subordinated to the gauge functiona(given by (1.5)) and kukm,A=kp T[u, u]k(A), k · k(A) being the Luxemburg norm onEA(Ω)

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∈W₀^mEA(Ω) andgα: Ω×R→R,|α|< m, be Carath´eodory functions.

We study the problem Jau= X

|α|<m

(−1)^|α|D^αgα(x, D^αu) in Ω,

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

W₀^mE_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 byW₀^mE_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^αgα(x, D^αu) in Ω, (1.2) D^αu= 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 W₀^mE_A(Ω), involving the only generalized derivatives of order mof the functions u, v∈W₀^mEA(Ω), satisfying

c₁ X

|α|=m

(D^αu)²≤T[u, u]≤c₂ X

|α|=m

(D^αu)² ∀u∈W₀^mL_A(Ω), (1.4) withc₁,c₂ being positive constants;

• kukm,A = kp

T[u, u]k(A) is a norm on W₀^mEA(Ω), k · k(A) designating the Luxemburg norm on the Orlicz spaceLA(Ω);

•Ja : W₀^mEA(Ω),k · km,A

→ W₀^mEA(Ω),k · km,A

∗

is the duality mapping on W₀^mE_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∈W₀^mEA(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 inW₀^1,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 W₀^mE_A(Ω),k · k_m,A

. Note that, in order to prove the uniform con- vexity of the space W₀^mEA(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 W₀^mE_A(Ω),k · km,A

allow us to show that the duality mapping on W₀^mEA(Ω),k · km,A

corresponding to the gauge functionais given by J_a(0) = 0,

J_au=a(k · k_m,A)k · k⁰_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∈W₀^mE_A(Ω) one hasD^αu

_∂Ω= 0,

|α| ≤ m−1, the approach is realized in W₀^mEA(Ω)-space. It is shown that if a pointu∈W₀^mEA(Ω) satisfies

Jau= (i^∗◦N◦i)u, or, equivalently,

u= (J_a⁻¹◦i^∗◦N◦i)u,

thenusatisfies (1.2) (in the sense of (W₀^mEA(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 inW₀^mEA(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 : W₀^mEA(Ω) → R, given by (7.13). In order to prove thatF possesses a critical point in W₀^mEA(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^−1pkuk_Lp(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 andt₀>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∈ L¹(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^mLA(Ω) W^mEA(Ω)

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

The spaces W^mLA(Ω) and W^mEA(Ω) are Banach spaces with respect to the norm

kukW^mL_A(Ω)= X

|α|≤m

kD^αuk²_(A)^1/2

. (2.4)

If Ω has the segment property, thenC^∞(Ω) is dense inW^mEA(Ω) [1, Theorem 8.28]. The spaceW₀^mEA(Ω) is defined as the norm-closure ofD(Ω) inW^mEA(Ω).

Now, let us suppose that the boundary∂Ω of Ω isC¹. Consider the “restriction to ∂Ω” mapping eγ : C^∞(Ω) → C(∂Ω), eγ(u) = u|∂Ω. This mapping is continuous from C^∞(Ω),k · kW¹L_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 W¹E_A(Ω),k · k_W1L_A(Ω)

to E_A(∂Ω),k · k_EA(∂Ω) . Theorem 2.8 ([19, Proposition 2.3]). The kernel of the trace mapping γ:W¹E_A(Ω)→E_A(∂Ω)isW₀¹E_A(Ω).

The following results are useful.

Theorem 2.9 ([7]). W^mLA(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^αu)dx≤cm

Z

Ω

X

|α|=m

A(cm,ΩD^αu)dx, for allu∈W₀^mLA(Ω).

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

|α|≤m

kD^αuk²_(A)^1/2

and X

|α|=m

kD^αuk²_(A)^1/2 are equivalent onW₀^mL_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+1N

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+1N

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+1N

dτ, t≥0. (2.9)

Theorem 2.12([1]). If theN-functionA satisfies (2.7)and (2.8), then W₀¹LA(Ω)→LA∗(Ω).

Moreover, if Ω0 is a bounded subdomain ofΩ, then the imbeddings W₀¹L_A(Ω)→L_B(Ω₀)

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 W₀¹LA(Ω)→EA(Ω)

holds.

3. Geometry and smoothness of the space (W₀^mE_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 W₀^mE_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 · k⁰_(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∈W₀^mEA(Ω), satisfying the inequalities (1.4). From these inequalities and taking into account Corollary 2.11, we obtain thatW₀^mEA(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 W₀^mE_A(Ω),k · k_m,A

is smooth. Thus, the normk · k_m,A is Gˆateaux-differentiable onW₀^mE_A(Ω). Foru6= 0_Wm

0 E_A(Ω), we have

hk · k⁰_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∈W₀^mEA(Ω). (3.3)

Moreover, if the N-function A satisfies the ∆2-condition, thenu→ k · k⁰_m,A(u)is continuous thusk · km,A is Fr´echet-differentiable.

Proof. Let u 6= 0 be in W₀^mEA(Ω), that is p

T[u, u] 6= 0_EA(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 :W₀^mEA(Ω) →EA(Ω) is given byP(u) =p

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

hQ⁰(v), hi=kvk⁰_(A)(h), (3.4) for all v, h∈E_A(Ω), v6= 0_EA(Ω). Simple computations show that the operator P is Gˆateaux differentiable atuand

P⁰(u)(h) = T[u, h]

pT[u, u], (3.5)

for all u, h ∈ W₀^mEA(Ω), u 6= 0W₀^mE_A(Ω). Combining (3.4) and (3.5), we obtain thatψ is Gˆateaux differentiable atuand

hψ⁰(u), hi=hQ⁰(P u), P⁰(u)(h)i

=hk · k⁰_(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→ ψ⁰(u) is continuous. In order to do that it is sufficient to show that any sequence (u_n)_n ⊂ W₀^mE_A(Ω) converging to u∈W₀^mE_A(Ω) contains a subsequence (u_nk)_k ⊂(u_n)_n such thatψ⁰(u_nk)→ψ⁰(u), ask→ ∞, in W₀^mE_A(Ω)∗

. We set hψ⁰(u), hi=hϕ(u), hi

q(u) , ∀h∈W₀^mE_A(Ω), whereϕ:W₀^mEA(Ω)→W₀^mEA(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:W₀^mE_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 W₀^mEA(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_nk)_k ⊂(u_n)_n andw∈K_A(Ω) =E_A(Ω), such that

a

pT[un_k, un_k](x) kun_kkm,A

→a

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

ask→ ∞, for a.e. x∈Ω (3.9) and

a

pT[u_nk, u_nk](x) kun_kkm,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_nk−u, u_nk−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_kkm,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_kkm,A

pT[u_nk, u_nk](x)≤w(x)·w₁(x), for a.e. x∈Ω.

Sincew·w1∈L¹(Ω), by using (3.8) and Lebesgue’s dominated convergence theorem, it follows that

Z

Ω

a

pT[un_k, un_k](x) ku_nkk_m,A

pT[un_k, un_k](x) ku_nkk_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 W₀^mEA(Ω)∗

. But

T[u, v] = X

|α|=|β|=m

c_αβ(x)D^αuD^βv, 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_αβh a

pT[un_k, un_k](x) kun_kkm,A

D^αu_nk pT[un_k, un_k](x)

−a

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

D^αu pT[u, u](x)

i

D^βhdx

≤M X

|α|=|β|=m

Z

Ω

h a

pT[un_k, un_k](x) kun_kkm,A

D^αu_nk pT[un_k, un_k](x)

−a

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

D^αu pT[u, u](x)

i D^βhdx

.

(3.14)

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

a

pT[un_k, un_k] kun_kkm,A

D^αu_nk

pT[un_k, un_k]−a

pT[u, u]

kukm,A

D^αu

pT[u, u] ∈L_A(Ω).

Moreover, we will show that a

pT[un_k, un_k] ku_nkk_m,A

D^αun_k

pT[u_nk, u_nk]−a

pT[u, u]

kuk_m,A

D^αu

pT[u, u] ∈E_A(Ω) =K_A(Ω).

(3.15) Indeed,a

√

T[u,u]

kukm,A

√D^αu

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^αu

pT[u, u]. ≤ 1

√c₁; therefore

a

pT[u, u]

kukm,A

D^αu

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^αu

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_nkk_m,A

D^αun_k

pT[u_nk, u_nk] ∈K_A(Ω) =E_A(Ω);

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

|(ϕ(u_nk)−ϕ(u)) (h)| ≤M₁ X

|α|=|β|=m

a

pT[un_k, un_k](x) kun_kkm,A

D^αu_nk pT[un_k, un_k](x)

−a

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

D^αu

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^αun_k

pT[u_nk, u_nk](x)

−a

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

D^αu

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

a

pT[u_nk, u_nk](x) ku_nkk_m,A

D^αun_k

pT[u_nk, u_nk](x)−a

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

D^αu 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_kkm,A

D^αu_nk

pT[un_k, un_k] ∈E_A(Ω), fe=a

pT[u, u]

kukm,A

D^αu

pT[u, u] ∈E_A(Ω) From (1.4) and (3.11), it follows

D^αun_k(x)→D^αu(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)

√c₁, for a.e. x∈Ω, withw∈K_A(Ω) =E_A(Ω). Setting

f = w

√c₁ ∈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 W₀^mEA(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)≥p₁(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_∗=

ra²+b²

2 −c². (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)≤_λ¹ <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^pu) ≤ k^pρ(u) = 0, thereforeρ(2^pu) = 0. Consequently k2^pukA ≤ρ(2^pu) + 1 = 1, therefore kuk(A)≤ kukA≤ ₂¹p < 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+v₂ 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, thenW₀^mEA(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 ¹₂ < 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 ∈W₀^mEA(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−v₂ km,A =kq

T[^u−v₂ ,^u−v₂ ]k_(A)> ^ε₂. From Proposition 3.12 it follows that there exists η > 0 such that R

ΩA q

T[^u−v₂ ,^u−v₂ ]

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+v₂ ,^u+v₂ ]

dx < 1−η.

From the above remark, we conclude that there is aδ >0 depending onεsuch that k^u+v₂ k_m,A=kq

T[^u+v₂ ,^u+v₂ ]k_(A)<1−δ.

4. Duality mapping on W₀^mE_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_ϕ0 ={0},

J_ϕx=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_ϕx6=∅}=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_ϕis 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 · k⁰(x) ∈ X^∗ such that, for any h∈X, the following equality

limt→0

kx+thk − kxk

t =hk · k⁰(x), hi holds. Since, at anyx6= 0, the gradient of the norm satisfies

kk · k⁰(x)k= 1, hk · k⁰(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_ϕx=ϕ(kxk)k · k⁰(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=ψ⁰(x) =

(0 ifx= 0

ϕ(kxk)k · k⁰(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_ϕx_n* J_ϕx.

Indeed, since (x_n)_n is bounded andkJϕx_nk=ϕ(kxnk), it follows that (Jϕx_n)_n is bounded inX^∗. SinceX^∗ is reflexive, in order to proveJ_ϕx_n * J_ϕxit is enough to prove thatJ_ϕxis the unique point in the weak closure of (J_ϕx_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_ϕx−J_ϕy, 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⁻¹J_ϕ^∗−1, (4.8)

where J_ϕ^∗−1 :X^∗→X^∗∗ is the duality mapping on X^∗ corresponding to the gauge functionϕ⁻¹andχ: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⁻¹x^∗=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⁻¹x^∗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⁻¹x^∗, x^∗i=hχ(x), x^∗i=hx^∗, xi=ϕ⁻¹(kx^∗k)kx^∗k,

kχJ_ϕ⁻¹x^∗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⁻¹x^∗_n=χ⁻¹J_ϕ^∗−1x^∗_n * χ⁻¹J_ϕ^∗−1x^∗=J_ϕ⁻¹x^∗. (4.13) On the other hand,

kJ_ϕ⁻¹x^∗_nk=kχ⁻¹J_ϕ^∗−1x^∗_nk=kJ_ϕ^∗−1x^∗_nk=ϕ⁻¹(kx^∗_nk)→ϕ⁻¹(kxk) =kJ_ϕ⁻¹x^∗k.

(4.14) From (4.13), (4.14) and the Kadeˇc-Klee property of X, we infer thatJ_ϕ⁻¹x^∗_n →

J_ϕ⁻¹x^∗.

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 (W₀^mEA(Ω), kukm,A) is the single valued operatorJϕ:W₀^mEA(Ω)→ W₀^mEA(Ω)^∗

defined by

J_ϕu=ψ⁰(u) =

(0 ifu= 0 ϕ(kukm,A)k · k⁰_m,A(u) ifu6= 0, where

ψ(u) =

Z kukm,A

0

ϕ(t)dt= Φ (kukm,A), ∀u∈W₀^mEA(Ω),

where Φ is the N-function generated by ϕ and, for any u 6= 0, k · k⁰_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^pu)≤k^pB(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_nk)_k ⊂(u_n)_n andh∈K_A(Ω) such that

k→∞lim u_nk(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_nk)(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_nk)(x))≤k

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

2 A(h(x)),