ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

VARIATIONAL AND TOPOLOGICAL METHODS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS

ON ORLICZ-SOBOLEV SPACES

GEORGE DINCA, PAVEL MATEI

Abstract. Leta :R→Rbe a strictly increasing odd continuous function with limt→+∞a(t) = +∞andA(t) =Rt

0a(s)ds,t∈R, theN-function gener-
ated bya. Let Ω be a bounded open subset ofR^{N},N ≥2,T[u, u] a nonneg-
ative quadratic form involving the only generalized derivatives of ordermof
the functionu∈W_{0}^{m}EA(Ω) andgα: Ω×R→R,|α|< m, be Carath´eodory
functions.

We study the problem Jau= X

|α|<m

(−1)^{|α|}D^{α}gα(x, D^{α}u) in Ω,

D^{α}u= 0 on∂Ω,|α| ≤m−1,
whereJa is the duality mapping on`

W_{0}^{m}E_{A}(Ω),k · k_{m,A}´

, subordinated to the gauge functiona(given by (1.5)) and

kuk_{m,A}=kp

T[u, u]k_{(A)},
k · k_{(A)} being the Luxemburg norm onEA(Ω).

By using the Leray-Schauder topological degree and the mountain pass theorem of Ambrosetti and Rabinowitz, the existence of nontrivial solutions is established. The results of this paper generalize the existence results for Dirichlet problems withp-Laplacian given in [12] and [13].

1. Introduction

Throughout this paper Ω denotes a bounded open subset ofR^{N},N ≥2. Leta:
R→Rbe a strictly increasing odd continuous function with lim_{t→+∞}a(t) = +∞.

Form∈N^{∗}, let us denote byW_{0}^{m}E_{A}(Ω) the Orlicz-Sobolev space generated by the
N−functionA, given by

A(t) = Z t

0

a(s)ds. (1.1)

2000Mathematics Subject Classification. 35B38, 35B45, 47J30, 47H11.

Key words and phrases. A priori estimate; critical points; Orlicz-Sobolev spaces;

Leray-Schauder topological degree; Duality mapping; Nemytskij operator;

Mountain Pass Theorem.

c

2007 Texas State University - San Marcos.

Submitted June 4, 2007. Published June 21, 2007.

G. Dinca was supported by Contract CERES 12/25.07.2006 from the CEEX programm, Romanian Ministry of Education and Research.

1

In this paper we study the existence of solutions of the boundary-value problem Jau= X

|α|<m

(−1)^{|α|}D^{α}gα(x, D^{α}u) in Ω, (1.2)
D^{α}u= 0 on∂Ω,|α| ≤m−1, (1.3)
in the following functional framework:

• T[u, v] is a nonnegative symmetric bilinear form on the Orlicz-Sobolev space
W_{0}^{m}E_{A}(Ω), involving the only generalized derivatives of order mof the functions
u, v∈W_{0}^{m}EA(Ω), satisfying

c_{1} X

|α|=m

(D^{α}u)^{2}≤T[u, u]≤c_{2} X

|α|=m

(D^{α}u)^{2} ∀u∈W_{0}^{m}L_{A}(Ω), (1.4)
withc_{1},c_{2} being positive constants;

• kukm,A = kp

T[u, u]k(A) is a norm on W_{0}^{m}EA(Ω), k · k(A) designating the
Luxemburg norm on the Orlicz spaceLA(Ω);

•Ja : W_{0}^{m}EA(Ω),k · km,A

→ W_{0}^{m}EA(Ω),k · km,A

∗

is the duality mapping on
W_{0}^{m}E_{A}(Ω),k · k_{m,A}

subordinated to the gauge functiona:

hJau, hi=

a(kukm,A)·R

Ωa

√

T[u,u]

kukm,A

√T[u,h]

T[u,u]dx R

Ωa

√

T[u,u]

kukm,A

√

T[u,u]

kukm,A dx

, u, h∈W_{0}^{m}EA(Ω); (1.5)

• gα: Ω×R→R, |α|< m, are Carath´eodory functions satisfying some appro- priate growth conditions.

The main existence results are contained in Theorems 6.4 and 7.4 and the tech- niques used are essentially based on Leray-Schauder topological degree and on the mountain pass theorem due to Ambrosetti and Rabinowitz, respectively.

Let us remark that for the particular choice of a(t) = |t|^{p−2}·t, 1 < p < ∞,
m= 1 andT[u, v] =∇u· ∇v, the existence results given by Theorems 6.4 and 7.4
reduce to the well known existence results of the weak solution inW_{0}^{1,p}(Ω) for the
Dirichlet problem

−∆pu=g0(x, u) in Ω u= 0 on∂Ω.

The plan of the paper is as follows: In section 2, some fundamental results concerning the Orlicz-Sobolev spaces are given; these results are taken from Adams [1], Gossez [19], Krasnosel’skij and Rutitskij [22], Tienari [30].

The main results of section 3 concern the smoothness and the uniform convexity
of the space W_{0}^{m}E_{A}(Ω),k · k_{m,A}

. Note that, in order to prove the uniform con-
vexity of the space W_{0}^{m}EA(Ω),k · km,A

, an inequality given by Proposition 3.9 and playing a similar role to that of Clarkson’s inequalities is used. This inequality is a corollary of a result due to Gr¨oger [20] (see, also Langenbach [23]).

The content of section 4 is as follows: the smoothness and the uniform convexity
of the space W_{0}^{m}E_{A}(Ω),k · km,A

allow us to show that the duality mapping on
W_{0}^{m}EA(Ω),k · km,A

corresponding to the gauge functionais given by
J_{a}(0) = 0,

J_{a}u=a(k · k_{m,A})k · k^{0}_{m,A}(u), u6= 0.

Moreover,J_{a} is bijective with a continuous inverse, J_{a}^{−1}.

Section 5 deals with the properties of the so called Nemytskij operator on Orlicz spaces. These properties will be used later coupled with compact imbeddings of Orlicz-Sobolev spaces in some Orlicz spaces (a prototype of such a theorem is Theorem 2.12, due to Donaldson and Trudinger [15] (see, also Adams [1]).

In section 6, the existence of a solution for problem (1.2), (1.3), reduces to a
fixed point existence theorem. Since for any u∈W_{0}^{m}E_{A}(Ω) one hasD^{α}u

_{∂Ω}= 0,

|α| ≤ m−1, the approach is realized in W_{0}^{m}EA(Ω)-space. It is shown that if a
pointu∈W_{0}^{m}EA(Ω) satisfies

Jau= (i^{∗}◦N◦i)u,
or, equivalently,

u= (J_{a}^{−1}◦i^{∗}◦N◦i)u,

thenusatisfies (1.2) (in the sense of (W_{0}^{m}EA(Ω))^{∗}), that isuis a weak solution for
(1.2), (1.3). In writing of compact operatorP =J_{a}^{−1}◦i^{∗}◦N◦i, i^{∗} is the adjoint
ofiand the meaning ofiandN are given by Propositions 6.2 and 6.3 respectively.

In order to prove thatP possesses a fixed point inW_{0}^{m}EA(Ω), an a priori estimate
method is used.

In section 7, the existence of a solution for problem (1.2), (1.3), reduces to
proving the existence of a critical point for the functional F : W_{0}^{m}EA(Ω) → R,
given by (7.13). In order to prove thatF possesses a critical point in W_{0}^{m}EA(Ω),
we show that F has a mountain-pass geometry. Consequently, the mountain pass
theorem of Ambrosetti and Rabinowitz applies.

In section 8, some examples of functions a for which existence results for the problem (1.2), (1.3) may be obtained are given. It would be notice that the same functionaappears in examples 8.3 and 8.4; however, the corresponding hypotheses being different, the existence results are obtained by using distinct techniques: the mountain-pass theorem for example 8.3 and a priori estimate method for exam- ple 8.4. The same is true for examples 8.6 and 8.7. The only a Leray-Schauder technique can be applied for example 8.8. A slight modification of functiona, ap- pearing in example 8.8, enables the use of the mountain-pass theorem, as example 8.10 shows.

2. Orlicz and Orlicz-Sobolev spaces

Definition 2.1. A function A :R→R+ is called an N-function if it admits the representation

A(t) = Z |t|

0

a(s)ds,

where the function a: R+ →R+ is right-continuous for t ≥0, positive for t >0
and non-decreasing which satisfies the conditionsa(0) = 0, lim_{t→∞}a(t) =∞.

It is assumed everywhere below that the functionais continuous.

Remark 2.2. In many applications, it will be convenient to extend the functiona for negative values of the argument. Thus, letea:R→R+be the function given by

ea(s) =

(a(t), ift≥0

−a(−t), ift <0.

Then, the functionA:R→R+, A(t) =

Z t

0 ea(s)ds,

is anN-function. Obviously, the functioneais continuous and odd.

Throughout this paper, we suppose that a:R→Ris a strictly increasing odd
continuous function with lim_{t→+∞}a(t) = +∞ andA is the N−function given by
(1.1).

Let us consider theOrlicz class

K_{A}(Ω) ={u: Ω→Rmeasurable;

Z

Ω

A(u(x))dx <∞}.

TheOrlicz space L_{A}(Ω) is defined as the linear hull ofK_{A}(Ω) and it is a Banach
space with respect to theLuxemburg norm

kuk(A)= inf{k >0;

Z

Ω

A u(x) k

dx≤1}.

Remark 2.3. Ifa(t) =|t|^{p−2}·t, 1< p <∞, thenA(t) = ^{|t|}_{p}^{p}, KA(Ω) =LA(Ω) =
L^{p}(Ω) and kuk_{(A)}=p^{−}^{1}^{p}kuk_{L}p(Ω).

GenerallyK_{A}(Ω)⊂L_{A}(Ω). Moreover,K_{A}(Ω) =L_{A}(Ω) if and only ifAsatisfies
the ∆_{2}-condition: there exist k >0 andt_{0}>0 such that

A(2t)≤kA(t), for allt≥t0. (2.1) Theorem 2.4 ([22, p. 24]). A necessary and sufficient condition for theN- func- tion A to satisfy the ∆2-condition is that there exists a constant α such that, for u >0,

ua(u)

A(u) < α. (2.2)

TheN-function given by

A(u) = Z |u|

0

a^{−1}(s)ds,
is called thecomplementary N-function to A.

Remark 2.5. Letp,qbe such thatp >1 andp^{−1}+q^{−1}= 1. IfA(t) = ^{|t|}_{p}^{p}, then
A(t) = ^{|t|}_{q}^{q}. Consequently K_{A}(Ω) =L_{A}(Ω) =L^{q}(Ω).

We recallYoung’s inequality

uv≤A(u) +A(v), ∀u, v∈R

with equality if and only ifu=a^{−1}(|v|)·signv orv=a(|u|)·signu.

The spaceLA(Ω) is also a Banach space with respect to the Orlicz norm kukA= sup

Z

Ω

u(x)v(x)dx

;v∈K_{A}(Ω),
Z

Ω

A(v(x))dx≤1 . Moreover [22, p. 80],

kuk(A)≤ kukA≤2kuk(A), ∀u∈L_{A}(Ω).

One also has aH¨older’s type inequality: ifu∈LA(Ω) andv ∈ L_{A}(Ω), then uv∈
L^{1}(Ω) and

Z

Ω

u(x)v(x)dx

≤2kuk(A)kvk_{(A)}. (2.3)

We shall denote the closure of L^{∞}(Ω) inLA(Ω) byEA(Ω). One has EA(Ω)⊂
KA(Ω) andEA(Ω) =KA(Ω) if and only ifA satisfies the ∆2 -condition. We shall
denote byQ

EA(Ω), r

the set of thoseufromLA(Ω) whose distance (with respect to the Orlicz norm) toEA(Ω) is strictly less thanr. If theN -functionAdoes not satisfy the ∆2-condition, then

Y(EA(Ω), r)⊂KA(Ω)⊂Y

(EA(Ω), r), the inclusions being proper.

Theorem 2.6 ([22, p. 79]). If u∈LA(Ω) andkuk(A) ≤1, then u∈ KA(Ω) and ρ(u;A) =R

ΩA(u(x))dx≤ kuk(A). If u∈ L_{A}(Ω) and kuk(A)>1, thenρ(u;A)≥
kuk(A).

Lemma 2.7 ([18]). Ifu∈EA(Ω), then a(|u|)∈KA(Ω).

The Orlicz-Sobolev spaceW^{m}LA(Ω) W^{m}EA(Ω)

is the space of allu∈LA(Ω)
whose distributional derivativesD^{α}uare inLA(Ω) (EA(Ω)) for anyα, with|α| ≤m;

The spaces W^{m}LA(Ω) and W^{m}EA(Ω) are Banach spaces with respect to the
norm

kukW^{m}L_{A}(Ω)= X

|α|≤m

kD^{α}uk^{2}_{(A)}^{1/2}

. (2.4)

If Ω has the segment property, thenC^{∞}(Ω) is dense inW^{m}EA(Ω) [1, Theorem
8.28]. The spaceW_{0}^{m}EA(Ω) is defined as the norm-closure ofD(Ω) inW^{m}EA(Ω).

Now, let us suppose that the boundary∂Ω of Ω isC^{1}. Consider the “restriction
to ∂Ω” mapping eγ : C^{∞}(Ω) → C(∂Ω), eγ(u) = u|∂Ω. This mapping is continuous
from C^{∞}(Ω),k · kW^{1}L_{A}(Ω)

to C(∂Ω),k · kL_{A}(∂Ω)

[19, p. 69]. Consequently, the mapping eγ can be extended into a continuous mapping, denoted γ and called the

”trace mapping”, from W^{1}E_{A}(Ω),k · k_{W}1L_{A}(Ω)

to E_{A}(∂Ω),k · k_{E}_{A}_{(∂Ω)}
.
Theorem 2.8 ([19, Proposition 2.3]). The kernel of the trace mapping
γ:W^{1}E_{A}(Ω)→E_{A}(∂Ω)isW_{0}^{1}E_{A}(Ω).

The following results are useful.

Theorem 2.9 ([7]). W^{m}LA(Ω) is reflexive if and only if the N-functions A and
A satisfy the∆_{2}-condition.

Proposition 2.10 ([18]). There exist constants cm andcm,Ωsuch that Z

Ω

X

|α|<m

A(D^{α}u)dx≤cm

Z

Ω

X

|α|=m

A(cm,ΩD^{α}u)dx,
for allu∈W_{0}^{m}LA(Ω).

Corollary 2.11 ([18]). The two norms X

|α|≤m

kD^{α}uk^{2}_{(A)}^{1/2}

and X

|α|=m

kD^{α}uk^{2}_{(A)}^{1/2}
are equivalent onW_{0}^{m}L_{A}(Ω).

We recall that, ifAandB are twoN-functions, we say thatBdominatesAnear infinity if there exist positive constantskandt0 such that

A(t)≤B(kt) (2.5)

for all t ≥t0. The two N-functions A and B are equivalent near infinity if each dominates the other near infinity. IfB dominatesAnear infinity andAandB are not equivalent near infinity, then we say that A increases essentially more slowly than B near infinity and we denote A ≺≺ B. This is the case if and only if for everyk >0

t→∞lim A(kt)

B(t) = 0. (2.6)

If theN-functionsAandBare equivalent near infinity, thenAandB define the same Orlicz space [1, p. 234].

Let us now introduce the Orlicz-Sobolev conjugateA_{∗} of theN-functionA. We
shall always suppose that

t→0lim Z 1

t

A^{−1}(τ)
τ^{N+1}^{N}

dτ <∞, (2.7)

replacing, if necessary,Aby anotherN-function equivalent toAnear infinity (which determines the same Orlicz space).

Suppose also that

t→∞lim Z t

1

A^{−1}(τ)
τ^{N+1}^{N}

dτ =∞. (2.8)

With (2.8) satisfied, we define theSobolev conjugate A∗ ofAby setting
A^{−1}_{∗} (t) =

Z t

0

A^{−1}(τ)
τ^{N+1}^{N}

dτ, t≥0. (2.9)

Theorem 2.12([1]). If theN-functionA satisfies (2.7)and (2.8), then
W_{0}^{1}LA(Ω)→LA∗(Ω).

Moreover, if Ω0 is a bounded subdomain ofΩ, then the imbeddings
W_{0}^{1}L_{A}(Ω)→L_{B}(Ω_{0})

exist and are compact for anyN-functionB increasing essentially more slowly than
A_{∗} near infinity.

Theorem 2.13([30, Theorem 2.7]). The compact imbedding
W_{0}^{1}LA(Ω)→EA(Ω)

holds.

3. Geometry and smoothness of the space (W_{0}^{m}E_{A}(Ω),k · km,A)
Definition 3.1. The space X is said to be smooth, if for each x∈ X, x6= 0_{X},
there exists a unique functionalx^{∗}∈X^{∗}, such thatkx^{∗}k= 1 andhx^{∗}, xi=kxk.

The following results will be useful.

Theorem 3.2([10]). Let(X,kk)be a real Banach space. The norm ofX is Gˆateaux differentiable if and only if X is smooth.

In order to study the smoothness of the space W_{0}^{m}E_{A}(Ω), we recall a result
concerning the differentiability of the norm on Orlicz spaces.

Theorem 3.3 ([22]). The Luxemburg norm k · k(A) is Gˆateaux-differentiable on EA(Ω). For u6= 0, we have

hk · k^{0}_{(A)}(u), hi=
R

Ωa _{kuk}^{u(x)}

(A)

h(x)dx R

Ωa _{kuk}^{u(x)}

(A)

u(x)
kuk_{(A)}dx

, for allh∈E_{A}(Ω). (3.1)

Moreover, if the N-functionA satisfies the∆2-condition, then the normk · k_{(A)}is
Fr´echet-differentiable onEA(Ω).

The following results will be also useful.

Lemma 3.4 ([30, Lemma 2.5]). If (un)n ⊂EA(Ω) with un →u in EA(Ω), then
there exists h∈KA(Ω)⊂LA(Ω) and a subsequence (un_{k})n_{k} such that |un_{k}(x)| ≤
h(x)a.e. and un_{k}(x)→u(x)a.e.

Lemma 3.5 ([22, Lemma 18.2]). Let A and A be mutually complementary N - functions the second of which satisfies the∆2-condition. Suppose that the derivative a of A is continuous. Then, the operator Na, defined by means of the equality Nau(x) =a(|u(x)|), acts from Q

(EA(Ω),1) intoK_{A}(Ω) =L_{A}(Ω) =E_{A}(Ω) and is
continuous.

Now, let T[u, v] be a nonnegative symmetric bilinear form involving the only
generalized derivatives of ordermof the functionsu, v∈W_{0}^{m}EA(Ω), satisfying the
inequalities (1.4). From these inequalities and taking into account Corollary 2.11,
we obtain thatW_{0}^{m}EA(Ω) may be (equivalent) renormed by using the norm

kuk_{m,A}=kp

T[u, u]k_{(A)}. (3.2)

Theorem 3.6. The space W_{0}^{m}E_{A}(Ω),k · k_{m,A}

is smooth. Thus, the normk · k_{m,A}
is Gˆateaux-differentiable onW_{0}^{m}E_{A}(Ω). Foru6= 0_{W}m

0 E_{A}(Ω), we have

hk · k^{0}_{m,A}(u), hi=
R

Ωa

√

T[u,u](x) kukm,A

√T[u,h](x) T[u,u](x)dx R

Ωa

√

T[u,u](x) kukm,A

√

T[u,u](x) kukm,A dx

, for allh∈W_{0}^{m}EA(Ω). (3.3)

Moreover, if the N-function A satisfies the ∆2-condition, thenu→ k · k^{0}_{m,A}(u)is
continuous thusk · km,A is Fr´echet-differentiable.

Proof. Let u 6= 0 be in W_{0}^{m}EA(Ω), that is p

T[u, u] 6= 0_{E}_{A}_{(Ω)}. Let us denote
ψ(u) =kp

T[u, u]k(A). It is obvious thatψ can be written as a productψ=QP,
where Q:EA(Ω)→Ris given byQ(v) =kvk(A) and P :W_{0}^{m}EA(Ω) →EA(Ω) is
given byP(u) =p

T[u, u]. The functionalQis Gˆateaux differentiable (see Theorem 3.3) and

hQ^{0}(v), hi=kvk^{0}_{(A)}(h), (3.4)
for all v, h∈E_{A}(Ω), v6= 0_{E}_{A}_{(Ω)}. Simple computations show that the operator P
is Gˆateaux differentiable atuand

P^{0}(u)(h) = T[u, h]

pT[u, u], (3.5)

for all u, h ∈ W_{0}^{m}EA(Ω), u 6= 0W_{0}^{m}E_{A}(Ω). Combining (3.4) and (3.5), we obtain
thatψ is Gˆateaux differentiable atuand

hψ^{0}(u), hi=hQ^{0}(P u), P^{0}(u)(h)i

=hk · k^{0}_{(A)}(P u), T[u, h]

pT[u, u]i

= R

Ωa

√

T[u,u](x)
kuk_{m,A}

√T[u,h](x) T[u,u](x)dx R

Ωa

√

T[u,u](x)
kuk_{m,A}

√

T[u,u](x)
kuk_{m,A} dx

.

Now, we will show that the mapping u7→ ψ^{0}(u) is continuous. In order to do
that it is sufficient to show that any sequence (u_{n})_{n} ⊂ W_{0}^{m}E_{A}(Ω) converging to
u∈W_{0}^{m}E_{A}(Ω) contains a subsequence (u_{n}_{k})_{k} ⊂(u_{n})_{n} such thatψ^{0}(u_{n}_{k})→ψ^{0}(u),
ask→ ∞, in W_{0}^{m}E_{A}(Ω)∗

. We set
hψ^{0}(u), hi=hϕ(u), hi

q(u) , ∀h∈W_{0}^{m}E_{A}(Ω),
whereϕ:W_{0}^{m}EA(Ω)→W_{0}^{m}EA(Ω) is defined by

hϕ(u), hi= Z

Ω

a

pT[u, u](x)
kuk_{m,A}

T[u, h](x)
pT[u, u](x)dx
andq:W_{0}^{m}E_{A}(Ω)→Ris given by

q(u) = Z

Ω

a

pT[u, u](x) kukm,A

pT[u, u](x) kukm,A

dx.

First, we show that if un →uin W_{0}^{m}EA(Ω), then the sequence (un)n contains a
subsequence (un_{k})k ⊂(un)n such thatq(un_{k})→q(u) ask→ ∞. Since

|p

T[u_{n}, u_{n}]−p

T[u, u]| ≤p

T[u_{n}−u, u_{n}−u], (3.6)
it follows from

kun−ukm,A=kp

T[un−u, un−u]k_{(A)}→0 asn→ ∞, (3.7)
that

pT[un, un]→p

T[u, u] asn→ ∞, in EA(Ω); (3.8) therefore

pT[u_{n}, u_{n}]
kunkm,A

→

pT[u, u]

kukm,A

as n→ ∞, in EA(Ω).

By applying Lemma 3.5, and obtain a

pT[un, un] kunkm,A

→a

pT[u, u]

kukm,A

as n→ ∞, in E_{A}(Ω).

Then, from Lemma 3.4, it follows that there exists a subsequence (u_{n}_{k})_{k} ⊂(u_{n})_{n}
andw∈K_{A}(Ω) =E_{A}(Ω), such that

a

pT[un_{k}, un_{k}](x)
kun_{k}km,A

→a

pT[u, u](x) kukm,A

ask→ ∞, for a.e. x∈Ω (3.9) and

a

pT[u_{n}_{k}, u_{n}_{k}](x)
kun_{k}km,A

≤w(x), for a.e. x∈Ω. (3.10)

Taking into account (3.7), written for (un_{k})k, and applying again Lemma 3.4, it
follows that there exists a subsequence (also denoted (un_{k})k), and w1 ∈ KA(Ω)
such that

pT[u_{n}_{k}−u, u_{n}_{k}−u](x)→0 as k→ ∞, for a.e. x∈Ω. (3.11)
and

pT[un_{k}, un_{k}](x)≤w1(x), for a.e. x∈Ω. (3.12)
Out of (3.11) and (3.6), we obtain

pT[un_{k}, un_{k}](x)→p

T[u, u](x) ask→ ∞, for a.e. x∈Ω. (3.13) Consequently

a

pT[un_{k}, un_{k}](x)
kun_{k}km,A

pT[un_{k}, un_{k}](x)

→a

pT[u, u](x) kukm,A

pT[u, u](x) ask→ ∞, for a.e. x∈Ω and

a

pT[un_{k}, un_{k}](x)
kun_{k}km,A

pT[u_{n}_{k}, u_{n}_{k}](x)≤w(x)·w_{1}(x), for a.e. x∈Ω.

Sincew·w1∈L^{1}(Ω), by using (3.8) and Lebesgue’s dominated convergence theorem,
it follows that

Z

Ω

a

pT[un_{k}, un_{k}](x)
ku_{n}_{k}k_{m,A}

pT[un_{k}, un_{k}](x)
ku_{n}_{k}k_{m,A} dx

→ Z

Ω

a

pT[u, u](x)
kuk_{m,A}

pT[u, u](x)

kuk_{m,A} dx, as k→ ∞,
which isq(un_{k})→q(u) ask→ ∞.

For the (un_{k})k obtained above, we shall show that

ϕ(un_{k})→ϕ(u), ask→ ∞, in W_{0}^{m}EA(Ω)∗

. But

T[u, v] = X

|α|=|β|=m

c_{αβ}(x)D^{α}uD^{β}v,
wherec_{αβ}∈ C(Ω), therefore they are bounded.

First let us remark that, for arbitraryh, one has

|(ϕ(un_{k})−ϕ(u)) (h)|

=

X

|α|=|β|=m

Z

Ω

c_{αβ}h
a

pT[un_{k}, un_{k}](x)
kun_{k}km,A

D^{α}u_{n}_{k}
pT[un_{k}, un_{k}](x)

−a

pT[u, u](x) kukm,A

D^{α}u
pT[u, u](x)

i

D^{β}hdx

≤M X

|α|=|β|=m

Z

Ω

h a

pT[un_{k}, un_{k}](x)
kun_{k}km,A

D^{α}u_{n}_{k}
pT[un_{k}, un_{k}](x)

−a

pT[u, u](x) kukm,A

D^{α}u
pT[u, u](x)

i
D^{β}hdx

.

(3.14)

We intend to apply H¨older’s inequality (2.3) in (3.14). SinceD^{β}h∈EA(Ω), for all
β with|β|=m, it is sufficient to show that

a

pT[un_{k}, un_{k}]
kun_{k}km,A

D^{α}u_{n}_{k}

pT[un_{k}, un_{k}]−a

pT[u, u]

kukm,A

D^{α}u

pT[u, u] ∈L_{A}(Ω).

Moreover, we will show that a

pT[un_{k}, un_{k}]
ku_{n}_{k}k_{m,A}

D^{α}un_{k}

pT[u_{n}_{k}, u_{n}_{k}]−a

pT[u, u]

kuk_{m,A}

D^{α}u

pT[u, u] ∈E_{A}(Ω) =K_{A}(Ω).

(3.15) Indeed,a

√

T[u,u]

kukm,A

√D^{α}u

T[u,u] ∈K_{A}(Ω), because

√

T[u,u]

kukm,A ∈EA(Ω), by Lemma 2.7, we obtaina

√

T[u,u]

kukm,A

∈K_{A}(Ω). On the other hand, sinceT satisfies inequalities (1.4),
we have

D^{α}u

pT[u, u]. ≤ 1

√c_{1};
therefore

a

pT[u, u]

kukm,A

D^{α}u

pT[u, u] ≤ 1

√c1

a

pT[u, u]

kukm,A

∈K_{A}(Ω) =E_{A}(Ω)
(theN-functionAsatisfies the ∆_{2}-condition). Consequently,

a

pT[u, u]

kukm,A

D^{α}u

pT[u, u] ∈K_{A}(Ω) =E_{A}(Ω). (3.16)
Now, using the same technique, we obtain

a

pT[un_{k}, un_{k}]
ku_{n}_{k}k_{m,A}

D^{α}un_{k}

pT[u_{n}_{k}, u_{n}_{k}] ∈K_{A}(Ω) =E_{A}(Ω);

therefore we have (3.15). Applying H¨older’s inequality in (3.14), we obtain

|(ϕ(u_{n}_{k})−ϕ(u)) (h)| ≤M_{1} X

|α|=|β|=m

a

pT[un_{k}, un_{k}](x)
kun_{k}km,A

D^{α}u_{n}_{k}
pT[un_{k}, un_{k}](x)

−a

pT[u, u](x) kukm,A

D^{α}u

pT[u, u](x)k_{(A)}
hkm,A.
Consequently,

kϕ(un_{k})−ϕ(u)k ≤M1

X

|α|=|β|=m

a

pT[un_{k}, un_{k}](x)
kunkkm,A

D^{α}un_{k}

pT[u_{n}_{k}, u_{n}_{k}](x)

−a

pT[u, u](x) kukm,A

D^{α}u

pT[u, u](x)k_{(A)}.
Finally, we show that

a

pT[u_{n}_{k}, u_{n}_{k}](x)
ku_{n}_{k}k_{m,A}

D^{α}un_{k}

pT[u_{n}_{k}, u_{n}_{k}](x)−a

pT[u, u](x)
kuk_{m,A}

D^{α}u
pT[u, u](x)

_{(A)}→0,
(3.17)
ask→ ∞.

We will use the following result [29, Theorem 14, p. 84]. An elementf ∈LA(Ω)
has an absolutely continuous norm if and only if for each measurablefn such that
fn →fea.e. and|fn| ≤ |f|, a.e., we havekfn−fek_{(A)}→0 asn→ ∞. The fact that
f ∈ LA(Ω) has an absolutely continuous norm means that for every ε > 0 there
exists a δ >0 such that kf ·χEkA< ε provided mes(E)< δ (E ⊂Ω). Moreover,
any function fromEA(Ω) has an absolutely continuous norm [22, Theorem 10.3].

Then, (3.17) follows from the above result with the following choices:

f_{k}=a

pT[un_{k}, un_{k}]
kun_{k}km,A

D^{α}u_{n}_{k}

pT[un_{k}, un_{k}] ∈E_{A}(Ω),
fe=a

pT[u, u]

kukm,A

D^{α}u

pT[u, u] ∈E_{A}(Ω)
From (1.4) and (3.11), it follows

D^{α}un_{k}(x)→D^{α}u(x), for a.e. x∈Ω;

therefore, taking into account (3.9), (3.13), we obtain

fk(x)→fe(x), as k→ ∞, for a.e. x∈Ω.

On the other hand, from (1.4) and (3.10), we have

|f_{k}(x)| ≤ w(x)

√c_{1}, for a.e. x∈Ω,
withw∈K_{A}(Ω) =E_{A}(Ω). Setting

f = w

√c_{1} ∈E_{A}(Ω),

it follows (3.17). It follows thatkϕ(un_{k})−ϕ(u)k →0 ask→ ∞.

Now, we will study the uniform convexity of the space W_{0}^{m}EA(Ω),k · km,A
.
To do it, we still need some prerequisites. We begin with a technical result due to
Gr¨oger ([20]) (see, also [23, p. 153]).

Lemma 3.7. Let A(u) =R|u|

0 p(t)dt andA1(u) =R|u|

0 p1(t)dtbe two N-functions, such that the functionspandp1 should satisfy the conditions

p(τ) τ ≥ p(t)

t , τ≥t >0, (3.18)

p(t+τ)−p(τ)≥p_{1}(t), τ ≥t >0. (3.19)

Then 1

2A(a) +1

2A(b)−A(c)≥A1(c_{∗}), (3.20)
where

a≥b≥0, a−b

2 ≤c≤a+b

2 , c_{∗}=

ra^{2}+b^{2}

2 −c^{2}. (3.21)
The next corollary is a direct consequence of the preceding lemma.

Corollary 3.8. LetA(u) =R|u|

0 p(t)dtbe anN-function. Suppose that the function p(t)/tis nondecreasing on(0,∞). Then

1

2A(a) +1

2A(b)−A(c)≥A(c_{∗}),
wherea,b,c andc_{∗} are as in (3.21).

Proposition 3.9. Let A(u) = R|u|

0 p(t)dt be an N-function. Suppose that the
function ^{p(t)}_{t} is nondecreasing on(0,∞). Then

1 2A p

T[u, u]

+1 2A p

T[v, v]

−A r

T[u+v 2 ,u+v

2 ]

≥A r

T[u−v 2 ,u−v

2 ] . Proof. We apply Corollary 3.8 witha=p

T[u, u], b=p T[v, v], c=

r T[u+v

2 ,u+v

2 ], c_{∗}=
r

T[u−v 2 ,u−v

2 ].

Proposition 3.10. Let A be an N-function. If the N-function A satisfies the

∆_{2}-condition, then

ρ:L_{A}(Ω) =E_{A}(Ω) =K_{A}(Ω)→R, ρ(u) =
Z

Ω

A(u(x))dx, is continuous.

Proof. Obviously,ρis convex, therefore it suffices to show thatρis upper bounded
on a neighborhood of 0. But, ifkuk_{(A)}<1, thenρ(u)≤ kuk_{(A)}<1.

Proposition 3.11. Let Abe an N-function. Then, one has:

(i) If ρ(u) =R

ΩA(u(x))dx= 1, thenkuk_{(A)}= 1;

(ii) if, in addition, A satisfies a ∆2-condition, then ρ(u) = R

ΩA(u(x))dx = 1
if and only ifkuk_{(A)}= 1.

Proof. (i) Indeed, we have

1 =ρ(u) = Z

Ω

A(u(x)

1 )dx≥ kuk(A),

the last inequality being justified by the definition of thek·k_{(A)}-norm. Ifkuk_{(A)}<1,
then (see Theorem 2.6), we have

Z

Ω

A(u(x))dx≤ kuk(A)<1, which is a contradiction.

(ii) Taking into account the result given by (i), the “only if” implication has to
be proved. Now, sincekuk_{(A)}= 1, we can write

ρ(u) = Z

Ω

A(u(x) 1 )dx=

Z

Ω

A( u(x)

kuk_{(A)})dx≤1.

The strict inequality cannot hold. Indeed, if for some u with kuk_{(A)} = 1, we
have R

ΩA(u(x))dx <1, then there exists ε > 0 such that R

ΩA(u(x))dx+ε < 1.

From Proposition 3.10, lim_{λ→1}_{+}ρ(λu) = ρ(u), therefore, there exists δ > 0, such
that for eachλwith|λ−1|< δ, we have

Z

Ω

A(λu(x))dx− Z

Ω

A(u(x))dx < ε.

It follows that, for 1< λ < 1 +δ,R

ΩA(λu(x))dx <R

ΩA(u(x))dx+ε <1. Since R

ΩA(λu(x))dx <1, we infer thatkuk_{(A)}≤_{λ}^{1} <1, which is a contradiction.

Proposition 3.12. Let A be an N-function which satisfies the ∆2-condition. If kuk(A)> ε, then there exists η >0such that R

ΩA(u(x))dx > η.

Proof. Letube such thatkuk(A)> ε. Assume that the assertion in the proposition is not true, therefore for eachηwe haveR

ΩA(u(x))dx≤η. This means thatρ(u) = R

ΩA(u(x))dx = 0. Then, the ∆_{2}-condition implies that, ρ(2^{p}u) ≤ k^{p}ρ(u) = 0,
thereforeρ(2^{p}u) = 0. Consequently k2^{p}ukA ≤ρ(2^{p}u) + 1 = 1, therefore kuk(A)≤
kukA≤ _{2}^{1}p < εforplarge enough, which is a contradiction.

Definition 3.13. The space (X,k · kX) is called uniformly convex if for eachε∈
(0,2] there exists δ(ε)∈(0,1] such that foru, v ∈X withkukX =kvkX = 1 and
ku−vkX≥ε, one hask^{u+v}_{2} kX ≤1−δ(ε).

Theorem 3.14. Let A(u) = R|u|

0 p(t)dt be an N-function. Suppose that the
function p(t)/t is nondecreasing on (0,∞). If the N-function A satisfies the ∆2-
condition, thenW_{0}^{m}EA(Ω) endowed with the norm

kukm,A=kp

T[u, u]k_{(A)}
is uniformly convex.

Proof. We start with the following technical remark: if theN-functionAsatisfies a

∆_{2}-condition andR

ΩA(u(x))dx <1−ηfor some 0< η <1, there isδ >0 such that
kuk_{(A)}<1−δ. In the contrary case, there isusatisfyingR

ΩA(u(x))dx <1−η for
whichkuk_{(A)}≥1−δfor any δ >0. In particular inequalityR

ΩA(u(x))dx <1−η
may be satisfied for someuwithkuk_{(A)}>1/2. On the other hand, everyusatisfying
R

ΩA(u(x))dx <1−η has to satisfykuk_{(A)}<1 (see Theorem 2.6 and Proposition
3.11). Puta= 1/kuk(A). Clearly 1< a <2,kauk(A) = 1 and R

ΩA(au(x))dx= 1 (again by Proposition 3.11).

Now, by the convexity ofAwe derive that 1 =

Z

Ω

A(au(x))dx

= Z

Ω

A(2(a−1)u(x) + (2−a)u(x))dx

≤(a−1) Z

Ω

A(2u(x))dx+ (2−a) Z

Ω

A(u(x))dx

≤(a−1)k Z

Ω

A(u(x))dx+ (2−a) Z

Ω

A(u(x))dx;

therefore

1≤[(a−1)k+ 2−a]· Z

Ω

A(u(x))dx <[(a−1)k+ 2−a]·(1−η).

On the other hand, from ^{1}_{2} < a < 1, 0 < η < 1 and k > 2, it follows that
[(a−1)k+ 2−a]·(1−η)<1, which is a contradiction.

Now, letε >0 be andu, v ∈W_{0}^{m}EA(Ω) such thatkukm,A =kp

T[u, u]k_{(A)}= 1
,kvkm,A=kp

T[v, v]k(A)= 1 andku−vkm,A=kp

T[u−v, u−v]k(A)> ε. Then
k^{u−v}_{2} km,A =kq

T[^{u−v}_{2} ,^{u−v}_{2} ]k_{(A)}> ^{ε}_{2}. From Proposition 3.12 it follows that there
exists η > 0 such that R

ΩA q

T[^{u−v}_{2} ,^{u−v}_{2} ]

dx > η. On the other hand, from Proposition 3.11, we have R

ΩA p T[u, u]

dx = R

ΩA(p

T[v, v])dx = 1. Taking

into account Proposition 3.9, we obtain that R

ΩAq

T[^{u+v}_{2} ,^{u+v}_{2} ]

dx < 1−η.

From the above remark, we conclude that there is aδ >0 depending onεsuch that
k^{u+v}_{2} k_{m,A}=kq

T[^{u+v}_{2} ,^{u+v}_{2} ]k_{(A)}<1−δ.

4. Duality mapping on W_{0}^{m}E_{A}(Ω),k · km,A

LetX be a real Banach space and letϕ:R+→R+ be a gauge function, i.e. ϕ is continuous, strictly increasing,ϕ(0) = 0 andϕ(t)→ ∞as t→ ∞.

By duality mapping corresponding to the gauge functionϕ we understand the
multivalued mappingJ_{ϕ}:X → P(X^{∗}), defined as follows:

J_{ϕ}0 ={0},

J_{ϕ}x=ϕ(kxk){u^{∗}∈X^{∗};ku^{∗}k= 1,hu^{∗}, xi=kxk}, ifx6= 0. (4.1)
According to the Hahn-Banach theorem it is easy to see that the domain of Jϕ is
the whole space:

D(J_{ϕ}) ={x∈X;J_{ϕ}x6=∅}=X.

Due to Asplund’s result [3],

Jϕ=∂ψ, ψ(x) = Z kxk

0

ϕ(t)dt, (4.2)

for any x ∈ X and ∂ψ stands for the subdifferential of ψ in the sense of convex analysis.

By the preceding definition, it follows thatJ_{ϕ}is single valued if and only ifX is
smooth, i.e. for anyx6= 0 there is a unique element u^{∗}(x)∈X^{∗} having the metric
properties

hu^{∗}(x), xi=kxk, ku^{∗}(x)k= 1 (4.3)
But it is well known (see, for example, Diestel [10]) that a real Banach spaceX is
smooth if and only if its norm is differentiable in the Gˆateaux sense, i.e. at any
point x ∈ X, x 6= 0 there is a unique element k · k^{0}(x) ∈ X^{∗} such that, for any
h∈X, the following equality

limt→0

kx+thk − kxk

t =hk · k^{0}(x), hi
holds. Since, at anyx6= 0, the gradient of the norm satisfies

kk · k^{0}(x)k= 1, hk · k^{0}(x), xi=kxk (4.4)
and it is the unique element in the dual space having these properties, we imme-
diately get that: if X is a smooth real Banach space, then the duality mapping
corresponding to a gauge function ϕis the single valued mapping Jϕ : X →X^{∗},
defined as follows:

Jϕ0 = 0,

J_{ϕ}x=ϕ(kxk)k · k^{0}(x), ifx6= 0. (4.5)
Remark 4.1. By coupling (4.5) with the Asplund’s result quoted above, we get:

ifX is smooth, then

Jϕx=ψ^{0}(x) =

(0 ifx= 0

ϕ(kxk)k · k^{0}(x) ifx6= 0, (4.6)
whereψis given by (4.2).

From (4.4) and (4.5), it follows that

kJϕxk=ϕ(kxk),

hJϕx, xi=ϕ(kxk)kxk, for allx∈X. (4.7) The following surjectivity result will play an important role in what follows:

Theorem 4.2. IfX is a real reflexive and smooth Banach space, then any duality
mapping Jϕ :X → X^{∗} is surjective. Moreover, if X is also strictly convex, then
Jϕ is a bijection of X ontoX^{∗}.

In proving the surjectivity ofJϕ, the main ideas are as follows: (for more details, see Browder [5], Lions [24], Deimling [9])

(i)Jϕ is monotone:

hJϕx−Jϕy, x−yi ≥(ϕ(kxk)−ϕ(kyk)) (kxk − kyk)≥0,∀x, y∈X.

The first inequality is a direct consequence of (4.7) while the second one follows fromϕbeing increasing.

(ii) Any duality mapping on a real smooth and reflexive Banach space is demi- continuous:

x_{n} →x⇒J_{ϕ}x_{n}* J_{ϕ}x.

Indeed, since (x_{n})_{n} is bounded andkJϕx_{n}k=ϕ(kxnk), it follows that (Jϕx_{n})_{n} is
bounded inX^{∗}. SinceX^{∗} is reflexive, in order to proveJ_{ϕ}x_{n} * J_{ϕ}xit is enough
to prove thatJ_{ϕ}xis the unique point in the weak closure of (J_{ϕ}x_{n})_{n}.

(iii)J_{ϕ} is coercive, in the sense that
hJϕx, xi

kxk =ϕ(kxk)→ ∞askxk → ∞.

According to a well-known surjectivity result due to Browder (see, for example,
Browder [5], Lions [24], Zeidler [31], Deimling [9]), ifX is a reflexive real Banach
space, then any monotone, demicontinuous and coercive operator T :X →X^{∗} is
surjective.

Consequently, from (i), (ii), (iii) and the Browder’s surjectivity result above
mentioned it follows that, under the hypotheses of Theorem 4.2,J_{ϕ} is surjective.

It can be shown that ifX is a strictly convex real Banach space, then any duality
mapping J_{ϕ}:X→ P(X^{∗}) is strictly monotone, in the following sense: ifx, y∈X
and x 6= y, then, for anyx^{∗} ∈ Jϕxand y^{∗} ∈ Jϕy one has hx^{∗}−y^{∗}, x−yi >0.

Clearly, the strict monotonicity implies the injectivity: ifx, y∈X andx6=y then
Jϕx∩Jϕy =∅. In particular, if the strictly convex real Banach space X is also a
smooth one, then any duality mappingJϕ:X →X^{∗} is strictly monotone:

hJ_{ϕ}x−J_{ϕ}y, x−yi>0, ∀x, y∈X, x6=y,
and, consequently, injective.

Corollary 4.3. IfX is a reflexive and smooth real Banach space having the Kadeˇc-
Klee property, then any duality mapping J_{ϕ} : X → X^{∗} is bijective and has a
continuous inverse. Moreover,

J_{ϕ}^{−1}=χ^{−1}J_{ϕ}^{∗}−1, (4.8)

where J_{ϕ}^{∗}−1 :X^{∗}→X^{∗∗} is the duality mapping on X^{∗} corresponding to the gauge
functionϕ^{−1}andχ:X →X^{∗∗}is the canonical isomorphism defined byhχ(x), x^{∗}i=
hx^{∗}, xi, for allx∈X, for allx^{∗} ∈X^{∗}.

Proof. The existence of J_{ϕ}^{−1} follows from Theorem 4.2. As far as formula (4.8) is
concerned, first we shall prove that, under the hypotheses of Corollary 4.3, any
duality mapping on X^{∗} (in particular, that corresponding to the gauge function
ϕ^{−1}) is single valued. This is equivalent with proving thatX^{∗} is smooth.

The smoothness of X^{∗} will be proved by using the (partial) duality between
strict convexity and smoothness given by the following theorem due to Klee (see
Diestel [10, Chapter 2,§2, Theorem 2]):

X^{∗} smooth (strictly convex) ⇒X strictly convex (smooth).

Clearly, ifX is reflexive, then

X^{∗} smooth (strictly convex) ⇔X strictly convex (smooth).

Now, by the hypotheses of Corollary 4.3,X is reflexive and smooth. Also, by the same hypotheses, X possesses the Kadeˇc-Klee property, that means: X is strictly convex and

[xn* xandkxnk → kxk]⇒xn →x. (4.9)
Consequently,X being reflexive, smooth and strictly convex so isX^{∗}.

Let us prove that equality (4.8) holds or, equivalently,

χJ_{ϕ}^{−1}x^{∗}=J_{ϕ}^{∗}−1x^{∗},∀x^{∗}∈X^{∗}. (4.10)
From the definition of duality mappings,J_{ϕ}^{∗}−1x^{∗} is the unique element in X^{∗∗}

having the metric properties

hJ_{ϕ}^{∗}−1x^{∗}, x^{∗}i=ϕ^{−1}(kx^{∗}k)kx^{∗}k,

kJ_{ϕ}^{∗}−1x^{∗}k=ϕ^{−1}(kx^{∗}k). (4.11)
We shall show thatχJ_{ϕ}^{−1}x^{∗}possesses the same metric properties and then the result
follows by unicity. Puttingx^{∗}=Jϕxit follows (by definition ofJϕ) that

x^{∗}=ϕ(kxk),

hx^{∗}, xi=ϕ(kxk)kxk=ϕ^{−1}(kx^{∗}k)kx^{∗}k
and, consequently, we deduce that

hχJ_{ϕ}^{−1}x^{∗}, x^{∗}i=hχ(x), x^{∗}i=hx^{∗}, xi=ϕ^{−1}(kx^{∗}k)kx^{∗}k,

kχJ_{ϕ}^{−1}x^{∗}k=kχ(x)k=kxk=ϕ^{−1}(kxk) (4.12)
Equality (4.10) follows by comparing (4.11) and (4.12) and using the uniqueness
result evoked above. Formula (4.8) is basic in proving the continuity ofJ_{ϕ}^{−1}. Indeed,
letx^{∗}_{n} →x^{∗} in X^{∗}. As any duality mapping on a reflexive Banach space, J_{ϕ}^{∗}−1 is
demicontinuous,J_{ϕ}^{∗}−1x^{∗}_{n}* J_{ϕ}^{∗}−1x^{∗}. Consequently, we deduce that

J_{ϕ}^{−1}x^{∗}_{n}=χ^{−1}J_{ϕ}^{∗}−1x^{∗}_{n} * χ^{−1}J_{ϕ}^{∗}−1x^{∗}=J_{ϕ}^{−1}x^{∗}. (4.13)
On the other hand,

kJ_{ϕ}^{−1}x^{∗}_{n}k=kχ^{−1}J_{ϕ}^{∗}−1x^{∗}_{n}k=kJ_{ϕ}^{∗}−1x^{∗}_{n}k=ϕ^{−1}(kx^{∗}_{n}k)→ϕ^{−1}(kxk) =kJ_{ϕ}^{−1}x^{∗}k.

(4.14)
From (4.13), (4.14) and the Kadeˇc-Klee property of X, we infer thatJ_{ϕ}^{−1}x^{∗}_{n} →

J_{ϕ}^{−1}x^{∗}.

Corollary 4.4. If X is a weakly locally uniformly convex, reflexive and smooth
real Banach space, then any duality mapping J_{ϕ} :X →X^{∗} is bijective and has a
continuous inverse given by (4.8).

Proof. Since any weakly locally uniformly convex Banach space has the Kadeˇc-Klee property (see Diestel[Chapter 2, §2, Theorems 3 and 4(iii)][10]) the result follows

by Corollary 4.3.

Theorem 4.5. Let ϕ be a gauge function. The duality mapping on (W_{0}^{m}EA(Ω),
kukm,A) is the single valued operatorJϕ:W_{0}^{m}EA(Ω)→ W_{0}^{m}EA(Ω)^{∗}

defined by

J_{ϕ}u=ψ^{0}(u) =

(0 ifu= 0
ϕ(kukm,A)k · k^{0}_{m,A}(u) ifu6= 0,
where

ψ(u) =

Z kukm,A

0

ϕ(t)dt= Φ (kukm,A), ∀u∈W_{0}^{m}EA(Ω),

where Φ is the N-function generated by ϕ and, for any u 6= 0, k · k^{0}_{m,A}(u) being
given by (3.3).

This result immediately follows by Theorem 3.6 and Remark 4.1.

5. Nemytskij operator on L_{A}(Ω)

We recall thatf : Ω×R→Ris aCarath´eodory function if it satisfies:

(i) for eachs∈R, the functionx→f(x, s) is Lebesgue measurable in Ω;

(ii) for a.e. x∈Ω, the functions→f(x, s) is continuous inR.

We make convention that in the case of a Carath´eodory function, the assertion x∈Ω to be understood in the sense a.e. x∈Ω.

Proposition 5.1([22, Theorem 17.1]). Suppose thatf : Ω×R→Ris a Carath´eo- dory function. Then, for each measurable function u, the function Nfu: Ω→R, given by

(Nfu)(x) =f(x, u(x)), for eachx∈Ω (5.1) is measurable in Ω.

Definition 5.2. Let M be the set of all measurable functions u : Ω → R, f :
Ω×R→ R be a Carath´eodory function. The operator N_{f} : M → M given by
(5.1) is calledNemytskij operator defined by Carath´eodory functionf.

Theorem here below states sufficient conditions when Nemytskij operator maps a Orlicz class KA(Ω) into another Orlicz class KB(Ω), being at the same time continuous and bounded. The following result is useful.

Theorem 5.3. LetAandBbe twoN-functions andf : Ω×R→Rbe a Carath´eo- dory function which satisfies the growth condition

|f(x, u)| ≤c(x) +bB^{−1}(A(u)), x∈Ω, u∈R, (5.2)
wherec∈KB(Ω) andb≥0 is a constant. Then the following statements are true:

(i) If B satisfies the ∆2-condition, then Nf is well-defined and mean bounded from KA(Ω) into KB(Ω) = EB(Ω). Moreover, Nf : EA(Ω),k · k(A)

→
EB(Ω),k · k_{(B)}

is continuous;

(ii) If both A and B satisfy the ∆2-condition, then Nf : EA(Ω),k · k(A)

→
EB(Ω),k · k_{(B)}

is norm bounded.

Proof. Let us first remark that the well-definedness ofNf as well as the continuity and the boundedness on every ballB(0, r)⊂LA(Ω), withr <1, may be obtained as consequences Theorem 17.6 in Krasnosel’skij and Rutickij ([22]). The proof of this theorem is quite complicated; that is why a direct proof of Theorem 5.3, including the supplementary result given by (ii), will be given below.

(i) Letu, v ∈R. SinceB is convex and satisfies the ∆_{2}-condition, one has
B(u+v) =B 2·1

2(u+v)

≤k

2(B(u) +B(v)). (5.3)
Letpbe such that 2^{p}≥b. SinceB satisfies the ∆2-condition, one has

B(bu)≤B(2^{p}u)≤k^{p}B(u). (5.4)
Now, letu∈K_{A}(Ω). By using (5.2), (5.3), ( 5.4) and integrating on Ω, we have

Z

Ω

B[Nf(u)(x)]dx= Z

Ω

B(|f(x, u(x))|)dx

≤ k 2 Z

Ω

B(c(x))dx+k^{p+1}
2

Z

Ω

A(u(x))dx <∞,

(5.5)

saying thenNf(KA(Ω))⊂LB(Ω) =EB(Ω).

From (5.5) it follows that, ifu∈KA(Ω) and R

ΩA(u(x))dx≤const., then Z

Ω

B[Nf(u)(x)]dx≤k 2

Z

Ω

B(c(x))dx+ const.;

thereforeNf transforms mean bounded sets in KA(Ω) into mean bounded sets in EB(Ω).

Now, let us consider u∈ EA(Ω). For the continuity ofNf, it suffices to show that every sequence (un)n⊂EA(Ω) such that

n→∞lim kun−ukA= 0

has a subsequence (un_{k})k such that Nf(un_{k}) → Nf(u) as k → ∞, in LB(Ω) =
EB(Ω).

Indeed, let (u_{n})_{n} be a sequence as above. By using Lemma 3.4, it follows that
there exists a subsequence (u_{n}_{k})_{k} ⊂(u_{n})_{n} andh∈K_{A}(Ω) such that

k→∞lim u_{n}_{k}(x) =u(x), a.e. x∈Ω (5.6)
and

|un_{k}(x)| ≤ |h(x)|, a.e. x∈Ω, k∈N. (5.7)
The functionf being a Carath´eodory function, it is clear that

k→∞lim Nf(un_{k})(x) =Nf(u)(x), a.e. x∈Ω,
therefore,

lim

k→∞B(N_{f}(u_{n}_{k})(x)−N_{f}(u)(x)) = 0, a.e. x∈Ω. (5.8)
On the other hand, from (5.2) it follows that

|Nf(un_{k})(x)|=|f(x, un_{k}(x))| ≤c(x) +bB^{−1}(A(h(x))), a.e. x∈Ω, k∈N.
Consequently, by using a similar argument to that in (5.5) and taking into account
(5.7), one obtains

B(N_{f}(u_{n}_{k})(x))≤k

2B(c(x)) +k^{p+1}

2 A(h(x)),