ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CONTINUOUS IMBEDDING IN MUSIELAK SPACES WITH AN APPLICATION TO ANISOTROPIC NONLINEAR

NEUMANN PROBLEMS

AHMED YOUSSFI, MOHAMED MAHMOUD OULD KHATRI

Abstract. We prove a continuous embedding that allows us to obtain a boundary trace imbedding result for anisotropic Musielak-Orlicz spaces, which we then apply to obtain an existence result for Neumann problems with non- linearities on the boundary associated to some anisotropic nonlinear elliptic equations in Musielak-Orlicz spaces constructed from Musielak-Orlicz func- tions on which and on their conjugates we do not assume the ∆2-condition.

The uniqueness of weak solutions is also studied.

1. Introduction

Let Ω be an open bounded subset ofR^{N}, (N ≥2). We denote by ~φ: Ω×R^{+}→
R^{N} the vector function φ~ = (φ1, . . . , φN) where for every i ∈ {1, . . . , N}, φi is a
Musielak-Orlicz function differentiable with respect to its second argument whose
complementary Musielak-Orlicz function is denoted byφ^{∗}_{i} (see preliminaries). We
consider the problem

−

N

X

i=1

∂_{x}_{i}a_{i}(x, ∂_{x}_{i}u) +b(x)ϕ_{max}(x,|u(x)|) =f(x, u) in Ω,
u≥0 in Ω,

N

X

i=1

a_{i}(x, ∂_{x}_{i}u)ν_{i}=g(x, u) on∂Ω,

(1.1)

where ∂_{x}_{i} = _{∂}^{∂}

xi and for every i = 1, . . . , N, we denote by ν_{i} the i^{th} component
of the outer normal unit vector and a_{i} : Ω×R→ R is a Carath´eodory function
such that there exist a locally integrable Musielak-Orlicz function (see definition 1.1
below)Pi : Ω×R^{+} →R^{+} with Pi φi, a positive constantci and a nonnegative
function di ∈ Eφ^{∗}_{i}(Ω) satisfying for all s, t ∈ R and for almost every x ∈ Ω the
following assumptions

|ai(x, s)| ≤ci di(x) + (φ^{∗}_{i})^{−1}(x, Pi(x, s))

, (1.2)

φ_{i}(x,|s|)≤a_{i}(x, s)s≤A_{i}(x, s), (1.3)

2010Mathematics Subject Classification. 46E35, 35J20, 35J25, 35B38, 35D30.

Key words and phrases. Musielak-Orlicz space; imbedding; boundary trace imbedding;

weak solution; minimizer.

c

2021 Texas State University.

Submitted April 12, 2019. Published April 5, 2021.

1

ai(x, s)−ai(x, t)

· s−t

>0, for alls6=t, (1.4)
the functionA_{i} : Ω×R→Ris defined by

A_{i}(x, s) =
Z s

0

a_{i}(x, t)dt.

Here and in what follows, we define φmin(x, s) = min

i=1,...,Nφi(x, s) and φmax(x, s) = max

i=1,...,Nφi(x, s).

Letϕ_{max}(x, y) = ^{∂φ}_{∂y}^{max}(x, y). We also assume that there exist a locally integrable
Musielak-Orlicz function R : Ω×R^{+} → R^{+} with R φmax and a nonnegative
functionD∈Eφ^{∗}_{max}(Ω), such that for alls,t∈Rand for almost everyx∈Ω,

|ϕ_{max}(x, s)| ≤D(x) + (φ^{∗}_{max})^{−1}(x, R(x, s)), (1.5)
whereφ^{∗}_{max} stands for the complementary function ofφmax defined below in (2.1).

As regards the data, we suppose that f : Ω×R^{+} →R^{+} and g :∂Ω×R^{+} →R^{+}
are Carath´eodory functions. We define the antiderivatives F : Ω×R → R and
G:∂Ω×R→Roff andgrespectively by

F(x, s) = Z s

0

f(x, t)dt, G(x, s) = Z s

0

g(x, t)dt.

We say that a Musielak-Orlicz functionφsatisfies the ∆2-condition, if there exists
a positive constantk >0 and a nonnegative functionh∈L^{1}(Ω) such that

φ(x,2t)≤kφ(x, t) +h(x).

Remark that the condition (∆2) is equivalent to the following condition: for all
α >1 there exists a positive constantk >0 and a nonnegative functionh∈L^{1}(Ω)
such that

φ(x, αt)≤kφ(x, t) +h(x).

We assume now that there exist two positive constantsk_{1} and k_{2} and two locally
integrable Musielak-Orlicz functionsM andH : Ω×R^{+} →R^{+} satisfying the ∆_{2}-
condition and differentiable with respect to their second arguments withM φ^{∗∗}_{min},
H φ^{∗∗}_{min} andH ψ_{min}, such that the functions f and g satisfy for all s∈R+

the following assumptions

|f(x, s)| ≤k1m(x, s), for a.e. x∈Ω, (1.6)

|g(x, s)| ≤k2h(x, s), for a.e. x∈∂Ω, (1.7) where

ψmin(x, t) =

(φ^{∗∗}_{min})∗(x, t)^{N−1}_{N}

, m(x, s) =∂M(x, s)

∂s , h(x, s) =∂H(x, s)

∂s .

(1.8) Finally, for the functionbinvolved in (1.1), we assume that there exists a constant b0>0 such thatbsatisfies the hypothesis

b∈L^{∞}(Ω) andb(x)≥b0, or a.e.x∈Ω. (1.9)
Observe that (1.4) and the relationai(x, ζ) =∇ζAi(x, ζ) imply in particular that
for anyi= 1, . . . , N, the function ζ→Ai(·, ζ) is convex.

Let us put ourselves in the particular case of φ~ = (φ_{i})_{i∈{1,...,N}_{}} where for i ∈
{1, . . . , N}, φ_{i}(x, t) = |t|^{p}^{i}^{(x)} with p_{i} ∈ C_{+}( ¯Ω) = {h∈ C( ¯Ω) : inf_{x∈Ω}h(x) > 1}.

Definingpmax(x) = maxi∈{1,...,N}pi(x) andpmin(x) = min_{i∈{1,...,N}_{}}pi(x), one has
φmax(x, t) = |t|^{p}^{M}^{(x)} and then ϕmax(x, t) = pM(x)|t|^{p}^{M}^{(x)−2}t, where pM is pmax

or pmin according to whether |t| ≥ 1 or |t| ≤ 1 and then the space W^{1}Lφ~(Ω) is
nothing but the anisotropic space with variable exponentW^{1,~}^{p(·)}(Ω), where~p(·) =
(p1(·), . . . , pN(·)) (see [7] for more details on this space). Therefore, the problem
(1.1) can be rewritten as

−

N

X

i=1

∂x_{i}ai(x, ∂x_{i}u) +b1(x)|u|^{p}^{M}^{(x)−2}u=f(x, u) in Ω,
u≥0 in Ω,

N

X

i=1

ai(x, ∂xiu)νi=g(x, u) on∂Ω,

(1.10)

where b1(x) =pM(x)b(x). Boureanu and Rˇadulescu [2] have proved the existence
and uniqueness of the weak solution of (1.10). They prove an imbedding and a
trace results which they use together with a classical minimization existence result
for functional reflexive framework (see [22, Theorem 1.2]). Problem (1.10) with
Dirichlet boundary condition and b1(x) = 0 was treated in [15]. The authors
proved that iff(·, u) =f(·)∈L^{∞}(Ω) then (1.10) admits a unique solution by using
[22, Theorem 1.2]. The problem (1.10) with for alli= 1, . . . , N

ai(x, s) =a(x, s) =s^{p(x)−1},

withp∈C^{1}(Ω) andb1 =g = 0 was treated in [12], where the authors proved the
three nontrivial smooth solutions, two of which have constant sign (one positive, the
other negative). In connection with Neumann problems, the authors [21] studied
the problem

−diva(∇u(z)) + ζ(z) +λ

u(z)^{p−1}=f(z, u(z)) in Ω,

∂u

∂n= 0 on∂Ω u >0, λ >0, 1< p <+∞,

(1.11)

where the function a : R^{N} → R^{N} is strictly monotone, continuous and satisfies
certain other regularity and growth conditions. The functionζ involved in (1.11)
changes its sign and is such that ζ ∈ L^{∞}(Ω). The reaction term f(z, x) is a
Carath´eodory function. They proved the existence of a critical parameter value
λ_{∗} >0 such that if λ > λ_{∗} problem (1.11) has at least two positive solutions, if
λ=λ_{∗} (1.11) has at least a positive solution and ifλ∈(0, λ_{∗}) problem (1.11) has
no positive solution.

Let us mention some related results in the framework of Orlicz-Sobolev spaces.

Le and Schmitt [17] proved an existence result for the boundary value problem

−div(A(|∇u|^{2})∇u) +F(x, u) = 0, in Ω,
u= 0 on∂Ω,

in W_{0}^{1}Lφ(Ω) where φ(s) =A(|s|^{2})s and F is a Carath´eodory function satisfying
some growth conditions. This result extends the one obtained in [11] withF(x, u) =

−λψ(u), whereψis an odd increasing homeomorphism ofRontoR. In [11, 17] the
authors assume that theN-functionφ^{∗}complementary to theN-functionφsatisfies

the ∆2 condition, which is used to prove that the functional u→R

ΩΦ(|∇u|)dx is
coercive and of classC^{1}, where Φ is the antiderivative of φvanishing at the origin.

Here we are interested in proving the existence and uniqueness of the weak
solutions for problem (1.1) without any additional condition on the Musielak-Orlicz
function φ_{i} or its complementary φ^{∗}_{i} for i = 1, . . . , N. Therefore, the resulting
Musielak-Orlicz spacesL_{φ}_{i}(Ω) are neither reflexive nor separable and thus classical
existence results can not be applied.

The approach we use consists in proving first a continuous imbedding and a trace result which we then apply to solve the problem (1.1). The results we prove extend to the anisotropic Musielak-Orlicz-Sobolev spaces the continuous imbedding obtained in [6] under some extra conditions and the trace result proved in [18]. The imbedding result we obtain extends to Musielak spaces a part of the one obtained in [19] in the anisotropic case and that of Fan [9] in the isotropic case (see Remark 3.2).

In the variable exponent Sobolev spaceW^{1,p(x)}(Ω) where 1< p_{+}= sup_{x∈Ω}p(x)<

N, other imbedding results can be found for instance in [3, 4, 16] while the case
1≤p_{−}≤p_{+} ≤N was investigated in [13].

To the best of our knowledge, the trace result we obtain here is new and does not exist in the literature. The main difficulty we found when we deal with problem (1.1) is the coercivity of the energy functional. We overcome this by using both our continuous imbedding and trace results. Then we prove the boundedness of a minimization sequence and by a compactness argument, we are led to obtain a minimizer which is a weak solution of problem (1.1).

Definition 1.1. Let Ω be an open subset ofR^{N}, (N ≥2). We say that a Musielak-
Orlicz functionφis locally-integrable, if for every compact subsetKof Ω and every
constantc >0, we have

Z

K

φ(x, c)dx <∞.

The article is organized as follows: Section 2 contains some definitions. In Sec- tion 3, we give and prove our main results, which we then apply in Section 4 to solve problem (1.1). In the last section we give an appendix which contains some important lemmas that are necessary for the accomplishment of the proofs of the results.

2. Preliminaries

2.1. Anisotropic Musielak-Orlicz-Sobolev spaces. Let Ω be an open subset
ofR^{N}. A real functionφ: Ω×R^{+}→R^{+} will be called a Musielak-Orlicz function
if it satisfies the following conditions

(i) φ(·, t) is a measurable function on Ω.

(ii) φ(x,·) is an N-function, that is a convex nondecreasing function with φ(x, t) = 0 if only if t = 0, φ(x, t) > 0 for all t > 0 and for almost ev- eryx∈Ω,

lim

t→0^{+}

φ(x, t)

t = 0 and lim

t→+∞inf

x∈Ω

φ(x, t)

t = +∞.

We will extend these Musielak-Orlicz functions into even functions on all Ω×R.
The complementary functionφ^{∗} of the Musilek-Orlicz functionφis defined by

φ^{∗}(x, s) = sup

t≥0

{st−φ(x, t)}. (2.1)

It can be checked that φ^{∗} is also a Musielak-Orlicz function (see [20]). Moreover,
for everyt,s≥0 and a.e.x∈Ω we have the so-called Young inequality (see [20])

ts≤φ(x, t) +φ^{∗}(x, s).

For any functionh:R→Rthe second complementary functionh^{∗∗}= (h^{∗})^{∗} (cf.

(2.1)), is convex and satisfies

h^{∗∗}(x)≤h(x), (2.2)

with equality whenhis convex. Roughly speaking,h^{∗∗} is a convex envelope ofh,
that is the biggest convex function smaller or equal toh.

Letφandψbe two Musielak-Orlicz functions. We say thatψgrows essentially more slowly thanφ, denoteψφ, if

t→+∞lim sup

x∈Ω

ψ(x, t) φ(x, ct) = 0,

for every constant c > 0 and for almost every x ∈ Ω. We point out that if ψ :
Ω×R^{+} → R^{+} is locally integrable then ψ φ implies that for all c > 0 there
exists a nonnegative functionh∈L^{1}(Ω) such that

ψ(x, t)≤φ(x, ct) +h(x), for allt∈Rand for a.e. x∈Ω.

The Musielak-Orlicz spaceLφ(Ω) is defined by
L_{φ}(Ω) =n

u: Ω→Rmeasurable : Z

Ω

φ x,u(x)

λ

<+∞for someλ >0o . Endowed with the so-called Luxemborg norm

kukφ= infn λ >0 :

Z

Ω

φ x,u(x)

λ

dx≤1o ,

(L_{φ}(Ω),k · kφ) is a Banach space. Observe that since lim_{t→+∞}inf_{x∈Ω}^{φ(x,t)}_{t} = +∞

and if Ω has finite measure then we have the following continuous imbedding

Lφ(Ω),→L^{1}(Ω). (2.3)

We will also use the space Eφ(Ω) =n

u: Ω→Rmeasurable : Z

Ω

φ x,u(x)

λ

<+∞for allλ >0o . Observe that for everyu∈Lφ(Ω) the following inequality holds

kuk_{φ}≤
Z

Ω

φ(x, u(x))dx+ 1. (2.4)

For two complementary Musielak-Orlicz functions φ and φ^{∗}, H¨older’s inequality
(see [20])

Z

Ω

|u(x)v(x)|dx≤2kuk_{φ}kvk_{φ}^{∗} (2.5)
holds for everyu∈Lφ(Ω) and v∈Lφ^{∗}(Ω). Defineφ^{∗−1} for everys≥0 by

φ^{∗−1}(x, s) = sup{τ≥0 :φ^{∗}(x, τ)≤s}.

Then, for almost everyx∈Ω and for everys∈Rwe have

φ^{∗}(x, φ^{∗−1}(x, s))≤s, (2.6)

s≤φ^{∗−1}(x, s)φ^{−1}(x, s)≤2s, (2.7)

φ(x, s)≤s∂φ(x, s)

∂s ≤φ(x,2s). (2.8)

Definition 2.1. Letφ~ : Ω×R+ −→R^{N} be the vector functionφ~ = (φ1, . . . , φN)
where for every i ∈ {1, . . . , N}, φi is a Musielak-Orlicz function. We define the
anisotropic Musielak-Orlicz-Sobolev space by

W^{1}L_{φ}_{~}(Ω) =n

u∈L_{φ}_{max}(Ω); ∂_{x}_{i}u∈L_{φ}_{i}(Ω) for all i= 1,· · ·, No
.

By the continuous imbedding (2.3), we obtain thatW^{1}L_{~}_{φ}(Ω) is a Banach space
with respect to the following norm

kuk_{W}1L_{~}_{φ}(Ω)=kukφ_{max}+

N

X

i=1

k∂x_{i}ukφ_{i}.

Moreover, we have the continuous embeddingW^{1}Lφ~(Ω),→W^{1,1}(Ω).

3. Main results

In this section we prove an imbedding theorem and a trace result. Let us assume the conditions

Z 1 0

(φ^{∗∗}_{min})^{−1}(x, t)

t^{1+}^{N}^{1} dt <+∞ and

Z +∞

1

(φ^{∗∗}_{min})^{−1}(x, t)

t^{1+}^{N}^{1} dt= +∞, ∀x∈Ω.

(3.1)
Thus, we define the Sobolev conjugate (φ^{∗∗}_{min})_{∗}

(φ^{∗∗}_{min})^{−1}_{∗} (x, s) =
Z s

0

(φ^{∗∗}_{min})^{−1}(x, t)

t^{1+}^{N}^{1} dt, forx∈Ω ands∈[0,+∞). (3.2)
It may readily be checked that (φ^{∗∗}_{min})_{∗} is a Musielak-Orlicz function. We assume
that there exist two positive constantsν < _{N}^{1} andc0>0 such that

∂(φ^{∗∗}_{min})∗

∂xi

(x, t) ≤c0

h

(φ^{∗∗}_{min})∗(x, t) + ((φ^{∗∗}_{min})∗(x, t))^{1+ν}i

, (3.3)

for allt∈Rand for almost everyx∈Ω, provided that for every i= 1, . . . , N the
derivative ^{∂(φ}_{∂x}^{∗∗}^{min}^{)}^{∗}

i (x, t) exists.

3.1. Imbedding theorem.

Theorem 3.1. Let Ω be an open bounded subset of R^{N}, (N ≥2), with the cone
property. Assume that (3.1)and (3.3)are fulfilled, (φ^{∗∗}_{min})_{∗}(·, t)is Lipschitz contin-
uous on Ωandφmax is locally integrable. Then, there is a continuous embedding

W^{1}Lφ~(Ω),→L(φ^{∗∗}_{min})_{∗}(Ω).

Some remarks about Theorem 3.1 are in order. We discuss how Theorem 3.1 include some previous results known in the literature when reducing to some par- ticular Musielak-Orlicz functions.

Remark 3.2. (1) Let M(x, t) =t^{p(x)} andm(x, t) = ^{∂M}_{∂t}^{(x,t)} =p(x)t^{p(x)−1}, where
p(·) is Lipschitz continuous on Ω, with 1 < p_{−} = inf_{x∈Ω}p(x) ≤ p(x) ≤ p_{+} =

sup_{x∈Ω}p(x)< N. SinceM(·, t) andm(·, t) are continuous on Ω, we can use Lemma
5.8 (given in Appendix) to define the following Musielak-Orlicz function

φ(x, t) =

t^{p(x)}_{1}

t^{α}_{1} t^{α} ift≤t1,
t^{p(x)} ift≥t1,

where t1 >1 andα > 1 are two constants mentioned in the proof of Lemma 5.8.

Let us now consider the particular case where for alli= 1, . . . , N, φi(x, t) =φ(x, t) =

t^{p(x)}_{1}

t^{α}_{1} t^{α} ift≤t1,
t^{p(x)} ift≥t1.

(3.4)
It is worth pointing out that since Ω is of finite Lebesgue measure, it can be seen
easily that W^{1}L~φ(Ω) =W^{1}Lφ(Ω) = W^{1,p(·)}(Ω). Thus, φ^{∗∗}_{min}(x, t) = φmin(x, t) =
φ(x, t) and

(φ^{∗∗}_{min})_{∗}(x, t) = (φ_{min})_{∗}(x, t) =φ_{∗}(x, t) =

(N−α)t
N αt_{1}

_{N−α}^{N α}
t

N p(x) N−α

1 ift≤t_{1},

1

p∗(x)t^{p}∗(x)

ift≥t1,
provided that α < N. Now we shall prove that (φ^{∗∗}_{min})_{∗} satisfies (3.3) and our
imbedding result include some previous result known in the literature. For every
t∈Rand for almost everyx∈Ω we have

∂(φ^{∗∗}_{min})_{∗}

∂x_{i} (x, t) =
( _{N}

N−α

∂p(x)

∂x_{i} log(t1)(φ^{∗∗}_{min})_{∗}(x, t) ift≤t1,

∂p∗(x)

∂x_{i} log _{ep}^{t}

∗(x)

(φ^{∗∗}_{min})_{∗}(x, t) ift≥t1.

•Ift≤t1, then

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t) = N

N−α

∂p

∂xi

(x)

log(t_{1})(φ^{∗∗}_{min})_{∗}(x, t).

Since p(·) is Lipschitz continuous on Ω there exists a constant C1 > 0 satisfying

_{∂x}^{∂p}

i(x)

≤C_{1} thus we obtain

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t)

≤C_{1} N

N−αlog(t_{1})(φ^{∗∗}_{min})_{∗}(x, t). (3.5)

•Ift≥t_{1}, then

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t) =

∂p_{∗}

∂xi

(x)

log t
ep_{∗}(x)

(φ^{∗∗}_{min})∗(x, t).

Sincep(·) is Lipschitz continuous on Ω, it can be seen easily thatp_{∗}(·) is also Lip-
schitz continuous on Ω. Then, there exists a constantC_{2}>0 satisfying

∂p∗

∂x_{i}(x)
≤
C2. So that we have

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t)
≤C_{2}

log t ep∗(x)

(φ^{∗∗}_{min})_{∗}(x, t).

Let 0< <1/N. For allt >0 we can easily check that log(t)≤ 1

^{2}N et^{}. (3.6)

Now, since the Musielak-Orlicz function (φ^{∗∗}_{min})∗ has a superlinear growth, we can
chooseA >0 for which there existst0>max{t1, e}(not depending onx) such that
At≤(φ^{∗∗}_{min})∗(x, t) whenevert≥t0. Therefore,

•Ift≥t_{0} then by (3.6) we obtain

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t)

≤C2

logt

e

+ log( N^{2}
N−p+

)

(φ^{∗∗}_{min})∗(x, t)

≤ C_{2}

^{2}N e^{1+}t^{}(φ^{∗∗}_{min})∗(x, t) +C2log( N^{2}
N−p+

)(φ^{∗∗}_{min})∗(x, t)

≤ C_{2}

^{2}N e^{1+}A^{}((φ^{∗∗}_{min})∗(x, t))^{1+}+C2log( N^{2}
N−p+

)(φ^{∗∗}_{min})∗(x, t).

(3.7)

•Ift1< t≤t0, then

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t) ≤C2

log(t0) + log eN^{2}
N−p+

(φ^{∗∗}_{min})∗(x, t). (3.8)
Therefore, from (3.5), (3.7) and (3.8), we obtain that for everyt≥0 and for almost
everyx∈Ω, there is a constantc0>0 such that

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t)
≤c_{0}

(φ^{∗∗}_{min})_{∗}(x, t) + ((φ^{∗∗}_{min})_{∗}(x, t))^{1+}

.

Before we show that our imbedding result includes some previous known results in
the literature, we remark that the proof of Theorem 3.1 relies to the application
of Lemma 5.4 in Appendix for the function g(x, t) = ((φ^{∗∗}_{min})_{∗}(x, t))^{α}, α ∈ (0,1),
where we have used the fact that Ω is bounded to ensure that max_{x∈Ω}g(x, t)<∞
for somet >0. In the case of the variable exponent Sobolev spaceW^{1,p(·)}(Ω) built
upon the Musielak-Orlicz function given in (3.4), we do not need Ω to be bounded,
since

φ_{∗}(x, t)≤max{t

N α N−α

1 , t

N2

N−p+}<∞, for some t >0.

Therefore, the embedding result in Theorem 3.1 can be seen as an extension to the Musielak-Orlicz framework of the one obtained in [9, Theorem 1.1].

(2) Let us consider the particular case where, fori∈ {1, . . . , N},
φ_{i}(x, t) =

t^{pi}_{1} ^{(x)}

t^{α}_{1} t^{α} ift≤t1,
t^{p}^{i}^{(x)} ift≥t1

wheret_{1}>1, 1< α < N andφ~ = (φ_{i})i∈{1,...,N} with
p_{i}∈C_{+}(Ω) ={h∈C(Ω) : inf

x∈Ω

h(x)>1},

1< pi(x)< N, N ≥3. We definep^{−}_{i} = inf_{x∈Ω}pi(x),pM(x) = max_{i∈{1,...,N}_{}}pi(x),
pm(x) = min_{i∈{1,...,N}_{}}pi(x). Then

φ^{∗∗}_{min}(x, t) =φ_{min}(x, t) =

t^{pm(x)}_{1}

t^{α}_{1} t^{α} ift≤t1,
t^{p}^{m}^{(x)} ift≥t1,

whose Sobolev conjugate function is
(φ^{∗∗}_{min})∗(x, t) =

(N−α)t N αt1

_{N−α}^{N α}
t

N pm(x) N−α

1 ift≤t1,

1

(pm)∗(x)t^{(p}m)_{∗}(x)

ift≥t_{1}.
Let us definep^{∗}_{−}=PN ^{N}

i=1 1 p−

i

−1. Notice thatp^{−}_{i} > p^{−}_{m}implies
p^{∗}_{−}> N p^{−}_{m}

N−p^{−}m

= (p^{−}_{m})_{∗}. (3.9)

Since Ω is of finite Lebesgue measure, it can be seen easily that W^{1}L_{~}_{φ}(Ω) =
W^{1,~}^{p(·)}(Ω). So, by Theorem 3.1 we have W^{1,~}^{p(·)}(Ω) ,→ L^{(p}^{m}^{)}^{∗}^{(·)}(Ω) and since
(p_{m})_{∗}(x)≥(p^{−}_{m})_{∗} for eachx∈Ω, we deduce thatW^{1,~}^{p(·)}(Ω),→L^{(p}^{−}^{m}^{)}^{∗}(Ω). There-
fore, by (3.9) the result we obtain can be found in [19, Theorem 1].

(3) Let us now consider the case where φi(x, t) =

t^{pi}_{1} ^{(x)}log(t_{1}+1)

t^{α}_{1} t^{α} ift≤t1,
t^{p}^{i}^{(x)}log(t+ 1) ift≥t1,

where t1 >1, 1 < α < N and for each i ∈ {1, . . . , N} the function pi(·) is Lip-
schitz continuous on Ω with 1 < inf_{x∈Ω}pi(x) ≤ pi(x) ≤ sup_{x∈Ω}pi(x) < N −1.

Define pM(x) = max_{i∈{1,...,N}_{}}pi(x), pm(x) = mini∈{1,...,N}pi(x) andφmin(x, t) =
min_{i∈{1,...,N}_{}}φi(x, t). Then

φ_{min}(x, t) =φ^{∗∗}_{min}(x, t) =

t^{pm(x)}_{1} log(t_{1}+1)

t^{α}_{1} t^{α} ift≤t1,
t^{p}^{m}^{(x)}log(t+ 1) ift≥t1.

SetA(x, t) =t^{p}^{m}^{(x)}log(t+ 1). By [18, Example 2] there existσ < _{N}^{1},C0>0 and
t0>0 such that

∂A_{∗}

∂xi

(x, t)

≤C_{0}(A_{∗}(x, t))^{1+σ},

forx∈Ω and t ≥t_{0}. Choosing thist_{0} >0 in Lemma 5.8 given in Appendix, we
can taket_{1}> t_{0}+ 1 obtaining

∂A_{∗}

∂x_{i} (x, t)

≤C0(A_{∗}(x, t))^{1+σ}, for allt≥t1. (3.10)
On the other hand, fort≤t1 we have

(φ^{∗∗}_{min})_{∗}(x, t) =(N−α)t
N αt1

_{N−α}^{N α} t^{p}_{1}^{m}^{(x)}
log(t1+ 1)

_{N−α}^{N}
.
Thus

∂(φ^{∗∗}_{min})_{∗}

∂xi

(x, t)

= Nlog(t_{1})
N−α

∂p_{m}

∂xi

(x)

(φ^{∗∗}_{min})∗(x, t).

Sincep_{m}(·) is Lipschitz continuous on Ω there exists a constant C_{3}>0 satisfying

∂p_{m}

∂x_{i}(x)

≤C3. So we have

∂(φ^{∗∗}_{min})_{∗}

∂x_{i} (x, t)

≤ C3Nlog(t1)

N−α (φ^{∗∗}_{min})_{∗}(x, t). (3.11)

Therefore, by (3.10) and (3.11) the function (φ^{∗∗}_{min})∗ satisfies the assertions of The-
orem 3.1 and then we obtain the continuous embedding

W^{1}L_{φ}_{~}(Ω),→L_{(φ}^{∗∗}

min)∗(Ω).

Proof ofTheorem 3.1. Let u ∈ W^{1}L_{φ}_{~}(Ω). Assume first that the function u is
bounded and u 6= 0. Defining f(s) = R

Ω(φ^{∗∗}_{min})_{∗} x,^{|u(x)|}_{s}

dx, for s > 0, one has
lim_{s→0}+f(s) = +∞and lim_{s→∞}f(s) = 0. Sincef is continuous on (0,+∞), there
exists λ >0 such thatf(λ) = 1. Then by the definition of the Luxemburg norm,
we obtain

kuk_{(φ}^{∗∗}

min)∗≤λ. (3.12)

On the other hand,
f(kuk_{(φ}^{∗∗}

min)∗) = Z

Ω

(φ^{∗∗}_{min})_{∗}

x, u(x)
kuk_{(φ}^{∗∗}

min)∗

dx≤1 =f(λ) and sincef is decreasing,

λ≤ kuk_{(φ}^{∗∗}

min)∗. (3.13)

So that by (3.12) and (3.13), we obtainλ=kuk_{(φ}^{∗∗}

min)∗ and Z

Ω

(φ^{∗∗}_{min})∗

x,u(x)

λ

dx= 1. (3.14)

From (3.2) we can easily check that (φ^{∗∗}_{min})∗ satisfies the differential equation
(φ^{∗∗}_{min})^{−1}(x,(φ^{∗∗}_{min})_{∗}(x, t))∂(φ^{∗∗}_{min})_{∗}

∂t (x, t) = ((φ^{∗∗}_{min})_{∗}(x, t))^{N+1}^{N} .
Hence, by (2.7) we obtain the inequality

∂(φ^{∗∗}_{min})_{∗}

∂t (x, t)≤((φ^{∗∗}_{min})∗(x, t))^{N}^{1}(φ^{∗∗}_{min})^{∗−1}(x,(φ^{∗∗}_{min})∗(x, t)), (3.15)
for a.e.x∈Ω. Let

h(x) =h

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

i^{N−1}_{N}

. (3.16)

Since (φ^{∗∗}_{min})_{∗}(·, t) is Lipschitz continuous on Ω and (φ^{∗∗}_{min})_{∗}(x,·) is locally Lipschitz
continuous on R^{+}, the function h is Lipschitz continuous on Ω. Hence, we can
compute using Lemma 5.6 (given in Appendix) forf =h, obtaining for a.e.x∈Ω,

∂h

∂xi

(x) = N−1 N

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

−_{N}^{1}h∂(φ^{∗∗}_{min})_{∗}

∂t

x,u(x) λ

∂x_{i}u
λ (x)
+∂(φ^{∗∗}_{min})∗

∂xi

x,u(x)

λ i

,
where∂_{x}_{i}u:= _{∂x}^{∂u}

i. Therefore,

N

X

i=1

∂h

∂xi

(x)

≤I_{1}+I_{2}, for a.e. x∈Ω, (3.17)
where we have set

I_{1}= N−1
N λ

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

^{−1}_{N} ∂(φ^{∗∗}_{min})_{∗}

∂t

x,u(x) λ

X^{N}

i=1

|∂_{x}_{i}u(x)|,

I2=N−1 N

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

^{−1}_{N} X^{N}

i=1

∂(φ^{∗∗}_{min})_{∗}

∂x_{i}

x,u(x) λ

. Now we estimate the two integrals R

ΩI1(x)dx and R

ΩI2(x)dx. By (3.15), we can write

I_{1}(x)≤N−1

N λ (φ^{∗∗}_{min})^{∗−1}

x,(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

X^{N}

i=1

|∂xiu(x)|. (3.18) By (2.6), we have

Z

Ω

(φ^{∗∗}_{min})^{∗}

x,(φ^{∗∗}_{min})^{∗−1}

x,(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

dx≤ Z

Ω

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

dx= 1.

Hence

(φ^{∗∗}_{min})^{∗−1}

·,(φ^{∗∗}_{min})∗

·,u(·) λ

_{(φ}∗∗

min)^{∗}≤1. (3.19)
From (2.5), (3.18) and (3.19) it follows that

Z

Ω

I1(x)dx

≤ 2(N−1) N λ

(φ^{∗∗}_{min})^{∗−1}

·,(φ^{∗∗}_{min})_{∗}

·,u(·) λ

_{(φ}∗∗

min)^{∗}
N

X

i=1

∂x_{i}u
_{φ}∗∗

min

≤ 2(N−1) N λ

N

X

i=1

∂_{x}_{i}u
_{φ}_{∗∗}

min

≤ 2 λ

N

X

i=1

∂x_{i}u
_{φ}∗∗

min

.

(3.20)

Recalling the definition ofφminand (2.2), we obtain k∂x_{i}u(x)kφ^{∗∗}_{min}≤ k∂x_{i}u(x)kφ_{i},
so that (3.20) implies

Z

Ω

I1(x)dx≤ 2 λ

N

X

i=1

∂x_{i}u(x)
_{φ}

i. (3.21)

Using (3.3) we can write I2(x)≤c1

h

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

1−_{N}^{1}

+

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

1−_{N}^{1}+νi
,
withc1=c0(N−1). Since (φ^{∗∗}_{min})_{∗}(·, t) is continuous on Ω andν < _{N}^{1}, we can apply
Lemma 5.4 (given in Appendix) with the functionsg(x, t) =^{((φ}^{∗∗}^{min}^{)}^{∗}^{(x,t))}

1−1 N+ν

t and

f(x, t) = ^{(φ}^{∗∗}^{min}^{)}_{t}^{∗}^{(x,t)} and= _{8c}^{1}

1c∗ obtaining for t= ^{|u(x)|}_{λ}
h

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

i1−_{N}^{1}+ν

≤ 1

8c1c_{∗}(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

+K0

|u(x)|

λ . (3.22)
Using again Lemma 5.4 with the functionsg(x, t) =^{((φ}^{∗∗}^{min}^{)}^{∗}^{(x,t))}

1−1 N

t andf(x, t) =

(φ^{∗∗}_{min})∗(x,t)

t and= _{8c}^{1}

1c∗, we obtain by substitutingtby ^{|u(x)|}_{λ}
h

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

i1−_{N}^{1}

≤ 1

8c_{1}c_{∗}(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

+K0

|u(x)|

λ , (3.23)

where c_{∗} is the constant in the continuous embeddingW^{1,1}(Ω),→L^{N−1}^{N} (Ω), that
is

kwkL^{N−1}^{N} (Ω)≤c∗kwkW^{1.1}(Ω), for allw∈W^{1,1}(Ω). (3.24)
By (3.22) and (3.23), we obtain

Z

Ω

I2(x)dx≤ 1

4c_{∗} +2K0c1

λ kuk_{L}1(Ω). (3.25)

Putting together (3.21) and (3.25) in (3.17) we obtain

N

X

i=1

k∂x_{i}hk_{L}1(Ω)≤ 1
4c_{∗} +2

λ

N

X

i=1

k∂x_{i}u(x)kφ_{i}+2K0c1

λ kuk_{L}1(Ω)

≤ 1 4c∗

+2 λ

N

X

i=1

k∂xiu(x)kφi+2K_{0}c_{1}c_{2}

λ kukφmax,

wherec2is the constant in the continuous embedding (2.3). Then it follows that

N

X

i=1

k∂x_{i}hk_{L}1(Ω)≤ 1
4c_{∗} +c3

λkuk_{W}1L_{φ}_{~}(Ω), (3.26)
withc3= max{2,2K0c1c2}. Now, using again Lemma 5.4 (in Appendix) with the
functions g(x, t) =

(φ^{∗∗}_{min})∗(x, t)1−_{N}^{1}

/tand f(x, t) = (φ^{∗∗}_{min})∗(x, t)/tand= _{4c}^{1}

∗, fort=|u(x)|/λ, we obtain

h(x)≤ 1

4c_{∗}(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

+K0

|u(x)|

λ , From (2.3), we obtain

khk_{L}1(Ω)≤ 1

4c_{∗} +K0c2

λ kuk_{L}_{φmax}_{(Ω)}. (3.27)
Thus, by (3.26) and (3.27) we obtain

khkW^{1,1}(Ω)≤ 1
2c_{∗} +c4

λkukW^{1}L_{~}_{φ}(Ω),

where c_{4} = c_{3}+K_{0}c_{2}, which shows that h ∈ W^{1,1}(Ω) and which together with
(3.24) yield

khkL^{N−1}^{N} (Ω)≤1
2 +c4c∗

λ kukW^{1}L_{φ}_{~}(Ω).
Having in mind (3.14), we obtain

Z

Ω

[h(x)]^{N−1}^{N} dx=
Z

Ω

(φ^{∗∗}_{min})_{∗}
x,u(x)

λ

dx= 1.

So that one has

kuk(φ^{∗∗}_{min})∗ =λ≤2c_{4}c_{∗}kukW^{1}L_{Φ}_{~}(Ω). (3.28)
We now extend the estimate (3.28) to an arbitraryu∈W^{1}L~φ(Ω). LetTn,n >0, be
the truncation function at levels±ndefined onRbyTn(s) = min{n,max{s,−n}}.

Since φmax is locally integrable, by [1, Lemma 8.34.] one has Tn(u)∈W^{1}Lφ~(Ω).

So that in view of (3.28)
kTn(u)k_{(φ}^{∗∗}

min)∗≤2c4c_{∗}kTn(u)k_{W}1L_{~}_{φ(Ω)} ≤2c4c_{∗}kuk_{W}1L_{φ(Ω)}_{~} . (3.29)

Letkn =kTn(u)k(φ^{∗∗}_{min})_{∗}. Thanks to (3.29), the sequence{kn}^{∞}_{n=1} is nondecreasing
and converges. If we denotek= limn→∞kn, by Fatou’s lemma we have

Z

Ω

(φ^{∗∗}_{min})_{∗}

x,|u(x)|

k

dx≤lim inf Z

Ω

(φ^{∗∗}_{min})_{∗}

x,|T_{n}(u)|

kn

dx≤1.

This implies thatu∈L(φ^{∗∗}_{min})_{∗}(Ω) and
kuk(φ^{∗∗}_{min})∗≤k= lim

n→∞kTn(u)k(φ^{∗∗}_{min})∗≤2c4c∗kukW^{1}L_{φ(Ω)}_{~} .

Inequality (3.28) trivially holds ifu= 0. Then proof is complete.

Corollary 3.3. Let Ω be an open bounded subset of R^{N}, N ≥ 2, with the cone
property. Assume that (3.1),(3.3)are fulfilled,(φ^{∗∗}_{min})∗(·, t)is Lipschitz continuous
on Ω and φ_{max} is locally integrable. Let A be a Musielak-Orlicz function where
the function A(·, t) is continuous on Ω and A (φ^{∗∗}_{min})_{∗}. Then, the embedding
W^{1}L_{φ}_{~}(Ω),→L_{A}(Ω) is compact.

Proof. Let {u_{n}} is a bounded sequence in W^{1}L_{φ}_{~}(Ω). By Theorem 3.1, {u_{n}} is
bounded inL_{(φ}^{∗∗}

min)∗(Ω). Since the embeddingW^{1}Lφ~(Ω),→W^{1,1}(Ω) is continuous
and the imbedding W^{1,1}(Ω) ,→ L^{1}(Ω) is compact, we deduce that there exists a
subsequence of{un} still denoted by{un}which converges in measure in Ω. Since
A(φ^{∗∗}_{min})∗, by Lemma 5.5 (in Appendix) the sequence{un} converges in norm

inLA(Ω).

3.2. Trace result. We prove here a trace result which is a useful tool to prove the
coercivity of some energy functionals. Recall thatψ_{min}(x, t) = [(φ^{∗∗}_{min})_{∗}(x, t)]^{N−1}^{N} is
a Musielak-Orlicz function. Indeed, we have

∂

∂t(ψmin)^{−1}(x, t) = ∂

∂t(φ^{∗∗}_{min})^{−1}_{∗} x, t^{N−1}^{N}
.
By (3.2), we obtain

∂

∂t(ψmin)^{−1}(x, t) = N

N−1t^{N−1}^{1} (φ^{∗∗}_{min})^{−1} x, t^{N−1}^{N}
t^{N−1}^{N} ^{+}^{N−1}^{1}

= N

N−1

(φ^{∗∗}_{min})^{−1} x, t^{N−1}^{N}
t^{N−1}^{N}

.
Being the inverse of a Musielak-Orlicz function, it is clear that (φ^{∗∗}_{min})^{−1} satisfies

τ→+∞lim

(φ^{∗∗}_{min})^{−1}(x, τ)

τ = 0 and lim

τ→0^{+}

(φ^{∗∗}_{min})^{−1}(x, τ)

τ = +∞.

Moreover, (φ^{∗∗}_{min})^{−1}(x,·) is concave so that if 0< τ < σ then we obtain
(φ^{∗∗}_{min})^{−1}(x, τ)

(φ^{∗∗}_{min})^{−1}(x, σ) ≥ τ
σ.
Hence, if 0< s1< s2, then

∂

∂t(ψ_{min})^{−1}(x, s_{1})

∂

∂t(ψmin)^{−1}(x, s2) = (φ^{∗∗}_{min})^{−1} x, s

N N−1

1

(φ^{∗∗}_{min})^{−1} x, s

N N−1

2

s

N N−1

2

s

N N−1

1

≥ s

N N−1

1

s

N N−1

2

s

N N−1

2

s

N N−1

1

= 1.

It follows that _{∂t}^{∂}(ψ_{min})^{−1}(x, t) is positive and decreases monotonically from +∞

to 0 astincreases from 0 to +∞and thusψ_{min} is a Musielak-Orlicz function.

Theorem 3.4. Let Ω be an open bounded subset of R^{N}, N ≥ 2, with the cone
property. Assume that (3.1),(3.3)are fulfilled,(φ^{∗∗}_{min})∗(·, t)is Lipschitz continuous
on Ωandφ_{max} is locally integrable. Let ψ_{min} the Musielak-Orlicz function defined
in (1.8). Then, the following boundary trace embedding W^{1}L_{φ}_{~}(Ω),→L_{ψ}_{min}(∂Ω)is
continuous.

Remark 3.5. In the case where for alli= 1, . . . , N,
φ_{i}(x, t) =φ(x, t) =

t^{p(x)}_{1}

t^{α}_{1} t^{α} ift≤t1,
t^{p(x)} ift≥t1,

for some t_{1} > 0, with p ∈ L^{∞}(Ω), 1 ≤ inf_{x∈Ω}p(x) ≤ sup_{x∈Ω}p(x) < N, |∇p| ∈
L^{γ(·)}(Ω), whereγ∈L^{∞}(Ω) and inf_{x∈Ω}γ(x)> N. It is worth pointing out that since
Ω is of finite Lebesgue measure, it can be seen easily thatW^{1}L_{~}_{φ}(Ω) =W^{1}L_{φ}(Ω) =
W^{1,p(·)}(Ω). Thenφ^{∗∗}_{min}(x, s) =φ_{min}(x, t) =φ(x, t) and so

(φ^{∗∗}_{min})_{∗}(x, t) = (φ_{min})_{∗}(x, t) =φ_{∗}(x, t) =

(N−α)t N αt1

_{N−α}^{N α}
t

N p(x) N−α

1 ift≤t_{1},

1

p∗(x)t^{p}∗(x)

ift≥t1.
As above we can prove that (φ^{∗∗}_{min})_{∗}satisfies the conditions of Theorem 3.4 and then
our trace result is an extension to Musielak-Orlicz framework of the one proved by
Fan in [8].

Proof of Theorem 3.4. Let u∈ W^{1}Lφ~(Ω). Because of the continuous embedding
W^{1}Lφ~(Ω),→L_{(φ}^{∗∗}

min)∗(Ω), the functionubelongs toL_{(φ}^{∗∗}

min)∗(Ω) and thenubelongs
to Lψmin(Ω). Clearly W^{1}Lφ~(Ω) ,→ W^{1,1}(Ω) and by the Gagliardo trace theorem
(see [10]) we have the embeddingW^{1,1}(Ω),→L^{1}(∂Ω). Hence, we conclude that for
allu∈ W^{1}L_{~}_{φ}(Ω) there holdsu|_{∂Ω} ∈L^{1}(∂Ω). Therefore, for every u∈ W^{1}L_{~}_{φ}(Ω)
the trace u|_{∂Ω} is well defined. Assume first that uis bounded and u6= 0. Since
(φ^{∗∗}_{min})_{∗}(·, t) is continuous on∂Ω, the functionubelongs toLψ_{min}(∂Ω). Let

k=kuk_{L}_{ψ}

min(∂Ω)= infn λ >0 :

Z

∂Ω

ψmin

x,u(x)

λ

dx≤1o
.
We distinguish the two cases: k≥ kukL_{(φ}∗∗

min)∗(Ω)andk <kukL_{(φ}∗∗

min)∗(Ω). Case 1: Assume that

k≥ kuk_{L}_{(φ}∗∗

min)∗(Ω). (3.30)

Going back to (3.16) we can repeat exactly the same lines withl(x) =ψ_{min} x,^{u(x)}_{k}
instead of the functionh, obtaining

klk_{W}1,1(Ω)≤1
4c +c_{3}

kkuk_{W}1L_{φ}_{~}(Ω)+klk_{L}1(Ω)

, (3.31)

wherec is the constant in the imbeddingW^{1,1}(Ω),→L^{1}(∂Ω), that is

kwk_{L}1(∂Ω)≤ckwk_{W}1,1(Ω), for allw∈W^{1,1}(Ω). (3.32)
Since (φ^{∗∗}_{min})_{∗}(·, t) is continuous on Ω, using Lemma 5.4 (in Appendix) with the
functionsf(x, t) =^{(φ}^{∗∗}^{min}^{)}_{t}^{∗}^{(x,t)} andg(x, t) = ^{l(x)}_{t} and=_{4c}^{1}, we obtain fort= ^{|u(x)|}_{k}

l(x)≤ 1

4c(φ^{∗∗}_{min})_{∗} x,u(x)
k

+K_{0}|u(x)|

k . (3.33)