor.  Low Rank Parity Check codes and their application to cryptography.  In L. Budaghyan, T. Helleseth, and M. G. Parker, editors, The International Workshop on Coding and Cryptography (WCC 13), page 13 p., Bergen, Norway, Apr. 2013.
  * [7] A. Hauteville and J.-P. Tillich.  New algorithms for decoding in the rank metric and an attack on the lrpc cryptosystem.  In 2015 IEEE International Symposium on Information Theory (ISIT), pages 2747–2751. IEEE, 2015.
  * [8] N. Makhijani, R. Sharma, and J. Srivastava.  The unit group of finite group algebra of a generalized dihedral group.  Asian-European Journal of Mathematics, 7(02):1450034, 2014.
  * [9] C. Melchor, N. Aragon, M. Bardet, S. Bettaieb, L. Bidoux, O. Blazy, J. Deneuville, P. Gaborit, A. Hauteville, A. Otmani, et al.  ROLLO (merger of Rank-Ouroboros, LAKE and LOCKER).  Second round submission to the NIST post-quantum cryptography call, 2020.
  * [10] M. Sahai and S. F. Ansari.  Unit groups of group algebras of certain dihedral groups-ii.  Asian-European Journal of Mathematics, 12(04):1950066, 2019.
  * [11] R. Sandling.  Units in the modular group algebra of a finite abelian p-group.  Journal of Pure and Applied Algebra, 33(3):337–346, 1984.
  * [12] A. K. Yazbek, I. El Qachchach, J.-P. Cances, and V. Meghdadi.  Low rank parity check codes and their application in power line communications smart grid networks.  International Journal of Communication Systems, 30(12):e3256, 2017\.

Martino Borello
Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA,
Université Sorbonne Paris Nord, CNRS, UMR 7539, France
<EMAIL_ADDRESS>

Paolo Santonastaso and Ferdinando Zullo
Università degli Studi della Campania “Luigi Vanvitelli”, Caserta, Italy
<EMAIL_ADDRESS>


# A direct proof of existence of weak solutions to fully anisotropic and
inhomogeneous elliptic problems

Iwona Chlebicka, Arttu Karppinen, and Ying Li  Iwona Chlebicka
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland
<EMAIL_ADDRESS>Arttu Karppinen
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland
<EMAIL_ADDRESS>Ying Li
Department of Mathematics, Shanghai University, Shanghai 200444, China
<EMAIL_ADDRESS>
###### Abstract.

We provide a direct proof of existence and uniqueness of weak solutions to a
broad family of strongly nonlinear elliptic equations with lower-order terms.
The leading part of the operator satisfies general growth conditions settling
the problem in the framework of fully anisotropic and inhomogeneous
Musielak–Orlicz spaces generated by an $N$-function
$M:\Omega\times{\mathbb{R}}^{d}\to{[0,\infty)}$. Neither $\nabla_{2}$ nor
$\Delta_{2}$ conditions are imposed on $M$. Our results cover among others
problems with anisotropic polynomial, Orlicz, variable exponent, and double
phase growth.

Keywords: existence, elliptic boundary value problems, second order partial
differential equations, Musielak-Orlicz spaces

MSC 2010: 35J25

Acknowledgements: IC and AK are supported by NCN Grant Sonata Bis
2019/34/E/ST1/00120. YL is supported by China Scholarship Council,
No.202106890038.

##  1\. Introduction

In this paper we investigate the following strongly nonlinear problem

(1.1)
$\left\\{\begin{array}[]{cl}-\operatorname{div}\big{(}{\mathcal{A}}(x,\nabla
u)+\Phi(u)\big{)}+b(x,u)=\operatorname{div}F&\qquad\mathrm{in}\qquad\Omega,\\\
u(x)=0&\qquad\mathrm{on}\qquad\partial\Omega,\end{array}\right.$

where $\Omega$ is a bounded Lipschitz domain in ${\mathbb{R}}^{d}$, $d>1$. The
growth and coercivity of the vector field ${\mathcal{A}}$ are assumed to be
controlled by a generalized anisotropic $N$-function
$M:\Omega\times{\mathbb{R}}^{d}\to{[0,\infty)}$, $\Phi\in
L^{\infty}(\Omega,{\mathbb{R}}^{d})$ is a continuous vector field, $b$ is a
function satisfying a sign condition without any extra growth conditions and
the vector field $F$ has growth dictated by the conjugate function to $M$. The
considered framework is described in [11]. We briefly recall the definitions
of the involved $N$-functions, conjugates, and Musielak–Orlicz spaces in
Section 2. In the present study we do not impose $\nabla_{2}$ nor $\Delta_{2}$
condition on $M$, but we assume the modular density of smooth functions in the
related Sobolev-type space that can be inferred from balanced behaviour of
$M$. In this setting we show the existence of weak solutions by a direct
reasoning that does not rely on a regularisation of isotropic equation. The
result is supplied with a short proof of uniqueness.

Study of nonlinear problems was initiated in 1960s by Browder [6], Hess [32],
Leray and Lions [34] and many others. In the beginning the considered
operators were assumed to satisfy polynomial growth and coercivity conditions.
The typical example of such an operator is the $p$-Laplacian
$-\operatorname{div}{\mathcal{A}}(x,\nabla
u)=-\Delta_{p}u=-\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ for
$1<p<\infty$. Henceforth, the problem was stated in the reflexive setting of
the classical Sobolev spaces. In recent years, there has been an increasing
interest in the study of existence of solutions for elliptic problems of
nonstandard growth. See [7, 30, 37] and references therein for a pretty
complete account of new developments. Nowadays, the field of nonlinear
elliptic boundary value problems aims to embrace investigations on problems
admitting

  * •

general growth – when the power function governing the growth of the operator
is substituted by an $N$-function $M(x,\xi)=M(|\xi|)$, which do not
necessarily satisfy the so-called $\Delta_{2}$-condition (being a necessary
condition for an Orlicz space $L_{M}$ to be reflexive);

  * •

inhomogeneity – when the growth of the operator could be controlled by an
$x$-dependent function e.g. $M(x,\xi)=|\xi|^{p(x)}$ (which results in the lack
of the density of smooth functions in $L^{p(\cdot)}$, if $p(\cdot)$ is not
regular enough);

  * •

anisotropy – when the growth of the operator is governed by a function
depending on the full vector of $\xi$, not just its length $|\xi|$. For
example, $M(x,\xi)=|\xi_{1}|^{p}\log(1+|\xi_{1}-\xi_{2}|)$ for
$\xi=(\xi_{1},\xi_{2})$.

There are a lot of important ideas introduced to overcome the abovementioned
challenges. The difficulty caused by the lack of reflexivity of $L_{M}$ under
non-doubling regime was avoided by the idea of the complementary systems in
Orlicz–Sobolev spaces. Contributions in this direction were initiated by
Donaldson [14] and continued by Gossez [19, 20, 21] and Mustonen and Tienari
[38]. We refer to [2, 3, 10, 26] for analysis of problems in anisotropic
Orlicz spaces governed by possibly fully anisotropic modular function, which
is independent of the spacial variable. None of these contributions, however,
provides a direct proof of existence to PDEs. Note that the complementary
systems are very homogeneous by their nature. In the typical inhomogeneous
case of variable exponent problems, the setting fall into the realm of
reflexive spaces, and consequently the general theory exposed e.g. in [33]
yields existence to the problems with regular data. In [18, 29, 36] isotropic,
separable, and reflexive Musielak–Orlicz spaces are employed and [15] concerns
separable, but not reflexive Musielak–Orlicz spaces. Existence to problems
that are in the same time of general growth, inhomogeneous, and fully
anisotropic were studied in [9, 11, 12, 24, 25, 31, 35, 42], but none of them
provide a direct proof. Anisotropic problems with lower-order terms are less
understood – we can only refer to [13, 27], but they do not cover our
generality of the problem.

We are interested in a short and compact proofs of the existence and
uniqueness of solutions to (1.1) under possibly general regime under the
framework of [11]. Let us lay out our structural conditions for capturing
inhomogeneous and anisotropic problems more precisely.

Assumptions.  We consider a vector field
${\mathcal{A}}:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ belonging to a
Musielak–Orlicz class. Namely, we assume that a function
${\mathcal{A}}:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies the
following conditions:

  * (A1)

${\mathcal{A}}$ is a Carathéodory’s function;

  * (A2)

${\mathcal{A}}(x,0)=0$ for almost every $x\in\Omega$ and there exists an
$N$-function $M:\Omega\times{\mathbb{R}}^{d}\to{[0,\infty)}$ and constants
$c_{1}^{\mathcal{A}},c_{2}^{\mathcal{A}},c_{3}^{\mathcal{A}},c_{4}^{\mathcal{A}}>0$
such that for all $\xi\in{\mathbb{R}}^{d}$ we have

${\mathcal{A}}(x,\xi)\cdot\xi\geq M(x,c_{1}^{\mathcal{A}}\xi)-h_{1}(x)$

and

$c_{2}^{\mathcal{A}}M^{\ast}(x,c_{3}^{\mathcal{A}}{\mathcal{A}}(x,\xi))\leq
M(x,c_{4}^{\mathcal{A}}\xi)+h_{2}(x),$

where $M^{\ast}$ is the conjugate to $M$ and $h_{1},h_{2}\in L^{1}(\Omega)$;

  * (A3)

For all $\xi,\eta\in{\mathbb{R}}^{d}$ and for almost every $x\in\Omega$ we
have

$({\mathcal{A}}(x,\xi)-{\mathcal{A}}(x,\eta))\cdot(\xi-\eta)\geq 0.$

Moreover, we assume

  * (P)

$\Phi:{\mathbb{R}}\to{\mathbb{R}}^{d}$ is bounded and continuous;

  * (b)

$b:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory’s function,
which is nondecreasing with respect to the second variable, and such that
$b(\cdot,s)\in L^{1}(\Omega)$ and $b(\cdot,s)\operatorname{sign}(s)\geq 0$ for
every $s\in{\mathbb{R}}$.

We point out that we do not control the growth of $M$ with respect to the
second variable by any kind of doubling condition or a power function.
Instead, we rely on the density of smooth functions in a relevant function
space to study problem (1.1), namely

$V_{0}^{1}L_{M}(\Omega):=\\{\phi\in W_{0}^{1,1}(\Omega):\ \ \nabla\phi\in
L_{M}(\Omega)\\}\,,$

where $L_{M}(\Omega)$ is the inhomogeneous and fully anisotropic
Musielak–Orlicz spaces defined in Section 2. The space $E_{M}$ is a closure in
$L_{M}$-norm of the set of bounded functions. If $M$ does not satisfy growth
condition of doubling type, $E_{M}$ is a proper subspace of $L_{M}$. There is
a clear condition proven to be sufficient in [4], but since our reasoning does
not involve the condition in any other way, we assume density and recall the
condition below the statement of our main theorem.

Our main theorem is an existence and uniqueness result for problem (1.1).

######  Theorem 1.1.

Let $\Omega$ be a bounded Lipschitz domain in ${\mathbb{R}}^{d}$. Suppose that
an $N$-function $M$ is regular enough so that the set of smooth functions is
dense in $V^{1}_{0}L_{M}(\Omega)$ in the modular topology. Assume further that
$F\in E_{M^{\ast}}(\Omega)$, ${\mathcal{A}}$ satisfies assumptions (A1), (A2)
and (A3), $\Phi$ satisfies (P), and $b$ satisfies (b). Then there exists at
least one weak solution to the problem (1.1). Namely, there exists a function
$u\in V^{1}_{0}L_{M}(\Omega)$ satisfying

$\int_{\Omega}{\mathcal{A}}(x,\nabla u)\cdot\nabla v+\Phi(u)\cdot\nabla
v+b(x,u)v\,dx=\int_{\Omega}F\cdot\nabla v\,dx$

for all $v\in V^{1}_{0}L_{M}(\Omega)\cap L^{\infty}(\Omega)$.

If, additionally, $s\mapsto b(\cdot,s)$ is strictly increasing and $\Phi$ is
Lipschitz continuous, then the weak solution is unique.

######  Remark 1.2.

Our result is valid in the case of bounded data. In fact, for each $g\in
L^{\infty}(\Omega)$, we know that there exists $F:\Omega\to{\mathbb{R}}^{d}$,
such that $g=\operatorname{div}F$ and $F\in E_{M^{\ast}}(\Omega)$. The fact
one can take $F\in E_{M^{\ast}}(\Omega)$ is a consequence of properties of
Bogovski operator. This is explained in [11, Remark 4.1.7] with the use of
[40, Lemma II.2.1.1].

In order to ensure the density one may assume regularity of $M$. Note that the
smooth functions are dense in $V^{1}_{0}L_{M}(\Omega)$ in the modular topology
if the following balance condition holds, see [4].

Condition $\mathsf{(B)}$.  Given an $N$-function
$M:\Omega\times{\mathbb{R}}^{d}\to{[0,\infty)}$ suppose there exists a
constant $C_{M}>1$ such that for every ball $B\subset\Omega$ with $|B|\leq 1$,
every $x\in B,$ and for all $\xi\in{\mathbb{R}}^{d}$ such that $|\xi|>1$ and
$M(x,C_{M}\xi)\in[1,\tfrac{1}{|B|}]$ there holds $\sup_{y\in B}M(y,\xi)\leq
M(x,C_{M}\xi)$.

Note that in the isotropic and doubling regime this condition is known to be
sufficient to the boundedness of the maximal operator. Moreover, when $d=1$
they are equivalent [28]. Condition $\mathsf{(B)}$ is essentially less
restrictive than the isotropic one from [1] or the anisotropic ones used in
[24, 11]. Following [4, 28], we give examples of $N$-functions satisfying the
above balance condition.

######  Example 1.1.

The following $N$-functions satisfy the balance condition $\mathsf{(B)}$.

  1. (1)

Variable exponent case: $M(x,\xi)=|\xi|^{p(x)}$, where
$p(x):\Omega\rightarrow[p^{-},p^{+}]$ is log-Hölder continuous and
$1<p^{-}\leq p(\cdot)\leq p^{+}\leq\infty$; see the proof of [28, Proposition
7.1.2].

  2. (2)

Double phase case: $M(x,\xi)=|\xi|^{p}+a(x)|\xi|^{q}$, with $1<p\leq
q<\infty$, $0\leq a\in C^{0,\alpha}(\Omega)$, $\alpha\in(0,1]$ and
$\frac{p}{q}\leq 1+\frac{\alpha}{n}$; see the proof of [28, Proposition
7.2.2].

  3. (3)

Anisotropic variable case: $M(x,\xi)=\sum_{i=1}^{n}|\xi_{i}|^{p_{i}(x)}$,
where $p_{i}(x):\Omega\rightarrow[p_{i}^{-},p_{i}^{+}]$ are log-Hölder
continuous and $1<p^{-}_{i}\leq p_{i}(\cdot)\leq p_{i}^{+}\leq\infty$; see [4,
Subsection 4.4].

  4. (4)

Anisotropic double phase case:
$M(x,\xi)=\sum_{i=1}^{n}\left(|\xi_{i}|^{p_{i}}+a_{i}(x)|\xi_{i}|^{q_{i}}\right)$,
where $1<p_{i}\leq q_{i}<\infty$, $0\leq a_{i}\in C^{0,\alpha_{i}}(\Omega)$,
$\alpha_{i}\in(0,1]$, and $\frac{p_{i}}{q_{i}}\leq 1+\frac{\alpha_{i}}{n}$;
for this example, as well as anisotropic multi-phase case (also with Orlicz
phases), see [4, Subsection 4.4].

Taking into account [10] and [4, Section 4] one can provide an explicit
condition that implies $\mathsf{(B)}$ even in the case when the anisotropic
function $M(x,\xi)$ does not admit a so-called orthotropic decomposition
$\sum_{i=1}^{d}M_{i}(x,\xi_{i})$ even after an affine change of variables.

Let us comment on the choice of method of the proof and its importance. We
point out that there are some related results in the literature. For instance
[23] is devoted to the existence of the weak solutions to a problem of the
form (1.1) with $\Phi\equiv 0$ and $b\equiv 0$. Since the equation is posed in
the inhomogeneous and fully anisotropic space setting, where no conditions of
doubling-type are assumed, the underlying space needs to be equipped with some
approximation property. For this a compatibility condition might be imposed.
The main idea of the proof of existence in [23] is to introduce a regularised
problem with solutions in the classical Orlicz–Sobolev space, make use of the
theory of pseudo-monotone operators due to [20, 22, 38], and pass to the
limit. This scheme was later applied also e.g. in [2, 11, 24] in the proofs of
existence to problems without lower-order terms. A direct proof was a puzzling
lack in this theory even in the absence of the lower-order terms.

Note that the framework of [20, 38] is isotropic and describes the admitted
behaviour of lower-order terms by the means of functions governing the
operator. It is hard to provide natural counterpart of assumptions on the
lower-order terms to get existence for possibly broad class of problems via
this approach. Let us point out that no ‘good’ embeddings of fully anisotropic
Musielak–Orlicz–Sobolev spaces into Musielak–Orlicz spaces are known. In turn,
so far one is not equipped with relevant growth conditions that can be applied
to the lower-order terms due to the embeddings.

Interested in the existence of solutions to as broad as possible class of
problems and in a straightforward reasoning that will be relevant for the
general properties of the function setting and problems with lower-order
terms, we provide a direct proof of existence of weak solutions to (1.1). Our
method is based on the Galerkin approximation. To be precise, we take a
sequence of solutions to finite dimensional problems and show that it has a
subsequence converging sufficiently well to our weak solution. Despite the
idea is classical, the proof is not trivial because of two reasons. The first
of them results from the fact that the understanding of the dual pairing is
complicated ($L_{M^{*}}$ is not dual to $L_{M}$, but only associate). The
second reason comes from the fact that in the general Orlicz–Sobolev spaces
the set of smooth functions is not dense in the norm topology. Assuming the
balanced behaviour of $M$ one is equipped with the best possible density in
the framework of Musielak–Orlicz–Sobolev spaces, namely with the modular
density of smooth functions, cf. [4].

Despite we follow the schemes of the proofs that can be found in [11, 27, 39],
the compactness arguments are much more delicate. We take into account the
anisotropy and inhomogeneity not considered in [39], introduce general lower-
order terms unlike [11], and relax the doubling growth condition of [27],
where assuming $M^{*}\in\Delta_{2}$ is essential. In turn, most of the limit
passages needed to be justified in a different way than in the above-mentioned
literature. Since no theory of operator classes or regularization of the
equation is used in our proof, we do not apply directly ideas of [13, 19, 20,
21, 38]. This enables us to dismiss imposing growth conditions on the lower-
order terms. Summing up, we provide a detailed direct proof of the existence
result supplied with uniqueness shown by choosing a relevant sequence of test
functions.

##  2\. Preliminaries

By $\Omega$ we always mean a bounded domain of ${\mathbb{R}}^{d}$ with
Lipschitz regular boundary. If not specified, a constant $C$ is a positive
constant, possibly changing line by line. For a measurable set $A$, we denote
by $\chi_{A}$ its characteristic function. By $C^{\infty}_{c}(\Omega)$ we mean
the set of compactly supported smooth functions over $\Omega$.

Our framework is anisotropic Musielak–Orlicz spaces based on $N$-functions.
Readers are advised to turn to [11] for a thorough resource. In order to
introduce the main definitions let us define a Young function in the classical
way. We say a function $m:[0,\infty)\to[0,\infty)$ is a Young function if it
is convex, satisfies $m(s)=0\Leftrightarrow s=0$ and is superlinear at zero
and infinity, i.e.

$\lim_{s\to
0^{+}}\frac{m(s)}{s}=0\quad\text{and}\quad\lim_{s\to\infty}\frac{m(s)}{s}=\infty.$

A function $M(x,\xi):\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ is
called an $N$-function if

  * •

it is a convex function with respect to ${\xi}$;

  * •

it is a Carathéodory function satisfying $M(x,0)=0$;

  * •

$M(x,\xi)=M(x,-\xi)$ for a.e. $x\in\Omega$;

  * •

there exist two Young functions $m_{1},m_{2}:[0,\infty)\to[0,\infty)$ such
that for a.e. $x\in\Omega$ it holds

$\displaystyle m_{1}(|\xi|)\leq M(x,\xi)\leq m_{2}(|\xi|).$

The conjugate function $M^{*}$ to an $N$-function $M$ is defined by

$M^{\ast}(x,\eta):=\displaystyle\sup_{\xi\in\mathbb{R}^{d}}\left[\xi\cdot\eta-M(x,\xi)\right]\quad\mbox{for
any}~{}~{}\eta\in\mathbb{R}^{d}~{}~{}\mbox{and}~{}~{}\mbox{a.e.}~{}~{}x\in\Omega.$

In that case, the Fenchel–Young inequality reads

$\displaystyle\xi\cdot\eta\leq M(x,\xi)+M^{\ast}(x,\eta)\quad\mbox{for
all}~{}~{}~{}\xi,\eta\in\mathbb{R}^{d}~{}~{}~{}\mbox{and}~{}~{}\mbox{a.e.}~{}~{}x\in\Omega.$

For an $N$-function we define the general Musielak–Orlicz class
$\mathcal{L}_{M}(\Omega)$ as the set of all measurable functions
$\xi:\Omega\rightarrow\mathbb{R}^{d}$ satisfying

$\int_{\Omega}M(x,\xi(x))\,dx<\infty\,.$

The Musielak–Orlicz space $L_{M}(\Omega)$ is the smallest linear hull of
$\mathcal{L}_{M}(\Omega)$ equipped with the Luxemburg norm

$\|\xi\|_{L_{M}(\Omega)}:=\inf\left\\{\lambda>0:\int_{\Omega}M\left(x,\frac{\xi(x)}{\lambda}\right)dx\leq
1\right\\}.$

The space $E_{M}(\Omega)$ is the closure in $L_{M}$-norm of the set of bounded
functions. Equivalently, $L_{M}(\Omega)$ and $E_{M}(\Omega)$ are defined as
sets of functions $\xi:\Omega\to{\mathbb{R}}^{d}$ satisfying

$\int_{\Omega}M(x,\lambda\xi(x))\,dx<\infty$

for some $\lambda\in{\mathbb{R}}$ and for every $\lambda\in{\mathbb{R}}$,
respectively [11, Lemma 3.1.8]. We also note that
$(E_{M}(\Omega))^{\ast}=L_{M^{\ast}}(\Omega)$ and
$(E_{M^{\ast}}(\Omega))^{\ast}=L_{M}(\Omega)$ [11, Theorem 3.5.3] but no other
duality relations are expected.

Next we define various notions of convergence. The first is called modular
convergence and it generates a more suitable topology to our possibly non-
reflexive spaces than the norm topology. A sequence
$\left\\{\xi_{i}\right\\}^{\infty}_{i=1}$ converges modularly to ${\xi}$ in
$L_{M}(\Omega)$, which we denote as ${\xi_{i}}\xrightarrow[]{M}\xi$, if

$\int_{\Omega}M\left(x,\frac{\xi_{i}-\xi}{\lambda}\right)dx\xrightarrow{i\rightarrow\infty}0$

for some $\lambda>0$.

Let us point out that smooth functions are not dense in our Sobolev space in
the norm topology, but they are dense in the modular topology.

######  Lemma 2.1 (Theorem 1 in [4]).

Assume that $\Omega$ is a Lipschitz domain and $M$ is an $N$-function
satisfying the Balance condition ($B$). Then for any $\phi\in
V^{1}_{0}L_{M}(\Omega)$, there exists a sequence
$\left\\{\phi_{\delta}\right\\}_{\delta>0}\in C^{\infty}_{c}(\Omega)$
satisfying $\phi_{\delta}\rightarrow{\phi}$ in $L^{1}(\Omega)$ and
$\nabla\phi_{\delta}\xrightarrow{M}\nabla\phi$. Additionally, if $\phi$ is
bounded, then $\|\phi_{\delta}\|_{L^{\infty}(\Omega)}\leq
C(\Omega)\|\phi\|_{L^{\infty}(\Omega)}$ for every $\delta>0$.

Again, the possible non-reflexivity forces us to consider replacements for
weak convergence. Let $X$ and $Y$ be subsets of $L^{1}(\Omega)$ not
necessarily related by duality. We say $f_{n}\to f$ for $\sigma(X,Y)$ if

$\int_{\Omega}f_{n}g\,dx\xrightarrow{n\to\infty}\int_{\Omega}fg\,dx$

for all $g\in Y$. If $X=L_{M}(\Omega)$ and $Y=E_{M^{\ast}}(\Omega)$, we
recover the weak-$\ast$ convergence and can also denote
$f_{n}\stackrel{{\scriptstyle\ast}}{{\rightharpoonup}}f$.

Another consequence of modular topology is the following fact.

######  Lemma 2.2 (Lemma 3.4.6 in [11]).

Let $M$ be an $N$-function and $\\{\xi_{n}\\}^{\infty}_{n=1}$, $\xi\in
L_{M}(\Omega)$. If $\xi_{n}\xrightarrow{M}\xi$ in $L_{M}(\Omega)$ then, up to
a subsequence, $\xi_{n}\xrightarrow{n\rightarrow\infty}\xi$ in
$\sigma(L_{M},L_{M^{\ast}})$.

Next, we shall give some definitions and preliminary lemmas related to
$N$-functions and Musielak–Orlicz spaces. The first one compares a modular and
a norm of a function.

######  Lemma 2.3 (Lemma 3.1.14 in [11]).

Let $M$ be an $N$-function.

  1. (1)

If $\xi\in L_{M}(\Omega)$ and $\|\xi\|_{L_{M}(\Omega)}\leq 1$, then
$\int_{\Omega}M(x,\xi(x))\,dx\leq\|\xi\|_{L_{M}(\Omega)}.$

  2. (2)

If $\xi\in L_{M}(\Omega)$ and $\|\xi\|_{L_{M}(\Omega)}>1$, then
$\int_{\Omega}M(x,\xi(x))\,dx\geq\|\xi\|_{L_{M}(\Omega)}.$

Throughout the paper we take advantage of the corollary of the Vitali
convergence theorem. For that we require the concept of uniform integrability.

######  Definition 2.4 (Uniform integrability).

Let $\Omega\subset{\mathbb{R}}^{d}$ have finite measure. A sequence
$\\{f_{n}\\}\subset L^{1}(\Omega)$ is uniformly integrable if for any
$\varepsilon>0$ there exists a $\delta>0$ such that for every measurable set
$E$, if $|E|<\delta$, then $\int_{E}|f_{n}|\,dx<\varepsilon$ for every
$n\in\mathbb{N}$.

The following consequence of in the Vitali convergence theorem results from
[16, Theorem 8.23]. Since we assume that $\Omega$ has a finite measure, we use
pointwise convergence instead of convergence in measure.

######  Lemma 2.5.

If $f_{n},f\in L^{1}(\Omega),$ $n=1,2,\dots$, and $|\Omega|<\infty$, $f_{n}\to
f$ almost everywhere in $\Omega$ and $\\{f_{n}\\}$ is uniformly integrable in
$L^{1}(\Omega)$, then $f_{n}\to f$ in $L^{1}(\Omega)$.

Since $N$-functions are superlinear, they allow us to detect uniform
integrability.

######  Lemma 2.6 (Lemma 3.4.2 in [11]).

Suppose $M$ is an $N$-function and let $\\{\xi_{n}\\}_{n=1}^{\infty}$ be a
sequence of measurable functions $\xi_{n}:\Omega\to{\mathbb{R}}^{d}$
satisfying

$\sup_{n\in\mathbb{N}}\int_{\Omega}M(x,\xi_{n}(x))dx<\infty.$

Then the sequence $\\{\xi_{n}\\}_{n=1}^{\infty}$ is uniformly integrable in
$L^{1}(\Omega,{\mathbb{R}}^{d})$.

Lastly we give a few miscellaneous and useful results.

######  Lemma 2.7 (Lemma 8.22 in [11]).

Suppose $f_{k},f\in L^{\infty}(\Omega)$ and $g_{k}\rightharpoonup g$ in
$L^{1}(\Omega)$. Assume further that there exists a $C>0$ such that
$\sup_{k\in\mathbb{N}}\|f_{k}\|_{L^{\infty}(\Omega)}<C$ and $f_{k}\to f$
almost everywhere in $\Omega$. Then

$\lim_{k\to\infty}\int_{\Omega}f_{k}g_{k}\,dx=\int_{\Omega}fg\,dx.$

######  Lemma 2.8.

Let $\Phi:{\mathbb{R}}\to{\mathbb{R}}^{d}$ be continuous and belong to
$L^{\infty}(\Omega,{\mathbb{R}}^{d})$. Let $u\in W^{1,1}_{0}(\Omega)$. Then

$\int_{\Omega}\Phi(u)\cdot\nabla u\,dx=0.$

###### Proof.

Since $\Phi$ is continuous, for each of its components $\Phi_{k}$ there exists
a function $G_{k}(t)=\int_{0}^{t}\Phi_{k}(s)\,ds$ so that
$(G_{k}(u))^{\prime}=\Phi_{k}(u)\dfrac{\partial u}{\partial x_{k}}$ and
$G_{k}(0)=0$. Since each $G_{k}$ is defined as an integral, all of them are
absolutely continuous. Therefore, by denoting as $dS$ the $(n-1)$-dimensional
Hausdorff measure we can use integration by parts and note that $u=0$ on
$\partial\Omega$ to get

$\displaystyle\int_{\Omega}\Phi(u)\cdot\nabla u\,dx$
$\displaystyle=\sum_{k=1}^{d}\int_{\Omega}\Phi_{k}(u)\frac{\partial
u}{\partial x_{k}}\,dx=\sum_{k=1}^{d}\int_{\Omega}\dfrac{\partial}{\partial
x_{k}}(G_{k}(u))\,dx$
$\displaystyle=\sum_{k=1}^{d}\left\\{\int_{\partial\Omega}G_{k}(u)\cdot
1\,dS-\int_{\Omega}G_{k}(u)\cdot 0\,dx\right\\}=0.$

∎

######  Lemma 2.9 (Section 9.1 in [17]).

Let $s:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be a continuous mapping and

$s(x)\cdot x\geq 0\quad\mbox{if}\quad|x|=r$

for some $r>0$. Then there is a point $x$ with $|x|\leq r$ such that $s(x)=0$.

######  Lemma 2.10 (Monotonicity trick, Theorem 4.1.1 in [11]).

Suppose ${\mathcal{A}}:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$
satisfies growth and coercivity conditions (A1) and (A2) for an $N$-function
$M:\Omega\times{\mathbb{R}}^{d}\to[0,\infty)$. Let $h\in L_{M^{\ast}}(\Omega)$
and $\xi\in L_{M}(\Omega)$ satisfy

$\int_{\Omega}\big{(}h-{\mathcal{A}}(x,\eta)\big{)}\cdot(\xi-\eta)\,dx\geq
0\quad\text{for all }\eta\in{\mathbb{R}}^{d}.$

Then $h={\mathcal{A}}(x,\xi)$ almost everywhere in $\Omega$.

##  3\. Galerkin solutions

In this section we discuss finite dimensional approximations of our problem
(1.1) and their solutions, called Galerkin solutions. In the proof of Theorem
1.1 the existence of weak solutions is found as a limit of Galerkin solutions
when the dimension of the approximating problem is increased. In the lemmas
below we show the existence and boundedness properties of Galerkin solutions
required for compactness arguments. The proofs follow similar scheme as in
[11, Theorem 4.1.2], but here we skip the structural assumption
$M^{*}\in\Delta_{2}$ that influence the functional analysis of the underlying
space, and include the analysis of lower order terms.

Since $C^{\infty}_{c}(\Omega)$ is separable and dense in $C^{1}_{c}(\Omega)$
we can extract a sequence of $\\{w_{i}\\}_{i=1}^{\infty}\subset
C^{\infty}_{c}(\Omega)$ such that
$\overline{\text{span}\\{w_{1},w_{2},\dots\\}}^{C^{1}_{c}}=C^{1}_{c}(\Omega)$.
We denote the finite dimensional spaces as $V_{n}:=\text{span}\\{w_{1},\dots
w_{n}\\}$.

######  Definition 3.1.

For every $n=1,2,\dots$ and spaces $V_{n}$ defined above, we say a function
$u_{n}\in V_{n}$ is called a Galerkin solution to the problem (1.1) if

(3.1) $\int_{\Omega}{\mathcal{A}}(x,\nabla
u_{n})\cdot\nabla\omega+\Phi(u_{n})\cdot\nabla\omega+b(x,u_{n})\omega\,dx=\int_{\Omega}F\cdot\nabla\omega\
dx.$

holds for every $\omega\in V_{n}$.

######  Lemma 3.2 (Existence of Galerkin solutions).

Let $\Omega\subset{\mathbb{R}}^{d}$ be a bounded Lipschitz domain. Assume
${\mathcal{A}}$ satisfies (A1), (A2) and (A3) and
$M:\Omega\times{\mathbb{R}}^{d}\to[0,\infty)$ is an $N$-function. Let $F\in
E_{M^{\ast}}(\Omega)$, $b:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ satisfies
condition (b) and $\Phi:{\mathbb{R}}\to{\mathbb{R}}^{d}$ satisfy (P). Then for
every $n\in\mathbb{N}$, there exists a Galerkin solution in the sense of
(3.1).

###### Proof.

Every $u_{n}\in V_{n}$ can be written in a form
$u_{n}=\sum_{k=1}^{n}\alpha_{k}w_{k}$ for $n\in\mathbb{N}$, where
$\alpha_{k}\in{\mathbb{R}}$. We show that there exist such
$\\{\alpha_{k}\\}_{k=1}^{n}$ such that

$\int_{\Omega}{\mathcal{A}}(x,\nabla u_{n})\cdot\nabla
w_{k}+\Phi(u_{n})\cdot\nabla
w_{k}+b(x,u_{n})w_{k}\,dx=\int_{\Omega}F\cdot\nabla w_{k}\,dx.$

We define
$\vec{\alpha}=(\alpha_{1},\alpha_{2},\dots\alpha_{n})\in{\mathbb{R}}^{n}$, a
mapping $s:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ given coordinate by the
formula

$s_{j}(\vec{\alpha}):=\int_{\Omega}{\mathcal{A}}\left(x,\sum_{k=1}^{n}\alpha_{k}\nabla
w_{k}\right)\cdot\nabla
w_{j}+\Phi\left(\sum_{k=1}^{n}\alpha_{k}w_{k}\right)\cdot\nabla
w_{j}+b\left(x,\sum_{k=1}^{n}\alpha_{k}w_{k}\right)w_{j}-F\cdot\nabla
w_{j}\,dx$

and

$\omega(\vec{\alpha}):=\sum_{k=1}^{n}\alpha_{k}w_{k}.$

In order to prove that $s$ takes the value zero, first we show that $s$ is
continuous and for that we choose a sequence
$\\{\vec{a}^{i}\\}_{i=1}^{\infty}$ such that $\vec{a}^{i}\to\vec{\alpha}$ in
${\mathbb{R}}^{n}$. Consider

$\begin{split}|s_{j}(\vec{a}^{i})-s_{j}(\vec{\alpha})|=&\bigg{|}\int_{\Omega}h_{i,j}(x)\,dx\bigg{|}\end{split}$

for all $j=1,\dots,n$ and every $i\in\mathbb{N}$, where

$h_{i,j}(x):=\big{(}{\mathcal{A}}(x,\nabla\omega(\vec{a}^{i}))-{\mathcal{A}}(x,\nabla\omega(\vec{\alpha}))\big{)}\cdot\nabla
w_{j}+\big{(}\Phi(\omega(\vec{a}^{i}))-\Phi(\omega(\vec{\alpha}))\big{)}\cdot\nabla
w_{j}+\big{(}b(x,\omega(\vec{a}^{i}))-b(x,\omega(\vec{\alpha}))\big{)}w_{j}.$

Since ${\mathcal{A}}$ and $b$ are Carathéodory functions and $\Phi$ is
continuous, we see that for almost every $x\in\Omega$ we have $h_{i,j}(x)\to
0$ as $i\to\infty$. We show uniform integrability of the three terms in
$h_{i,j}$ one by one starting with ${\mathcal{A}}$. Assumption (A2) shows that

$c_{2}M^{\ast}(x,c_{3}{\mathcal{A}}(x,\nabla\omega(\vec{a}^{i})))\leq
M(x,c_{4}\nabla\omega(\vec{a}^{i}))+h_{2}(x)$

and for $\vec{a}^{i}=(a^{i}_{1},\dots,a^{i}_{n})$

$\displaystyle\int_{\Omega}M(x,c_{4}\nabla\omega(\vec{a}^{i}))\,dx$
$\displaystyle\leq\sum_{k=1}^{n}\frac{|a^{i}_{k}|}{|\vec{a}^{i}|}\int_{\Omega}M(x,c_{4}|\vec{a}^{i}|\nabla
w_{k})\,dx\leq
n\max_{k\in\\{1,\dots,n\\}}\int_{\Omega}M(x,c_{4}|\vec{a}^{i}|\nabla
w_{k})\,dx$

and this is finite since $\\{\vec{a}^{i}\\}_{i=1}^{\infty}$ is bounded
${\mathbb{R}}^{n}$ and $w_{k}\in C_{c}^{\infty}(\Omega)$, see [11, Lemma
3.1.8]. Now Lemma 2.6 shows the uniform integrability of
$\\{{\mathcal{A}}(\cdot,\nabla\omega(\vec{a}^{i}))\\}_{i=1}^{\infty}$ in
$L^{1}(\Omega,{\mathbb{R}}^{d})$. As
${\mathcal{A}}(\cdot,\nabla\omega(\vec{\alpha}))\in
L^{1}(\Omega,{\mathbb{R}}^{d})$ and $\nabla w_{k}\in
L^{\infty}(\Omega,{\mathbb{R}}^{d})$ and it follows that
$\big{(}{\mathcal{A}}(x,\nabla\omega(\vec{a}^{i}))-{\mathcal{A}}(x,\nabla\omega(\vec{\alpha}))\big{)}\cdot\nabla
w_{j}$ is uniformly integrable in $L^{1}(\Omega)$.

Moreover, we notice that $|\Phi(\cdot)|\in L^{\infty}({\mathbb{R}})$ and
$|\nabla w_{k}|\in L^{\infty}(\Omega)$. Thus for any set $E\subset\Omega$

$\left|\int_{E}\Phi(\omega(\vec{a}^{i}))\cdot\nabla
w_{k}\,dx\right|\leq\max_{k\in\\{1,\dots,n\\}}\int_{E}\||\Phi(\omega(\vec{a}^{i}))|\|_{L^{\infty}({\mathbb{R}})}\||\nabla
w_{k}|\|_{L^{\infty}(\Omega)}\,dx<C|E|.$

Furthermore by the fact that $s\mapsto b(\cdot,s)$ is non-decreasing,
convergence of $\vec{a}^{i}$ and boundedness of $\\{w_{k}\\}$ we get

$\displaystyle\left|\int_{E}\big{(}b(x,\omega(\vec{a}^{i}))\big{)}w_{k}\,dx\right|\leq
n\max_{k\in\\{1,\dots,n\\}}\left\\{\|w_{k}\|_{L^{\infty}(\Omega)}\int_{E}|b(x,|\vec{a}^{i}|\,w_{k})|\,dx\right\\}<C|E|.$

Thus all the terms in $h_{i,j}$ are uniformly integrable in $L^{1}(\Omega)$
and $h_{i,j}\to 0$ almost everywhere, so it follows from Vitali convergence
theorem (Lemma 2.5) that

$|s(\vec{a}^{i})-s(\vec{\alpha})|\leq
n\max_{j\in\\{1,\dots,n\\}}\int_{\Omega}|h_{i,j}|\,dx\to 0$

as $i\to\infty$. Thus $s$ is continuous.

Our next aim is to show that there exists a vector
$\vec{\alpha}\in{\mathbb{R}}^{n}$ such that $s(\vec{\alpha})=0$. For this we
will apply Lemma 2.9. Therefore we need to prove that
$s(\vec{\alpha})\cdot\vec{\alpha}\geq 0$ if $|\vec{\alpha}|=R$ for some $R>0$.
From (A2), Young’s inequality, the facts that
$\Phi(\omega(\vec{\alpha}))\cdot\nabla\omega(\vec{\alpha})=0$ and $b$
satisfies (b), and Lemma 2.3(2) we get

$\displaystyle s(\vec{\alpha})\cdot\vec{\alpha}$
$\displaystyle=\int_{\Omega}{\mathcal{A}}(x,\nabla\omega(\vec{\alpha}))\cdot\nabla\omega(\vec{\alpha})+b(x,\omega(\vec{\alpha}))\omega(\vec{\alpha})-F\cdot\nabla\omega(\vec{\alpha})\,dx$
$\displaystyle\geq\frac{1}{2}\int_{\Omega}M(x,c_{1}^{\mathcal{A}}\nabla\omega(\vec{\alpha}))\,dx-
h_{1}(x)\,dx-\int_{\Omega}M^{\ast}\left(x,\frac{2}{c_{1}}F\right)\,dx$
$\displaystyle\geq\frac{1}{2}(c_{1}^{\mathcal{A}}\|\nabla\omega(\vec{\alpha})\|_{L_{M}(\Omega)}-1)-C.$

Note that $\vec{\alpha}\mapsto\|\nabla\omega(\vec{\alpha})\|_{L_{M}(\Omega)}$
is a continuous function and therefore it attains its minimum on the unit
sphere $S_{1}\subset{\mathbb{R}}^{n}$ at some
$\vec{\beta}=(\beta_{1},\dots,\beta_{n})\in S_{1}$.

We show that $\|\nabla\omega(\vec{\beta})\|_{L_{M}(\Omega)}>0$. Assume the
contrary that $\|\nabla\omega(\vec{\beta})\|_{L_{M}(\Omega)}=0$. Thus
$\|\sum_{k=1}^{n}\beta_{k}\nabla w_{k}\|_{L^{1}(\Omega)}=0$ and by Poincaré
inequality we also have $\|\sum_{k=1}^{n}\beta_{k}w_{k}\|_{L^{1}(\Omega)}=0$
(recall that $w_{k}\in C_{c}^{\infty}(\Omega)$). This implies that
$\sum_{k=1}^{n}\beta_{k}w_{k}=0$ almost everywhere in $\Omega$, i.e.
$\beta_{k}=0$ for each $k=1,\dots,n$ since functions $w_{k}$ are linearly
independent. Since $\vec{\beta}\in S_{1}$ this is contradiction so
$\|\nabla\omega(\vec{\beta})\|_{L_{M}(\Omega)}>0$.

For our choice of $\vec{\beta}$ we have

$\|\nabla\omega(\vec{\alpha})\|_{L_{M}(\Omega)}\geq|\alpha|\left\|\nabla\omega\left(\frac{\vec{\alpha}}{|\vec{\alpha}|}\right)\right\|_{L_{M}(\Omega)}\geq|\vec{\alpha}|\|\nabla\omega(\vec{\beta})\|_{L_{M}(\Omega)}>0.$

This means that $\|\nabla\omega(\vec{\alpha})\|_{L_{M}(\Omega)}\to\infty$ when
$|\vec{\alpha}|\to\infty$. Thus $s(\vec{\alpha})\cdot\vec{\alpha}\geq 0$ if
$|\vec{\alpha}|=R$ for some $R>0$. From the definition of $s$ we see that for
$u_{n}=\sum_{k=1}^{n}\alpha_{k}w_{k}$ we have

$\int_{\Omega}{\mathcal{A}}(x,\nabla
u_{n})\cdot\nabla\omega+\Phi(u_{n})\cdot\nabla\omega+b(x,u_{n})\omega\,dx=\int_{\Omega}F\cdot\nabla\omega\,dx$

and thus the Galerkin solution $u_{n}$ exists. ∎

######  Lemma 3.3 (Uniform boundedness of Galerkin solutions).

Let $\Omega\subset{\mathbb{R}}^{d}$ be a bounded Lipschitz domain. Assume
${\mathcal{A}}$ satisfies (A1), (A2), (A3) and
$M:\Omega\times{\mathbb{R}}^{d}\to[0,\infty)$ is an $N$-function. Let $F\in
E_{M^{\ast}}(\Omega)$, $b:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ satisfy
condition (b) and $\Phi:{\mathbb{R}}\to{\mathbb{R}}^{d}$ satisfy (P). Then
there exists a constant $C$ independent of $n$ such that for every Galerkin
solution $u_{n}$ it holds

$\int_{\Omega}{\mathcal{A}}(x,\nabla u_{n})\cdot\nabla u_{n}\,dx\leq
C;\quad\|\nabla u_{n}\|_{L_{M}(\Omega)}\leq
C;\quad\int_{\Omega}b(x,u_{n})u_{n}\,dx\leq C.$

###### Proof.

We test the equation (3.1) by $u_{n}$, which is an admissible test function
due to Lemma 3.2, to get

$\int_{\Omega}{\mathcal{A}}(x,\nabla u_{n})\cdot\nabla
u_{n}+\Phi(u_{n})\cdot\nabla
u_{n}+b(x,u_{n})u_{n}\,dx=\int_{\Omega}F\cdot\nabla u_{n}\,dx.$

Using condition (A2) we get an estimate

$\frac{1}{2}\int_{\Omega}M(x,c_{1}^{\mathcal{A}}\nabla
u_{n})-h_{1}(x)\,dx\leq\frac{1}{2}\int_{\Omega}A(x,\nabla u_{n})\cdot\nabla
u_{n}\,dx.$

Moreover, Fenchel–Young inequality and the definition of $N$ functions allow
us to infer that

$\displaystyle\int_{\Omega}\frac{4}{c_{1}^{\mathcal{A}}}F\cdot\frac{c_{1}^{\mathcal{A}}}{4}\nabla
u_{n}dx$
$\displaystyle\leq\int_{\Omega}M\left(x,\frac{c_{1}^{\mathcal{A}}}{4}\nabla
u_{n}\right)\,dx+\int_{\Omega}M^{\ast}\left(x,\frac{4}{c_{1}^{\mathcal{A}}}F\right)\,dx$
$\displaystyle\leq\frac{1}{4}\int_{\Omega}M(x,c_{1}^{\mathcal{A}}\nabla
u_{n})dx+\int_{\Omega}M^{\ast}\left(x,\frac{4}{c_{1}^{\mathcal{A}}}F\right)\,dx.$

And thus, according to $M(x,\xi)\geq 0$ for almost every $x\in\Omega$ and
every $\xi\in{\mathbb{R}}^{d}$, $b(x,u_{n})u_{n}\geq 0$ as $b$ satisfies
condition (b), and $\Phi(u_{n})\cdot\nabla u_{n}=0$ by Lemma 2.8 we have

$\displaystyle\frac{1}{4}\int_{\Omega}M(x,c_{1}^{\mathcal{A}}\nabla
u_{n})dx+\frac{1}{2}\int_{\Omega}{\mathcal{A}}(x,\nabla u_{n})\cdot\nabla
u_{n}dx+\int_{\Omega}\Phi(u_{n})\cdot\nabla
u_{n}dx+\int_{\Omega}b(x,u_{n})u_{n}\,dx$
$\displaystyle\quad\leq\int_{\Omega}M^{\ast}\left(x,\frac{4}{c_{1}^{\mathcal{A}}}F\right)+\frac{1}{2}\|h_{1}(x)\|_{L^{1}(\Omega)}.$

Since $F\in E_{M^{\ast}}(\Omega)$ by assumption, the right-hand side of the
latter inequality is finite and we infer that

$\int_{\Omega}{\mathcal{A}}(x,\nabla u_{n})\cdot\nabla u_{n}\,dx\leq C$

and

$\int_{\Omega}b(x,u_{n})u_{n}\,dx\leq C.$

Furthermore, by Lemma 2.3

$\|\nabla
u_{n}\|_{L_{M}(\Omega)}\leq\frac{1}{c_{1}^{\mathcal{A}}}\left(\int_{\Omega}M(x,c_{1}^{\mathcal{A}}\nabla
u_{n})\,dx+1\right)\leq C,$

where $C$ is independent of $n$. ∎

The proof of the next fact follows almost the same lines as [11, Lemma 3.8.2],
but we correct some flaws and include it for readers’ convenience.

######  Lemma 3.4.

Suppose $M$ is an $N$-function and
${\mathcal{A}}:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ satisfies
(A1), (A2), (A3) and suppose
$\|{\mathcal{A}}(\cdot,\xi)\cdot\xi\|_{L^{1}(\Omega)}\leq\tilde{c}$. Then
there exists a constant $C>0$ depending only on the parameters from (A1), (A2)
and $\tilde{c}$ such that
$\|{\mathcal{A}}(\cdot,\xi)\|_{L_{M^{\ast}}(\Omega)}<C$.

###### Proof.

Let $w\in E_{M}(\Omega)$. Since ${\mathcal{A}}$ is a Carathéodory’s function,
we see that ${\mathcal{A}}(x,w)$ is bounded and thus belongs to
$E_{M^{\ast}}(\Omega)$. Now by a direct calculation and (A3)

$\displaystyle\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot w\,dx$
$\displaystyle=-\int_{\Omega}({\mathcal{A}}(x,\xi)-{\mathcal{A}}(x,w))\cdot(\xi-w)\,dx+\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot\xi\,dx$
$\displaystyle\quad-\int_{\Omega}{\mathcal{A}}(x,w)\cdot(\xi-w)\,dx$
$\displaystyle\leq\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot\xi\,dx-\int_{\Omega}{\mathcal{A}}(x,w)\cdot(\xi-w)\,dx.$

The first term on the right-hand side is bounded by assumption. For the second
term we use Fenchel–Young inequality, (A2) and convexity to estimate

$\displaystyle-\int_{\Omega}{\mathcal{A}}(x,w)\cdot(\xi-w)\,dx$
$\displaystyle=-\frac{2}{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}\int_{\Omega}c_{3}^{\mathcal{A}}{\mathcal{A}}(x,w)\cdot\left(\frac{c_{1}^{\mathcal{A}}(\xi-w)}{2}\right)\,dx$
$\displaystyle\leq\frac{2}{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}\int_{\Omega}{M^{\ast}}(x,c_{3}^{\mathcal{A}}{\mathcal{A}}(x,w))+M\left(x,\frac{c_{1}^{\mathcal{A}}(\xi-w)}{2}\right)\,dx$
$\displaystyle\leq\frac{2}{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}\int_{\Omega}\frac{1}{c_{2}^{\mathcal{A}}}M(x,c_{4}^{\mathcal{A}}w)+\frac{1}{c_{2}^{\mathcal{A}}}h_{2}(x)+M(x,c_{1}^{\mathcal{A}}\xi)+M(x,c_{1}^{\mathcal{A}}w)\,dx$
$\displaystyle\leq\frac{2}{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}\left[\left(\frac{1}{c_{2}^{\mathcal{A}}}+1\right)\int_{\Omega}M(x,\max\\{{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}}\\}w)\,dx\right.$
$\displaystyle\left.\qquad\qquad\
+\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot\xi+h_{1}(x)+\frac{1}{c_{2}^{\mathcal{A}}}h_{2}(x)\,dx\right].$

In turn,

$\displaystyle\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot
w\,dx\leq\frac{2}{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}$
$\displaystyle\left[\left(\frac{1}{c_{2}^{\mathcal{A}}}+1\right)\int_{\Omega}M(x,\max\\{{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}}\\}w)\,dx\right.$
(3.2) $\displaystyle\left.\
+\left(\frac{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}{2}+1\right)\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot\xi+h_{1}(x)+\frac{1}{c_{2}^{\mathcal{A}}}h_{2}(x)\,dx\right]\,.$

Provided
$\max\\{{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}}\\}\|w\|_{L_{M}(\Omega)}\leq
1$, by Lemma 2.3(1), we know that

$\displaystyle\int_{\Omega}M(x,\max\\{{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}}\\}w)\,dx\leq\max\\{{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}}\\}\|w\|_{L_{M}(\Omega)}\leq
1.$

Using it in (3.2) we obtain

$\displaystyle\|{\mathcal{A}}(x,\xi)\|_{(E_{M}(\Omega))^{\ast}}$
$\displaystyle=\max\\{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}\\}\sup\left\\{\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot
w\,dx:\quad\max\\{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}\\}\|w\|_{L_{M}(\Omega)}\leq
1\right\\}$
$\displaystyle\leq\frac{2\max\\{c_{1}^{\mathcal{A}},c_{4}^{\mathcal{A}}\\}}{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}\left[\left(\frac{1}{c_{2}^{\mathcal{A}}}+1\right)+\left(\frac{c_{1}^{\mathcal{A}}c_{3}^{\mathcal{A}}}{2}+1\right)\int_{\Omega}{\mathcal{A}}(x,\xi)\cdot\xi+h_{1}(x)+\frac{1}{c_{2}^{\mathcal{A}}}h_{2}(x)\,dx\right]$
$\displaystyle\leq C.$

Thus ${\mathcal{A}}(x,\xi)$ stays bounded in $L_{M^{\ast}}(\Omega)$. ∎

##  4\. Existence and uniqueness for problem (1.1)

We are ready to prove our main Theorem.

###### Proof of Theorem 1.1.

We base our proof on Galerkin approximations. We fix spaces $V_{n}$ and
functions $u_{n}$ as in Definition 3.1. We shall prove that there exists a
weak solution of the problem (1.1) being a limit of a subsequence of the
Galerkin solutions $\\{u_{n}\\}$. We divide our proof into four steps.

Step 1 According to the Lemma 3.2, we get

(4.1) $\int_{\Omega}{\mathcal{A}}(x,\nabla
u_{n})\cdot\nabla\omega+\Phi(u_{n})\cdot\nabla\omega+b(x,u_{n})\omega\,dx=\int_{\Omega}F\cdot\nabla\omega\,dx\quad\text{
for all $\omega\in V_{k}$, $k\leq n$.}$

By Lemma 3.3 we have that $\|\nabla u_{n}\|_{L_{M}(\Omega)}\leq C$ and
$\|{\mathcal{A}}(\cdot,\nabla u_{n})\cdot\nabla u_{n}\|_{L^{1}(\Omega)}\leq
C$, where $C$ is independent of $n$. By the Banach–Alaoglu Theorem, we know
that $\\{\nabla u_{n}\\}_{n}$ is weakly-$\ast$ compact in $L_{M}(\Omega)$. By
Lemma 2.6 and the fact that $M$ is an $N$-function, we see that $\\{\nabla
u_{n}\\}_{n}$ is uniformly integrable in $L^{1}(\Omega,{\mathbb{R}}^{d})$. The
Dunford–Pettis Theorem [5, Theorem 4.3] implies that $\left\\{\nabla
u_{n}\right\\}_{n}$ is relatively compact in the weak topology of
$L^{1}(\Omega,{\mathbb{R}}^{d})$. Taking into accont the Rellich-Kondrashev
theorem and Poincaré inequality there exists a function $u\in
W^{1,1}_{0}(\Omega)$ such that

$u_{n}\rightharpoonup u\quad$ in $\quad W^{1,1}(\Omega)$.

Since the weak-$\ast$ limit is unique, we see that additionally $\nabla u\in
V^{1}_{0}L_{M}(\Omega)$ and

$\nabla u_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nabla u\quad$ for
$\quad\sigma(L_{M},E_{M^{\ast}})$.

Also, by Lemma 3.4, we see that ${\mathcal{A}}(\cdot,\nabla u_{n})$ is bounded
in $L_{M^{\ast}}(\Omega)$ and a similar argument shows the existence of $h\in
L_{M^{\ast}}(\Omega)$ such that

(4.2) ${\mathcal{A}}(x,\nabla
u_{n})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}h\quad\text{ for
}\quad\sigma(L_{M^{\ast}},E_{M}).$

As $b$ is a Carathéodory’s function, we get $b(x,u_{n})\to b(x,u)$ almost
everywhere in $\Omega$. Fix $\varepsilon>0$ and choose $\ell$ large enough to
guarantee $\frac{C}{\ell}<\frac{\varepsilon}{2}$ for $C$ from Lemma 3.3. Since
$b(\cdot,\ell)\in L^{1}(\Omega)$ is a given function independent of $n$, for
any subset $E\subset\Omega$ with $|E|$ small enough, we have
$\|b(\cdot,\ell)\|_{L^{1}(E)}<\frac{\varepsilon}{2}$. Now

$\displaystyle\int_{E}b(x,u_{n})\,dx$
$\displaystyle=\int_{E\cap\\{|u_{n}|<\ell\\}}b(x,u_{n})\,dx+\int_{E\cap\\{|u_{n}|\geq\ell\\}}b(x,u_{n})\,dx$
$\displaystyle\leq\int_{E\cap\\{|u_{n}|<\ell\\}}|b(x,\ell)|\,dx+\int_{E\cap\\{|u_{n}|\geq\ell\\}}\frac{1}{|u_{n}|}|b(x,u_{n})u_{n}|\,dx$
$\displaystyle\leq\int_{E}|b(x,\ell)|\,dx+\int_{\\{|u_{n}|\geq\ell\\}}\frac{1}{\ell}|b(x,u_{n})u_{n}|\,dx$
$\displaystyle\leq\int_{E}|b(x,\ell)|\,dx+\frac{C}{\ell}<\varepsilon.$

Therefore $\\{b(\cdot,u_{n})\\}$ is uniformly integrable. By Vitali
convergence theorem (Lemma 2.5) we have

(4.3) $b(\cdot,u_{n})\to b(\cdot,u)\text{ in }L^{1}(\Omega).$

As $\Phi$ is continuous and belongs to $L^{\infty}(\Omega,{\mathbb{R}}^{d})$,
we immediately see that

(4.4) $\Phi(u_{n})\to\Phi(u)$ weakly in $L^{1}(\Omega,{\mathbb{R}}^{d})$

due to dominated convergence theorem.

Having (4.2), (4.4), and (4.3), we can pass to the limit in (4.1) with $n$. In
turn we get

(4.5) $\langle
F,\nabla\omega\rangle=\int_{\Omega}h\cdot\nabla\omega+\Phi(u)\cdot\nabla\omega+b(x,u)\omega\,dx\quad\text{for
any $\omega\in V_{k}$, $k\in\mathbb{N}$}.$

Step 2 (extending the class of test functions): Let $\varphi\in
C_{c}^{\infty}(\Omega)$ be arbitrary and $\varphi_{j}\in V_{j}$ be a sequence
of smooth function such that $\varphi_{j}\to\varphi$ in $C^{1}_{c}(\Omega)$.
Due to (4.5) we have

(4.6) $\displaystyle\begin{split}\langle
F,\nabla\varphi_{j}\rangle&=\int_{\Omega}h\cdot\nabla\varphi_{j}+\Phi(u)\cdot\nabla\varphi_{j}+b(x,u)\varphi_{j}\,dx\quad\text{for
any $j$.}\end{split}$

Now $h\cdot\nabla\varphi_{j}\to h\cdot\nabla\varphi$ almost everywhere and by
uniform convergence of $\\{\nabla\varphi_{j}\\}$ we have for large enough $j$

$\int_{\Omega}h\cdot\nabla\varphi_{j}\
dx\leq\int_{\Omega}h\cdot\nabla\varphi+1\,dx<\infty.$

Since (4.6), $\Phi(u)$ is bounded, $b(\cdot,u)\in L^{1}(\Omega)$, by dominated
convergence theorem applied similarly also to all the other terms in we have

(4.7) $\displaystyle\begin{split}\langle
F,\nabla\varphi\rangle=\lim_{j\to\infty}\langle
F,\nabla\varphi_{j}\rangle&=\lim_{j\to\infty}\int_{\Omega}h\cdot\nabla\varphi_{j}+\Phi(u)\cdot\nabla\varphi_{j}+b(x,u)\varphi_{j}\,dx\\\
&=\int_{\Omega}h\cdot\nabla\varphi+\Phi(u)\cdot\nabla\varphi+b(x,u)\varphi\,dx.\end{split}$

Let $v\in V^{1}_{0}L_{M}(\Omega)\cap L^{\infty}(\Omega)$ and
$\\{v_{k}\\}\subset C_{c}^{\infty}(\Omega)$ be a sequence such that $\nabla
v_{k}\to\nabla v$ modularly in $L_{M}(\Omega)$, $v_{k}\to v$ in
$L^{1}(\Omega)$ and $\|v_{k}\|_{L^{\infty}(\Omega)}\leq
c\|v\|_{L^{\infty}(\Omega)}$ with $c$ independent of $k$. This is possible
because of the modular density assumption (Lemma 2.1).

The modular convergence of $\\{\nabla v_{k}\\}$ implies that $\nabla
v_{k}\to\nabla v$ for $\sigma(L_{M},L_{M^{\ast}})$ by Lemma 2.2. Note that

(4.8) $F\in E_{M^{\ast}}(\Omega)\subset L_{M^{\ast}}(\Omega)$, $\quad h\in
L_{M^{\ast}}(\Omega)$, $\quad\Phi(u)\in
L^{\infty}(\Omega,{\mathbb{R}}^{d})\subset E_{M^{\ast}}(\Omega)$

and since $b(\cdot,u)\in L^{1}(\Omega)$, Lemma 2.7 implies that

(4.9)
$\lim_{k\to\infty}\int_{\Omega}b(x,u)v_{k}\,dx\to\int_{\Omega}b(x,u)v\,dx.$

Using (4.8) and (4.9) we can extend (4.7) as

(4.10) $\displaystyle\begin{split}\langle F,\nabla
v\rangle=\lim_{k\to\infty}\langle F,\nabla
v_{k}\rangle&=\lim_{k\to\infty}\int_{\Omega}h\cdot\nabla
v_{k}+\Phi(u)\cdot\nabla v_{k}+b(x,u)v_{k}\,dx\\\ &=\int_{\Omega}h\cdot\nabla
v+\Phi(u)\cdot\nabla v+b(x,u)v\,dx\end{split}$

for test functions $v\in V^{1}_{0}L_{M}(\Omega)\cap L^{\infty}(\Omega)$.

Step 3 ($h={\mathcal{A}}(x,\nabla u)$ almost everywhere): Let $w\in
L^{\infty}(\Omega,{\mathbb{R}}^{d})$ be arbitrary. By (A3) we have

$\displaystyle 0$
$\displaystyle\leq\limsup_{n\to\infty}\int_{\Omega}({\mathcal{A}}(x,\nabla
u_{n})-{\mathcal{A}}(x,w))\cdot(\nabla u_{n}-w)\,dx$
$\displaystyle=\limsup_{n\to\infty}\bigg{(}\int_{\Omega}{\mathcal{A}}(x,\nabla
u_{n})\cdot\nabla u_{n}\,dx-\int_{\Omega}{\mathcal{A}}(x,\nabla u_{n})\cdot
w\,dx$ $\displaystyle\qquad\qquad-\int_{\Omega}{\mathcal{A}}(x,w)\cdot\nabla
u_{n}\,dx+\int_{\Omega}A(x,w)\cdot w\,dx\bigg{)}$ $\displaystyle\leq
I_{1}+I_{2}+I_{3}+I_{4}.$

Let us handle the integrals one by one. Before we estimate the $I_{1}$, we
would like to point out that, as $u_{n}\to u$ a.e. in $\Omega$,
$b:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory’s function
satisfying the sign condition, using the Fatou lemma, we have

$\int_{\Omega}b(x,u)u\,dx\leq\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx\leq
C.$

In the case of $I_{1}$, as $u_{n}$ is a Galerkin solution, $F\in
E_{M^{\ast}}(\Omega)$, taking the test function as $u_{n}$ in (3.1) and
combining with Lemma 2.8, we deduced that

$\displaystyle\begin{split}I_{1}&=\limsup_{n\to\infty}\int_{\Omega}{\mathcal{A}}(x,\nabla
u_{n})\cdot\nabla u_{n}\,dx\\\
&=\limsup_{n\to\infty}\int_{\Omega}\bigg{(}F\cdot\nabla
u_{n}-b(x,u_{n})u_{n}-\Phi(u_{n})\cdot\nabla u_{n}\bigg{)}\,dx\\\
&\leq\limsup_{n\to\infty}\int_{\Omega}F\cdot\nabla
u_{n}\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx\\\
&=\int_{\Omega}F\cdot\nabla
u\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx.\end{split}$

According to the fact that $\nabla T_{k}(u)\to\nabla u$ modularly in
$L_{M}(\Omega)$, we obtain

$\displaystyle\begin{split}&\int_{\Omega}F\cdot\nabla
u\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx=\lim_{k\to\infty}\int_{\Omega}F\cdot\nabla
T_{k}(u)\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx.\end{split}$

In addition, in the step $2$, we have used the approximation theory (lemma
2.1) to extend the test function of (4.10) to hold for any $v\in
V^{1}_{0}L_{M}(\Omega)\cap L^{\infty}(\Omega)$, so taking the test function as
$T_{k}(u)$ in (4.10) and combining with Lemma 2.8, we obtain

$\displaystyle\begin{split}&\lim_{k\to\infty}\int_{\Omega}F\cdot\nabla
T_{k}(u)\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx\\\
&\quad=\lim_{k\to\infty}\int_{\Omega}h\cdot\nabla
T_{k}(u)+b(x,u)T_{k}(u)+\Phi(u)\nabla
T_{k}(u)\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx\\\
&\quad=\int_{\Omega}h\cdot\nabla
u\,dx+\int_{\Omega}b(x,u)u\,dx-\liminf_{n\to\infty}\int_{\Omega}b(x,u_{n})u_{n}\,dx\\\
&\quad\leq\int_{\Omega}h\cdot\nabla u\,dx.\end{split}$

Therefore, we deduce that

(4.11) $\displaystyle
I_{1}=\limsup_{n\to\infty}\int_{\Omega}{\mathcal{A}}(x,\nabla
u_{n})\cdot\nabla u_{n}\,dx\leq\int_{\Omega}h\cdot\nabla u\,dx.$

For $I_{2}$, recalling the fact (4.2), it is easy to check that

(4.12) $\ I_{2}=\limsup_{n\to\infty}\int_{\Omega}-{\mathcal{A}}(x,\nabla
u_{n})\cdot w\,dx=-\int_{\Omega}h\cdot w\,dx.$

In addition, according to the fact that ${\mathcal{A}}(\cdot,w)\in
L^{\infty}(\Omega,{\mathbb{R}}^{d})$, we have

(4.13) $I_{3}=\limsup_{n\to\infty}\int_{\Omega}-{\mathcal{A}}(x,w)\cdot\nabla
u_{n}\,dx=-\int_{\Omega}{\mathcal{A}}(x,w)\cdot\nabla u\,dx.$

Lastly, $I_{4}$ does not depend on $n$ or $k$.

All in all, combining (4.11)–(4.13) we see that

$\displaystyle 0\leq\int_{\Omega}(h-{\mathcal{A}}(x,w))\cdot(\nabla u-w)\,dx.$

Now the monotonicity trick (Lemma 2.10) yields that $h={\mathcal{A}}(x,\nabla
u)$ almost everywhere in $\Omega$. Plugging this to (4.10) we see that $u_{n}$
is a weak solution to equation (1.1).

Step 4 (uniqueness): We prove uniqueness of the weak solution under the
additional assumptions that $s\mapsto b(x,s)$ is strictly increasing for
almost every $x\in\Omega$ and $\Phi$ is Lipschitz continuous. We take an
approximation of a Heaviside function and pass to the limit, see [8, 27, 41]
for similar proofs.

Assume that there exists two weak solutions $u^{1}$ and $u^{2}$. Let us take
an approximation of a Heaviside function $H(t)=\chi_{\\{t\geq 0\\}}$ given by

$\displaystyle H_{\delta}(t):=\begin{cases}0,&\ t<0,\\\ \tfrac{1}{\delta}t,&\
0\leq t\leq\delta,\\\ 1,&\ t>\delta.\end{cases}$

Let $k\in\mathbb{N}$ be arbitrary and test the equation (1.1) with
$\varphi=H_{\delta}(u^{1}-u^{2})\in V^{1}_{0}L_{M}(\Omega)\cap
L^{\infty}(\Omega)$ to get

$\int_{\Omega}\big{(}{\mathcal{A}}(x,\nabla
u^{i})+\Phi(u^{i})\big{)}\cdot\nabla\varphi+b(x,u^{i})\varphi\,dx=\int_{\Omega}F\cdot\nabla\varphi\,dx,\quad
i=1,2\,.$

Subtracting the equations we receive

$\displaystyle J_{1}+J_{2}+J_{3}$
$\displaystyle:=\frac{1}{\delta}\int_{\\{0\leq
u^{1}-u^{2}\leq\delta\\}}\big{(}{\mathcal{A}}(x,\nabla
u^{1})-{\mathcal{A}}(x,\nabla u^{2})\big{)}\cdot\nabla(u^{1}-u^{2})\,dx$
$\displaystyle\quad+\frac{1}{\delta}\int_{\\{0\leq
u^{1}-u^{2}\leq\delta\\}}\big{(}\Phi(u^{1})-\Phi(u^{2})\big{)}\cdot\nabla(u^{1}-u^{2})\,dx$
$\displaystyle\quad+\int_{\Omega}(b(x,u^{1})-b(x,u^{2}))H_{\delta}(u^{1}-u^{2})\,dx=0.$

We note that $J_{1}\geq 0$ for each $\delta$ due to (A3). For $J_{2}$ we
estimate with Lipschitz continuity of $\Phi$

$\displaystyle J_{2}$ $\displaystyle\leq\frac{1}{\delta}\int_{\\{0\leq
u^{1}-u^{2}\leq\delta\\}}|\Phi(u^{1})-\Phi(u^{2})||\nabla(u^{1}-u^{2})|\,dx\leq\frac{1}{\delta}\int_{\\{0\leq
u^{1}-u^{2}\leq\delta\\}}L_{\Phi}\delta|\nabla(u^{1}-u^{2})|\,dx\to 0$

as $\delta\to 0$ as we integrate over a shrinking domain. Lastly we note that
due to Lemma 2.7 we have

$\displaystyle\lim_{\delta\to
0}J_{3}=\int_{\Omega}(b(x,u^{1})-b(x,u^{2}))H(u^{1}-u^{2})\,dx.$

Dropping the non-negative $J_{1}$ and taking into account limits of $J_{2}$
and $J_{3}$ we see that

$\displaystyle\int_{\Omega}(b(x,u^{1})-b(x,u^{2}))H(u^{1}-u^{2})\,dx\leq 0$

so that $(b(x,u^{1})-b(x,u^{2}))H(u^{1}-u^{2})=0$ almost everywhere due to
sign condition of $b$. Since $b$ is strictly increasing with respect to the
second variable we see that $u^{1}\leq u^{2}$. Considering
$\varphi=H_{\delta}(u^{2}-u^{1})$ yields the opposite inequality and thus
$u^{1}=u^{2}$. ∎

## References

  * [1] Youssef Ahmida, Iwona Chlebicka, Piotr Gwiazda and Ahmed Youssfi  “Gossez’s approximation theorems in Musielak–Orlicz–Sobolev spaces”  In _J. Funct. Anal._ 275.9, 2018, pp. 2538–2571
  * [2] Angela Alberico, Iwona Chlebicka, Andrea Cianchi and Anna Zatorska-Goldstein  “Fully anisotropic elliptic problems with minimally integrable data”  In _Calc. Var. Partial Differential Equations_ 58:186, 2019
  * [3] Giuseppina Barletta and Andrea Cianchi  “Dirichlet problems for fully anisotropic elliptic equations”  In _Proc. Roy. Soc. Edinburgh Sect. A_ 147.1, 2017, pp. 25–60
  * [4] Michał Borowski and Iwona Chlebicka  “Modular density of smooth functions in inhomogeneous and fully anisotropic Musielak–Orlicz–Sobolev spaces”  In _J. Funct. Anal._ , 2022
  * [5] Haim Brézis  “Functional analysis, Sobolev spaces and partial differential equations” 2.3, Universitext  Springer, New York, 2011
  * [6] Felix Earl Browder  “Nonlinear elliptic boundary value problems”  In _Bulletin of the American Mathematical Society_ 69.6, 1963, pp. 530–550
  * [7] Iwona Chlebicka  “A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces”  In _Nonl. Analysis_ 175, 2018, pp. 1–27
  * [8] Iwona Chlebicka, Flavia Giannetti and Anna Zatorska-Goldstein  “A note on uniqueness for $L^{1}$-data elliptic problems with Orlicz growth”  In _Colloq. Math._ 168.2, 2022, pp. 199–209
  * [9] Iwona Chlebicka, Piotr Gwiazda and Anna Zatorska–Goldstein  “Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev’s phenomenon”  In _Annales de l’Institut Henri Poincaré C, Analyse non linéaire_ 36.5, 2019, pp. 1431–1465  Elsevier
  * [10] Iwona Chlebicka and Piotr Nayar  “Essentially fully anisotropic Orlicz functions and uniqueness to measure data problem”  In _Math. Methods Appl. Sci._ 45.14, 2022, pp. 8503–8527
  * [11] Iwona Chlebicka, Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda and Aneta Wróblewska-Kamińska  “Partial differential equations in anisotropic Musielak-Orlicz spaces”, Springer Monographs in Mathematics  Springer, Cham, 2021, pp. xiii+389
  * [12] Anna Denkowska, Piotr Gwiazda and Piotr Kalita  “On renormalized solutions to elliptic inclusions with nonstandard growth”  In _Calc. Var. Partial Differential Equations_ 60.1, 2021, pp. 21:52
  * [13] Agnese Di Castro  “Anisotropic elliptic problems with natural growth terms”  In _Manuscripta Math_ 135.3-4, 2011, pp. 521–543
  * [14] Thomas Donaldson  “Nonlinear elliptic boundary value problems in Orlicz–Sobolev spaces”  In _Journal of Differential Equations_ 10.3 Academic Press, 1971, pp. 507–528
  * [15] Ge Dong and Xiaochun Fang  “Differential equations of divergence form in separable Musielak–Orlicz–Sobolev spaces”  In _Boundary Value Problems_ 2016.1 Springer, 2016, pp. 1–19
  * [16] Jürgen Elstrodt  “Maß- und Integrationstheorie”, Springer-Lehrbuch. [Springer Textbook]  Springer-Verlag, Berlin, 2005, pp. xvi+434
  * [17] Lawrence Craig Evans  “Partial differential equations, volume 19 of Graduate Studies in Mathematics”  In _American Mathematical Society, Providence, RI_ , 1998
  * [18] Xianling Fan  “Differential equations of divergence form in Musielak–Sobolev spaces and a sub-supersolution method”  In _Journal of Mathematical Analysis and Applications_ 386.2 Elsevier, 2012, pp. 593–604
  * [19] Jean-Pierre Gossez  “Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients”  In _Transactions of the American Mathematical Society_ 190, 1974, pp. 163–205
  * [20] Jean-Pierre Gossez  “Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems”  In _Nonlinear analysis, function spaces and applications_ BSB BG Teubner Verlagsgesellschaft, 1979, pp. 59–94
  * [21] Jean-Pierre Gossez  “Some approximation properties in Orlicz–Sobolev spaces”  In _Studia Mathematica_ 74 Instytut Matematyczny Polskiej Akademii Nauk, 1982, pp. 17–24
  * [22] Jean-Pierre Gossez and Vesa Mustonen  “Variational inequalities in Orlicz-Sobolev spaces”  In _Nonlinear Anal._ 11.3, 1987, pp. 379–392
  * [23] Piotr Gwiazda, Piotr Minakowski and Aneta Wróblewska-Kamińska  “Elliptic problems in generalized Orlicz–Musielak spaces”  In _Central European Journal of Mathematics_ 10.6 Springer, 2012, pp. 2019–2032
  * [24] Piotr Gwiazda, Iwona Skrzypczak and Anna Zatorska-Goldstein  “Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space”  In _Journal of Differential Equations_ 264.1 Elsevier, 2018, pp. 341–377
  * [25] Piotr Gwiazda and Agnieszka Świerczewska-Gwiazda  “On non-Newtonian fluids with a property of rapid thickening under different stimulus”  In _Mathematical Models and Methods in Applied Sciences_ 18.07 World Scientific, 2008, pp. 1073–1092
  * [26] Piotr Gwiazda and Agnieszka Świerczewska-Gwiazda  “Parabolic equations in anisotropic Orlicz spaces with general N-functions”  In _Parabolic problems_ Springer, 2011, pp. 301–311
  * [27] Piotr Gwiazda, Petra Wittbold, Aneta Wróblewska and Aleksandra Zimmermann  “Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces”  In _Journal of Differential Equations_ 253.2 Elsevier, 2012, pp. 635–666
  * [28] Petteri Harjulehto and Peter Hästö  “Generalized Orlicz Spaces”  In _Orlicz Spaces and Generalized Orlicz Spaces_ Springer, 2019, pp. 47–78
  * [29] Petteri Harjulehto, Peter Hästö and Riku Klén  “Generalized Orlicz spaces and related PDE”  In _Nonlinear Analysis: Theory, Methods $\&$ Applications_ 143 Elsevier, 2016, pp. 155–173
  * [30] Petteri Harjulehto, Peter Hästö, Út V. Lê and Matti Nuortio  “Overview of differential equations with non-standard growth”  In _Nonlinear Analysis: Theory, Methods $\&$ Applications_ 72.12 Elsevier, 2010, pp. 4551–4574
  * [31] Peter Hästö  “A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces”  In _Journal of Geometric Analysis_ , 2022
  * [32] Peter Hess  “On nonlinear mappings of monotone type with respect to two Banach spaces”  In _Journal de Mathématiques Pures et Appliqués_ 52, 1973, pp. 13–26
  * [33] David Kinderlehrer and Guido Stampacchia  “An introduction to variational inequalities and their applications” 88, Pure and Applied Mathematics  Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980, pp. xiv+313
  * [34] Jean Leray and Jacques-Louis Lions  “Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder”  In _Bull. Soc. Math. France_ 93, 1965, pp. 97–107
  * [35] Ying Li, Fengping Yao and Shulin Zhou  “Entropy and renormalized solutions to the general nonlinear elliptic equations in Musielak–Orlicz spaces”  In _Nonlinear Analysis: Real World Applications_ 61 Elsevier, 2021, pp. 103330
  * [36] Duchao Liu and Peihao Zhao  “Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces”  In _Nonlinear Analysis: Real World Applications_ 26 Elsevier, 2015, pp. 315–329
  * [37] Giuseppe Mingione and Vincentiu D. Radulescu  “Recent developments in problems with nonstandard growth and nonuniform ellipticity”  In _J. Math. Anal. Appl._ 501.1, 2021, pp. Paper No. 12519741
  * [38] Vesa Mustonen and Matti Tienari  “On monotone-like mappings in Orlicz–Sobolev spaces”  In _Mathematica Bohemica_ 124.2-3, 1999, pp. 255–271
  * [39] Rainer Schumann  “The Galerkin approximation for quasilinear elliptic equations with rapidly (or slowly) increasing coefficients”  In _Zeitschrift für Analysis und ihre Anwendungen_ 1.4, 1982, pp. 73–85
  * [40] Hermann Sohr  “The Navier-Stokes equations: An elementary functional analytic approach”  Springer Science $\&$ Business Media, 2012
  * [41] Petra Wittbold and Aleksandra Zimmermann  “Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and $L^{1}$-data”  In _Nonlinear Analysis: Theory, Methods $\&$ Applications_ 72.6 Elsevier, 2010, pp. 2990–3008
  * [42] Aneta Wróblewska  “Steady flow of non-Newtonian fluids—monotonicity methods in generalized Orlicz spaces”  In _Nonlinear Analysis: Theory, Methods $\&$ Applications_ 72.11 Elsevier, 2010, pp. 4136–4147