Existence of multiple solutions for a class of nonhomogeneous problems with critical growth

Existence of multiple solutions for a class of nonhomogeneous problems with critical growth

Mohammed Massar

Mohamed First University, Faculty of Sciences & Techniques, Al Hoceima, Morocco Received 27 April 2016, appeared 6 April 2017

Communicated by Gabriele Bonanno

Abstract. In this paper, we study the existence and multiplicity of solutions for (p₁(x),p2(x))-equation with critical growth. The technical approach is mainly based on the variational method combined with the genus theory.

Keywords: nonhomogeneous, multiple solutions, critical growth.

2010 Mathematics Subject Classification: 35J62, 35J66, 35D30.

1 Introduction

In this article, we are concerned with the following problem

(−_∆p1(x)u−_∆p2(x)u−a(x)|u|^m(x)−2u=λ|u|^q(x)−2u+ f(x,u) in Ω

u=0 on ∂Ω, (P_λ)

where Ω⊂ _R^N(N ≥ 1)is a bounded smooth domain, λis a positive parameter and f : Ω× R →_R is a continuous function which satisfies some assumptions provided later. Moreover, p₁,p2,q∈C(_Ω)andm(x) =max(p₁(x),p2(x))for all x∈Ω, such that

1< p⁻_i =min

x∈_Ωp_i(x)≤ p⁺_i =max

x∈_Ω

p_i(x)< N, i=1, 2, m⁺=max

x∈Ωm(x)<q⁻ =min

x∈_Ωq(x)≤q(x)≤m^∗(x) ∀x ∈_Ω, ^(mq1) wherem^∗(x) = _N^Nm_−m⁽₍^x_x⁾₎ for allx ∈_Ωand the set

B={x ∈_Ω:q(x) =m^∗(x)}is nonempty. (mq₂) In recent years, the study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. A more and more impor- tant number of surveys and books dealing with this type of problems and their corresponding functional spaces setting have been published (see [1–4,12,16–20]). We also have to mention

BEmail: massarmed@hotmail.com

the books [13] and [21] as important references in this field. This great interest may be jus- tified by their various physical applications. In fact, there are applications concerning elastic mechanics [25], electrorheological fluids [23,24], image restoration [9], dielectric breakdown, electrical resistivity and polycrystal plasticity [6,7] and continuum mechanics [5].

It is well known that although most of the materials can be accurately modeled with the help of the classical Lebesgue and Sobolev spacesL^pandW^1,p, where pis a fixed constant, but there are some nonhomogeneous materials, for which this is not adequate, e.g. the rheological fluids mentioned above, which are characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field. Thus it is necessary for the exponentpto be variable, hence the need for spaces with variable exponents.

This leads, on the one hand,to many interesting applications, and, on the other hand,to the study of much more mathematically complicated problems.

In [19], Mih˘ailescu considered the problem (−_∆p

1(x)u−_∆p

2(x)u=±(−λ|u|^m(x)−2u+|u|^q(x)−2u) _{in Ω}

u=0 on ∂Ω,

where m(x) = max{p₁(x),p2(x)} for any x ∈ _Ω or m(x) < q(x) < _N^Nm_−m⁽₍^x_x⁾₎ for any x ∈ _Ω.

In the first case, using mountain pass theorem, he established the existence of infinity many solutions. In the second case, by simple variational arguments, he proved that the problem has a solution forλlarge enough. The novelty of this paper lies in the fact we consider problem (P_λ), with growth q(x)which is critical in a set with positive measure. The difficulty in this case, is due to the lack of compactness of the imbedding W₀^1,m(x)(_Ω),→L^m∗(x)(_Ω) and the Palais–Smale condition for the corresponding energy functional could not be checked directly.

To deal with this difficulty, we use a version of the concentration compactness lemma due to Lions for variable exponents [8].

Here, we are interested in the existence and multiplicity of weak solutions under the following hypotheses ona(x)and f.

(a₁) a(x)∈ L^∞(_Ω)and there exists α>0 such that Z

Ω

|∇u|^p1(x)

p₁(x) +|∇u|^p2(x)

p₂(x) −a(x)|u|^m(x) m(x)

!

dx≥α Z

Ω

|u|^m(x)

m(x) ^dx, ∀u∈W₀^1,m(x)(_Ω); (a₂) m(x) =m⁺for all x∈_Γ⁺:={x∈ _Ω:a(x)>0};

(f₁) f ∈C(_Ω×_R,R), odd with respect tot such that limt→0

f(x,t)

|t|^m(x)−1 =0, uniformly inx∈_Ω,

|t|→+lim_∞

f(x,t)

|t|^q(x)−1 =_0, uniformly inx∈_Ω;

(f₂) F(x,t)≤ _m¹+f(x,t)t, ∀t∈ _{Rand a.e.} x∈_{Ω, where}F(x,t) =Rt

0 f(x,s)ds.

Example 1.1. In this example we just exhibit a function a(x) satisfying assumption (a₁). Let Ω = B(0, 2) := {x ∈ _R^N : |x| < 2}(N ≥ 3), p₁(x) = 2− ¹₄x₁− ₁₆¹|x|² and p₂(x) = 2− ¹₄x₂− ₁₆¹|x|² _{for all} x = (x₁,x₂, . . . ,x_N) ∈ _{Ω, where} |x|² = _∑^N_i=1x²_i. Then p_i ∈ C(_Ω)

and 1 < p_i⁻ ≤ p⁺_i < N, i = 1, 2. Let e₁ = (1, 0, . . . , 0). Then for any x ∈ Ω, the function h₁:t 7→ p₁(x+te₁)is monotone in Ix ={t:x+te₁∈_Ω}. In fact, we have

h₁(t) =2−¹

4(x₁+t)− ¹

16(x₁+t)²− ¹ 16

∑

x²_i,

thus h₁⁰(t) = −¹₈(2+x₁+t). Since |x₁+t| ≤ |x+te₁| < 2, h₁⁰(t) < 0 for all t ∈ I_x, and henceh₁is decreasing inIx. Similarly, fore₂= (0, 1, . . . , 0), the functionh₂:t 7→ p₂(x+te₂)is monotone inI_x ={t: x+te₂ ∈_Ω}. Thereforep_i,i=1, 2, satisfying conditions of [15, Theorem 3.3], thus the modular Poincaré inequality holds:

Z

Ω|∇u|^pi(x)dx≥ C_i Z

Ω|u|^pi(x)dx (C_i >0, i=1, 2), which is equivalent to

Z

Ω

|∇u|^pi(x)

p_i(x) ^dx≥C⁰_i Z

Ω

|u|^pi(x)

p_i(x) ^dx (C_i⁰ >0, i=1, 2), since 1< p⁻_i ≤ p_i(x)≤ p⁺_i < N.

Now, letV ∈ L^∞(_Ω) such that 1+V(x) > c > 0. Note that p_i, i = 1, 2, are log-Hölder continuous functions. Indeed, letx,y∈_Ωsuch that|x−y| ≤ ¹₂, then

|p₁(x)−p₁(y)| ≤ ¹

4|x₁−y₁|+ ¹ 16

|x|²− |y|²

≤ ¹

4|x−y|+ ¹

16||x| − |y||(|x|+|y|)

≤ ¹

4|x−y|+ ¹

16|x−y|(|x|+|y|)

≤ ¹ 2|x−y|

≤ ¹

2 log

1

|x−y|

.

In the same way, we get|p₂(x)−p₂(y)| ≤ ¹

2 log _|x−¹_y|. Applying [11, Theorem 2.2], we can find α₁,α₂>0 such that

Z

Ω

|∇u|^p1(x)

p₁(x) ^dx≥α₁ Z

Ω(1+V(x))|u|^p1(x) p₁(x) ^dx and

Z

Ω

|∇u|^p2(x)

p₂(x) ^dx≥α₂ Z

Ω(1+V(x))|u|^p2(x) p₂(x) ^dx.

Observe that|u|^p1(x)+|u|^p2(x) ≥ |u|^m(x), thus ^|u_p^|p1(x)

1(x) + ^|u_p^|p2(x)

2(x) ≥ ^|u_m^|m_(x^(x₎⁾. It follows that Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

!

dx≥ min(α₁,α₂)

Z

Ω(1+V(x)) |u|^p1(x)

p₁(x) + |u|^p2(x) p₂(x)

! dx

≥ min(α₁,α₂)

Z

Ω(1+V(x))|u|^m(x) m(x) ^dx.

Therefore Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p2(x)

! dx−

Z

Ωmin(α₁,α₂)V(x)|u|^m(x)

m(x) ^dx≥_min(α₁,α₂)

Z |u|^m(x) m(x) ^dx, and hence the functiona(x):=min(α₁,α₂)V(x)satisfies condition(a₁).

2 Preliminaries

Here, we state some interesting properties of the variable exponent Lebesgue and Sobolev spaces that will be useful to discuss problem (P_λ). Every where below we considerΩ ⊂ _R^N to be a bounded domain with smooth boundary andp(x)∈C+(_Ω), where

C+(_Ω) ={h∈ C(_Ω):h(x)>1 for allx ∈_Ω}. Define the variable exponent Lebesgue space by

L^p(x)(_Ω) =

u:Ω→_Rmeasurable : Z

Ω|u(x)|^p(x)dx <_∞

. This space endowed with the Luxemburg norm,

kuk_Lp(x)(_Ω) =_inf (

τ>_{0 :}

Z

Ω

u(x) τ

p(x)

dx≤₁ )

is a separable and reflexive Banach space. Denoting by L^p0(x)(_Ω) the conjugate space of L^p(x)(_Ω) where _p(¹_x) + _p0¹_(x) = 1; for any u ∈ L^p(x)(_Ω) and v ∈ L^p0(x)(_Ω) we have the fol- lowing Hölder type inequality

Z

Ω|uv|dx≤ 1

p⁻ + ¹ p⁰⁻

kuk_Lp(x)(_Ω)kvk

L^p0(x)(_Ω).

Now, we introduce the modular of the Lebesgue–Sobolev space L^p(x)(_Ω) as the mapping ρ_p(x): L^p(x)(_Ω)→R, defined by

ρ_p(x)(u) =

Z

Ω|u|^p(x)dx, ∀u∈ L^p(x)(_Ω).

In the following proposition, we give some relations between the Luxemburg norm and the modular.

Proposition 2.1([14]). If u,u_n∈ L^p(x)(_Ω),then following properties hold true:

(1) kuk_Lp(x)(_Ω) ≤1⇒ kuk^p+

L^p(x)(_Ω) ≤ρ_p(x)(u)≤ kuk^p−

L^p(x)(_Ω); (2) kuk_Lp(x)(_Ω) ≥1⇒ kuk^p−

L^p(x)(_Ω) ≤ρ_p(x)(u)≤ kuk^p+

L^p(x)(_Ω); (3) lim

n→_∞ku_nk_Lp(x)(_Ω) =0⇔ lim

n→_∞ρ_p(x)(u_n) =0;

(4) lim

n→_∞kunk_Lp(x)(_Ω) =_∞⇔ lim

n→_∞ρ_p(x)(un) =_∞.

Next, we define the variable exponent Sobolev spaceW^1,p(x)(_Ω)by W^1,p(x)(_Ω) =ⁿu∈ L^p(x)(_Ω)_:|∇u| ∈ L^p(x)(_Ω)^o_. endowed with the norm

kuk_W1,p(x)(_Ω) =kuk_Lp(x)(_Ω)+k∇uk_Lp(x)(_Ω),

where byk∇uk_Lp(x)(_Ω) we understandk |∇u| k_Lp(x)(_Ω). We denote byW₀^1,p(x)(_Ω)the closure of C₀^∞(_Ω) in W^1,p(x)(_Ω) with respect to the above norm. The spaceW^1,p(x)(_Ω)and W₀^1,p(x)(_Ω) are separable and reflexive Banach spaces.

Proposition 2.2 ([14]).

(₁) If r∈ C+(_Ω)and r(x)< p^∗(x)for all x∈_Ω,then the embedding W^1,p(x)(_Ω),→ L^r(x)(_Ω) is compact and continuous.

(2) There is a constant C>0such that

kuk_Lp(x)(_Ω)≤Ck∇uk_Lp(x)(_Ω) for all u∈W₀^1,p(x)(_Ω).

3 Main result

From now on, we considerm(x) =max(p₁(x),p₂(x))for all x ∈_{Ω. By}(2)of Proposition2.2, we know that kuk := k∇uk_Lm(x)(_Ω) and kuk_W1,m(x)(_Ω) are equivalent norms inW₀^1,m(x)(_Ω). In the following, we will usekukinstead ofkuk_W1,m(x)(_Ω) onW₀^1,m(x)(_Ω).

Definition 3.1. We say thatu ∈W₀^1,m(x)(_Ω)is a weak solution of problem (P_λ) if Z

Ω

|∇u|^p1(x)−2+|∇u|^p2(x)−2∇u∇vdx−

Z

Ωa(x)|u|^m(x)−2uvdx

=λ Z

Ω|u|^q(x)−2uvdx+

Z

Ω f(x,u)vdx, for all v∈W₀^1,m(x)(_Ω).

Our main result is the following theorem.

Theorem 3.2. Assume that(a₁)–(a₂),(f₁)–(f₂)and(mq₁)–(mq₂)hold. Then, there exits a sequence {λ_k} ⊂ (0,+_∞)withλ_k > λ_k+₁,such that for anyλ ∈(λ_k+₁,λ_k],problem(P_λ)has at least k pairs of nontrivial solutions.

4 Proof of main result

We will start by recalling an important abstract theorem involving genus theory, which will be used in the proof of Theorem3.2.

Theorem 4.1 ([22]). Let E be an infinite dimensional Banach space with E = V^LX, where V is finite dimensional and let I∈C¹(E,R)be a even function with I(0) =0and satisfying

(i) There are constantsβ,$>0such that I(u)≥βfor all u∈∂B_$∩X;

(ii) There isτ>0such that I satisfies the(PS)_ccondition, for0<c< τ;

(iii) For each finite dimensional subspaceEe⊂ E,there is R= R(Ee)> 0such that I(u) ≤0for all u∈Ee\B_R(0).

Suppose that V is k dimensional and V = span{e₁, . . . ,e_k}. For n ≥ k, inductively choose e_n+1 6∈

E_n:=span{e₁, . . . ,e_n}. Let R_n =R(E_n)and D_n= B_Rn∩E_n. Define

Gn ={h∈C(Dn,E):h is odd and h(u) =u, ∀u∈ ∂BRn∩En} and

Γj =ⁿh

D_n\Y

:h∈G_n, n≥ j, Y∈ _Σ, andγ(Y)≤n−jo , where

Σ={Y⊂E\{0}:Y is closed in E and Y=−Y} andγ(Y)is the genus of Y∈Σ. For each j ∈_N,let

c_j = inf

K∈_Γjmax

u∈K I(u).

Then0 < β ≤ c_j ≤ c_j+1for j > k,and if j > k and c_j < τ,we have that c_j is the critical value of I.

Moreover, if c_j =_cj+1=· · · =_cj+l = _c<τfor j>_k,thenγ(_Kc)≥l+_1,where K_c={u∈ E: I(u) =c and I⁰(u) =0}.

In the sequel, we derive some results related to the above theorem and the Palais–Smale compactness condition.

Since we will rely on the critical point theory, we define the energy functional correspond- ing to problem (P_λ) as I_λ :W₀^1,m(x)(_Ω)7→_R,

I_λ(u) =

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

! dx−

Z

Ωa(x)|u|^m(x) m(x) −λ

Z

Ω

|u|^q(x) q(x) ^dx−

Z

ΩF(x,u)dx.

Clearly,I_λ isC¹functional and the critical points of it are weak solutions of problem (P_λ).

Lemma 4.2. Assume that(a₁), (f₁)and(mq₁) hold. Then for each λ > 0, I_λ satisfies condition (i) given in Theorem4.1.

Proof. Letδ>0. By (a₁), we have Z

Ω

|∇u|^p1(x)

p₁(x) +|∇u|^p2(x)

p2(x) −a(x)|u|^m(x) m(x)

! dx

= ¹ 1+δ

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x)

p2(x) −a(x)|u|^m(x) m(x)

! dx + ^δ

1+δ Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x)

p2(x) −a(x)|u|^m(x) m(x)

! dx

= ¹ 1+δ

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x)

p₂(x) −a(x)|u|^m(x) m(x)

! dx + ^δ

1+δ Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

!

dx− ^δ 1+δ

Z

Ωa(x)|u|^m(x) m(x) ^dx

≥ ^α 1+δ

Z

Ω

|u|^m(x)

m(x) ^dx+ ^δ 1+δ

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

!

dx− ^δkak_∞ (1+δ)m⁻

Z

Ω|u|^m(x)dx

≥ ^δ 1+δ

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

!

dx+ ¹ 1+δ

α

m⁺− ^δkak_∞ m⁻

Z

Ω|u|^m(x)dx.

We can chooseδ>0 such thatC₀:= ₁₊¹

δ α

m⁺ −^δk_m^a₋^k∞ >_{0. So}

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x)

p₂(x) −a(x)|u|^m(x) m(x)

!

dx≥ ^δ 1+δ

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

! dx +C0

Z

Ω|u|^m(x)dx.

(4.1)

By the seconde part of (f₁), there isη_ε >0 such that

|f(x,t)| ≤ε|t|^q(x)−1 for all|t| ≥ηε and for allx ∈_Ω.

Thanks to the continuity of f, there is Aε >0, such that

|f(x,t)| ≤ A_ε for all|t| ≤η_ε and for allx∈_Ω.

Therefore

|f(x,t)| ≤ε|t|^q(x)−1+A_ε for all(x,t)∈_Ω×_{R. (4.2)} On the other hand, by the first part of(f₁), for eachε>0 there exists 0<δε <1 such that

|f(x,t)| ≤ε|t|^m(x)−1 for all|t| ≤δ_ε and for allx∈_Ω. (4.3) For |t| ≥δ_ε, it follows from (4.2) that

|f(x,t)|

|t|^q(x)−1 ≤ε+ ^Aε δ_ε^q(x)−1

≤ ε+ ^Aε δ^q

+−1 ε

=C_ε, that is

|f(x,t)| ≤C_ε|t|^q(x)−1 _{for all}|t| ≥δ_ε and for allx ∈_{Ω. (4.4)}

Combining (4.3)–(4.4), we obtain

|f(x,t)| ≤ε|t|^m(x)−1+C_ε|t|^q(x)−1 for all(x,t)∈ _Ω×_R.

By integrating this last inequality, we get

|F(x,t)| ≤ ^ε

m(x)|t|^m(x)+ ^Cε

q(x)|t|^q(x) _{for all}(x,t)∈_Ω×_{R. (4.5)} Therefore

I_λ(u)≥ ^δ 1+δ

Z

Ω

|∇u|^p1(x)

p₁(x) + |∇u|^p2(x) p₂(x)

!

dx+C₀− ^ε m⁻

^Z

Ω|u|^m(x)dx−^λ+C_ε q⁻

Z

Ω|u|^q(x)dx

≥ ^δ

(1+δ)max(p⁺₁,p⁺₂)

Z

Ω

|∇u|^p1(x)+|∇u|^p2(x)dx+C₀− ^ε m⁻

^Z

Ω|u|^m(x)dx

− ^λ+Cε

q⁻ Z

Ω|u|^q(x)dx.

Hence forεsufficiently small, I_λ(u)≥ ^δ

(1+δ)max(p⁺₁,p⁺₂)

Z

Ω

|∇u|^p1(x)+|∇u|^p2(x)dx− ^λ+Cε

q⁻ Z

Ω|u|^q(x)dx.

Using the fact that

|∇u|^p1(x)+|∇u(x)|^p2(x)≥ |∇u(x)|^m(x)for allx∈ _Ω, (4.6) it follows that

I_λ(u)≥ ^δ

(₁+δ)_max(p⁺₁,p⁺₂)

Z

Ω|∇u|^m(x)dx− ^λ+C_ε q⁻

Z

Ω|u|^q(x)dx.

By the continuous embeddingW₀^1,m(x)(_Ω),→ L^q(x)(_Ω), there existsC₁>0 such that kuk_Lq(x)(_Ω)≤ C₁kuk.

Consequently, by Proposition2.1, forkuk= $, with 0<$<1, I_λ(u)≥ ^δ

(1+δ)max(p⁺₁,p⁺₂)kuk^m+−(λ+Cε)C2

q⁻ kuk^q−.

Since m⁺ < q⁻, there exists β > 0 such that I_λ(u) ≥ β for kuk = _{$, where} $ is chosen sufficiently small.

Lemma 4.3. Assume that (a₁)_,(f₁)and(mq₁) hold. Then I_λ satisfies condition (iii) given in Theo- rem4.1.

Proof. LetEto be a finite dimensional subspace ofW₀^1,m(x)(_Ω)_{. By}(_4.2)_,

|f(x,t)| ≤ε|t|^q(x)−1+A_ε for all(x,t)∈_Ω×_R.

Integrating this inequality, we entail

|F(x,t)|

|t|^q(x) ≤ ^ε

q(x)+ ^Aε

|t|^q(x)−1

≤ ^ε

q⁻+ ^Aε

|t|^{q−−1 as}|t| →+_∞, hence forε>0, there isδε >0, such that

|F(x,t)| ≤ε|t|^q(x) for all|t| ≥δε and for allx∈_Ω. (4.7) By continuity, there is Mε >0 such that

|F(x,t)| ≤Mε for all|t| ≤δε and for allx∈_Ω. (4.8) This and (4.7) imply that

F(x,t)≥ −Mε−ε|t|^q(x) for all(x,t)∈ _Ω×_R. (4.9) Thus

I_λ(u)≤ ¹ min(p⁻₁,p⁻₂)

Z

Ω

|∇u|^p1(x)dx+|∇u|^p2(x)dx+kak_∞ m⁻

Z

Ω|u|^m(x)dx +

ε− ^λ

q⁺ Z

Ω|u|^q(x)dx+M_ε|_Ω|_. By choosingε= _2q^λ+, we obtain

I_λ(u)≤ ¹ min(p⁻₁,p⁻₂)

Z

Ω

|∇u|^p1(x)dx+|∇u|^p2(x)dx+kak_∞ m⁻

Z

Ω|u|^m(x)dx

− ^λ 2q⁺

Z

Ω|u|^q(x)dx+Mε|_Ω|

≤ ¹

min(p⁻₁,p⁻₂)

2|_Ω|+2 Z

Ω|∇u|^m(x)dx+kak_∞

Z

Ω|u|^m(x)dx

− ^λ 2q⁺

Z

Ω|u|^q(x)dx+M_ε|_Ω|.

Since dimE<∞, the normsk · kandk · k_Lq(x)(_Ω)are equivalent inE. According to Proposition 2.1, for min kuk,kuk_Lm(x)(_Ω),kuk_Lq(x)(_Ω)

>1, I_λ(u)≤ ¹

min(p⁻₁,p⁻₂)

2|_Ω|+2kuk^m++kak_∞kuk^m+

L^m(x)(_Ω

− ^λ

2q⁺kuk^q−

L^q(x)(_Ω+M_ε|_Ω|

≤ ¹

min(p⁻₁,p⁻₂) ²|_Ω|+2kuk^m++C⁰kak_∞kuk^m+− ^λC3

2q⁺kuk^q−+Mε|_Ω|.

Using the fact thatm⁺< q⁻, we conclude that I_λ(u)<_{0 for}kuk ≥ R>_{1, where} Ris chosen large enough.

4.1 Palais–Smale condition

Lemma 4.4. Assume that(a₁), (f₁)–(f₂)and(mq₁)hold. Then any (PS)sequence of I_λ is bounded in W₀^1,m(x)(_Ω).

Proof. Let {un} ⊂ W₀^1,m(x)(_Ω) such that I_λ(un) → c and I_λ⁰(un) → 0. By (f₂), for n large enough,

c+₁+kunk ≥Iλ(_un)− ¹

m⁺hI_λ⁰(_un)_,uni

=

Z

Ω

1

p₁(x)− ¹ m⁺

|∇un|^p1(x)dx+

Z

Ω

1

p₂(x)− ¹ m⁺

|∇un|^p2(x)dx +

Z

Ω

1

m⁺ − ¹ m(x)

a(x)|u_n|^m(x)dx+λ Z

Ω

1

m⁺− ¹ q(x)

|u_n|^q(x)dx +

Z

Ω

1

m⁺f(x,u_n)u_n−F(x,u_n)

dx

≥

Z

Ω

1

p₁(x)− ¹ m⁺

|∇un|^p1(x)dx+

Z

Ω

1

p₂(x)− ¹ m⁺

|∇un|^p2(x)dx +

Z

Ω

1

m⁺ − ¹ m(x)

a(x)|u_n|^m(x)dx+λ Z

Ω

1

m⁺− ¹ q(x)

|u_n|^q(x)dx.

Therefore λ

1 m⁺− ¹

q⁻ _Z

Ω|u_n|^q(x)dx≤λ Z

Ω

1

m⁺ − ¹ q(x)

|u_n|^q(x)dx

≤c+1+kunk+

Z

Ω

1

m(x)− ¹ m⁺

a(x)|un|^m(x)dx

≤c+1+ku_nk+kak_∞ 1

m⁻ − ¹ m⁺

_Z

Ω|u_n|^m(x)dx.

On the other hand, by (mq₁), for anyε>0, there existsCε >0 such that

|t|^m(x)≤ ε|t|^q(x)+Cε for all(x,t)∈_Ω×_R.

It follows that λ

1 m⁺ − ¹

q⁻ _Z

Ω|u_n|^q(x)dx≤c+1+ku_nk+εkak_∞ 1

m⁻ − ¹ m⁺

_Z

Ω|u_n|^q(x)dx +kak_∞

1 m⁻ − ¹

m⁺

Cε|_Ω|, hence

λ

1 m⁺ − ¹

q⁻

−εkak_∞ 1

m⁻ − ¹ m⁺

Z

Ω|u_n|^q(x)dx

≤c+1+ku_nk+kak_∞ 1

m⁻ − ¹ m⁺

Cε|_Ω|. Choosingε= _2k^λ_ak

∞

1 m⁺ −_q¹−

₁

m⁻ − _m¹+

, we obtain λ

2 1

m⁺ − ¹ q⁻

_Z

Ω|u_n|^q(x)dx≤c+1+ku_nk+kak_∞ 1

m⁻− ¹ m⁺

Cε|_Ω|.

Thus,

Z

Ω|un|^q(x)dx≤C₄(1+kunk). (4.10) By(4.7)and (4.8), forε>0, there existsCε >0 such that

|F(x,t)| ≤ε|t|^q(x)+Cε for all (x,t)∈_Ω×_R. (4.11) (4.1), (4.6), (4.10) and (4.11) imply that fornlarge enough

δ

(1+δ)max(p⁺₁,p⁺₂)

Z

Ω|∇u_n|^m(x)dx≤ ^δ 1+δ

Z

Ω

|∇u_n|^p1(x)

p₁(x) +|∇u_n|^p2(x) p₂(x)

! dx

≤ I_λ(u_n) + ^λ q⁻

Z

Ω|u_n|^q(x)dx+

Z

ΩF(x,u_n)dx

≤C5+ λ

q⁻ +ε Z

Ω|un|^q(x)dx+Cε|_Ω|

≤ λ

q⁻ +ε

C₄(1+ku_nk) +C₅+C_ε|_Ω|.

Hence _Z

Ω|∇un|^m(x)dx≤C6(1+kunk) and so

min ku_nk^m−,ku_nk^m+ ≤C(1+ku_nk). Consequently {u_n}is bounded inW₀^1,m(x)(_Ω).

In view of Lemma 4.4, {u_n} is bounded in W₀^1,m(x)(_Ω). So, up to subsequence, we may assume that

un*_{u inW0}^1,m(x)(_Ω)_, u_n*u in L^q(x)(_Ω),

u_n→u in L^r(x)(_Ω)_, r∈ C+(_Ω)_, r(x)< m^∗(x)∀x∈ _Ω.

Taking in to account (mq₂), from the concentration compactness lemma of Lions [8], there exist tow nonnegative measures µ,ν ∈ M(_Ω), a countable set J, points {x_j}_j∈J in Ω and sequences{µ_j}_j∈J,{ν_j}_j∈J ⊂[0,+_∞), such that

|∇u_n|^m(x)*µ≥ |∇u|^m(x)+

∑

j∈J

µ_jδ_xj inΩ

|u_n|^q(x)*ν ≥ |u|^q(x)+

∑

j∈J

ν_jδ_xj in Ω (4.12)

Sν

1 m∗(_xj)

j ≤µ

1 m(_xj)

j for allj∈ J_, where

S= inf

φ∈C^∞₀(_Ω)

k∇φk_Lm(x)(_Ω)

kφk_Lq(x)(_Ω)

. Letφ∈C₀^∞ R^N

such that

0≤φ≤_1, φ≡_{1 in}B₁(₀)_, φ=_{0 inR}^N\B₂(₀)_.