# Positive solutions of elliptic systems with superlinear terms on the
critical hyperbola

Mabel Cuesta  Department of Mathematics, Université du Littoral Côte d’Opale
(ULCO), Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
(LMPA), 62100 Calais (France<EMAIL_ADDRESS>,  Rosa Pardo
Departamento de Análisis Matemático y Matemática Aplicada, Universidad
Complutense de Madrid, 28040-Madrid, Spain<EMAIL_ADDRESS>and  Angela Pistoia
Dipartimento SBAI, Sapienza Università di Roma, via Antonio Scarpa 16, 00161
Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider a slightly subcritical elliptic system with Dirichlet boundary
conditions and a non-power nonlinearity in a bounded smooth domain. For this
problem, standard compact embeddings cannot be used to guarantee the existence
of solutions as in the case of power-type nonlinearities. Instead, we use the
dual method on Orlicz spaces, showing that our problem possesses a mountain
pass type solution.

Keywords: mountain pass solutions, critical Sobolev exponent, dual method

MSC2020: 35B38, 35B33, 35J47, 35J67.

The second author is supported by grants PID2019-103860GB-I00, and
PID2022-137074NB-I00, MICINN, Spain, and by UCM-BSCH, Spain, GR58/08, Grupo
920894. The third author is partially supported by GNAMPA-INdAM.

##  1\. Introduction

Let us consider the system

(1.1) $\left\\{\begin{aligned} &-\Delta
v=\frac{u^{p}}{\big{(}\ln(e+u)\big{)}^{\alpha}}\ \hbox{in}\ \Omega\\\ &-\Delta
u=\frac{v^{q}}{\big{(}\ln(e+v)\big{)}^{\beta}}\ \hbox{in}\ \Omega\\\
&u>0,\,v>0\ \hbox{in}\ \Omega\\\ &u=v=0\ \hbox{on}\
\partial\Omega\end{aligned}\right.$

where $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, is a bounded domain of class
$C^{2},$ $p,q>0$, $\alpha\leq p,\ \beta\leq q,$ and $(p,q)$ may belong to the
critical hyperbola; specifically, either

(1.2) $1>\frac{1}{p+1}+\frac{1}{q+1}>\frac{N-2}{N},$

or

(1.3)
$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N},\qquad\frac{\alpha}{p+1}+\frac{\beta}{q+1}>0.$

We want to find a solution $(u,v)$ of (1.1), positive in both components.

Problems of type (1.1) has been considered by several authors, we refer to [3,
9, 11, 17]. In [3] the authors study the case (1.2) and $\alpha=\beta=0$.
Whereas in [9, Theorem 2.7] the authors study related nonlinearities when the
pair of exponents $(p,q)$ lies below the critical Sobolev hyperbola (1.2),
using variational approaches. In [11, Theorem 1.3], using an Orlicz-space
approach, the authors study the existence of solutions of (1.1) when the pairs
of exponents $(p,q)$, $(\alpha,\beta)$ satisfy (1.3), in particular $(p,q)$
lies on the critical Sobolev hyperbola, with $p,q>1$; they do not cover the
case $p,q>0$. While in [17] the authors establish a-priori $L^{\infty}$ bounds
when the pair of exponents $(p,q)$ lies on the critical Sobolev hyperbola
(1.3), $1<p,q<\infty$, and $\displaystyle\alpha,\,\beta>2/(N-2).$ Concerning
one single equation, we can mention [5] establishing $L^{\infty}$ a priori
bounds when $\alpha>2/(N-2)$, [8] analyzing the asymptotic behavior of the
solutions as $\alpha\to 0$, and [10] including a changing sign weight.

Our main result is the following:

######  Theorem 1.1.

Assume $p,q>0$, $\alpha\leq p,\ \beta\leq q,$ and either $(p,q)$ satisfy (1.2)
or $(p,q)$, $(\alpha,\beta)$ satisfy (1.3). Then Problem (1.1) possesses a
solution of mountain pass type.

The proof relies on the dual method which allows to reduce the existence of
solutions to the problem (1.1) to finding critical points of an energy
functional defined on a suitable Orlicz space, whose accurate choice is the
main novelty of the present paper.

######  Remark 1.2.

We would like to comment on condition (1.3). At this aim we point out that
Theorem 1.1 also applies to the case of the single equation, namely
$p=q=\frac{N+2}{N-2}$ and $\alpha=\beta>0$. In particular, the problem

(1.4) $\left\\{\begin{aligned} &-\Delta
u=\frac{u^{p}}{\big{(}\ln(e+u)\big{)}^{\alpha}}\ \hbox{in}\ \Omega\\\ &u>0\
\hbox{in}\ \Omega\\\ &u=0\ \hbox{on}\ \partial\Omega\end{aligned}\right.$

has always a positive solution. On the other hand, it is known that Pohozaev’s
identity [21] ensures the non-existence of positive solutions on star-shaped
domains to equation (1.4) whenever $\alpha\leq 0,$ see Remark (4.3).
In a similar way, it could be interesting to prove the criticality of our
condition 1.3. Indeed, a question naturally arises: if $p$ and $q$ lie on the
critical hyperbola and

$\frac{\alpha}{p+1}+\frac{\beta}{q+1}\leq 0$

does the system (1.1) have any positive solutions on a star-shaped domain? In
Theorem 4.2 we give a partial answer when both $\alpha$ and $\beta$ are not
positive. The general case remains open.

The paper is organized as follows. In Section 2 we describe the dual method
which allows us to consider $p,\ q>0$. Roughly speaking, this method consists
in taking the inverse of the Laplace operator, and defining the inverse of the
nonlinearities. In Section 3 we prove our main result, that the associated
energy functional, defined in a suitable product of Orlicz spaces, has a
mountain pass geometry. Section 4 is devoted to analyze the criticality of the
condition 1.3. Finally, Section 5 summarize the theory of Orlicz spaces needed
for our purposes.

##  2\. The dual method

###  2.1. The variational formulation

Given any $s\in(1,\infty)$, we introduce the operator
$\mathtt{K}:=(-\Delta)^{-1}:L^{s}(\Omega)\to W^{2,s}(\Omega)\cap
W_{0}^{1,s}(\Omega)$ which is defined as

$\mathtt{K}(f):=u\qquad\mbox{if and only if}\quad-\Delta u=f\ \hbox{in}\
\Omega,\ u=0\ \hbox{on}\ \partial\Omega.$

Next, we set

(2.1)
$a(t):=\frac{t^{p}}{\big{(}\ln(e+t)\big{)}^{\alpha}},\qquad\mbox{and}\quad
b(t):=\frac{t^{q}}{\big{(}\ln(e+t)\big{)}^{\beta}}\ \hbox{ with}\ t\geq 0,$

and consider their primitive functions:

(2.2) $A(t):=\int\limits_{0}^{t}a(s)\,ds\qquad\mbox{and}\quad
B(t):=\int\limits_{0}^{t}b(s)\,ds\qquad\mbox{for all}\quad t\geq 0.$

If $\alpha\leq p$ and $\beta\leq q$, the functions $a$, $b$ are strictly
increasing and so invertible in $[0,+\infty)$. Let us denote by

(2.3) $\tilde{a}:=a^{-1}\qquad\mbox{and}\quad\tilde{b}:=b^{-1}\quad\text{the
inverse functions.}$

If we set $f=a(u)$ and $g=b(v)$ or equivalently $u=\tilde{a}(f)$ and
$v=\tilde{b}(g),$ then problem (1.1) can be rewritten as

(2.4) $\left\\{\begin{aligned} &\tilde{b}(g)=\mathtt{K}(f)\ \hbox{in}\
\Omega\\\ &\tilde{a}(f)=\mathtt{K}(g)\ \hbox{in}\ \Omega\\\ &f>0,\,g>0\
\hbox{in}\ \Omega.&\end{aligned}\right.$

Let us introduce their primitive functions:

(2.5)
$\widetilde{A}(t):=\int\limits_{0}^{t}\tilde{a}(s)\,ds\qquad\mbox{and}\quad\widetilde{B}(t):=\int\limits_{0}^{t}\tilde{b}(s)\,ds,$

for all $t\geq 0$, and denote $L^{\tilde{A}}(\Omega)$ and
$L^{\tilde{B}}(\Omega)$ the Orlicz spaces associated to the functions
${\widetilde{A}}$ and ${\widetilde{B}}$ respectively, cf. definition 5.3 and
Remark 5.7. Observe that, since $\tilde{a}$ and $\tilde{b}$ are continuous and
increasing the functions $\tilde{A}$ are $\tilde{B}$ are $\mathcal{N}$\-
functions, cf. definition 5.1. We endowed the space $L^{\tilde{A}}(\Omega)$
(resp. $L^{\tilde{B}}(\Omega)$) with either the Luxembourg norm
$\|\cdot\|_{(\tilde{A})}$ or the Orlicz norm  $\|\cdot\|_{\tilde{A}}$ (resp.
$\|\cdot\|_{(\tilde{B})}$ and $\|\cdot\|_{\tilde{B}}$), see definitions (5.3)
and (5.4) for those norms. From Lemma 5.5 and Proposition 5.10(i), the spaces
$L^{\tilde{A}}(\Omega)$ and $L^{\tilde{B}}(\Omega)$ are complete with both the
Orlicz or the Luxemburg norm.

Problem (2.4) has a variational structure. Indeed, let us denote by $X$ the
Banach space

$X=L^{\tilde{A}}(\Omega)\times L^{\tilde{B}}(\Omega)$

endowed with the norm $\|(f,g)\|_{X}:=\|f\|_{\tilde{A}}+\|g\|_{\tilde{B}}$.

By the regularity result Theorem 5.29, the operator ${\mathtt{K}}$ sends
$L^{\tilde{A}}(\Omega)$ into $W_{0}^{1,{\widetilde{A}}}(\Omega)\cap
W^{2,{\tilde{A}}}(\Omega)$, cf. Definition 5.19 of the Orlicz-Sobolev spaces
$W_{0}^{1,H}(\Omega)$ and $W^{2,H}(\Omega)$, when $H$ is equal to either
$\tilde{A}$ or $\tilde{B}$. Thus for any $(f,g)\in X$,
$({\mathtt{K}}(f),{\mathtt{K}}(g))\in W^{2,{\tilde{A}}}(\Omega)\times
W^{2,{\tilde{B}}}(\Omega).$ By a solution $(u,v)$ of (1.1) we mean a pair
$(u,v)\in\big{(}W_{0}^{1,{\widetilde{A}}}(\Omega)\cap
W^{2,{\tilde{A}}}(\Omega)\big{)}\times\big{(}W_{0}^{1,{\widetilde{B}}}(\Omega)\cap
W^{2,{\tilde{B}}}(\Omega)\big{)}$ satisfying (1.1) in the weak sense.

Moreover, for $(p,q),\ (\alpha,\beta)$ satisfying either (1.2), or (1.3), we
have that the embeddings $W^{2,{\tilde{A}}}(\Omega)\hookrightarrow
L^{B}(\Omega)$ and $W^{2,{\tilde{B}}}(\Omega)\hookrightarrow L^{A}(\Omega)$,
are continuous and compact, see Lemma 2.8. Consequently,
$f\,{\mathtt{K}}(g)\in L^{1}(\Omega)$, $g\,{\mathtt{K}}(f)\in L^{1}(\Omega)$,
see the 2nd Holder’s inequality (5.5) in Proposition 5.10. Furthermore, it
follows from the definition of $\mathtt{K}$, that

(2.6)
$\int\limits_{\Omega}f\,{\mathtt{K}}(g)\,dx=\int_{\Omega}\nabla\,{\mathtt{K}}(f)\cdot\nabla\,{\mathtt{K}}(g)\,dx=\int\limits_{\Omega}g\,{\mathtt{K}}(f)\,dx.$

For convenience, let us extend $a$, $\tilde{a}$, $b$, $\tilde{b}$ in the whole
$\mathbb{R}$ as odd functions, and extend $A$, $\widetilde{A}$, $B$ and
$\widetilde{B}$ as even functions. We next introduce the $C^{1}$-functional
$J:X\to\mathbb{R}$ defined by

(2.7)
$J(f,g):=\int\limits_{\Omega}\widetilde{A}(f)\,dx+\int\limits_{\Omega}\widetilde{B}(g)\,dx-\frac{1}{2}\int\limits_{\Omega}\big{(}f\,\mathtt{K}(g)+g\,\mathtt{K}(f)\big{)}\,dx$

whose derivative at $(f,g)\in X$ is equal to

$J^{\prime}(f,g)[\psi_{1},\psi_{2}]=\left(\int\limits_{\Omega}\tilde{a}(f)\psi_{1}\,dx+\int\limits_{\Omega}\widetilde{b}(g)\psi_{2}\,dx\right)-\left(\int\limits_{\Omega}\mathtt{\psi}_{1}K(g)+\int\limits_{\Omega}\psi_{2}\,\mathtt{K}(f)\,dx\right),$

for all $(\psi_{1},\psi_{2})\in X.$

Trivially, the equations (2.4) are the Euler-Lagrange equations associated to
the action functional $J$.

###  2.2. On the nonlinearities

######  Lemma 2.1.

(i) There exists two constants $c_{1},\ c_{2}>0$, only dependent of $p,$ and
$\alpha,$ such that

(2.8)
$c_{1}t^{\frac{1}{p}}\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\leq\tilde{a}(t)\leq
c_{2}t^{\frac{1}{p}}\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}},\qquad\mbox{for
all}\quad t\geq 0.$

(ii) There exists two constants $C_{1},\ C_{2}>0$ only dependent of $p,$ and
$\alpha,$ such that

$C_{1}\,t^{\frac{p+1}{p}}\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\leq\widetilde{A}(t)\leq
C_{2}\,t^{\frac{p+1}{p}}(\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}},\qquad\mbox{for
all}\quad t\geq 0.$

###### Proof of Lemma 2.1.

(i) By definition,

$a\Big{(}t\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\Big{)}=t^{p}\big{(}\ln(e+t)\big{)}^{\alpha}\left[\ln\left(e+t\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\right)\right]^{-\alpha}.$

Now, checking that

$\lim_{t\to
0}\frac{\displaystyle\ln\left(e+t\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\right)}{\ln(e+t)}=1,\qquad\lim_{t\to\infty}\frac{\displaystyle\ln\left(e+t\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\right)}{\ln(e+t)}=1,$

and since that quotient of logarithms is a continuous, non vanishing function
for $t\geq 0$, we can conclude that there exists two constants $d_{1},\
d_{2}>0$ such that

$d_{1}t^{p}\leq a\Big{(}t\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\Big{)}\leq
d_{2}t^{p},\qquad\mbox{for all}\quad t\geq 0,$

and since $\tilde{a}$ is increasing,

$\tilde{a}(d_{1}t^{p})\leq
t\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}\leq\tilde{a}(d_{2}t^{p}),\qquad\mbox{for
all}\quad t\geq 0.$

Now, firstly denoting by $s=d_{1}t^{p}$, secondly using that

$\lim_{t\to
0}\frac{\displaystyle\ln\left(e+t^{1/p}\right)}{\ln(e+t)}=1,\qquad\lim_{t\to\infty}\frac{\displaystyle\ln\left(e+t^{1/p}\right)}{\ln(e+t)}=1,$

and thirdly, denoting by $s=d_{2}t^{p}$, we obtain (2.8).

(ii) Given $\gamma>0,$ and $\nu\geq-\gamma$, set

$\sigma_{\gamma,\nu}(t)=\sigma(t):=t^{\gamma}\,\big{(}\ln(e+t)\big{)}^{\nu},\qquad\mbox{for}\quad
t>0,$

then

$\sigma^{\prime}_{\gamma,\nu}(t)=\sigma^{\prime}(t)=t^{\gamma-1}\,\big{(}\ln(e+t)\big{)}^{\nu}\left[\gamma+\nu\frac{1}{\ln(e+t)}\,\frac{t}{(e+t)}\right].$

Observe that

$a(t)=\sigma_{p,-\alpha}(t),$

and that

(2.9) $\lim_{t\to\infty}\frac{ta^{\prime}(t)}{a(t)}=p.$

Moreover, if $\nu\geq-\gamma,$ then $\sigma$ is increasing.

Set

$h(t):=\frac{1}{\ln(e+t)}\frac{t}{(e+t)}>0,\qquad\mbox{for}\quad t>0.$

Trivially, $\lim_{t\to 0}h(t)=0,$ $\lim_{t\to\infty}h(t)=0,$ and there exists
a unique $t^{*}>0$ such that

$\ h(t)\leq h(t^{*})=\frac{e}{e+t^{*}}<1,\qquad\forall t\geq 0.$

Hence

(2.10)
$\left(\gamma+\frac{|\nu|\,e}{e+t^{*}}\right)^{-1}\sigma^{\prime}_{\gamma,\nu}(t)\leq
t^{\gamma-1}\,\big{(}\ln(e+t)\big{)}^{\nu}\leq\left(\gamma-\frac{|\nu|\,e}{e+t^{*}}\right)^{-1}\sigma^{\prime}_{\gamma,\nu}(t),$

and observe that whenever $\gamma>1$,
$t^{\gamma-1}\,\big{(}\ln(e+t)\big{)}^{\nu}=\sigma_{\gamma-1,\nu}(t)$, so

$\left(\gamma+\frac{|\nu|\,e}{e+t^{*}}\right)^{-1}\sigma^{\prime}_{\gamma,\nu}(t)\leq\sigma_{\gamma-1,\nu}(t)\leq\left(\gamma-\frac{|\nu|\,e}{e+t^{*}}\right)^{-1}\sigma^{\prime}_{\gamma,\nu}(t).$

By definition of $\widetilde{A}$ (see (2.2)), using (2.8), and (2.10), we can
write

$\widetilde{A}(t)=\int\limits_{0}^{t}\tilde{a}(s)\,ds\leq
c_{2}\int\limits_{0}^{t}s^{\frac{1}{p}}\big{(}\ln(e+s)\big{)}^{\frac{\alpha}{p}}\,ds\leq
C_{2}\sigma_{\frac{p+1}{p},\frac{\alpha}{p}}(t)=C_{2}t^{\frac{p+1}{p}}\,\big{(}\ln(e+t)\big{)}^{\frac{\alpha}{p}}.$

Likewise is obtained the reverse inequality. ∎

######  Remark 2.2.

Likewise, there exists two constants $C^{\prime}_{1},\ C^{\prime}_{2}>0$ only
dependent of $p,$ and $\alpha,$ such that

$C^{\prime}_{1}\,t^{p+1}\big{(}\ln(e+t)\big{)}^{-\alpha}\leq A(t)\leq
C^{\prime}_{2}\,t^{p+1}(\big{(}\ln(e+t)\big{)}^{-\alpha},\qquad\mbox{for
all}\quad t\geq 0.$

In fact, from (2.10), for all $t\geq 0$,

$\left(p+1+\frac{|\alpha|\,e}{e+t^{*}}\right)^{-1}\,t\,a(t)\leq
A(t)\leq\left(p+1-\frac{|\alpha|\,e}{e+t^{*}}\right)^{-1}\,t\,a(t).$

######  Lemma 2.3.

(i) Assume that $\alpha\geq 0$. Then, for all $f\in
L^{\widetilde{A}}(\Omega),$ and for the constants $C_{1},\ C_{2}>0$ provided
by Lemma 2.1, the following hold

(2.11)
$\int_{\Omega}{\widetilde{A}}(f)\,dx\geq\frac{C_{1}}{C_{2}}\,\,\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}},\qquad\mbox{whenever}\quad\|f\|_{(\tilde{A})}\geq
1,$

(2.12)
$\int_{\Omega}{\widetilde{A}}(f)\,dx\leq\frac{C_{2}}{C_{1}}\,\,\,\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}},\qquad\mbox{whenever}\quad\|f\|_{(\tilde{A})}\leq
1,$

and

(2.13) $\int_{\Omega}{\widetilde{A}}(f)\,dx\geq
C_{1}\,\|f\|_{\frac{p+1}{p}}^{\,\frac{p+1}{p}}.$

(ii) Assume now that $\alpha<0$. Then,

(2.14)
$\int_{\Omega}{\widetilde{A}}(f)\,dx\leq\frac{C_{2}}{C_{1}}\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}},\qquad\mbox{whenever}\quad\|f\|_{(\tilde{A})}\geq
1,$ (2.15)
$\int_{\Omega}{\widetilde{A}}(f)\,dx\geq\frac{C_{1}}{C_{2}}\,\,\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}},\qquad\mbox{whenever}\quad\|f\|_{(\tilde{A})}\leq
1,$

and if $f\in L^{\frac{p+1}{p}}(\Omega)$, then

$\int_{\Omega}{\widetilde{A}}(f)\,dx\leq
C_{2}\,\|f\|_{\frac{p+1}{p}}^{\,\frac{p+1}{p}}.$

######  Remark 2.4.

Likewise, for all $f\in L^{A}(\Omega),$ and for the constants
$C^{\prime}_{1},\ C^{\prime}_{2}>0$ provided by Remark 2.2, the following hold

(i) Assume that $\alpha\geq 0$. Then,

$\int_{\Omega}{A}(f)\,dx\leq\frac{C^{\prime}_{2}}{C^{\prime}_{1}}\,\,\|f\|_{(A)}^{p+1},\qquad\mbox{whenever}\quad\|f\|_{(A)}\geq
1,$

and

$\int_{\Omega}{A}(f)\,dx\geq\frac{C^{\prime}_{1}}{C^{\prime}_{2}}\,\,\,\|f\|_{(A)}^{p+1},\qquad\mbox{whenever}\quad\|f\|_{(A)}\leq
1.$

and if $f\in L^{p+1}(\Omega)$, then

(2.16) $\int_{\Omega}A(f)\,dx\leq C^{\prime}_{2}\,\|f\|_{p+1}^{\,p+1}.$

(ii) Assume now that $\alpha<0$. Then

$\int_{\Omega}A(f)\,dx\geq\frac{C^{\prime}_{1}}{C^{\prime}_{2}}\,\,\|f\|_{(A)}^{p+1},\qquad\mbox{whenever}\quad\|f\|_{(A)}\geq
1,$

(2.17)
$\int_{\Omega}A(f)\,dx\leq\frac{C^{\prime}_{2}}{C^{\prime}_{1}}\,\,\,\|f\|_{(A)}^{p+1},\qquad\mbox{whenever}\quad\|f\|_{(A)}\leq
1.$

and

$\int_{\Omega}{A}(f)\,dx\geq C^{\prime}_{1}\,\|f\|_{p+1}^{\,p+1}.$

###### Proof of Lemma 2.3.

Let

${\widetilde{A}_{1}}(s):=s^{\frac{p+1}{p}}\big{(}\ln(e+s)\big{)}^{\frac{\alpha}{p}}.$

Since Lemma 2.1

$C_{1}\,\int_{\Omega}{\widetilde{A}_{1}}(f)\,dx\leq\int_{\Omega}{\widetilde{A}}(f)\,dx\leq
C_{2}\,\int_{\Omega}{\widetilde{A}_{1}}(f)\,dx.$

(i) Assume that $\alpha>0$. If $\|f\|_{(\tilde{A})}\geq 1$, then by Lemma
5.5(iv)

$\displaystyle 1$
$\displaystyle=\int_{\Omega}{\widetilde{A}}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)\,dx\leq
C_{2}\int_{\Omega}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{\frac{\alpha}{p}}\left(\frac{\ln\left(e+\frac{|f|}{\|f\|_{(\tilde{A})}}\right)}{\ln(e+|f|)}\right)^{{\frac{\alpha}{p}}}$
$\displaystyle\leq\frac{C_{2}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}|f|^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{\frac{\alpha}{p}}=\frac{C_{2}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}_{1}}(f)\,dx\leq\frac{C_{2}}{C_{1}}\,\,\frac{1}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}}(f)\,dx.$

Consequently, (2.11) holds.

Moreover, for all $f\in L^{\widetilde{A}}(\Omega),$ with
$\|f\|_{(\tilde{A})}\leq 1$,

$\displaystyle 1$
$\displaystyle=\int_{\Omega}{\widetilde{A}}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)\,dx\geq
C_{1}\int_{\Omega}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{\frac{\alpha}{p}}\left(\frac{\ln\left(e+\frac{|f|}{\|f\|_{(\tilde{A})}}\right)}{\ln(e+|f|)}\right)^{{\frac{\alpha}{p}}}$
$\displaystyle\geq\frac{C_{1}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}|f|^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{\frac{\alpha}{p}}=\frac{C_{1}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}_{1}}(f)\,dx\geq\frac{C_{1}}{C_{2}}\,\,\frac{1}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}}(f)\,dx.$

So (2.12) holds.

Besides, for all $f\in L^{\widetilde{A}}(\Omega),$

$\int_{\Omega}{\widetilde{A}}(f)\,dx\geq
C_{1}\,\int_{\Omega}{\widetilde{A}_{1}}(f))\,dx=C_{1}\int_{\Omega}|f|^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{\frac{\alpha}{p}}\,dx\geq
C_{1}\|f\|_{\frac{p+1}{p}}^{\,\frac{p+1}{p}}.$

(ii) Assume now that $\alpha<0$, and let us write now
${\widetilde{A}_{2}}(s):=s^{\frac{p+1}{p}}\big{(}\ln(e+s)\big{)}^{-\frac{|\alpha|}{p}}$.
Assume now $\|f\|_{(\tilde{A})}\leq 1$, then

$\displaystyle 1$
$\displaystyle=\int_{\Omega}{\widetilde{A}}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)\,dx\leq
C_{2}\int_{\Omega}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{-\frac{|\alpha|}{p}}\left(\frac{\ln(e+|f|)}{\ln\left(e+\frac{|f|}{\|f\|_{(\tilde{A})}}\right)}\right)^{{\frac{|\alpha|}{p}}}$
$\displaystyle\leq\frac{C_{2}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}|f|^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{-\frac{|\alpha|}{p}}=\frac{C_{2}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}_{1}}(f)\,dx\leq\frac{C_{2}}{C_{1}}\,\,\frac{1}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}}(f)\,dx.$

So (2.15) holds.

Moreover, for all $f\in L^{\widetilde{A}}(\Omega),$ with
$\|f\|_{(\tilde{A})}\geq 1$,

$\displaystyle 1$
$\displaystyle=\int_{\Omega}{\widetilde{A}}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)\,dx\geq
C_{1}\int_{\Omega}\left(\frac{|f|}{\|f\|_{(\tilde{A})}}\right)^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{-\frac{|\alpha|}{p}}\left(\frac{\ln(e+|f|)}{\ln\left(e+\frac{|f|}{\|f\|_{(\tilde{A})}}\right)}\right)^{{\frac{|\alpha|}{p}}}$
$\displaystyle\geq\frac{C_{1}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}|f|^{\frac{p+1}{p}}\big{(}\ln(e+|f|)\big{)}^{-\frac{|\alpha|}{p}}=\frac{C_{1}}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}_{1}}(f)\,dx\geq\frac{C_{1}}{C_{2}}\,\frac{1}{\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}}\,\int_{\Omega}{\widetilde{A}}(f)\,dx.$

Consequently, (2.14) holds.

Finally, Lemma 2.1(ii) ends the proof. ∎

###  2.3. On the space $X=L^{\tilde{A}}(\Omega)\times L^{\tilde{B}}(\Omega)$

We prove in this section that $X$ is reflexive.

To do so, in Proposition 5.10(v) is stated that we only need to check that the
${\mathcal{N}}$-functions satisfy the $\Delta_{2}$-condition at infinity (see
(5.1) for a definition of the $\Delta_{2}$-condition at infinity).

######  Proposition 2.5.

$A$, $B$, ${\widetilde{A}}$, ${\widetilde{B}}$ satisfy the
$\Delta_{2}$-condition, and $X$ is a reflexive Banach space. Moreover, the
dual space

(2.18) $X^{\prime}:=\left(L^{\tilde{A}}(\Omega)\times
L^{\tilde{B}}(\Omega)\right)^{\prime}=\left(L^{\tilde{A}}(\Omega)\right)^{\prime}\times\left(L^{\tilde{B}}(\Omega)\right)^{\prime}=L^{A}(\Omega)\times
L^{B}(\Omega)$

###### Proof.

From Lemma 5.5 and Proposition 5.10(i), the spaces $L^{\tilde{A}}(\Omega)$ and
$L^{\tilde{B}}(\Omega)$ are complete with both the Orlicz or the Luxemburg
norm. From Proposition 5.10(v) it is enough to check that $A$, $B$,
${\widetilde{A}}$, ${\widetilde{B}}$ satisfy the $\Delta_{2}$-condition. To do
that, we apply Lemma 5.4 (iii), see condition (5.2).

Indeed, notice that by l’Hôpital rule

$\lim_{t\to+\infty}\frac{ta(t)}{A(t)}=\lim_{t\to+\infty}\frac{ta^{\prime}(t)}{a(t)}+1=p+1,\qquad\mbox{and}\quad\lim_{t\to+\infty}\frac{tb(t)}{B(t)}=q+1,$

see (2.9) in the proof of Lemma 2.1. Now, changing the the variable
$s=\tilde{a}(t)$, using Young equality(3.1), and l’Hôpital rule

$\lim_{t\to+\infty}\frac{t\tilde{a}(t)}{\tilde{A}(t)}=\lim_{s\to+\infty}\frac{sa(s)}{\tilde{A}(a(s))}=\lim_{s\to+\infty}\frac{sa(s)}{sa(s)-A(s)}=\lim_{s\to+\infty}\frac{a(s)+sa^{\prime}(s)}{sa^{\prime}(s)}=\frac{1}{p}+1,$

see also (2.9) in the proof of Lemma 2.1. Likewise,

$\lim_{t\to+\infty}\frac{t\tilde{b}(t)}{\tilde{B}(t)}=\lim_{s\to+\infty}\frac{sb(s)}{\tilde{B}(b(s))}=\lim_{s\to+\infty}\frac{sb(s)}{sb(s)-B(s)}=\lim_{s\to+\infty}\frac{b(s)+sb^{\prime}(s)}{sb^{\prime}(s)}=\frac{1}{q}+1.$

Finally, Proposition 5.10 (iv) and the fact that
$\widetilde{(\widetilde{A})}=A$, and $\widetilde{(\widetilde{A})}=A$ ends the
proof.

∎

n order to prove some geometric properties of the functional $J$, we will made
use of the following continuous embeddings :

######  Lemma 2.6.

If $\alpha,\ \beta\geq 0,$

(2.19) $L^{\tilde{A}}(\Omega)\hookrightarrow L^{\frac{p+1}{p}}(\Omega)\
\hbox{and}\ L^{\tilde{B}}(\Omega)\hookrightarrow L^{\frac{q+1}{q}}(\Omega),$

and if $\alpha<0,$ (respectively, $\beta<0$), for all $\varepsilon>0$ small
enough,

(2.20) $L^{\tilde{A}}(\Omega)\hookrightarrow
L^{\frac{p+1}{p}-\varepsilon}(\Omega),\qquad\big{(}\text{respectively,}\quad
L^{\tilde{B}}(\Omega)\hookrightarrow
L^{\frac{q+1}{q}-\varepsilon}(\Omega)\big{)}.$

Likewise, if $\alpha,\ \beta\geq 0$, for all $\varepsilon>0$ small enough,

(2.21) $L^{A}(\Omega)\hookrightarrow L^{p+1-\varepsilon}(\Omega)\ \hbox{and}\
L^{B}(\Omega)\hookrightarrow L^{q+1-\varepsilon}(\Omega),$

and if $\alpha<0,$ (respectively, $\beta<0$),

(2.22) $L^{A}(\Omega)\hookrightarrow
L^{p+1}(\Omega),\qquad\big{(}\text{respectively,}\quad
L^{B}(\Omega)\hookrightarrow L^{q+1}(\Omega)\big{)}.$

###### Proof.

We remark that whenever $\alpha,\ \beta\geq 0,$ (2.19) holds using that

(2.23) $a(t)\leq t^{p}\Longrightarrow\tilde{a}(t)\geq
t^{1/p}\Longrightarrow\widetilde{A}(t)\geq\frac{p}{p+1}t^{\frac{p+1}{p}}.$

Moreover, as for all $u\in L^{\tilde{A}}(\Omega)$, $u\not\equiv 0$, we have
from (5.6) in Proposition 5.10(iii),

$1\geq\int_{\Omega}\tilde{A}\bigg{(}\frac{u}{\|u\|_{\tilde{A}}}\bigg{)}dx\geq\frac{p}{p+1}\int_{\Omega}\bigg{|}\frac{u}{\|u\|_{\tilde{A}}}\bigg{|}^{\frac{p+1}{p}}dx=\frac{p}{p+1}\frac{\|u\|_{\frac{p+1}{p}}^{\frac{p+1}{p}}}{\|u\|_{\tilde{A}}^{\frac{p+1}{p}}}\,,$

so, the embedding (2.20) is proved.

Besides, if $\alpha<0,$ (respectively, $\beta<0$), for all $\varepsilon>0$
small enough, (2.20) holds. Indeed, fix $\varepsilon>0$ small, there exists
$c=c_{\varepsilon}>0$ such that

$a(t)\leq\left\\{\begin{array}[]{lc}ct^{p}&\hbox{ if }t\leq 1\\\
ct^{p+\varepsilon}&\hbox{ if }t\geq 1,\end{array}\right.$

then

$\tilde{a}(s)\geq\left\\{\begin{array}[]{lc}c^{-1/p}s^{1/p}&\hbox{ if }s\leq
c\\\ c^{-1/(p+\varepsilon)}s^{1/(p+\varepsilon)}&\hbox{ if }s\geq
c,\end{array}\right.$

and

$\tilde{A}(s)\geq\left\\{\begin{array}[]{lc}\frac{p}{p+1}c^{-1/p}s^{(p+1)/p}&\hbox{
if }s\leq c\\\
\frac{p+\varepsilon}{p+\varepsilon+1}c^{-1/(p+\varepsilon)}s^{(p+\varepsilon+1)/(p+\varepsilon)}+(\frac{p}{p+1}-\frac{p+\varepsilon}{p+\varepsilon+1})c&\hbox{
if }s\geq c.\end{array}\right.$

In particular

$\tilde{A}(s)\geq C_{1}s^{\frac{p+1}{p}-\varepsilon^{\prime}}-C_{2},$

for

$C_{1}:=\frac{p+\varepsilon}{p+\varepsilon+1}c^{-1/(p+\varepsilon)},\quad
C_{2}:=\max\left\\{\frac{p+\varepsilon}{p+\varepsilon+1}c,\frac{\varepsilon\,c}{(p+\varepsilon+1)(p+1)}\right\\},\quad\varepsilon^{\prime}=\frac{\varepsilon}{p(p+\varepsilon)}.$

with $C_{1},C_{2}>0$. From the above and by definition, if $u\in
L^{\tilde{A}}(\Omega)$, $u\not\equiv 0$, then putting
$v=\frac{u}{\|u\|_{\tilde{A}}}$

$1\geq\int_{\Omega}\tilde{A}\big{(}v(x)\big{)}\,dx\geq\int_{\Omega}C_{1}\,|v|^{\frac{p+1}{p}-\varepsilon^{\prime}}dx-
C_{2}|\Omega|,$

so,

$\big{(}1+C_{2}|\Omega|\big{)}\,\|u\|_{\tilde{A}}^{\frac{p+1}{p}-\varepsilon^{\prime}}\geq
C_{1}\|u\|_{\frac{p+1}{p}-\varepsilon^{\prime}}^{\frac{p+1}{p}-\varepsilon^{\prime}}$

and the embedding (2.20) is proved.

The proofs of (2.21) and (2.22) are similar. ∎

The following Lemma is a technical one, that will be useful to prove
Proposition 2.10.

######  Lemma 2.7.

Let $a,\ A,\ \tilde{a},\ \tilde{A}$ be defined by (2.1), (2.2), (2.3), and
(2.5) respectively. Here $\sim$ means equivalent at infinity, see definition
5.15. Then

  1. (i)

$A$ is equivalent to $\tilde{A}~{}\raisebox{1.0pt}{ \leavevmode\hbox
to2.8pt{\vbox to2.8pt{\pgfpicture\makeatletter\hbox{\hskip
1.40001pt\lower-1.40001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}a$ at infinity,
$(A\sim\tilde{A}~{}\raisebox{1.0pt}{ \leavevmode\hbox to2.8pt{\vbox
to2.8pt{\pgfpicture\makeatletter\hbox{\hskip 1.40001pt\lower-1.40001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}a)$, and
$L^{A}(\Omega)$ is isomorphic to $L^{\tilde{A}~{}\raisebox{1.0pt}{
\leavevmode\hbox to2.8pt{\vbox to2.8pt{\pgfpicture\makeatletter\hbox{\hskip
1.40001pt\lower-1.40001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}a}(\Omega).$

  2. (ii)

$\tilde{A}$ is equivalent to $A~{}\raisebox{1.0pt}{ \leavevmode\hbox
to2.8pt{\vbox to2.8pt{\pgfpicture\makeatletter\hbox{\hskip
1.40001pt\lower-1.40001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a}$ at
infinity, $(\tilde{A}\sim A~{}\raisebox{1.0pt}{ \leavevmode\hbox to2.8pt{\vbox
to2.8pt{\pgfpicture\makeatletter\hbox{\hskip 1.40001pt\lower-1.40001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a})$, and
$L^{\tilde{A}}(\Omega)$ is isomorphic to $L^{A~{}\raisebox{1.0pt}{
\leavevmode\hbox to2.8pt{\vbox to2.8pt{\pgfpicture\makeatletter\hbox{\hskip
1.40001pt\lower-1.40001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a}}(\Omega)$.

###### Proof.

(i) Clearly, since Lemma 2.1, and Proposition 5.17, for any $\psi\in
L^{A}(\Omega),$

$A(\psi)=A\big{(}|\psi|\big{)}\sim|\psi|^{p+1}\big{(}\ln|\psi|\big{)}^{-\alpha},\qquad\mbox{and}\quad\int_{\Omega}|\psi|^{p+1}\big{(}\ln|\psi|\big{)}^{-\alpha}\,dx\leq
C.$

Moreover,

$a(\psi)\sim|\psi|^{p}(\ln|\psi|)^{-\alpha},$
$\tilde{A}\big{(}a(\psi)\big{)}\sim\left(|\psi|^{p}(\ln|\psi|)^{-\alpha}\right)^{\frac{p+1}{p}}\big{(}\ln|\psi|\big{)}^{\frac{\alpha}{p}}=|\psi|^{p+1}\big{(}\ln|\psi|\big{)}^{-\alpha},$

so $A\sim\tilde{A}~{}\raisebox{1.0pt}{ \leavevmode\hbox to2.8pt{\vbox
to2.8pt{\pgfpicture\makeatletter\hbox{\hskip 1.40001pt\lower-1.40001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}a$ at infinity,
and

$\int_{\Omega}\tilde{A}\big{(}a(\psi)\big{)}\,dx\leq
C,\qquad\mbox{consequently}\quad a(\psi)\in L^{\tilde{A}}(\Omega).$

(ii) Likewise, for any $\psi\in L^{\tilde{A}}(\Omega),$

$\tilde{A}(\psi)=\tilde{A}\big{(}|\psi|\big{)}\sim|\psi|^{\frac{p+1}{p}}\big{(}\ln|\psi|\big{)}^{\frac{\alpha}{p}},\qquad\mbox{and}\quad\int_{\Omega}|\psi|^{\frac{p+1}{p}}\big{(}\ln|\psi|\big{)}^{\frac{\alpha}{p}}\,dx\leq
C.$

Moreover,

$\tilde{a}(\psi)\sim|\psi|^{\frac{1}{p}}(\ln|\psi|)^{\frac{\alpha}{p}},$

and

$A\big{(}\tilde{a}(\psi)\big{)}\sim\left(|\psi|^{\frac{1}{p}}(\ln|\psi|)^{\frac{\alpha}{p}}\right)^{p+1}\big{(}\ln|\psi|\big{)}^{-\alpha}=|\psi|^{\frac{p+1}{p}}\big{(}\ln|\psi|\big{)}^{\frac{\alpha}{p}},$

so $\tilde{A}\sim A~{}\raisebox{1.0pt}{ \leavevmode\hbox to2.8pt{\vbox
to2.8pt{\pgfpicture\makeatletter\hbox{\hskip 1.40001pt\lower-1.40001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a}$ at
infinity, and

$\int_{\Omega}A\big{(}\tilde{a}(\psi)\big{)}\,dx\leq
C,\qquad\mbox{consequently}\quad\tilde{a}(\psi)\in L^{A}(\Omega).$

∎

###  2.4. On the functional $J$

Let us write $J^{\prime}=\Phi-\Upsilon$, where $\Phi$ and $\Upsilon$ are
defined, for all $(f,g)\in X$ and for all $(\psi_{1},\psi_{2})\in X$, as

(2.24)
$\Phi(f,g)[\psi_{1},\psi_{2}]:=\int\limits_{\Omega}\tilde{a}(f)\psi_{1}\,dx+\int\limits_{\Omega}\widetilde{b}(g)\psi_{2}\,dx,$
(2.25)
$\Upsilon(f,g)[\psi_{1},\psi_{2}]:=\int\limits_{\Omega}\psi_{1}\,\mathtt{K}(g)+\int\limits_{\Omega}\psi_{2}\,\mathtt{K}(f)\,dx.$

We will see that $\Upsilon$ is a continuous compact operator, see Proposition
2.9, and that $\Phi$ is an homeomorphism, see Proposition 2.10 .

We start by proving that $\Upsilon$ is a compact operator. Let us recall that,
by using the regularity result Theorem 5.29,
$\big{(}\mathtt{K}(f),\mathtt{K}(g)\big{)}\in W^{2,\tilde{A}}(\Omega)\times
W^{2,\tilde{B}}(\Omega)$ for all $(f,g)\in X$. Furthermore, we have

######  Lemma 2.8.

Let $p,q>0,\ \alpha,\beta\in\mathbb{R}$ satisfying either (1.2) or (1.3).
Then, the embeddings $W^{2,{\tilde{A}}}(\Omega)\hookrightarrow L^{B}(\Omega)$
and $W^{2,{\tilde{B}}}(\Omega)\hookrightarrow L^{A}(\Omega)$ are compact.

###### Proof.

We use Theorem 5.27. Assume first that $\frac{p+1}{p}\geq\frac{N}{2}.$ In
particular, we observe that if $\frac{p+1}{p}>\frac{N}{2}$, or
$\frac{p+1}{p}=\frac{N}{2}$ and
$\alpha>\left(1-\frac{1}{N}\right)p\left(N-\frac{p+1}{p}\right)$, then
$W^{2,{\tilde{A}}}(\Omega)\hookrightarrow C_{b}(\Omega)$ is compact, where
$C_{b}(\Omega)=C(\Omega)\cap L^{\infty}(\Omega)$. Hence the embedding

$W^{2,{\tilde{A}}}(\Omega)\hookrightarrow L^{B}(\Omega)$

is also compact.

Moreover, if $\frac{p+1}{p}=\frac{N}{2}$ and
$\alpha\leq\left(1-\frac{1}{N}\right)p\left(N-\frac{p+1}{p}\right)$, then

$W^{2,{\tilde{A}}}(\Omega)\hookrightarrow L^{A_{1}}(\Omega)$

is compact for all ${\mathcal{N}}$-function $A_{1}$ such that

$A_{1}\prec\prec\big{(}{\widetilde{A}}^{*}\big{)}^{*},$

where $\prec\prec$ is defined in 5.14(iii), and
$\big{(}{\widetilde{A}}^{*}\big{)}^{*}$, at least of exponential type, is
defined by (5.13). Let us verify that
$B\prec\prec\big{(}{\widetilde{A}}^{*}\big{)}^{*}$ using that $B(s)\sim
s^{q+1}(\ln s)^{-\beta},$ see Lemma 2.1 and Example 5.18. Indeed, for all
$c>0$ fixed, we have that

$\lim\limits_{s\to+\infty}\frac{{B}(s)}{\big{(}{\widetilde{A}}^{*}\big{)}^{*}(cs)}=0,$

holds trivially.

Likewise, if $\frac{q+1}{q}\geq\frac{N}{2}$, then the embedding
$W^{2,{\tilde{B}}}(\Omega)\hookrightarrow L^{A}(\Omega)$, where $A(s)\sim
s^{p+1}(\ln s)^{-\alpha},$ is compact.

Let us now study the situation appearing when $\frac{p+1}{p}<\frac{N}{2}$. We
now remark (see Lemma 2.1 and Example 5.25) that for
$\frac{p+1}{p}<\frac{N}{2}$,

$\widetilde{A}(s)\sim s^{\frac{p+1}{p}}(\ln s)^{\frac{\alpha}{p}},\ \hbox{ so
then }\ W^{2,{\tilde{A}}}(\Omega)\hookrightarrow
L^{A_{1}}(\Omega)\qquad\forall
A_{1}\prec\prec\big{(}({\widetilde{A}})^{*}\big{)}^{*},$
$\qquad\mbox{with}\quad\big{(}({\widetilde{A}})^{*}\big{)}^{*}\sim
s^{\frac{N(p+1)}{Np-2(p+1)}}\,\big{[}\log(s)\big{]}^{\frac{\alpha
N}{Np-2(p+1)}}.$

Let us verify that $B\prec\prec\big{(}({\widetilde{A}})^{*}\big{)}^{*}$, in
other words, that for all $\delta>0$,

$\lim\limits_{s\to+\infty}\frac{\big{(}({\widetilde{A}})^{*}\big{)}^{*}(\delta
s)}{B(s)}=+\infty.$

Indeed,

$\frac{\big{(}({\widetilde{A}})^{*}\big{)}^{*}(\delta s)}{B(s)}\sim
s^{\frac{N(p+1)}{Np-2(p+1)}-(q+1)}\big{[}\log(s)\big{]}^{\frac{\alpha
N}{Np-2(p+1)}+\beta}\to\infty,$

if

$\frac{N(p+1)}{Np-2(p+1)}>q+1,\qquad\mbox{or}\quad\frac{N(p+1)}{Np-2(p+1)}=q+1,\text{
and }\frac{\alpha N}{Np-2(p+1)}+\beta>0,$

i.e. if

$\frac{p}{p+1}-\frac{2}{N}<\frac{1}{q+1},\qquad\mbox{or}\quad\frac{N}{Np-2(p+1)}=\frac{q+1}{p+1},\text{
and }\frac{\alpha}{p+1}+\frac{\beta}{q+1}>0.$

This last inequalities are equivalent to

$\frac{N-2}{N}-\frac{1}{p+1}<\frac{1}{q+1},\qquad\mbox{or}\quad\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N},\text{
and }\frac{\alpha}{p+1}+\frac{\beta}{q+1}>0.$

The first inequality holds under hypothesis (1.2), the second one holds by
hypothesis (1.3).

Likewise if $\frac{q+1}{q}<\frac{N}{2}$, then the embedding
$W^{2,{\tilde{B}}}(\Omega)\hookrightarrow L^{A}(\Omega)$ is compact, and the
proof is achieved. ∎

######  Proposition 2.9.

The linear operator $\Upsilon:X\to X^{\prime}$ defined by

$\Upsilon(f,g)[\psi_{1},\psi_{2}]:=\int\limits_{\Omega}\psi_{1}\,\mathtt{K}(g)\,dx+\int\limits_{\Omega}\psi_{2}\,\mathtt{K}(f)\,dx,\qquad\forall(\psi_{1},\psi_{2})\in
X$

is a continuous compact map.

###### Proof.

Let us prove the continuity. Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ be a sequence
in $L^{\tilde{A}}(\Omega)$ converging to some $f\in L^{\tilde{A}}(\Omega)$ and
$\\{g_{n}\\}_{n\in\mathbb{N}}$ a sequence in $L^{\tilde{B}}(\Omega)$
converging to some $g\in L^{\tilde{B}}(\Omega)$. Let $\psi_{1}\in
L^{\tilde{A}}(\Omega)$. We have

$\big{|}\Upsilon(f_{n},g_{n})[\psi_{1},\psi_{2}]-\Upsilon(f,g)[\psi_{1},\psi_{2}]\big{|}\leq\int_{\Omega}\Big{(}\big{|}{\mathtt{K}}(f_{n})\psi_{2}-\mathtt{K}(f)\psi_{2}\big{|}+\big{|}{\mathtt{K}}(g_{n})\psi_{1}-\mathtt{K}(g)\psi_{1}\big{|}\Big{)}\,dx$
$\leq
C\Big{(}\big{\|}{\mathtt{K}}(f_{n})-\mathtt{K}(f)\big{\|}_{B}\,\|\psi_{2}\|_{\tilde{B}}+\big{\|}{\mathtt{K}}(g_{n})-\mathtt{K}(g)\big{\|}_{A}\,\|\psi_{1}\|_{\tilde{A}}\Big{)}$

We only have to prove that
$\big{\|}\mathtt{K}(f_{n})-\mathtt{K}(f)\big{\|}_{B}\to 0$, the proof of
$\big{\|}\mathtt{K}(g_{n})-\mathtt{K}(g)\big{\|}_{A}\to 0$ is analogous. This
result follows from the regularity results of Theorem 5.29 and the above Lemma
2.8.

Let us now prove the compactness. Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ be a
bounded sequence in $L^{\tilde{A}}(\Omega)$ and $\\{g_{n}\\}_{n\in\mathbb{N}}$
a bounded sequence in $L^{\tilde{B}}(\Omega)$. From (5.17) we have

$\|{\mathtt{K}}(f_{n})\|_{W^{2,{\widetilde{A}}}}\leq
C,\quad\|{\mathtt{K}}(g_{n})\|_{W^{2,{\widetilde{B}}}}\leq C$

for some constant $C$ independent of $n$. Then, using Lemma 2.8 and the fact
that $X$ is reflexive (cf. Proposition 2.5, and Proposition 5.10) we infer the
existence of $g\in L^{B}(\Omega)$ and $f\in L^{A}(\Omega)$ such that, up to a
subsequence, $\|{\mathtt{K}}(f_{n})-g\|_{B}\to
0,\|{\mathtt{K}}(g_{n})-f\|_{A}\to 0$ as $n\to\infty$. Hence, using the 2nd
Hölder’s inequality, for any $(\psi_{1},\psi_{2})\in X$, defining
${\mathcal{I}}_{f,g}[\psi_{1},\psi_{2}]:=\int_{\Omega}(f\psi_{1}+g\psi_{2})dx$,

$\big{|}\Upsilon(f_{n},g_{n})[\psi_{1},\psi_{2}]-{\mathcal{I}}_{f,g}[\psi_{1},\psi_{2}]\big{|}\leq\int_{\Omega}\Big{(}\big{|}{\mathtt{K}}(f_{n})\psi_{2}-g\psi_{2}\big{|}+\big{|}{\mathtt{K}}(g_{n})\psi_{1}-f\psi_{1}\big{|}\Big{)}\,dx$
$\leq
C\Big{(}\big{\|}{\mathtt{K}}(f_{n})-g\big{\|}_{B}\,\|\psi_{2}\|_{\tilde{B}}+\big{\|}{\mathtt{K}}(g_{n})-f\big{\|}_{A}\,\|\psi_{1}\|_{\tilde{A}}\Big{)}$

so then $\|\Upsilon(f_{n},g_{n})-{\mathcal{I}}_{f,g}\|_{X^{\prime}}\to 0$ as
$n\to\infty$. ∎

We end this Section, proving that $\Phi$ is an homeomorphism.

######  Proposition 2.10.

The non-linear operator $\Phi:X\to X^{\prime}$ defined by

$\Phi(f,g)[\psi_{1},\psi_{2}]:=\int\limits_{\Omega}\tilde{a}(f)\psi_{1}\,dx+\int\limits_{\Omega}\tilde{b}(g)\psi_{2}\,dx,\qquad\mbox{for
all}\quad(\psi_{1},\psi_{2})\in X.$

is an homeomorphism.

###### Proof.

To prove that $\Phi:X\to X^{\prime}$ is an homeomorphism, we use Proposition
5.10(iii)-(iv), Remark 5.7, and Lemma 2.1 and Example 5.18.

Step 1. $\Phi$ is injective.

Assume that $\Phi(f_{1},g_{1})=\Phi(f_{2},g_{2})$, so for any
$(\psi_{1},\psi_{2})\in X$

$0=\big{(}\Phi(f_{1},g_{1})-\Phi(f_{2},g_{2})\big{)}[\psi_{1},\psi_{2}]=\int\limits_{\Omega}\big{[}\tilde{a}(f_{1})-\tilde{a}(f_{2})\big{]}\psi_{1}\,dx+\int\limits_{\Omega}\big{[}\tilde{b}(g_{1})-\tilde{b}(g_{2})\big{]}\psi_{2}\,dx,$

then

$\tilde{a}(f_{1})=\tilde{a}(f_{2}),\qquad\tilde{b}(g_{1})=\tilde{b}(g_{2})\qquad\mbox{a.e.}\quad
x\in\Omega.$

or equivalently, since $\tilde{a}:=a^{-1}$ and $\tilde{b}:=b^{-1}$,

$f_{1}=f_{2},\qquad g_{1}=g_{2}\qquad\mbox{a.e.}\quad x\in\Omega.$

Step 2. $\Phi$ is surjective.

Let us check that for any $(\psi_{1},\psi_{2})\in
X^{\prime}=L^{A}(\Omega)\times L^{B}(\Omega)$ (see (2.18)),
$\big{(}a(\psi_{1}),b(\psi_{2})\big{)}\in X,$ and

(2.26) $\Phi\big{(}a(\psi_{1}),b(\psi_{2})\big{)}=(\psi_{1},\psi_{2}).$

Since Lemma 2.7, $a(\psi_{1})\in L^{\tilde{A}}(\Omega)$. Likewise
$b(\psi_{2})\in L^{\tilde{B}}(\Omega),$ and
$\big{(}a(\psi_{1}),b(\psi_{2})\big{)}\in X.$

Since $\tilde{a}:=a^{-1}$ and $\tilde{b}:=b^{-1}$, it is straightforward to
check that (2.26) holds.

Consequently,

$\Phi^{-1}(\psi_{1},\psi_{2})=\big{(}a(\psi_{1}),b(\psi_{2})\big{)}.$

Step 3. $\Phi$ is continuous.

Let $(f_{n},g_{n})\to(f,g)$ be a convergent sequence in
$L^{\tilde{A}}(\Omega)\times L^{\tilde{B}}(\Omega)$, in other words

$\|f_{n}-f\|_{\tilde{A}}+\|g_{n}-g\|_{\tilde{B}}\to 0.$

Claim.

$\|f_{n}-f\|_{\tilde{A}}\to
0\implies\big{\|}\tilde{a}(f_{n})-\tilde{a}(f)\big{\|}_{A}\to 0.$

Once proved the claim

$\big{\|}\tilde{a}(f_{n})-\tilde{a}(f)\big{\|}_{A}+\big{\|}\tilde{b}(g_{n})-\tilde{b}(g)\big{\|}_{B}\to
0.$

Then, since Proposition 5.10(i)-(ii), and Remark 5.11, for all
$(\psi_{1},\psi_{2})\in X,$

$\displaystyle\Big{|}\big{(}\Phi(f_{n},g_{n})-\Phi(f,g)\big{)}[\psi_{1},\psi_{2}]\Big{|}$
$\displaystyle=\left|\int\limits_{\Omega}\big{[}\tilde{a}(f_{n})-\tilde{a}(f)\big{]}\psi_{1}\,dx+\int\limits_{\Omega}\big{[}\tilde{b}(g_{n})-\tilde{b}(g)\big{]}\psi_{2}\,dx\right|$
$\displaystyle\leq\big{\|}\tilde{a}(f_{n})-\tilde{a}(f)\big{\|}_{A}\|\psi_{1}\|_{\tilde{A}}+\big{\|}\tilde{b}(g_{n})-\tilde{b}(g)\big{\|}_{B}\|\psi_{2}\|_{\tilde{B}}\to
0,$

and the continuity is achieved.

Proof of the Claim.

Since Theorem 5.13, to conclude the claim, we need to check that
$\tilde{a}(f_{n})\to\tilde{a}(f)$ in $A$-mean, in other words

$\int_{\Omega}A\big{(}\big{|}\tilde{a}(f_{n})-\tilde{a}(f)\big{|}\big{)}\,dx\to
0.$

On the one hand since Lemma 2.7, $\tilde{A}\sim A~{}\raisebox{1.0pt}{
\leavevmode\hbox to2.8pt{\vbox to2.8pt{\pgfpicture\makeatletter\hbox{\hskip
1.40001pt\lower-1.40001pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a}$ at
infinity, and $L^{\tilde{A}}(\Omega)$ is isomorphic to
$L^{A~{}\raisebox{1.0pt}{ \leavevmode\hbox to2.8pt{\vbox
to2.8pt{\pgfpicture\makeatletter\hbox{\hskip 1.40001pt\lower-1.40001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a}}(\Omega)$,
so, for any $f\in L^{\tilde{A}}(\Omega)$

$\int_{\Omega}A\big{(}\tilde{a}(f)\big{)}\,dx\leq C,\qquad\mbox{and
consequently}\quad\tilde{a}(f)\in L^{A}(\Omega).$

Likewise, $\\{\tilde{a}(f_{n})\\}\subset L^{A}(\Omega)$.

On the other hand since Theorem 5.13, $f_{n}\to f$ in ${\tilde{A}}$-mean, so

(2.27)
$\int_{\Omega}{\tilde{A}}\big{(}f_{n}-f\big{)}\,dx=\int_{\Omega}{\tilde{A}}\big{(}|f_{n}-f|\big{)}\,dx\to
0.$

Assume by contradiction that the claim do not hold. Then, there exist
$\varepsilon_{0}>0$ and a subsequence $\\{f_{n_{k}}\\}_{k}$ satisfying

(2.28)
$\int_{\Omega}A\left(\tilde{a}(f_{n_{k}})-\tilde{a}(f)\right)dx>\varepsilon_{0}\quad\mbox{
for all }k\in\mathbb{N}.$

Put $z_{k}:=\widetilde{A}(|f_{n_{k}}-f|)$. Since (2.27), $z_{k}\to 0$ in
$L^{1}(\Omega)$, so by the Lebesgue reverse dominated converge theorem, there
exists a subsequence $\\{z_{k_{j}}\\}_{j}$ and there exists $h\in
L^{1}(\Omega)$ such that

$z_{k_{j}}\leq h,\quad z_{k_{j}}\to 0\hbox{ a.e.},\hbox{ and also
}\tilde{A}^{-1}z_{k_{j}}=f_{n_{k_{j}}}-f\to 0\hbox{ a.e.}$

We have

$|f_{n_{k_{j}}}|\leq\widetilde{A}^{-1}(h)+|f|\in L^{\widetilde{A}}(\Omega).$

By monotonicity of $\widetilde{a}$, and since Lemma 2.7, $\tilde{A}\sim
A~{}\raisebox{1.0pt}{ \leavevmode\hbox to2.8pt{\vbox
to2.8pt{\pgfpicture\makeatletter\hbox{\hskip 1.40001pt\lower-1.40001pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.1pt}{0.0pt}\pgfsys@curveto{1.1pt}{0.60751pt}{0.60751pt}{1.1pt}{0.0pt}{1.1pt}\pgfsys@curveto{-0.60751pt}{1.1pt}{-1.1pt}{0.60751pt}{-1.1pt}{0.0pt}\pgfsys@curveto{-1.1pt}{-0.60751pt}{-0.60751pt}{-1.1pt}{0.0pt}{-1.1pt}\pgfsys@curveto{0.60751pt}{-1.1pt}{1.1pt}{-0.60751pt}{1.1pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\tilde{a}$ at
infinity, so

$\widetilde{a}(f_{n_{k_{j}}})\leq\widetilde{a}\left(\widetilde{A}^{-1}(h)+|f|\right):=\ell\in
L^{A}(\Omega),$

moreover

$A\left(|\widetilde{a}(f_{n_{k_{j}}})-\widetilde{a}(f)|\right)\leq
A(\ell+\widetilde{a}(f))\in L^{1}(\Omega).$

Since pointwise convergence
$A\left(|\widetilde{a}(f_{n_{k_{j}}})-\widetilde{a}(f)|\right)\to 0$ a.e., and
from Lebesgue dominate convergence theorem we conclude that

$\int_{\Omega}A\left(|\widetilde{a}(f_{n_{k_{j}}})-\widetilde{a}(f)|\right)dx\to
0,$

which contradits (2.28). This concludes the proof of the claim. ∎

##  3\. Proof of Theorem 1.1

First we star by proving that $J$ satisfies the $(PS)$ condition. To this aim
we will use of the following (trivial) lemma:

######  Lemma 3.1 ([13], Lemma 3.1).

Let $X$ be a reflexive Banach space and $F\in C^{1}(X)$ be such that

  1. (1)

any Palais-Smale sequence of $F$ is bounded;

  2. (2)

for all $u\in X$,

$F^{\prime}(u)=L(u)-T(u),$

where $L:X\to X^{\prime}$ is a homeomorphism and $T:X\to X^{\prime}$ is a
continuous compact map.

Then, $F$ satisfies the Palais-Smale condition.

######  Proposition 3.2.

$J$ satisfies the $(PS)$ condition.

###### Proof.

Let us first check (2) for the space $X=L^{\widetilde{A}}(\Omega)\times
L^{\widetilde{B}}(\Omega)$ and the functional $F=J$ defined (2.7). Observe
that $X$ is a reflexive Banach space by Proposition 2.5 and $J\in
C^{1}(X;\mathbb{R})$ is such that $J^{\prime}=\Phi-\Upsilon$ where $\Phi$ is
defined in (2.24) and $\Upsilon$ is defined in (2.25). Thus, (2) follows from
Proposition 2.5 and Proposition 2.9.

Now we check (1). Let $\\{(f_{n},g_{n})\\}_{n\in\mathbb{N}}$ be a Palais-Smale
sequence for the functional $J$, that is, we have for some $C>0$ and some
sequence of positive real numbers $\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}\to
0$,

(PS1):$\quad\quad\ J(f_{n},g_{n})\leq C\hbox{ for all }n\in\mathbb{N},$

(PS2):
$\quad\quad\left|J^{\prime}(f_{n},g_{n})[\psi_{1},\psi_{2}]\right|\leq\varepsilon_{n}\|(\psi_{1},\psi_{2})\|_{X}\qquad\forall(\psi_{1},\psi_{2})\in
X.$

Since (2.6), for every $(f,g)\in X$, $\theta\in\mathbb{R}$ we have

$\frac{1}{2}\Upsilon(f,g)[f,g]=\frac{1}{2}\left[\int\limits_{\Omega}f\,\mathtt{K}(g)+\int\limits_{\Omega}g\,\mathtt{K}(f)\,dx\right]=\Upsilon(f,g)\left[\theta
f,(1-\theta)g\right],$

and since the identity

$J(f_{n},g_{n})-J^{\prime}(f_{n},g_{n})\left[\theta
f,(1-\theta)g\right]=\int_{\Omega}\big{(}\tilde{A}(f_{n})+\tilde{B}(g_{n})-\theta
f_{n}\tilde{a}(f_{n})-(1-\theta)g_{n}\tilde{b}(g_{n})\big{)}\,dx$

it follows from $(PS1)$-$(PS2)$,

$\int_{\Omega}\left(\tilde{A}(f_{n})-\theta
f_{n}\tilde{a}(f_{n})\right)dx+\int_{\Omega}\left(\tilde{B}(g_{n})-(1-\theta)g_{n}\tilde{b}(g_{n})\right)\,dx\leq\varepsilon_{n}\|(f_{n},g_{n})\|_{X}+2C.$

Assume by contradiction that $\|(f_{n},g_{n})\|_{X}\rightarrow+\infty$. Using
the Young equality, see Proposition 5.8 (iii),

(3.1) $\widetilde{A}(t)=t\tilde{a}(t)-A\big{(}\tilde{a}(t)\big{)}$

Integrating by parts $A(t)=\int_{0}^{t}\frac{s^{p}}{(\ln(e+s))^{\alpha}}\,ds,$
we have that

$\displaystyle
A(t)=\frac{t\,a(t)}{p+1}+\frac{\alpha}{p+1}\int^{t}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s},$

and consequently,

$\displaystyle
A\big{(}\tilde{a}(t)\big{)}=\frac{1}{p+1}\,t\,\tilde{a}(t)+\frac{\alpha}{p+1}\int^{\tilde{a}(t)}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s}.$

Likewise

$B(t)=\frac{1}{q+1}\,t\,b(t)+\frac{\beta}{q+1}\int^{t}_{0}\frac{s^{q+1}}{\big{(}\ln(e+s)\big{)}^{\beta+1}}\,\frac{ds}{e+s},$

and

$\displaystyle
B\big{(}\tilde{b}(t)\big{)}=\frac{1}{q+1}\,t\,\tilde{b}(t)+\frac{\beta}{q+1}\int^{\tilde{b}(t)}_{0}\frac{s^{q+1}}{\big{(}\ln(e+s)\big{)}^{\beta+1}}\,\frac{ds}{e+s}.$

Hence

$\displaystyle\widetilde{A}(t)-\theta t\tilde{a}(t)$
$\displaystyle=\left(\frac{p}{p+1}-\theta\right)\,t\,\tilde{a}(t)-\frac{\alpha}{p+1}\int^{\tilde{a}(t)}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s},$

and

$\displaystyle\widetilde{B}(t)-(1-\theta)t\,\tilde{b}(t)$
$\displaystyle=\left(\frac{q}{q+1}-(1-\theta)\right)\,t\,\tilde{b}(t)-\frac{\beta}{q+1}\int^{\tilde{b}(t)}_{0}\frac{s^{q+1}}{\big{(}\ln(e+s)\big{)}^{\beta+1}}\,\frac{ds}{e+s}.$

Since l’Hôpital rule,

$\lim_{t\to\infty}\frac{\displaystyle\int^{t}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s}}{\displaystyle\frac{t^{p+1}}{\big{(}\ln(e+t)\big{)}^{\alpha+1}}}=\frac{1}{p+1},\qquad\mbox{moreover}\quad\lim_{t\to\infty}\frac{\displaystyle\frac{t^{p+1}}{(\ln(e+t))^{\alpha+1}}}{ta(t)}=0,$

then, for each $\varepsilon>0$ there exists a $C_{\varepsilon}$ such that

$\frac{\alpha}{p+1}\int^{t}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s}\leq\varepsilon\,ta(t)+C_{\varepsilon},\qquad\mbox{for
all}\quad t\geq 0,$

and

$\int^{\tilde{a}(t)}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s}\leq\varepsilon\,t\tilde{a}(t)+C_{\varepsilon},\qquad\mbox{for
all}\quad t\geq 0.$

Choosing

$\theta=\frac{1}{2}\left(1-\frac{1}{p+1}+\frac{1}{q+1}\right)=\frac{1}{2}\left(\frac{p}{p+1}+\frac{1}{q+1}\right)\in[0,1],$

we get that

$\frac{p}{p+1}-\theta=\frac{q}{q+1}-(1-\theta)=\frac{1}{2}\left(1-\frac{1}{p+1}-\frac{1}{q+1}\right)>0.$

Consequently

$\displaystyle\widetilde{A}(t)-\theta t\tilde{a}(t)$
$\displaystyle\geq\left[\frac{1}{2}\left(1-\frac{1}{p+1}-\frac{1}{q+1}\right)-\varepsilon\right]\,t\,\tilde{a}(t)-C_{e},$

and

$\displaystyle\widetilde{B}(t)-(1-\theta)t\,\tilde{b}(t)$
$\displaystyle\geq\left[\frac{1}{2}\left(1-\frac{1}{p+1}-\frac{1}{q+1}\right)-\varepsilon\right]\,t\,\tilde{b}(t)-C_{e}.$

Hence

$\displaystyle\int_{\Omega}f_{n}\,\tilde{a}(f_{n})\,dx+\int_{\Omega}g_{n}\,\tilde{b}(g_{n})\,dx\leq\varepsilon_{n}\|(f_{n},g_{n})\|_{X}+C,$

and as $t\tilde{a}(t)>0$ for all $t$, we then get

$\frac{1}{\|(f_{n},g_{n})\|_{X}}\int_{\Omega}f_{n}\,\tilde{a}(f_{n})\,dx\to
0,\qquad\frac{1}{\|(f_{n},g_{n})\|_{X}}\,\int_{\Omega}g_{n}\,\tilde{b}(g_{n})\,dx\to
0.$

Thus, using that for all $u\in L^{H}(\Omega)$

$\|u\|_{H}\leq 2\|u\|_{(H)}\leq 2\max\left\\{\int_{\Omega}H(u)\,dx,1\right\\}$

(cf. Lemma 5.5(iii) and Proposition 5.10(i)) for $H=\tilde{A}$ and
$H=\tilde{B}$, we reach a contradiction. ∎

Proof of Theorem 1.1. We wish to apply a mountain pass type theorem. We divide
the proof in 2 steps. Recall that the functional $J$ is defined in (2.7).

Step 1. We prove that $(0,0)$ is a local minimum of $J$. By Young’s inequality
for both $A$ and $B$ implies

$\displaystyle J(f,g)=$
$\displaystyle\int\limits_{\Omega}\widetilde{A}(f)\,dx+\int\limits_{\Omega}\widetilde{B}(g)\,dx-\frac{1}{2}\int\limits_{\Omega}\left(f\,\mathtt{K}(g)+g\,\mathtt{K}(f)\right)\,dx$
$\displaystyle\geq$
$\displaystyle\int\limits_{\Omega}\widetilde{A}(f)\,dx+\int\limits_{\Omega}\widetilde{B}(g)\,dx-\frac{1}{2}\int_{\Omega}\left(\widetilde{A}(f)+A(\mathtt{K}(g))\right)\,dx-\frac{1}{2}\int_{\Omega}\left(\widetilde{B}(g)+B(\mathtt{K}(f))\right)\,dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int\limits_{\Omega}\widetilde{A}(f)\,dx+\frac{1}{2}\int\limits_{\Omega}\widetilde{B}(g)\,dx-\frac{1}{2}\int\limits_{\Omega}A(\mathtt{K}(g))\,dx-\frac{1}{2}\int_{\Omega}B(\mathtt{K}(f))\,dx.$

Let us distinguish 3 cases according to the signs of $\alpha$ and $\beta$.

Case 1: Assume that $\alpha\geq 0$ and $\beta\geq 0$. First observe that,
since

$\widetilde{A}(t)\geq\frac{p}{p+1}t^{\frac{p+1}{p}},\quad\widetilde{B}(t)\geq\frac{q}{q+1}t^{\frac{q+1}{q}}$

for all $t\geq 0$ (see (2.23)), then $X\subset L^{\frac{p+1}{p}}(\Omega)\times
L^{\frac{q+1}{q}}(\Omega)$. Moreover, using that

$A(t)\leq\frac{1}{p+1}t^{p+1},\quad B(t)\leq\frac{1}{q+1}t^{q+1}$

for all $t\geq 0$, we get

$\int_{\Omega}A(\mathtt{K}(g))dx\leq\frac{1}{p+1}\|\mathtt{K}(g)\|_{p+1}^{p+1},\quad\int_{\Omega}B(\mathtt{K}(f))dx\leq\frac{1}{q+1}\|\mathtt{K}(f)\|_{q+1}^{q+1}.$

Besides, using Sobolev embeddings, either the second inequality of (1.2) or
the equality in (1.3) and the regularity of $\mathtt{K}$, there exists a
constant $C>0$ independent of $f$ and $g$ such that

$\|\mathtt{K}(g)\|_{p+1}\leq C\|\mathtt{K}(g)\|_{W^{2,\frac{q+1}{q}}}\leq
C\|g\|_{\frac{q+1}{q}},\quad\|\mathtt{K}(f)\|_{q+1}\leq
C\|\mathtt{K}(f)\|_{W^{2,\frac{p+1}{p}}}\leq C\|f\|_{\frac{p+1}{p}}.$

Thus

$2J(f,g)\geq\bigg{(}\frac{p}{p+1}\|f\|_{\frac{p+1}{p}}^{\frac{p+1}{p}}-C^{q+1}\|f\|_{\frac{p+1}{p}}^{q+1}\bigg{)}+\bigg{(}\frac{q}{q+1}\|g\|_{\frac{q+1}{q}}^{\frac{q+1}{q}}-C^{p+1}\|g\|_{\frac{q+1}{q}}^{p+1}\bigg{)}.$

Finally, using the first inequality of (1.2) and (1.3) (i.e.
$\frac{p+1}{p}<q+1$, or equivalently, $\frac{q+1}{q}<p+1$) and that
$\|f\|_{\frac{p+1}{p}}\leq\left(\frac{p+1}{p}\right)^{\frac{p}{p+1}}\|f\|_{\widetilde{A}},\quad\|g\|_{\frac{q+1}{q}}\leq\left(\frac{q+1}{q}\right)^{\frac{q}{q+1}}\|g\|_{\widetilde{B}}$
, we conclude that $J(f,g)>0$ for all $(f,g)\in X$ with $0\not=\|(f,g)\|_{X}$
sufficiently small.

Case 2: Assume that $\alpha\geq 0$ and $\beta<0$ (the case $\alpha<0$ and
$\beta\geq 0$ is analogous).

Using Sobolev embeddings, the regularity of $\mathtt{K}$, and since
$q+1\leq\left(\frac{N-2}{N}-\frac{1}{p+1}\right)^{-1}=\left(\frac{p}{p+1}-\frac{2}{N}\right)^{-1},$
we have that

$\|\mathtt{K}(f)\|_{q+1}\leq
C\|\mathtt{K}(f)\|_{\left(\frac{p}{p+1}-\frac{2}{N}\right)^{-1}}\leq
C\|\mathtt{K}(f)\|_{W^{2,\frac{p+1}{p}}}\leq C\|f\|_{\frac{p+1}{p}}\,\,.$

On the one hand, since (2.16), and the above

$\int_{\Omega}B(\mathtt{K}(f))\,dx\leq\frac{1}{q+1}\,\|\mathtt{K}(f)\|_{q+1}^{q+1}\leq
C\|f\|_{\frac{p+1}{p}}^{q+1}\,\,.$

On the other hand, since (2.13)

$\int_{\Omega}{\widetilde{A}}(f)\,dx\geq
C\,\|f\|_{\frac{p+1}{p}}^{\,\frac{p+1}{p}}\,\,.$

Hence, using the Young inequality

$\frac{1}{2}\int\limits_{\Omega}\left(f\,\mathtt{K}(g)+g\,\mathtt{K}(f)\right)\,dx=\int\limits_{\Omega}f\,\mathtt{K}(g)\,dx\leq\int_{\Omega}\tilde{B}(g)dx+\int_{\Omega}B(K(f))dx$

so

$J(f,g)\geq\int_{\Omega}\tilde{A}(f)dx-\int_{\Omega}B(K(f))dx$

and using (2.41)

$J(f,g)\geq
C\|f\|_{\frac{p+1}{p}}^{\frac{p+1}{p}}-D\|f\|_{\frac{p+1}{p}}^{q+1}>0,$

for $0\not=\|f\|_{\tilde{A}}$ small enough. Observe that if $f=0$, then
$J(f,g)=\int_{\Omega}\tilde{B}(g)dx>0$ for $\|g\|_{\tilde{B}}\neq 0.$

Case 3: Assume that $\alpha<0$ and $\beta<0$.

Notice that if $\alpha<0,\,\beta<0$ then necessarilly (1.2) holds. Using
Orlicz-Sobolev embeddings (see Lemma 2.8), and the regularity of $\mathtt{K}$
(see Theorem 5.29), we have that

$\|\mathtt{K}(g)\|_{A}\leq C\|\mathtt{K}(g)\|_{W^{2,\tilde{B}}}\leq
C\|g\|_{\tilde{B}},\quad\|\mathtt{K}(f)\|_{B}\leq
C\|\mathtt{K}(f)\|_{W^{2,\tilde{A}}}\leq C\|f\|_{\tilde{A}}.$

On the one hand since (2.17), and the above

$\int_{\Omega}A(\mathtt{K}(g))\,dx\leq C\,\|\mathtt{K}(g)\|_{A}^{p+1}\leq
C\|g\|_{\tilde{B}}^{p+1},\quad\int_{\Omega}B(\mathtt{K}(f))dx\leq
C\|\mathtt{K}(f)\|_{B}^{q+1}\leq C\|f\|_{\tilde{A}}^{q+1}.$

whenever $\|\mathtt{K}(g)\|_{A}\leq 1,\ \|\mathtt{K}(f)\|_{B}\leq 1.$

On the other hand since (2.15) in Lemma 2.3, and Lemma 2.6,

$\int_{\Omega}{\widetilde{A}}(f)\,dx\geq
C\,\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}},\quad\int_{\Omega}{\widetilde{B}}(g)\,dx\geq
C\,\|g\|_{\tilde{B}}^{\frac{q+1}{q}},$

whenever $\|f\|_{(\tilde{A})}\leq 1,\ \|g\|_{\tilde{B}}\leq 1.$

Then, for all $(f,g)\in X$ with $0\not=\|(f,g)\|_{X}$ sufficiently small

$2J(f,g)\geq
d_{1}\|f\|_{(\tilde{A})}^{\,\frac{p+1}{p}}-d_{2}\|f\|_{\tilde{A}}^{q+1}+\bigg{(}d_{3}\|g\|_{\tilde{B}}^{\frac{q+1}{q}}-d_{4}\|g\|_{\tilde{B}}^{p+1}\bigg{)}>0.$

Step 2. There exists $(f_{1},g_{1})\in X$ such that $J(f_{1},g_{1})<0$.
Indeed, choose $s\in(1/p,q)$ (this is possible as $qp>1$, which follows from
(1.2) or (1.3)). We shall show that there exists $t\in\mathbb{R}$ big enough
such that $J(t\phi_{1},t^{s}\phi_{1})<0$. Indeed, since
$\mathtt{K}(\phi_{1})=\phi_{1}/\lambda_{1},$ we can write

$\displaystyle J(t\phi_{1},t^{s}\phi_{1})=$
$\displaystyle\int\limits_{\Omega}\widetilde{A}(t\phi_{1})\,dx+\int\limits_{\Omega}\widetilde{B}(t^{s}\phi_{1})\,dx-\frac{t^{1+s}}{\lambda_{1}}\int\limits_{\Omega}(\phi_{1})^{2}\,dx.$

Since Lemma 2.1 and Example 5.18,
$\widetilde{A}(t\phi_{1})\sim{(t\phi_{1})^{\frac{p+1}{p}}(\ln(t\phi_{1}))^{\frac{\alpha}{p}}}$,
and
$\widetilde{B}(t^{s}\phi_{1})\sim{(t^{s}\phi_{1})^{\frac{q+1}{q}}(\ln(t^{s}\phi_{1}))^{\frac{\beta}{q}}}$.
Since $\frac{p+1}{p}<1+s$, and also $s\frac{q+1}{q}<1+s$, hence
$J(t\phi_{1},t^{s}\phi_{1})\to-\infty$ as $t\to\infty.$

##  4\. A non-existence result

The following lemma provides a Rellich-Pohozaev-Mitidieri type identity, see
[21, 22, 18].

######  Lemma 4.1.

(Rellich-Pohozaev-Mitidieri type identity)
Let $u$ and $v$ be in $C^{2}(\bar{\Omega})$, where $\Omega$ is a $C^{1}$
domain in $\mathbb{R}^{N}$, and $u=v=0$ on $\partial{\Omega}$. Then

$\displaystyle\displaystyle\int_{\Omega}\Delta u\;(x\cdot\nabla v)+\Delta
v\;(x\cdot\nabla u)$ $\displaystyle=(N-2)\int_{\Omega}(\nabla u\cdot\nabla
v)+\int_{\partial\Omega}\frac{\partial u}{\partial\nu}\;(x\cdot\nabla v)$
$\displaystyle+\int_{\partial\Omega}\frac{\partial
v}{\partial\nu}\;(x\cdot\nabla u)-\int_{\partial\Omega}(\nabla u,\nabla
v)\;(x\cdot\nu),$

where $\nu$ denotes the exterior normal, and $(x\cdot\nu)$ denotes the inner
product.

For the proof we refer to [18].

By definition, a solution $(u,v)$ of (1.1) belongs to
$\big{(}W_{0}^{1,{\widetilde{A}}}(\Omega)\cap
W^{2,{\tilde{A}}}(\Omega)\big{)}\times\big{(}W_{0}^{1,{\widetilde{B}}}(\Omega)\cap
W^{2,{\tilde{B}}}(\Omega)\big{)}$. Since an estimate of Brezis-Kato [4], based
on Moser iteration techniques [19], $u,v\in C^{1,\nu}(\overline{\Omega})\cap
W^{2,s}(\Omega)$ for any $\nu<1,\ s<+\infty$, see for instance [20, Lemma 2.1
(iv)]. If, in particular $\Omega$ is $C^{2,\mu}$, then $u,v\in
C^{2,\mu}(\overline{\Omega}).$

######  Theorem 4.2 (Non existence of non-negative non-trivial classical
solutions).

Let $(u,v)\in\big{(}C^{2}(\overline{\Omega})\big{)}^{2}$ be a pair of non-
negative solutions to the problem (1.1).

Assume that

(4.1) $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$

Assume also that $\Omega$ is strongly star-shaped with respect to $0$ (the
inner product $(x\cdot\nu(x))\geq 0$ for all $x\in\partial\Omega$, and
$(x\cdot\nu(x))\not\equiv 0$) and that $\partial\Omega$ is $C^{1}$. If
$\alpha,\beta\leq 0$, then

$u\equiv 0,\quad v\equiv 0,\qquad\mbox{in}\quad\Omega.$

Assume now that

(4.2) $\frac{1}{p+1}+\frac{1}{q+1}<\frac{N-2}{N}.$

and that $\Omega$ is star-shaped with respect to $0$ ($(x\cdot\nu(x))\geq 0$
for all $x\in\partial\Omega$) and that $\partial\Omega$ is $C^{1}$. If
$\alpha,\beta\leq 0$, then

$u\equiv 0,\quad v\equiv 0,\qquad\mbox{in}\quad\Omega.$

######  Remark 4.3.

In particular, for the single equation, i.e. $p=q$ and $\alpha=\beta\leq 0$,
we have the following:

Assume that $p=2^{*}-1$, and also that $\Omega$ is strongly star-shaped with
respect to $0$. If $\alpha\leq 0$, then

$u\equiv 0,\qquad\mbox{in}\quad\Omega.$

Assume now that $p>2^{*}-1$, and that $\Omega$ is star-shaped with respect to
$0$. If $\alpha\leq 0$, then

$u\equiv 0,\qquad\mbox{in}\quad\Omega.$

###### Proof of Theorem 4.2.

If we set $W(u,v):=A(u)+B(v)$ then $W_{u}=a(u)$ and $W_{v}=b(v)$. Therefore,
for solutions $u>0$ and $v>0$ of (1.1),

$\displaystyle\displaystyle\int_{\Omega}-\left[\Delta u\;(x\cdot\nabla
v)\right.+\left.\Delta v\;(x\cdot\nabla
u)\right]=\int_{\Omega}\sum_{j}x_{j}\left(\frac{\partial W}{\partial
v}\frac{\partial v}{\partial x_{j}}+\frac{\partial W}{\partial
u}\frac{\partial u}{\partial
x_{j}}\right)=\int_{\Omega}\sum_{j}x_{j}\frac{\partial W}{\partial x_{j}}$
$\displaystyle\qquad=-N\int_{\Omega}W+\int_{\Omega}{\rm{div}}(W\,x)=-N\int_{\Omega}\left[A(u)+B(v)\right]+\int_{\partial\Omega}(x\cdot\nu)\;W(u,v),$

and

(4.3) $\displaystyle\int_{\Omega}\nabla u\nabla
v=\int_{\Omega}ua(u)=\int_{\Omega}vb(v).$

Applying Lemma 4.1 (Pohozaev-Rellich-Mitidieri type identity) we get that

(4.4) $\displaystyle\displaystyle
N\int_{\Omega}\left[A(u)+B(v)\right]-{(N-2)}\int_{\Omega}\,u\,a(u)$
$\displaystyle=\int_{\partial\Omega}(x\cdot\nu)\;W(u,v)-\int_{\partial\Omega}(\nabla
u\cdot\nabla v)\;(x\cdot\nu)+\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}\;(x\cdot\nabla v)+\int_{\partial\Omega}\frac{\partial
v}{\partial\nu}\;(x\cdot\nabla u).$

On the one hand, $u=v=0$ on the boundary, so the first integral in the r.h.s.
vanishes. Moreover, since $u=0$ on $\partial\Omega$, the tangential component
of $\nabla u$ vanishes and $\nabla u$ is parallel to the normal vector
$\nu(x)$ at each point $x\in\Omega$, in other words $\nabla u(x)=\pm|\nabla
u(x)|\,\nu(x)$.

On the other hand, since Hopf’s Lemma, $\frac{\partial u}{\partial\nu}(x)<0$
for all $x\in\partial\Omega$, so $\nabla u(x)\cdot\nu=\frac{\partial
u}{\partial\nu}(x)=-|\nabla u(x)|$, and consequently, $\nabla u(x)=-|\nabla
u(x)|\,\nu(x).$ Likewise, $\frac{\partial v}{\partial\nu}(x)=-|\nabla v(x)|$,
and $\nabla v(x)=-|\nabla v(x)|\,\nu(x).$ Consequently, the r.h.s. of (4.4) is
reduced to

$\displaystyle\Big{(}-|\nabla u(x)||\nabla v(x)|+|\nabla u(x)||\nabla
v(x)|+|\nabla u(x)||\nabla v(x)|\Big{)}\,(x\cdot\nu)$
$\displaystyle\qquad=|\nabla u(x)||\nabla v(x)|\,(x\cdot\nu)=\frac{\partial
u}{\partial\nu}\frac{\partial v}{\partial\nu}\,(x\cdot\nu),$

hence

(4.5) $\displaystyle\displaystyle
N\int_{\Omega}\left[A(u)+B(v)\right]-{(N-2)}\int_{\Omega}\,u\,a(u)=\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}\frac{\partial v}{\partial\nu}\,(x\cdot\nu).$

Integrating by parts $A(t)=\int_{0}^{t}\frac{s^{p}}{(\ln(e+s))^{\alpha}}\,ds,$
we have that

(4.6) $\displaystyle
A(t)-\frac{1}{p+1}ta(t)=\frac{\alpha}{p+1}\int^{t}_{0}\frac{s^{p+1}}{\big{(}\ln(e+s)\big{)}^{\alpha+1}}\,\frac{ds}{e+s}=\frac{\alpha}{p+1}\int^{t}_{0}\frac{a(s)}{\ln(e+s)}\frac{s}{e+s}\,ds.$

Likewise

(4.7) $\displaystyle
B(t)-\frac{1}{q+1}\,t\,b(t)=\frac{\beta}{q+1}\int^{t}_{0}\frac{b(s)}{\ln(e+s)}\frac{s}{e+s}\,ds.$

Substituting (4.6)-(4.7) into (4.5), and since (4.3),

$\displaystyle\left[N\left(\frac{1}{p+1}+\frac{1}{q+1}\right)-(N-2)\right]\int_{\Omega}\,u\,a(u)+\frac{\alpha
N}{p+1}\int_{\Omega}\int_{0}^{u(x)}\frac{a(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\,\,dx$
(4.8) $\displaystyle\qquad+\frac{\beta
N}{q+1}\int_{\Omega}\int^{v(x)}_{0}\frac{b(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\,\,dx=\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}\frac{\partial v}{\partial\nu}\,(x\cdot\nu).$

Introducing (4.1) in (4), we deduce

(4.9) $\displaystyle\frac{\alpha
N}{p+1}\int_{\Omega}\int_{0}^{u(x)}\frac{a(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\,\,dx$
$\displaystyle\qquad+\frac{\beta
N}{q+1}\int_{\Omega}\int^{v(x)}_{0}\frac{b(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\,\,dx=\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}\frac{\partial v}{\partial\nu}\,(x\cdot\nu).$

Assume $\alpha,\beta\leq 0$, if $u\not\equiv 0$, then $v\not\equiv 0$, and the
l.h.s. of (4.9) is non-positive. On the other hand, if $\Omega$ is strongly
star-shaped with respect to $0$, the r.h.s. of (4.9) is strictly positive,
giving a contradiction, and necessarily, $u\equiv 0$, and $v\equiv 0$.

Assume $u\not\equiv 0$, then $v\not\equiv 0$, and introducing (4.2) in (4), we
deduce

(4.10) $\displaystyle\frac{\alpha
N}{p+1}\int_{\Omega}\int_{0}^{u(x)}\frac{a(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\,\,dx$
$\displaystyle\qquad+\frac{\beta
N}{q+1}\int_{\Omega}\int^{v(x)}_{0}\frac{b(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\,\,dx>\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}\frac{\partial v}{\partial\nu}\,(x\cdot\nu).$

Assume $\alpha,\beta\leq 0$, then the l.h.s. of (4.10) is non-positive. On the
other hand, if $\Omega$ is star-shaped with respect to $0$, the r.h.s. of
(4.10) is strictly positive, giving a contradiction, and necessarily, $u\equiv
0$, and $v\equiv 0$.

∎

We point out that, in general if $p$ and $q$ lie on the critical hyperbola, it
would be interesting to check if the condition

$\frac{\alpha}{p+1}+\frac{\beta}{q+1}\leq 0$

ensures that

$\displaystyle\alpha\left(\int_{\Omega}\left[\int_{0}^{u(x)}\frac{a(s)}{\ln(e+s)}\frac{s}{e+s}\,ds-\int^{v(x)}_{0}\frac{b(s)}{\ln(e+s)}\frac{s}{e+s}\,ds\right]\,dx\right)\leq
0$

or equivalently

$N\int_{\Omega}\left[A(u)+B(v)\right]-{(N-2)}\int_{\Omega}\,\left[\theta\,u\,a(u)+(1-\theta)\,v\,b(v)\right]\leq
0.$

##  5\. On Orlicz spaces

Let us sumarise in this section some basics results on Orlicz spaces (cf.
([1],[15] and [23]).

###  5.1. ${\mathcal{N}}$-functions

######  Definition 5.1.

(${\mathcal{N}}$-functions) A function $H:[0,\infty)\to[0,\infty)$ is said to
be a ${\mathcal{N}}$-function if an only if $H$ is continuous, convex,
$H(t)=0$ if and only if $t=0$ and

$\lim_{t\to 0}H(t)/t=0,\quad\lim_{t\to+\infty}H(t)/t=+\infty.$

Any ${\mathcal{N}}$-function is extended to $\mathbb{R}$ as an even function.
We denote by $h$ the left derivative of $H$.

###  5.2. Orlicz classes, Orlicz spaces and the Luxembourg norm

Associated to the ${\mathcal{N}}$-function $H$ we have the following class of
functions. Let $\Omega\subset\mathbb{R}^{N}$ be here an arbitrary open set.

For our application, $\Omega\subset\mathbb{R}^{N}$ is a bounded set.

######  Definition 5.2.

(Orlicz class) The Orlicz class $K^{H}(\Omega)$ is defined by

$K^{H}(\Omega):=\left\\{u:\Omega\to\mathbb{R}:u\hbox{ is measurable and
}\int_{\Omega}H\big{(}u(x)\big{)}\,dx<+\infty\right\\}.$

Orlicz classes are convex sets, but in general not linear spaces.

######  Definition 5.3.

(Orlicz space). We say that $u\in L^{H}(\Omega)$ if and only if there exists a
constant $c>0$ such that $cu\in K^{H}(\Omega).$ $L^{H}(\Omega)$ is a vector
space, and it is called the Orlicz space associated to $H$.

Let us recall the following result:

######  Lemma 5.4.

Let $H$ be an ${\mathcal{N}}$-function. Then,

  1. (i)

$L^{H}(\Omega)=K^{H}(\Omega)$ if and only if $H$ satisfies the so called
$\Delta_{2}$-condition:

$\exists k\geq 1,\quad H(2t)\leq kH(t)\quad\forall t\geq 0.$

  2. (ii)

If $\Omega$ is bounded then $L^{H}(\Omega)=K^{H}(\Omega)$ if and only if $H$
satisfies the so called $\Delta_{2}$-condition at infinity :

(5.1) $\exists k\geq 1,\,\exists t_{0}\geq 0,\quad H(2t)\leq kH(t)\quad\forall
t\geq t_{0}.$

  3. (iii)

Assume $H$ is derivable and its derivative $h$ is continuous and strictly
increasing. If

(5.2) $\lim_{t\to+\infty}\frac{th(t)}{H(t)}=\alpha\in(1,+\infty)$

then both $H$ and $\widetilde{H}$ satisfy the $\Delta_{2}$-condition at
infinity, where $\widetilde{H}$ is defined below in Definition 5.6.111[15,
Theorem 4.2] clarify that $H$ and $\widetilde{H}$ satisfies the
$\Delta_{2}$-condition if and only if $H$ satisfies the $\Delta_{2}$-condition
and the $\nabla_{2}$-condition, in other words, there exist numbers $h>1$ and
$t_{1}\geq 0$ such that  $H(t)\leq\frac{1}{2h}H(ht)\quad\text{for}\ t\geq
t_{1}.$

###### Proof.

(i) It follows from [15, Theorem 8.2], (ii) from [23, Theorem 2] and (iii)
from [15, Theorem 4.1 and Theorem 4.3]. ∎

We can provide the Orlicz space $L^{H}(\Omega)$ with the Luxemburg norm :

(5.3) $\|u\|_{(H)}:=\inf\left\\{\lambda>0\ :\
\int\limits_{\Omega}H\left(\frac{u(x)}{\lambda}\right)\,dx\leq 1\right\\}.$

######  Lemma 5.5.

  1. (i)

$\left(L^{H}(\Omega),\|\cdot\|_{(H)}\right)$ is a Banach space.

  2. (ii)

If $u\in L^{H}(\Omega)$, $u\not\equiv 0$, then
$\int_{\Omega}H\left({\frac{u}{\|u\|_{(H)}}}\right)\,dx\leq 1$.

  3. (iii)

$\|u\|_{(H)}\leq\max\\{\int_{\Omega}H(u)\,dx,\,1\\}$.

  4. (iv)

If $H$ satisfies the $\Delta_{2}$-condition then, for all $u\not=0$, $u\in
L^{H}(\Omega)$, it holds

$\int_{\Omega}H\left({\frac{u}{\|u\|_{(H)}}}\right)\,dx=1.$

If $\Omega$ is bounded, the conclusion holds if $H$ satisfies the
$\Delta_{2}$\- condition at infinity.

###### Proof.

The proof of (i) can be found in [23][Theorem 10, p. 67]. The proof of (ii)
and (iii) follow trivially form the definition of the Luxembourg norm. The
result of (iv) is proved in [23][Proposition 6, p. 77]. ∎

###  5.3. The Young-conjugate of a ${\mathcal{N}}$-function

######  Definition 5.6.

Let $H$ be an $\mathcal{N}$-function and denote $\widetilde{H}$ its Legendre
transform, i.e.

$\widetilde{H}(s):=\sup\limits_{\sigma\in\mathbb{R}}\\{s\sigma-H(\sigma)\\}.$

Then, $\widetilde{H}$ is called the Young-conjugate of $H$.

Some authors call $\widetilde{H}$ the complementary function of $H$, see [15,
p. 11-13]. We denote by $\tilde{h}$ the right derivative of $\widetilde{H}$.

######  Remark 5.7.

If $h$ is strictly increasing, then $\tilde{h}(t)=h^{-1}(t)$.

The following properties are trivial to prove.

######  Proposition 5.8.

Let $H$ be an $\mathcal{N}$-function.

  1. (i)

$\widetilde{(\widetilde{H})}=H$.

  2. (ii)

$\widetilde{H}$ is an ${\mathcal{N}}$-function.

  3. (iii)

$H$ and $\widetilde{H}$ satisfy the Young inequality :

$\forall s,t\in(0,+\infty),\quad st\leq H(s)+\widetilde{H}(t)$

and the equality holds if and only if $t=h(s)$ or $s=\tilde{h}(t).$

  4. (iv)

The following 1st Hölder’s inequality holds :

$\forall f\in L^{H}(\Omega),\,\forall g\in
L^{\tilde{H}}(\Omega),\quad\int\limits_{\Omega}|f(x)g(x)|\,dx\leq
2\|f\|_{(H)}\|g\|_{(\tilde{H})}.$

###### Proof.

The proof of (i)-(ii) can be found for instance in [15][Chapter 1]. For the
proof of (iv), see [23, Proposition 1 and Remark in p. 58]. ∎

###  5.4. Dual norm, the dual of an Orlicz space and reflexivity

######  Definition 5.9.

Let $H$ be an $\mathcal{N}$-function. For all $u\in L^{H}(\Omega)$ we define
the dual norm of $u$ or Orlicz norm of $u$ as

(5.4) $\|u\|_{H}:=\sup\left\\{\int_{\Omega}uv\,dx:\,\|v\|_{(\tilde{H})}\leq
1\right\\}.$

We have

######  Proposition 5.10.

  1. (i)

For all $u\in L^{H}(\Omega)$

$\|u\|_{(H)}\leq\|u\|_{H}\leq 2\|u\|_{(H)}.$

  2. (ii)

2nd Hölder’s inequality :

(5.5) $\forall u\in L^{H}(\Omega),\,\forall v\in
L^{\tilde{H}}(\Omega),\quad\int\limits_{\Omega}|u(x)v(x)|\,dx\leq\|u\|_{H}\|v\|_{(\tilde{H})}.$

  3. (iii)

For all $u\in L^{H}(\Omega)$ , $u\not=0$,

(5.6) $\int_{\Omega}H\bigg{(}\frac{u}{\|u\|_{H}}\bigg{)}dx\leq 1.$

  4. (iv)

If $\widetilde{H}$ satisfies the $\Delta_{2}$-condition then the dual space
$\big{(}L^{H}(\Omega),\|\cdot\|_{H}\big{)}^{\prime}$ coincides with
$\big{(}L^{\tilde{H}}(\Omega),\|\cdot\|_{(\tilde{H})}\big{)}$.

  5. (v)

$L^{H}(\Omega)$ is a reflexive Banach space if and only if $H$ and
$\widetilde{H}$ satisfy the $\Delta_{2}$-condition.

If $\Omega$ is bounded, the conclusions (iv) and (v) hold if $H$ and
$\tilde{H}$ satisfy the $\Delta_{2}$\- condition at infinity.

###### Proof.

The proof of (i), (ii), (iii) follows from [15, Lemma 9.2 and Theorem 9.3, p.
74]. For the proof of (iv) and (v) see [23][Theorem 10, p. 113]. ∎

######  Remark 5.11.

Since Proposition 5.10 (ii) and (i), 2nd Hölder’s inequality can be written:

$\forall u\in L^{H}(\Omega),\,\forall v\in
L^{\tilde{H}}(\Omega),\quad\int\limits_{\Omega}|u(x)v(x)|\,dx\leq\|u\|_{H}\|v\|_{\tilde{H}}.$

Observe that Proposition 5.10 (iv) is a Riesz representation theorem for
Orlicz spaces, whenever $\widetilde{H}$ satisfies the $\Delta_{2}$-condition.

######  Definition 5.12.

We say that the sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ converges in $H$-mean
to $u$ whenever

$\lim_{n\rightarrow\infty}\int_{\Omega}H\Big{(}|u_{n}(x)-u(x)|\Big{)}\,dx=0.$

The folowing Theorem states that convergence in $H$-mean is equivalent to the
convergence with respect to the Orlicz-norm, provided that the
$\Delta_{2}$-condition is satisfied, see [15, Theorem 9.4].

######  Theorem 5.13.

Let $H$ be an $\mathcal{N}$-function satisfying the $\Delta_{2}$-condition.
Then, the convergence in $H$-mean is equivalent to the convergence with
respect to the $\|\cdot\|_{H}$ norm.

###  5.5. Comparison of ${\mathcal{N}}$-functions

It is possible to consider different partial ordering relations between
${\mathcal{N}}$-functions, and they imply continuous embedding into Orlicz
spaces.

First let us introduce the following ordering relations. The following partial
ordering relation between functions is involved in embedding theorems between
Orlicz space associated with different ${\mathcal{N}}$-functions:

######  Definition 5.14.

Let $H$ and $H_{1}$ be two ${\mathcal{N}}$-functions.

  1. (i)

The function $H_{1}$ is said to dominate the function $H$ globally
(respectively near infinity), denoted by $H\prec H_{1},$ if there exists a
positive constant $c$ such that

$H(s)\leq H_{1}(cs),\qquad\mbox{for}\quad s\geq 0\qquad(s\geq s_{0}).$

  2. (ii)

The functions $H$ and $H_{1}$ are called equivalent globally (near infinity),
denoted by $H\sim H_{1},$ if each dominates the other globally (near
infinity).

  3. (iii)

If for every $c>0$, there exists a number $s_{c}\geq 0$ such that

$H(s)\leq H_{1}(cs),\qquad\mbox{for}\quad s\geq 0\qquad(s\geq s_{c}),$

then $H$ is said to increase essentially more slowly than $H_{1}$ (at
infinity), and in this case we write $H\prec\prec H_{1}$.

######  Remark 5.15.

There are different partial ordering relations between
${\mathcal{N}}$-functions.

  1. (i)

It is said that $H$ growths more slowly than $H_{1}$ at infinity, and it is
written $H\ll H_{1}$, if and only if

$\forall r>0,\quad\lim_{s\to+\infty}\frac{H(rs)}{H_{1}(s)}=0.$

In [23, p. 15, Definition 1, (ii)] is said that $H_{1}$ is essentially
stronger than $H$. This definition is equivalent to 5.15 (i). See [23, p. 16,
Theorem 2.b, (i) $\iff$ (v)].

  2. (ii)

In particular, if

$\exists c>0,\quad\lim_{s\to+\infty}\frac{H(s)}{H_{1}(s)}=c,$

then $H$ is equivalent to $H_{1}$ at infinity.

######  Proposition 5.16.

Let $H$ and $H_{1}$ be two ${\mathcal{N}}$-functions.

The continuous embedding $L^{H_{1}}(\Omega)\to L^{H}(\Omega)$ holds if either
$H_{1}$ dominates $H$ globally, or $|\Omega|<\infty$ and $H_{1}$ dominates $H$
near infinity.

Moreover, $L^{H}(\Omega)\subset L^{H_{1}}(\Omega)$ if and only if there exists
a constant $C>0$ such that $\|u\|_{H_{1}}\leq C\|u\|_{H}$ for all $u\in
L^{H}(\Omega).$

###### Proof.

The proof of the first statement, follows directly from definition 5.3 of
Orlicz space, and from definition 5.14(i).

For the proof of the second estatement, see [16, Theorem 4, p. 51]. ∎

######  Proposition 5.17.

Let $H$ and $H_{1}$ be two ${\mathcal{N}}$-functions. If $H\sim H_{1}$ then
$L^{H}(\Omega)$ is isomorphic to $L^{H_{1}}(\Omega)$.

######  Example 5.18.

[Some useful facts] Let $p>0$, $\alpha<p$, and

$a(t):=\frac{t^{p}}{(\ln(e+t))^{\alpha}},\ t\geq 0.$

Then, we have

  1. (i)

$A(s)\sim s^{p+1}(\ln s)^{-\alpha}$,

  2. (ii)

$\tilde{a}(s):=a^{-1}(s)\sim{s^{\frac{1}{p}}(\ln s)^{\frac{\alpha}{p}}}$,

  3. (iii)

$\widetilde{A}(s)\sim{s^{\frac{p+1}{p}}(\ln s)^{\frac{\alpha}{p}}}$,

  4. (iv)

$\widetilde{A}^{\,-1}(s)\sim s^{\frac{p}{p+1}}(\ln s)^{-\frac{\alpha}{p+1}}$.

See Lemma 2.1.

###  5.6. The Orlicz-Sobolev spaces $W_{0}^{1,H}(\Omega)$ and
$W^{m,H}(\Omega)$.

Let $H$ be an $\mathcal{N}$-function and let $m\in\mathbb{N}^{*}$.

######  Definition 5.19.

  1. (i)

The space $W^{m,H}(\Omega)$ is defined as

$W^{m,H}(\Omega):=\big{\\{}u:\Omega\to\mathbb{R}\,:D^{\alpha}u\in
L^{H}(\Omega)\quad\forall\,|\alpha|\in\\{0,1,,\dots,m\\}\big{\\}}$

where $D^{\alpha}$ stands for the weak partial derivative of $u$.

  2. (ii)

The Luxemburg norm for the elements $u$ of this space is

$\|u\|_{m,(H)}:=\max_{0\leq|\alpha|\leq m}\|D^{\alpha}u\|_{(H)}$

and $\big{(}W^{m,H}(\Omega),\|\cdot\|_{m,(H)}\big{)}$ is a Banach space.

  3. (iii)

The space $W_{0}^{1,H}(\Omega)$ is defined as

$W_{0}^{1,H}(\Omega):=\overline{C_{0}^{\infty}(\Omega)}^{\|\cdot\|_{1,(H)}}.$

  4. (iv)

The Orlicz norm for $u\in W_{0}^{1,H}(\Omega)$ is defined by

$\|u\|_{1,H}:=\sup\left\\{\int_{\Omega}uv\,\,dx\,:v\in
W_{0}^{1,H}(\Omega),\;\|v\|_{1,(H)}\leq 1\right\\}.$

The Orlicz norm is equivalent to the Luxemburg norm, see Proposition 5.10(i).

###  5.7. Optimal Embedding Theorems for Orlicz-Sobolev spaces

We present in this section some known results concerning the optimal
Orlicz–Sobolev embedding. For that purpose, we define for any any
${\mathcal{N}}$-function $H$, the auxiliary function

(5.7)
$\Phi_{H}(s):=\int_{0}^{s}\frac{\widetilde{H}(\tau)}{\tau^{1+N^{\prime}}}\,d\tau$

where $N^{\prime}=\frac{N}{N-1}$. We will denote by $\to$ a continuous
embedding, and by $\hookrightarrow$ a compact embedding.

######  Remark 5.20.

We can always assume that $\Phi_{H}$ is well defined. Indeed, since $\Omega$
is of finite measure (it is bounded), we can assume without loss of generality
that

$\int_{0}\frac{\tilde{H}(\tau)}{\tau^{1+N^{\prime}}}d\tau<\infty$

otherwise $H$ can be replaced by any ${\mathcal{N}}$-function which is
equivalent to the original one near infinity and makes the previous integral
converge. Such a replacement does not affect the result since the
corresponding Orlicz-Sobolev norm is equivalent to the original one.

It turns out that one finds different optimal embeddings depending on whatever

(5.8)
$i_{H}:=\int^{\infty}\frac{\widetilde{H}(\tau)}{\tau^{1+N^{\prime}}}d\tau$

is finite or not. Notice that if $i_{H}=+\infty$, in particular $\Phi_{H}$ is
strictly increasing (at least for $s$ large) and $\Phi_{H}^{-1}$ is well
defined. Consequently we distinguish two cases :

Case 1 : $i_{H}<+\infty$. The following Theorem is a compact Orlicz–Sobolev
embedding for the spaces $W_{0}^{1,H}(\Omega)$ and $W^{1,H}(\Omega)$, in terms
of continuous bounded functions, see [12], and [6, Corollary 1]. Recall that a
bounded open set $\Omega$ is called strongly Lipschitz if, for each
$x\in\partial\Omega$, there exist a neighbourhood $U_{x}$ of $x$, a coordinate
system $(y_{1},...,y_{N})$ centered at $x$ and a Lipschitz continuous function
$\phi$ of $(y_{1},...,y_{N-1})$ such that

$\Omega\cap U_{x}=\\{(y_{1},...,y_{N}):\ y_{N}>\phi(y_{1},...,y_{N-1})\\}.$

######  Theorem 5.21.

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$ be a bounded open set. Let $H$ be
an $\mathcal{N}$-function satisfying $i_{H}<+\infty$, where $i_{H}$ is defined
in (5.8). Let $C_{b}(\Omega)$ be the space of continuous bounded functions on
$\Omega$.

  1. (1)

Then, the embedding

$W^{1,H}_{0}(\Omega)\hookrightarrow C_{b}(\Omega)$

is compact.

  2. (2)

If in addition $\Omega$ has the strong Lipschitz property then, the embedding

$W^{1,H}(\Omega)\hookrightarrow C_{b}(\Omega)$

is compact.

Case 2 : $i_{H}=+\infty$. The following theorem gives the optimal
Orlicz–Sobolev continuous embedding for the space $W^{1,H}_{0}(\Omega)$, in
terms of Orlicz spaces, see [7, Theorem 1].

######  Theorem 5.22.

Let $N\geq 2$. Let $H$ be any ${\mathcal{N}}$-function and let $\Phi_{H}$ be
the auxiliary function defined in (5.7) Assume that $i_{H}=+\infty$, where
$i_{H}$ is defined in (5.8). Define for all $s\geq 0$

(5.9)
$H^{*}(s)=H_{N}^{*}(s)=\int_{0}^{s}t^{N^{\prime}-1}\left(\Phi_{H}^{-1}\big{(}t^{N^{\prime}}\big{)}\right)^{N^{\prime}}dt$

Then $H^{*}$ is an ${\mathcal{N}}$-function, and the following continuous
embedding holds:

(5.10) $W_{0}^{1,H}(\Omega)\to L^{H^{*}}(\Omega).$

Furthermore $L^{H^{*}}(\Omega)$ is the smallest Orlicz space that renders
(5.10) true.

The following theorem is a also a continuous Orlicz–Sobolev embedding, this
time for the space $W^{1,H}(\Omega)$, in terms of Orlicz spaces, see [7,
Theorem 2]. We will say that $\Omega\subset\mathbb{R}^{N}$ satisfies the cone
property, if there exists a cone $\Sigma$ such that for any $x\in\Omega$, the
set $\Omega$ contains a cone congruent with $\Sigma$ and whose vertex is $x$.

######  Theorem 5.23.

Let $N\geq 2$. Assume that $\Omega\subset\mathbb{R}^{N}$ is any open bounded
connected set, satisfying the cone property. Let $H$ be any
$\mathcal{N}$-function, and let $H^{*}$ be the function defined by (5.9).
Assume that $i_{H}=+\infty$, where $i_{H}$ is defined in (5.8). Then, the
following holds:

  1. (i)

There exists a constant $C$, depending only on $H$, $|\Omega|$ and $N$, such
that

$\|u-u_{\Omega}\|_{L^{H^{*}}(\Omega)}\leq C\|\nabla u\|_{(H)}\qquad\text{for
all }u\in W^{1,H}(\Omega).$

Here,

$u_{\Omega}:=\frac{1}{|\Omega|}\int_{\Omega}u(x)\,\,dx$

is the mean value of $u$ over $\Omega$.

  2. (ii)

The continuous embedding

$W^{1,H}(\Omega)\to L^{H^{*}}(\Omega)$

holds, where possibly $H^{*}$ is replaced at zero, in the sens of Remark 5.20.

Finally we give in Theorem 5.24 below a compact Orlicz–Sobolev embedding for
the space $W^{1,H}(\Omega)$, this time in terms of Orlicz spaces, see [7,
Theorem 3].

######  Theorem 5.24.

Let $N\geq 2$ and let $\Omega$ be any open, bounded, connected set, and
satisfying the cone property. Let $H$ be any ${\mathcal{N}}$\- function.
Assume that $i_{H}=+\infty$, where $i_{H}$ is defined in (5.8) and $H_{1}$ a
${\mathcal{N}}$\- function increasing essentially more slowly near infinity
than $H^{*}$ defined by (5.9). Then the embedding

$W^{1,H}(\Omega)\hookrightarrow L^{H_{1}}(\Omega)$

is compact.

######  Example 5.25.

Some useful continuous and compact embeddings

Consider any ${\mathcal{N}}$-function $H$ such that

$H(s)\sim s^{p}\big{[}\log(s)\big{]}^{\alpha}$

where either $p>1$ and $\alpha\in\mathbb{R}$ or $p=1$ and $\alpha>0.$

Let $\Omega\subset\mathbb{R}^{N}$ be an bounded open set.

Case 1.  The following compact embedding holds :

$W^{1,H}_{0}(\Omega)\hookrightarrow C_{b}(\Omega)\qquad\mbox{if
either}\quad\begin{cases}p>N&\qquad\mbox{and}\quad\alpha\in\mathbb{R},\\\
\text{or}\quad p=N&\qquad\mbox{and}\quad\alpha>N-1.\end{cases}$

Case 2.  The following continuous embedding holds :

$W^{1,H}_{0}(\Omega)\to L^{H^{*}}(\Omega)\qquad\mbox{if}\quad p\leq N,$

see Theorem 5.22, where

(5.11)
$H^{*}(s)\sim\begin{cases}\Big{(}s^{p}\big{[}\log(s)\big{]}^{\alpha}\Big{)}^{\frac{N}{N-p}}&\qquad\mbox{if}\quad
1\leq p<N,\\\\[5.69054pt] e^{s^{N/(N-1-\alpha)}}&\qquad\mbox{if}\quad p=N,\
\alpha<N-1,\\\\[5.69054pt] e^{e^{s^{N^{\prime}}}}&\qquad\mbox{if}\quad p=N,\
\alpha=N-1.\end{cases}$

By Theorem 5.23, the same embeddings are true with $W^{1,H}(\Omega)$ replacing
$W^{1,H}_{0}(\Omega)$, provided that $\Omega\subset\mathbb{R}^{N}$ has finite
measure and satisfies the cone property.

Moreover, if $p\leq N,$ since Theorem 5.24

$W^{1,H}(\Omega)\hookrightarrow L^{H_{1}}(\Omega)\qquad\mbox{for any}\quad
H_{1}\prec\prec H^{*}.$

Notice that if $\alpha=0$, (5.11) agrees with Sobolev’s theorem when $p\neq
N$, and with Trudinger’s theorem when $p=N$.

######  Example 5.26.

Consider a ${\mathcal{N}}$-functions $\ H(s)\ $ which are equivalent near
infinity to

$s^{p}\Big{(}\log\big{[}\log(s)\big{]}\Big{)}^{\alpha},\qquad\mbox{where
either}\quad\begin{cases}p>1&\text{ and }\alpha\in\mathbb{R},\\\
\text{or}\quad p=1&\text{ and }\alpha>0.\end{cases}$

Then, Theorem 5.22, and Theorem 5.21 imply that

$W^{1,H}(\Omega)\hookrightarrow C_{b}(\Omega)\qquad\mbox{if}\quad p>N,$

and

$W^{1,H}(\Omega)\to L^{H^{*}}(\Omega)\qquad\mbox{if}\quad p\leq N,$

where

$H^{*}(s)\sim\begin{cases}\Big{(}s^{p}\big{[}\log\log(s)\big{]}^{\alpha}\Big{)}^{\frac{N}{N-p}}&\qquad\mbox{if}\quad
1\leq p<N,\\\\[8.53581pt]
e^{\left(s^{N}\,[\log(s)]^{\alpha}\right)^{\frac{1}{N-1}}}\,&\qquad\mbox{if}\quad
p=N.\end{cases}$

######  Theorem 5.27.

Assume that $\Omega$ has the strong Lipschitz property. Let
$C_{b}^{\,1}(\Omega)$ be the set of $C^{1}$ functions such that their
derivatives are bounded in $L^{\infty}(\Omega)$. Let $\tilde{A}$ the
${\mathcal{N}}$-function defined in (2.5). The following Orlicz–Sobolev
continuous embeddings hold :

(5.12)
$W^{2,{\tilde{A}}}(\Omega)\subset\begin{cases}L^{({\tilde{A}}^{*})^{*}}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}\leq\frac{N}{2},\
\alpha\leq\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)}\\\\[5.69054pt]
C_{b}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}=\frac{N}{2},\
\alpha>\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)},\\\
&\quad\qquad\mbox{or }\quad\frac{p+1}{p}>\frac{N}{2},\
\alpha\in\mathbb{R};\\\\[5.69054pt]
C_{b}^{\,1}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}=N,\
\alpha>p(N-1),\quad\text{or }\frac{p+1}{p}>N,\
\alpha\in\mathbb{R};\end{cases}$

with $\big{(}{\widetilde{A}}^{*}\big{)}^{*}$ defined by

(5.13)
$\big{(}{\widetilde{A}}^{*}\big{)}^{*}\sim\begin{cases}s^{\frac{N(p+1)}{Np-2(p+1)}}\,\big{[}\log(s)\big{]}^{\frac{\alpha
N}{Np-2(p+1)}}&\text{if }\frac{p+1}{p}<\frac{N}{2},\\\\[5.69054pt]
e^{s^{N/\big{(}N-1-\alpha\frac{N}{Np-(p+1)}\big{)}}}&\text{if
}\frac{p+1}{p}=\frac{N}{2},\
\alpha<\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)}\\\\[8.53581pt]
e^{e^{s^{N^{\prime}}}}&\text{if }\frac{p+1}{p}=\frac{N}{2},\
\alpha=\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)}.\end{cases}$

Moreover, the following Orlicz–Sobolev compact embeddings hold

(5.14)
$W^{2,{\tilde{A}}}(\Omega)\hookrightarrow\begin{cases}L^{A_{1}}(\Omega)&\qquad\forall
A_{1}\prec\prec\big{(}{\widetilde{A}}^{*}\big{)}^{*},\qquad\mbox{if}\quad\frac{p+1}{p}\leq\frac{N}{2},\\\
&\quad\qquad\mbox{and}\quad\alpha\leq\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)};\\\\[5.69054pt]
C_{b}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}=\frac{N}{2},\
\alpha>\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)},\\\
&\quad\qquad\mbox{or }\quad\frac{p+1}{p}>\frac{N}{2},\
\alpha\in\mathbb{R};\\\\[5.69054pt]
C_{b}^{\,1}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}=N,\
\alpha>p(N-1),\quad\text{or }\frac{p+1}{p}>N,\
\alpha\in\mathbb{R};\end{cases}$

for any $A_{1}$ increasing essentially more slowly than
$\big{(}({\widetilde{A}})^{*}\big{)}^{*}$, denoted by
$A_{1}\prec\prec\big{(}({\widetilde{A}})^{*}\big{)}^{*}$, see definition
5.14(iii).

###### Proof of Theorem 5.27.

Since Lemma 2.1 and Example 5.18,

$\widetilde{A}(s)\sim s^{\frac{p+1}{p}}(\ln s)^{\frac{\alpha}{p}}.$

Moreover, by Orlicz–Sobolev continuous embeddings,

$W^{2,{\tilde{A}}}(\Omega)\to\begin{cases}W^{1,({\tilde{A}})^{*}}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}\leq
N,\quad\alpha\leq p(N-1)\\\\[5.69054pt]
C_{b}^{\,1}(\Omega)&\qquad\mbox{if}\quad\frac{p+1}{p}=N,\
\alpha>p(N-1),\quad\text{or }\frac{p+1}{p}>N,\
\alpha\in\mathbb{R};\end{cases}$

with $({\widetilde{A}})^{*}$ defined specifically by

$({\widetilde{A}})^{*}\sim\begin{cases}\Big{(}s^{\frac{p+1}{p}}\big{[}\log(s)\big{]}^{\frac{\alpha}{p}}\Big{)}^{\frac{N}{N-\frac{p+1}{p}}},&\qquad\mbox{if}\quad\frac{p+1}{p}<N,\\\\[5.69054pt]
e^{s^{N/(N-1-\frac{\alpha}{p})}},&\qquad\mbox{if}\quad\frac{p+1}{p}=N,\
\alpha<p(N-1)\\\ e^{e^{s^{N^{\prime}}}},&\qquad\mbox{if}\quad\frac{p+1}{p}=N,\
\alpha=p(N-1),\end{cases}$

cf. Theorem 5.23, Theorem 5.21, example 5.25, and definition (5.11). Observe
that
$\frac{N}{N-\frac{p+1}{p}}\frac{p+1}{p}=\frac{N(p+1)}{Np-(p+1)}=\left(\frac{p}{p+1}-\frac{1}{N}\right)^{-1}$.

Iterating the above procedure we obtain (5.12), with
$\big{(}{\widetilde{A}}^{*}\big{)}^{*}$ defined, by

$\big{(}{\widetilde{A}}^{*}\big{)}^{*}\sim\begin{cases}\Big{(}s^{\frac{N(p+1)}{Np-(p+1)}}\big{[}\log(s)\big{]}^{\frac{\alpha
N}{Np-(p+1)}}\Big{)}^{\frac{Np-(p+1)}{Np-2(p+1)}}&\text{if
}\frac{p+1}{p}<\frac{N}{2},\\\\[5.69054pt]
e^{s^{N/\big{(}N-1-\alpha\frac{N}{Np-(p+1)}\big{)}}}&\text{if
}\frac{p+1}{p}=\frac{N}{2},\
\alpha<\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)}\\\\[8.53581pt]
e^{e^{s^{N^{\prime}}}}&\text{if }\frac{p+1}{p}=\frac{N}{2},\
\alpha=\left(1-\frac{1}{N}\right)\big{(}Np-(p+1)\big{)},\end{cases}$

which is obtained iterating twice (5.11). It can be equivalently rewritten as
(5.13). Observe that
$\frac{N(p+1)}{Np-(p+1)}\frac{Np-(p+1)}{Np-2(p+1)}=\frac{N(p+1)}{Np-2(p+1)}=\left(\frac{p}{p+1}-\frac{2}{N}\right)^{-1}$.

Moreover, from Theorem 5.24, the Orlicz–Sobolev compact embeddings described
in (5.14) hold, see definition 5.14(iii) for $A_{1}$ increasing essentially
more slowly than $\big{(}({\widetilde{A}})^{*}\big{)}^{*}$, denoted by
$A_{1}\prec\prec\big{(}({\widetilde{A}})^{*}\big{)}^{*}$. ∎

######  Corollary 5.28.

In particular, under the conditions of Theorem 5.27, assume that one of the
following two conditions holds:

(i) either (1.2) is satisfied, for any $\alpha\in\mathbb{R}$

(ii) either (1.3) is satisfied, and $\alpha>0$.

Then, the following Orlicz–Sobolev compact embedding holds

(5.15) $W^{2,{\tilde{A}}}(\Omega)\hookrightarrow L^{q+1}(\Omega).$

If (1.3) is satisfied, and $\alpha=0,$ then the above embedding is continuous.

###### Proof.

Using Theorem 5.27, we only have to realize that $A_{1}(s)=s^{q+1}$ increases
essentially more slowly than $\big{(}{\widetilde{A}}^{*}\big{)}^{*}$, i.e.,
$A_{1}\prec\prec\big{(}{\widetilde{A}}^{*}\big{)}^{*}$, see definition
5.14(iii).

We first check that $L^{({\tilde{A}}^{*})^{*}}(\Omega)\subset
L^{q+1}(\Omega)$. Indeed,

$\frac{N(p+1)}{Np-2(p+1)}\geq
q+1\iff\frac{p}{p+1}-\frac{2}{N}\leq\frac{1}{q+1}\iff\frac{N-2}{N}\leq\frac{1}{p+1}+\frac{1}{q+1}.$

Moreover, since $\alpha>0$ in case (ii), then
$A_{1}\prec\prec\big{(}({\widetilde{A}})^{*}\big{)}^{*}$, and so (5.15) holds.

If (1.3) is satisfied, and $\alpha=0,$ then
$A_{1}=\big{(}{\widetilde{A}}^{*}\big{)}^{*},$ and the embedding is
continuous. ∎

###  5.8. The Dirichlet problem

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain of class $C^{2}$, $H$ a
${\mathcal{N}}$-function and $f\in L^{H}(\Omega)$. Let us consider the
Dirichlet problem

(5.16) $-\Delta u=f\ \hbox{in}\ \Omega,\qquad u=0\ \hbox{on}\ \partial\Omega.$

Solutions are understood in the weak sense, i.e.

$\forall\varphi\in C_{0}^{1}(\Omega),\quad\int_{\Omega}\nabla
u\cdot\nabla\varphi\,dx=\int_{\Omega}f\varphi\,dx.$

We have the following regularity result (cf. [2, Theorem XI.8], and [14,
Theorem 4]).

######  Theorem 5.29.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain of class $C^{2}$. Assume
that $H$ and $\widetilde{H}$ satisfies the $\Delta_{2}$-condition. Then, the
unique solution $u=K(f)$ of problem (5.16) belongs to $W^{2,H}(\Omega)\cap
W_{0}^{1,H}(\Omega)$, $-\Delta u=f$ a.e. and

(5.17)
$\|u\|_{H}+\sum_{i}\big{\|}\partial_{x_{i}}u\big{\|}_{H}+\sum_{ij}\big{\|}\partial^{2}_{x_{i}x_{j}}u\big{\|}_{H}\leq
C\|f\|_{H}$

for some $C=C(N,H,\Omega)$.

See Lemma 5.4 (iii) for having a sufficient condition guarantying that $H$ and
$\widetilde{H}$ satisfies the $\Delta_{2}$-condition.

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