3 Existence of one weak solution

Weak solutions to Dirichlet boundary value problem driven by p ( x ) -Laplacian-like operator

Calogero Vetro

Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 - Palermo, Italy

Received 23 July 2017, appeared 4 January 2018 Communicated by Gabriele Bonanno

Abstract. We prove the existence of weak solutions to the Dirichlet boundary value problem for equations involving the p(x)-Laplacian-like operator in the principal part, with reaction term satisfying a sub-critical growth condition. We establish the existence of at least one nontrivial weak solution and three weak solutions, by using variational methods and critical point theory.

Keywords: Dirichlet boundary value problem, p(x)-Laplacian-like operator, variable exponent Sobolev space.

2010 Mathematics Subject Classification: 35D30, 35J60.

1 Introduction

In this article we consider the following Dirichlet boundary value problem:







−_∆^l_p(x)u(x) +|u(x)|^p(x)−2u(x) =λg(x,u(x)) _inΩ,

u =0 on ∂Ω, (P_λ)

where

∆^l_p(x)u:=div







1+ |∇u|^p(x) q

1+|∇u|^2p(x)



|∇u|^p(x)−2∇u





is the p(x)-Laplacian-like, Ω ⊂ _Rⁿ is an open bounded domain with smooth boundary, p ∈ C(_Ω)is a function with some regularity satisfying

1< p⁻ := inf

x∈_Ωp(x)≤ p(x)≤ p⁺:=sup

x∈_Ω

p(x)<+_∞.

The function g:Ω×_R→_Ris Carathéodory (that is, for allz∈_R, x→ g(x,z)is measurable and for a.a.x∈ _Ω,z →g(x,z)is continuous) andλis a real positive parameter. In the sequel of this article, we assume that the reaction term g(x,z)satisfies the hypothesis:

BEmail: calogero.vetro@unipa.it

(g₁) there exista₁,a₂∈[0,+_∞[andα∈C(_Ω)with 1<α(x)< p^∗(x)for all x∈ Ω, such that

|g(x,z)| ≤a₁+a₂|z|^α(x)−1 for all (x,z)∈_Ω×_R, where p^∗(x) = ^np(x)

n−p(x) ^{if p}(x)<nand p^∗(x) = +_∞if p(x)≥n.

Now, let W₀^1,p(x)(_Ω)be the closure of C₀^∞(_Ω) in the generalized Lebesgue–Sobolev space W^1,p(x)(_Ω) given in Section 2. For a weak solution of problem (Pλ), we mean a function u∈W₀^1,p(x)(_Ω)such that

Z

Ω|∇u(x)|^p(x)−2∇u(x)∇v(x)dx+

Z

Ω

|∇u(x)|^2p(x)−2∇u(x) q

1+|∇u(x)|^2p(x)

∇v(x)dx

+

Z

Ω|u(x)|^p(x)−2u(x)v(x)dx=λ Z

Ωg(x,u(x))v(x)dx, for allv∈W₀^1,p(x)(_Ω).

Existence and multiplicity results for problems involving the p(x)-Laplacian-like were ob- tained by Rodrigues [13] (Dirichlet boundary condition), Afrouzi–Kirane–Shokooh [1] (Neu- mann boundary condition). For other problems driven by the p(x)-Laplacian operator, there are the works of Fan–Zhang [9], Bonanno–Chinnì [3] (Dirichlet boundary condition), and Deng–Wang [7], Pan–Afrouzi–Li [12] (Neumann boundary condition). Also, we mention the comprehensive book on nonlinear boundary value problems by Motreanu–Motreanu–

Papageorgiou [11].

Here, we prove the existence of weak solutions to the Dirichlet boundary value problem (P_λ), by using variational methods and critical point theory. Precisely, we apply a result of Bonanno [2] for functionals satisfying the Palais–Smale condition cut off upper at r (the (PS)^[r]-condition for short), to obtain the existence of at least one nontrivial weak solution.

Then, we use a result of Bonanno–Marano [4] to obtain the existence of three weak solutions.

The motivation of this study comes from the use of such problems to model the behaviour of electrorheological fluids in physics (as discussed in Diening–Harjulehto–Hästö–R ˚užiˇcka [8]) and, in particular, the phenomenon of capillarity which depends on solid and liquid interfacial properties such as surface tension, contact angle, and solid surface geometry.

2 Mathematical background

LetXbe a real Banach space andX^∗its topological dual. In developing our study, we consider both the variable exponent Lebesgue space L^p(x)(_Ω) and the generalized Lebesgue–Sobolev spaceW^1,p(x)(_Ω). Indeed, these spaces, in respect to the norms defined below, are separable, reflexive and uniformly convex Banach spaces (see Fan–Zhang [9]). So, we have the variable exponent Lebesgue spaceL^p(x)(_Ω)given as

L^p(x)(_Ω) =

u:Ω→_R : u is measurable and Z

Ω|u(x)|^p(x)dx< +_∞

, where we consider the following norm

kuk_Lp(x)(_Ω):=inf (

λ>0 : Z

Ω

u(x) λ

p(x)

dx ≤1 )

(i.e., Luxemburg norm).

On the other hand, the generalized Lebesgue–Sobolev spaceW^1,p(x)(_Ω)is defined by W^1,p(x)(_Ω):=ⁿu∈L^p(x)(_Ω):|∇u| ∈ L^p(x)(_Ω)^o.

Also, we take the norm

kuk_W1,p(x)(_Ω)=kuk_Lp(x)(_Ω)+k |∇u| k_Lp(x)(_Ω), which is equivalent to the norm

kuk_:=_inf (

λ>_{0 :}

Z

Ω

u(x) λ

p(x)

+

∇u(x) λ

p(x)!

dx≤₁ )

,

(see D’Aguì–Sciammetta [6]). In the following, we will use the normkuk_{instead of}kuk_W1,p(x)(_Ω)

on W₀^1,p(x)(_Ω). In the proofs of our theorems, we use a Sobolev embedding result; precisely we refer to the following proposition due to Fan–Zhao [10].

Proposition 2.1. Let p∈ C(_Ω)with p(x)> 1for each x ∈_Ω. Then, there exists a continuous and compact embedding W₀^1,p(x)(_Ω),→ L^α(x)(_Ω), provided thatα∈ C(_Ω)and1< α(x)< p^∗(x)for all x∈_Ω.

Another useful theorem, which links kuk_Lp(x)(_Ω) to R

Ω|u(x)|^p(x)dx (respectively, kuk to R

Ω |u(x)|^p(x)+|∇u(x)|^p(x)dx), can be stated as follows (Fan–Zhao [10, Theorem 1.3] and Cammaroto–Chinnì–Di Bella [5, Proposition 2.1]).

Theorem 2.2. Let u∈ L^p(x)(_Ω) (resp., u∈W₀^1,p(x)(_Ω))and putkuk_∗ =kuk_Lp(x)(_Ω)(resp.,kuk_∗ = kuk) andρ∗(u) = R

Ω|u(x)|^p(x)dx (resp., ρ∗(u) = R

Ω |u(x)|^p(x)+|∇u(x)|^p(x)dx). Then, we have:

(i) kuk_∗ <1(=1, >1)⇔ρ∗(u)<1(=1, >1); (ii) ifkuk_∗ >1, thenkuk_∗^p− ≤ρ∗(u)≤ kuk_∗^p+; (iii) ifkuk_∗ <1, thenkuk_∗^p+ ≤ρ∗(u)≤ kuk_∗^p−.

Next, letG:Ω×_R→_Rbe the function defined by G(x,t) =

Z _t

0 g(x,z)dz for allt ∈_R, x ∈_Ω, and consider the functionalΨ :W₀^1,p(x)(_Ω)→_Rdefined by

Ψ(u) =

Z

ΩG(x,u(x))dx, for allu∈W₀^1,p(x)(_Ω).

By using(g₁), we getΨ∈ C¹(W₀^1,p(x)(_Ω),R). Also, by Proposition2.1we deduce thatΨhas a compact derivative given as

Ψ⁰(u)(v) =

Z

Ωg(x,u(x))v(x)dx, for all u,v∈W₀^1,p(x)(_Ω).

Moreover, letΦ:W₀^1,p(_Ω)→_Rbe the functional defined by Φ(u) =

Z

Ω

1

p(x)|∇u(x)|^p(x)dx+

Z

Ω

1 p(x)

q

1+|∇u(x)|^2p(x)−1

dx+

Z

Ω

1

p(x)|u(x)|^p(x)dx for all u ∈ W₀^1,p(x)(_Ω), so that Φ is in C¹(W₀^1,p(x)(_Ω),R). We recall that Φ is Gâteaux dif- ferentiable and sequentially weakly lower semicontinuous and its Gâteaux derivative Φ⁰ : W₀^1,p(x)(_Ω)→(W₀^1,p(x)(_Ω))^∗ is

Φ⁰(u)(v) =

Z

Ω|∇u(x)|^p(x)−2∇u(x)∇v(x)dx+

Z

Ω

|∇u(x)|^2p(x)−2∇u(x) q

1+|∇u(x)|^2p(x)

∇v(x)dx

+

Z

Ω|u(x)|^p(x)−2u(x)v(x)dx

for allu,v∈W₀^1,p(x)(_Ω). From Rodrigues [13], we recall the following proposition.

Proposition 2.3. The functional Φ⁰ : W₀^1,p(x)(_Ω) → W₀^1,p(x)(_Ω)^∗ is a strictly monotone and bounded homeomorphism.

Finally, consider the functional I_λ :W₀^1,p(x)(_Ω)→_Rdefined byI_λ(u) =_Φ(u)−λΨ(u)for allu∈W₀^1,p(x)(_Ω)_{. We have}

inf

u∈W₀^1,p(x)(_Ω)

Φ(u) =_Φ(0) =_Ψ(0) =0.

We conclude this section with the following notion.

Definition 2.4. Let X be a real Banach space and X^∗ its topological dual. Then, I_λ : X → _R satisfies the Palais–Smale condition cut off upper at r, with fixed r ∈ ]−_∞,+_∞], if any sequence{un}such that

(i) {I_λ(u_n)}is bounded;

(ii) limn→+_∞kI_λ⁰(un)k_X^∗ =0;

(iii) Φ(u_n)<r,

has a convergent subsequence.

3 Existence of one weak solution

In this section we establish an existence theorem producing at least one nontrivial weak so- lution of (P_λ). To this aim, we apply a theorem proved by Bonanno [2, Theorem 2.3], which reads as follows.

Theorem 3.1. Let X be a real Banach space and let Φ,Ψ : X → _R be two continuously Gâteaux differentiable functionals such thatinfu∈XΦ(u) = _Φ(0) =_Ψ(0) = 0.Assume that there exist r > 0 andu¯ ∈ X, with0<_Φ(u¯)<r, such that

(i) σ= ¹_rsup_φ(u)≤rΨ(u)< ^Ψ_Φ⁽₍^u¯)

¯ u) =ρ;

(ii) for eachλ∈¹

ρ,¹_σ

the functional I_λ:=_Φ−_λΨsatisfies the(P.S.)^[r]-condition.

Then, for eachλ∈_Λr :=¹

ρ,¹_σ

, there is u_0,λ ∈_Φ⁻¹(]0,r[)such that I_λ⁰(u_0,λ)≡ϑ_X^∗and I_λ(u_0,λ)≤ I_λ(u)for all u∈_Φ⁻¹(]0,r[).

Here, we need the function δ : Ω → _R given as δ(x) = d(x,∂Ω), with d to denote the Euclidean distance. Let x₀ ∈ _Ω be a point of maximum for δ and let D = δ(x₀), then B(x0,D) = {x ∈ _Rⁿ : d(x0,x) < D} ⊂ Ω. Now, we fix s ∈ ]1,+_∞[ and put sD = ¹_s and κ_D = _(s−^s_1)D. Clearly(1−s_D)Dκ_D =1. Then, forβ>0 andh∈C(_Ω)with 1< h⁻, we put

[β]^h :=maxn β^h

−,β^h

+o . The hypothesis on the functionG:Ω×_R→_Ris as follows:

(g₂) inf_x∈_ΩG(x,t)≥0 for allt ∈[0, 1] and lim sup_t→0+ infx∈ΩG(x,t)

t^p− = +_∞.

Letλ^∗:=a₁k₁(p⁺)^1/p− +_α^a−²[kα]^α(p⁺)^α+/p−−1, wherek₁andkαare the best constants for the compact embeddings W₀^1,p(x)(_Ω),→ L¹(_Ω)andW₀^1,p(x)(_Ω) ,→ L^α(x)(_Ω), respectively. We establish the following result.

Theorem 3.2. If hypotheses (g₁), (g₂) hold, then problem (P_λ) admits at least one nontrivial weak solution, for each λ∈ ]0,λ^∗[.

Proof. We consider the functionalsΦandΨgiven in Section2on the Banach spaceW₀^1,p(x)(_Ω), and prove that all the hypotheses of Theorem 3.1 hold true with r = 1. Since Φ,Ψ ∈ C¹(W₀^1,p(x)(_Ω),R)and Ψ⁰ is compact, the functional I_λ satisfies the (P.S.)^[r]-condition for all r >0 (see, Afrouzi–Kirane–Shokooh [1, Theorem 3.1]). We deduce that Theorem3.1(ii) holds true. Then, fixed λ∈]0,λ^∗[, by(g₂)we get

0<δ_λ <min (

1,

p⁻

mDⁿ(2[κ_D]^p(1−sⁿ_D) +1)

1/p⁻)

so that

p⁻sⁿ_Dinf_x∈_ΩG(x,δ_λ)

(2[κ_D]^p(1−sⁿ_D) +1)(δ_λ)^p− > ¹ λ. Now, we consider the function u_λ :Ω→_Rgiven as

u_λ(x) =











0, x∈ _Ω\B(x0,D), δ_λ, x∈ B(x₀,s_DD),

δ_λκ_D(D− |x−x₀|), x∈ B(x₀,D)\B(x₀,s_DD), where| · | is the Euclidean norm onRⁿ. We obtain

p⁻Φ(u_λ)≤

Z

Ω|∇u_λ(x)|^p(x)dx+

Z

Ω

q

1+|∇u_λ(x)|^2p(x)−1

dx+

Z

B(x₀,D)

|u_λ(x)|^p(x)dx

≤

Z

Ω2|∇u_λ(x)|^p(x)dx+

Z

B(x0,D)

(δ_λ)^p(x)dx

≤mDⁿ(2[κ_D]^p(1−sⁿ_D) +1) (δ_λ)^p−

⇒ _Φ(u_λ)<1,

wherem:= _n^2π_Γ(n/2^n/2₎ denotes the measure of unit ball ofRⁿandΓis the Gamma function. We get

Ψ(u_λ)≥

Z

B(x0,sDD)G(x,u_λ)dx≥ inf

x∈_ΩG(x,δ_λ)msⁿ_DDⁿ (by left part of(g₂))

⇒ ^Ψ(u_λ)

Φ(u_λ) ≥ ^p

−msⁿ_DDⁿinf_x∈_ΩG(x,δ_λ)

mDⁿ(2[κD]^p(1−sⁿ_D) +1) (δ_λ)^p− = ^p

−sⁿ_Dinf_x∈_ΩG(x,δ_λ)

(2[κD]^p(1−sⁿ_D) +1) (δ_λ)^p− > ¹ λ. Letr=1. For eachu∈ _Φ⁻¹(]−_{∞, 1}]), we can use Theorem2.2 and conclude that

kuk ≤ _Z

Ω

|∇u(x)|^p(x)+|u(x)|^p(x)dx 1/p

≤p⁺Φ(u)^1/p ≤(p⁺)^1/p−,

⇒ kuk ≤(p⁺)^1/p−. (3.1)

Next, Proposition2.1and Theorem2.2imply that Z

Ω|u(x)|^α(x)dx= ρα(u)≤^hkuk_Lα(x)(_Ω)

iα

≤ [kαkuk]^α (3.2) for allu∈W₀^1,p(x)(_Ω), wherek_αis the best constant for the compact embeddingW₀^1,p(x)(_Ω),→ L^α(x)(_Ω). Moreover, the compact embedding W₀^1,p(x)(_Ω) ,→ L¹(_Ω) (with best constant k₁), (g₁), (3.1) and (3.2) imply that, for eachu∈_Φ⁻¹(]−_{∞, 1}]), we have

Ψ(u)≤ a₁ Z

Ω|u(x)|dx+ ^a2 α⁻

Z

Ω|u(x)|^α(x)dx≤ a₁k₁kuk+ ^a2

α⁻[k_α]^α[kuk]^α

≤ a₁k₁(p⁺)^1/p− + ^a2

α⁻[k_α]^α(p⁺)^α+/p−

⇒ sup

Φ(u)≤1

Ψ(u)≤a₁k₁(p⁺)^1/p−+ ^a2

α⁻[kα]^α(p⁺)^α+/p− = ¹ λ^∗ < ¹

λ

⇒ sup

Φ(u)≤1

Ψ(u)< ¹

λ < ^Ψ(u_λ) Φ(_uλ)^.

It follows that Theorem3.1(i) holds true. Sinceλ ∈ ^Φ_Ψ⁽₍^u_u^λ)

λ),_sup ¹

Φ(u)≤rΨ(u)

, by an application of Theorem3.1 withu = u_λ andr = 1, we obtain the existence of a local minimum point v_λ of the functionalI_λsuch that 0 <_Φ(v_λ)<1. This means thatv_λ is a nontrivial weak solution of problem (P_λ).

4 Existence of three weak solutions

In this section we prove a theorem producing at least three weak solutions of (P_λ). To this aim, we apply a theorem proved by Bonanno–Marano [4, Theorem 3.6], which run as follows.

Theorem 4.1. Let X be a reflexive real Banach space and let Φ : X → _R be a coercive, continu- ously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X^∗, Ψ : X → _R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such thatinf_XΦ(x) = _Φ(0) = _Ψ(0) = 0. Assume that there exist r>0andu¯ ∈X, with0<r<_Φ(u¯), such that

(i) σ= ¹_rsup_φ(u)≤rΨ(u)< ^Ψ_Φ⁽₍^u¯)

¯ u) =ρ;

(ii) for eachλ∈¹

ρ,¹_σ

the functional I_λ:=_Φ−_λΨis coercive.

Then, for eachλ∈_Λr := ¹

ρ,_σ¹

, the functional I_λ :=_Φ−λΨhas at least three distinct critical points in X.

The hypotheses on the functionG:Ω×_R→_Rare as follows:

(g3) there existc∈[0,+_∞[andγ∈C(_Ω)with 1<γ⁻ ≤γ⁺ < p⁻ such that G(x,t)≤c

1+|t|^γ(x) for all(x,t)∈_Ω×_R; (g₄) G(x,t)≥0 for all(x,t)∈_Ω×[0,+_∞[;

(g₅) there existr>0 andδ >0 withr< _p¹+mDⁿ min

κ^p

− D ,κ^p

+

D (1−sⁿ_D) +1 δ^p

+ such that

ω:=¹ r

a₁k₁(p⁺)^1/p−[r]^1/p+ ^a2

α⁻[kα]^α(p⁺)^α+/p−[[r]^1/p]^α

< ^p

−sⁿ_Dinfx∈_ΩG(x,δ) (2[κ_D]^p(1−sⁿ_D) +1)δ^p−. So, we establish the following result.

Theorem 4.2. If hypotheses(g₁), (g₃),(g₄),(g₅)hold, then problem(P_λ)admits at least three weak solutions, for eachλ∈_Λr,δ :=^{(2[κD]p(1−snD)+1)δp}

− p⁻sⁿ_Dinfx∈_ΩG(x,δ) ,_ω¹

.

Proof. We adapt the proof of Theorem 3.2 to the new situation. So, we consider the same working space W₀^1,p(x)(_Ω) with the norm k · k and the functionals Φ,Ψ : W₀^1,p(x)(_Ω) → _R.

This means that the regularity assumptions of Theorem 4.1hold true.

Again, let s_D and κ_D as in Section 3. Let r and δ as in (g₅) and consider the function w:Ω→_Rgiven as

w(x) =











0, x ∈_Ω\B(x₀,D), δ, x ∈B(x₀,s_DD),

δκ_D(D− |x−x₀|), x ∈B(x₀,D)\B(x₀,s_DD).

Following the same arguments in the proof of Theorem3.2(by taking in mind(g₄)), we obtain Ψ(w)

Φ(w) ≥ ^p

−sⁿ_Dinf_x∈ΩG(x,δ) (2[κ_D]^p(1−sⁿ_D) +1)δ^p−. On the other hand, it turns out that

Φ(w)≥ ¹ p⁺

Z

Ω

|∇w(x)|^p(x)+|w(x)|^p(x)dx≥ ¹

p⁺mDⁿh minn

κ^p

− D ,κ^p

+ D

o

(1−sⁿ_D) +1i δ^p

+.

Fromr < _p¹+mDⁿ min

κ^p

− D ,κ^p

+

D (1−sⁿ_D) +1 δ^p

+, we deducer<_Φ(w). Thus, Proposition2.1 and Theorem 2.2imply that

Z

Ω|u(x)|^α(x)dx=ρα(u)≤ ^hkuk_Lα(x)(_Ω)

iα

≤[kαkuk]^α (4.1)

for allu∈W₀^1,p(x)(_Ω), wherek_αis the best constant for the compact embeddingW₀^1,p(x)(_Ω),→ L^α(x)(_Ω). For eachu∈_Φ⁻¹(]−_∞,r]), by Theorem2.2we have

kuk ≤p⁺Φ(u)^1/p ≤[p⁺r]^1/p= (p⁺)^1/p−[r]^1/p,

⇒ kuk ≤(p⁺)^1/p−[r]^1/p. (4.2)

Moreover, the compact embedding W₀^1,p(x)(_Ω) ,→ L¹(_Ω) (with best constant k₁), (g₁), (4.1) and (4.2) imply that, for eachu ∈_Φ⁻¹(]−_∞,r]), we have

Ψ(u)≤ a₁ Z

Ω|u(x)|dx+ ^a2 α⁻

Z

Ω|u(x)|^α(x)dx≤ a₁k₁kuk+ ^a2

α⁻[kα]^α[kuk]^α

≤ a₁k₁(p⁺)^1/p−[r]^1/p+ ^a2

α⁻[kα]^α(p⁺)^α+/p−[[r]^1/p]^α

⇒ ¹ r sup

Φ(u)≤r

Ψ(u)≤ ¹ r

a₁k₁(p⁺)^1/p−[r]^1/p+ ^a2

α⁻[kα]^α(p⁺)^α+/p−[[r]^1/p]^α

⇒ ¹ r sup

Φ(u)≤r

Ψ(u)< ^Ψ(w) Φ(w)^.

It follows that Theorem4.1(i) holds true. Finally, we prove that Theorem4.1(ii) holds true too (i.e., Iλ :=_Φ−λΨis coercive for eachλ> 0). In fact, Proposition2.1and Theorem2.2imply

that _Z

Ω|u(x)|^γ(x)dx =ρ_γ(u)≤^hkuk_Lγ(x)(_Ω)

iγ

≤[k_γkuk]^γ (4.3) for allu∈W₀^1,p(x)(_Ω), wherekγis the best constant for the compact embeddingW₀^1,p(x)(_Ω),→ L^γ(x)(_Ω). Consequently, for eachu∈W₀^1,p(x)(_Ω)withkuk ≥max{1,k⁻_γ¹}, using(g₃)and (4.3), we get

Ψ(u) =

Z

ΩG(x,u(x))dx≤

Z

Ωc

1+|u(x)|^γ(x)dx

≤c(|_Ω|+ [kγkuk]^γ) =c

|_Ω|+ [kγ]^γkuk^γ+. It follows that

I_λ(u)≥

Z

Ω

1

p(x)|∇u(x)|^p(x)dx+

Z

Ω

1

p(x)|u(x)|^p(x)dx−λc

|_Ω|+ [k_γ]^γkuk^γ+

≥ ¹

p⁺kuk^p−−λc

|_Ω|+ [k_γ]^γkuk^γ+

⇒ I_λ is coercive.

SinceΛr,δ ⊂ ^Φ_Ψ⁽_(w^w)₎,_sup ^r

Φ(u)≤rΨ(u)

, by an application of Theorem4.1 with u= w, we have that, for eachλ∈_Λr,δ,Iλadmits at least three critical points inW₀^1,p(x)(_Ω). Obviously, these critical points are three weak solutions of (P_λ).

We conclude this article by dealing with a reaction term satisfying the hypotheses (g₁), (g₃),(g₄). Based on the sub-critical growth condition(g₁), we take the functiong :Ω×_R→_R given by

g(x,z) =

(1+|z|^q(x)−1_{, if}(x,z)∈_Ω×]−_∞,r]_, 1+z^γ(x)−1r^q(x)−γ(x), if(x,z)∈_Ω×]r,+_∞[,

where r is a real positive number greater than 1, and q,γ ∈ C(_Ω) with 1 < γ⁻ ≤ γ⁺ <

min{q⁻,p⁻} < p^∗(x) for all x ∈ Ω. Trivially, g is a Carathéodory function satisfying (g₁). Next, consider the function G : Ω×_R → _R given as G(x,t) = Rt

0 g(x,z)dzfor all t ∈ _R and x∈_{Ω, so}(g₄)holds true as g(x,z)≥1 for all (x,z)∈_Ω×_{R. From}

G(x,t) =





 t+ ^tq(x)

q(x), if(x,t)∈_Ω×[0,r],

t+ ^tγ(x)

γ(x)r^q(x)−γ(x)+r^q(x)

1 q(x) − ¹

γ(x)

, if(x,t)∈_Ω×]r,+_∞[,

by routine calculations, we get G(x,t) ≤ (r+r^q+/γ⁻)(1+t^γ(x)) for all (x,t) ∈ _Ω×[0,+_∞[ and so (g₃)holds true (indeed,G(x,t)≤0 for all(x,z)∈_Ω×]−_{∞, 0}]).

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