Existence results for a nonlinear elliptic transmission problem of p ( x ) -Kirchhoff type

Existence results for a nonlinear elliptic transmission problem of p ( x ) -Kirchhoff type

Eugenio Cabanillas Lapa

, Felix León Barboza, Juan B. Bernui Barros and Benigno Godoy Torres

Universidad Nacional Mayor de San Marcos, Facultad de Ciencias Matemáticas Instituto de Investigación, Lima, Perú

Received 24 April 2016, appeared 21 November 2016 Communicated by Maria Alessandra Ragusa

Abstract. In this article, we establish the existence of weak solutions for a nonlinear transmission problem involving nonlocal coefficients ofp(x)-Kirchhoff type in two dif- ferent domains, which are connected by a nonlinear transmission condition at their interface. We get our results by means of the monotone operator theory and the (S+)mapping theory; the weak formulation takes place in suitable variable exponent Sobolev spaces.

Keywords: nonlinear transmission problem, p(x)-Laplacian, monotone operator.

2010 Mathematics Subject Classification: 35J50, 35J57, 35J70.

1 Introduction

Let Ω, Ω1, Ω2 ⊆ _R² be a bounded polygonal domains with their boundaries ∂Ω, ∂Ω1, ∂Ω2

and closures Ω,Ω1 ,Ω2 satisfying the relationsΩ= _Ω1∪_Ω2,Ω1∩_Ω2=φ.

We denote

Γ3=_∂Ω1∩_∂Ω2, Γi =_∂Ωi\ _Γ3, i=1, 2. (see Figure1.1).

We are concerned with the existence of solutions to the following nonlinear elliptic system

−M_i^Z

Ωi

A(x,∇u_i)dx

div(a(x,∇u_i)) =div~_f +h_i(x,u_i) in Ωi, i=1, 2.

M_i ^Z

Ωi

A(_x,∇u_i)_dxa(_x,∇u_i)_.~ν+_k|u_i|^α(x)−2u_i =~_f.~ν on Γi, i=_{1, 2.}

M₁^Z

Ω1

A(x,∇u₁)dx

a(x,∇u₁).~ν¹ =−M₂^Z

Ω2

A(x,∇u₂)dx

a(x,∇u₂).~ν²

=k|u₂−u₁|^α(x)−2(u₂−u₁) +~_f.ν¹ _onΓ

3

(1.1)

where

BCorresponding author. Emails: cleugenio@yahoo.com (E. Cabanillas), fleonb@unmsm.edu.pe (F. León), jbernuib@unmsm.edu.pe (J. B. Bernui), bgodoyt@unmsm.edu.pe (B. Godoy)

Figure 1.1: DomainΩ

(M0) M_i :[0,+_∞[−→[m_0i,+_∞[, are non decreasing locally Lipschitz continuous functions, (H0) h_i :Ωi×_R−→_R are Carathéodory functions and satisfy the growth condition

|h_i(x,t)| ≤c_i0+c_i1|t|^β(x)−1 , for anyx ∈_Ω, t ∈_R,i=1, 2,

(A1) a(x,ξ):Ω×_R^N →_R^N is the continuous derivative with respect toξ of the continuous mappingA:Ω×_R^N →_R,A= _A(_x,ξ)_{, i.e.a}(_x,ξ) =∇_ξA(_x,ξ); there exist two positive constantsΥ1≤_Υ2 such thatΥ1|ξ|^p(x)≤ a(x,ξ)ξ for all x∈_Ω,ξ ∈_R^N and

(A2) |a(x,ξ)| ≤_Υ2|ξ|^p(x)−1 for allx ∈_Ω,ξ ∈_R^N, (A3) A(x, 0) =0 for all x∈_Ω,

(A4) A(x,·)is strictly convex inR^N,

m_0i, i=1, 2, kare positive numbers, pandαare continuous functions onΩsatisfying appro- priate conditions,~_f = (f₁,f2) is a given vector field (determined from Maxwell’s equations),

~ν = (ν₁,ν₂) and~νⁱ = (ν₁ⁱ,ν₂ⁱ) denote a unit outer normal to ∂Ω and to ∂Ωi, respectively; of course~ν¹ = −~ν² and a(x,∇.).~ν¹ = − a(x,∇.).~ν² on Γ3,~ν = ~νⁱ on Γi, i = 1, 2. We confine ourselves to the case where M₁ = M2 with m₀₁ = m02 = m0 for simplicity. Notice that the results of this work remain valid for M₁ 6= M₂.

The study of problems in differential equations and variational problems involving vari- able exponents has been an interesting topic in recent decades. The interest for such problems is based on the multiple possibilities to apply them. There are applications in nonlinear elas- ticity, theory of image restoration, electrorheological fluids and so on (see [1,13,46]). We refer the readers to [15,31,42] for an overview of this subject, to [11,19,20] for the p-Laplacian and [23–27,34] for the study of p(x)-Laplacian equations and the corresponding variational problems.

Transmission problems problems arise in several applications in physics and biology (see [36]). Some results are available for linear parabolic equations with linear and nonlinear conditions at interfaces, for biological models for the transfer of chemicals through semiper- meable thin membranes (see [8,39,43]). There are cases where transmission conditions can allow to deal with models including chemical phenomena in materials with different poros- ity and diffusivity, and chemotaxis phenomena in regions with different substrate properties (see [30]).

Kirchhoff in 1883 [32] investigated an equation

ρ∂²u

∂t² − ^ρ0 h + ^E

2L Z _L

0

∂u

∂x

2

dx

!

∂²u

∂x² =0,

which is called the Kirchhoff equation, where ρ, ρ₀, h, E, Lare all constants, moreover, this equation contains a nonlocal coefficient

ρ₀ h + ^E

2L Z _L

0

∂u

∂x

2

dx which depends on the average

1 2L

Z _L

0

∂u

∂x

2

dx

hence the equation is no longer a pointwise identity, and therefore is often called nonlocal problem. Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [35], where a functional analysis framework for the problem was proposed; see e.g. [3,9,29] for some interesting results and further references. In recent years, various Kirchhoff-type problems have been discussed in many papers. The Kirchhoff model is an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings, which takes into account the changes in length of the string produced by transverse vibra- tions; while purely longitudinal motions of a viscoelastic bar of uniform cross section and its generalizations can be found in [12,33,40,41]. In particular, in a recent article, existence and multiplicity of nontrivial radial solutions are obtained via variational methods [34]. The study of nonlocal elliptic problem has already been extended to the case involving the p-Laplacian (for details, see [11,19,20]) and p(x)-Laplacian (see [14,17,22,29]).

More recently, Cabanillas L. et al. [7], have dealt with the p(x)-Kirchhoff type equation

−M _Z

Ω(A(x,∇u) + ¹

p(x)|u|^p(x))dx h

div(a(x,∇u))− |u|^p(x)−2ui

= f(x,u)|u|^t(x)

s(x) inΩ, u=constant on∂Ω,

Z

∂Ωa(x,∇u)_.νdΓ=0.

by topological methods. Our work is motivated by the ones of Feistauer et al. [28] and Cecik et al. [10].

The aim of this article is to study the existence of a solution to the problem (1.1) in the Sobolev spaces with variable exponents; we use the well-known theorem named as Browder–

Minty theorem and the degree theory of (S+)type mappings to attack it.

This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. Section 3 is devoted to the proof of our general existence results.

2 Preliminaries

To discuss problem (1.1), we need some theory onW^1,p(x)(_Ω)which is called variable exponent Sobolev space (for details, see [26]). Denote byS(_Ω)the set of all measurable real functions

defined onΩ. Two functions inS(_Ω)are considered as the same element ofS(_Ω)when they are equal almost everywhere. Write

C+(_Ω) ={h:h∈ C(_Ω),h(x)>1 for any x∈_Ω}, h⁻:=min

Ω h(x), h⁺ :=max

Ω h(x) for every h∈C+(_Ω). Define

L^p(x)(_Ω) =

u∈S(_Ω): Z

Ω|u(x)|^p(x)dx <+_∞for p∈ C+(_Ω)

with the norm

|u|_p(x),Ω =_inf (

λ>_{0 :}

Z

Ω

u(x) λ

p(x)

dx≤₁ )

, and

W^1,p(x)(_Ω) ={u∈ L^p(x)(_Ω):|∇u| ∈L^p(x)(_Ω)}

with the norm

kuk_1,p(x),Ω=|u|_p(x),Ω+|∇u|_p(x),Ω.

Proposition 2.1 ([26]). The spaces L^p(x)(_Ω) and W^1,p(x)(_Ω) are separable and reflexive Banach spaces.

Proposition 2.2([26]). Setρ(u) =R

Ω|u(x)|^p(x)dx. For any u∈ L^p(x)(_Ω), then (1) for u 6=0,|u|_p(x),Ω =λif and only ifρ(^u

λ) =1;

(2) |u|_p(x),Ω <₁(=_1;>₁)if and only ifρ(u)<₁(=_1;>₁); (3) if|u|_p(x),Ω>1, then|u|^p−

p(_x)_,Ω≤ρ(u)≤ |u|^p+

p(_x)_,Ω; (4) if|u|_p(x),Ω<1, then|u|^p+

p(x)_,Ω≤ρ(u)≤ |u|^p−

p(x)_,Ω;

(5) lim_k→+∞|u_k|_p(x),Ω =0if and only iflim_k→+∞ρ(u_k) =0;

(6) lim_k→+∞|u_k|_p(x),Ω = +_∞if and only iflim_k→+∞ρ(u_k) = +_∞.

Proposition 2.3([23,26]). If q∈ C+(_Ω)and q(x)≤ p^∗(x)(q(x)< p^∗(x)) for x ∈Ω, then there is a continuous (compact) embedding W^1,p(x)(_Ω),→L^q(x)(_Ω), where

p^∗(x) =

( _{N p(x)}

N−p(x) if p(x)< N, +_∞ if p(x)≥ N.

Proposition 2.4([23,24]). The conjugate space of L^p(x)(_Ω)is L^q(x)(_Ω), where _q(¹_x)+ ¹

p(x) =1holds a.e. inΩ. For any u ∈L^p(x)(_Ω)and v∈ L^q(x)(_Ω), we have the Hölder-type inequality

Z

Ωuv dx

≤ 1

p⁻ + ¹ q⁻

|u|_p(x)|v|_q(x). (2.1) Proposition 2.5([21]). If q ∈C+(_Ω)and q(x)≤ p^∂(x)(q(x)< p^∂(x)) for x∈∂Ω, then there is a continuous (compact) embedding W^1,p(x)(_Ω),→ L^q(x)(_∂Ω), where

p^∂(x) =

(_(N−1)p(x)

N−p(x) if p(x)< N, +_∞ if p(x)≥ N.

Proposition 2.6 ([26]). If h:Ω×_R→_Ris a Caratheodory function and satisfies

|h(x,t)| ≤a(x) +b|t|^p1(x)/p2(x), for any x ∈_Ω, t∈_R,

where p₁(x),p2(x) ∈ C+(_Ω),a(x) ∈ L^p2(x)(_Ω),a(x) ≥ 0, and b ≥ 0 is a constant, then the Nemytsky operator L^p1(x)(_Ω)to L^p2(x)(_Ω)defined by(N_h(u))(x) =h(x,u(x))is a continuous and bounded operator.

In the sequel we shall assume that~_f ∈L^p0(x)(_Ω)².

Let us define the Banach spaceE=W^1,p(x)(_Ω1)×W^1,p(x)(_Ω2)equipped with the norm kuk_E= ku₁k_1,p(x),Ω1+ku2k_1,p(x),Ω2, ∀u= (u₁,u2)∈ E

where ku_ik_1,p(x),Ωi is the norm ofu_i inW^1,p(x)(_Ωi),i =1, 2. By|u|_E we denote the seminorm in E

|u|_E =|∇u₁|_p(x),Ω

1+|∇u₂|_p(x),Ω

2. It is obvious that

|∇u_i|_p(x),Ω

i ≤ |u|_E ≤ kuk_E, ∀u= (u₁,u₂)∈E.

Remark 2.7. From the assumptions on A, arguing as in [37], we get after some computations that

Υ1

p⁺minn

|∇u_i|^p−

p(x)_,Ωi,|∇u_i|^p+

p(x)_,Ωi

o≤_Υ1

Z

Ωi

1

p(x)|∇u_i|^p(x)dx≤

Z

Ωi

A(x,∇u_i)dx

≤_Υ2

Z

Ωi

1

p(x)|∇u_i|^p(x)dx

≤ ^Υ2

p⁻maxn

|∇u_i|^p_p⁻_(x),Ω

i,|∇u_i|^p_p⁺_(x),Ω

i

o .

3 Existence of solutions

In this section, we shall state and prove the main result of the paper. For simplicity, we use c, c_i,i=1, 2, . . . to denote the general positive constants (the exact value may change from line to line).

Let us define the forms b(u,v) =

∑

M^Z

Ωi

A(x,∇u_i)dx^Z

Ωi

a(x,∇u_i)∇v_idx,

c(u,v) =k

∑

Z

Γi

|u_i|^α(x)−2u_iv_idS, α(x)>2, d(u,v) =k

Z

Γ3

|u₂−u₁|^α(x)−2(u₂−u₁)(v₂−v₁)dS, l(u,v) =−

∑

Z

Ωi

h(x,u_i)v_idx, 1< β(x)< p^∗(x)_, L(v) =

∑

Z

Ωi

~_f.∇v_idx,

g(u,v) =b(u,v) +c(u,v) +d(u,v) +l(u,v), u= (u₁,u₂), v= (v₁,v₂)∈ E.

We say thatu= (u₁,u₂)∈Eis a weak solution of problem (1.1) if g(u,v) = L(v)_{, for all}v∈ E.

Here we recall how the theory of monotone operators is used to prove existence of solutions to (1.1) . For this, it will be useful to consider the differential operator as a mapping from E into its dual space, i.e.,

G:E→E⁰, hG(u),vi= g(u,v)

provided of course that for fixed u this indeed defines a bounded linear functional onE.

The following lemma states the rather obvious relation between the operator G and the differential equation (1.1)

Lemma 3.1. For each u ∈ E the forms g(u,·), b(u,·), c(u,·), d(u,·), l(u,·) and L are linear and continuous on E .

Proof. The boundedness of the forms is an easy consequence of Hölder’s inequality, Re- mark2.7, Propositions2.2–2.5 and monotonicity of M. Indeed,

|b(u,v)| ≤c

∑

M Υ2

p⁻ Z

Ωi

|∇u_i|^p(x)dx Z

Ωi

|a(x,∇u_i)||∇v_i|dx

≤c

∑

M Υ2

p⁻|u|^γ_E⁺¹

maxn

|∇u_i|^p+−1

p(x)_,Ωi,|∇u_i|^p−−1

p(x)_,Ωi

o|∇v_i|_p(x),Ω

i

≤cM 1

p⁻kuk^γ_E⁺¹

kuk^γ_Ekvk_E where

γ=

(p⁺−1 if |∇u_i|_p(x),Ωi >1 p⁻−1 if |∇u_i|_p(x),Ωi <1,

|c(u,v)| ≤k

∑

Z

Γi

|u_i|^α(x)−1|v_i|dx≤kc

∑

|u_i|^α(x)−1

α⁰(x)_,Γi|v_i|_α(x),Γi

≤kc

∑

|u_i|^θ

α(x)_,Γi|v_i|_α(x),Γ

i ≤ kckuk^θ_Ekvk_E,

where

θ = (

α⁺−1 if|∇u_i|_α(x),Γi >1 α⁻−1 if|∇u_i|_α(x),Γ

i <1,

and

|d(u,v)| ≤k Z

Γ3

|u₁−u2|^α(x)−1|v₁−v2|dx≤kc

|u₁−u2|^α(x)−1

α⁰(x)_,Γ3|v₁−v2|_α(x),Γ3

≤kc^θ|u₁−u₂|^θ_α(x),Γ

3|v₁−v₂|_α(x),Γ3 ≤kc^0θkuk^θ_Ekvk_E, where

θ = (

α⁺−1 if|u₁−u₂|_α(x),Γ3 >1, α⁻−1 if|u₁−u2|_α(x),Γ3 <1.

and

|l(u,v)| ≤

∑

Z

Ωi

|h(x,u_i)||v_i|dx≤c

∑

Z

Ωi

|u_i|^β(x)−1|v_i|dx+

Z

Ωi

|v_i|dx

≤c

∑

|u_i|^δ

β(x)_,Ωi|v_i|_β(x),Ω

i +|v_i|_p(x),Ω

i

≤kc(kuk^δ_E+1)kvk_E,

where

δ = (

β⁺−1 if |u_i|_β(x),Ωi >1, β⁻−1 if |u_i|_β(x),Ωi <1.

By Lemma3.1 we can define the mappingG : E → E⁰ and the functional ϕ ∈ E⁰ by the identities

hG(u),vi=g(u,v) hϕ,vi=L(v) for each u,v∈ E.

Now, it is clear that solving (1.1) is the same as findingu∈ Esuch that G(u) =ϕ.

Theorem 3.2. Assume that(M0), (H0)and(A1)–(A4)hold. In addition, suppose that

(H1)h(x, 0) =₀and h :Ω×_R→_Ris a decreasing function with respect to the second variable, i.e.

h(x,s₁)≤h(x,s₂) for a.e. x∈ _Ωand s₁,s₂ ∈_R, s₁ ≥s₂. If p⁺ <α⁻, problem(1.1)has precisely one weak solution .

For the proof of our theorem we need to establish some lemmas.

Lemma 3.3. Let r,s≥1, β>1. Then there exists a positive constant c₂=c₂(r,s)such that

|u|^β_E+ kuk^β_E min{kuk^r_E,kuk^s_E}

∑

|u_i|^r

α(x)_,Γi+|u₁−u₂|^s

α(x)_,Γ3

!

≥ c₂kuk^β_E. (3.1) Proof. Firstly, we prove that there existsc₂ >0 such that

|u|^β_E+

∑

|u_i|^r

α(x)_,Γi +|u₁−u₂|^s

α(x)_,Γ3 ≥c₂ (3.2) for allu = (u₁,u₂)∈Ewithkuk_E =1 . Let us assume that (3.2) is not valid. Then there exists a sequence{u^ν} ⊂Esuch that

a) ku^νk_E =₁

b) u^ν*u = (u₁,u₂)weakly inE, c) |u^ν|^β_E+

∑

|u^ν_i|^r_α(x),Γ

i +|u^ν₁−u^ν₂|^s_α(x),Γ

3 ≤ ¹

ν. From Proposition2.5andb)it follows that

u^ν →u= (u₁,u₂)strongly inL^α(x)(_Γ1)×L^α(x)(_Γ2). (3.3) Using (3.3) , the weak lower semicontinuity of the seminorm|u|_E andc)_{we get}

|u|^β_E+

∑

|u_i|^r

α(x)_,Γi+|u₁−u₂|^s

α(x)_,Γ3 =_0.

Then,u_i =k_i ≡constant fori=1, 2. Sou_i |_Γi= k_i. As|u_i|_α(x),Γ

i =0 we havek_i =0 fori=1, 2, thereforeu=0. This is a contradiction toa).

Finally to prove (3.1) letu∈ E,u6=0 andw= _ku^u_k

E. From (3.2) we have

|u|^β_E

ku^νk_E^β + ¹

min{kuk^r_E,kuk^s_E}

∑

|u_i|^r_α(x),Γ

i +|u₁−u₂|^s_α(x),Γ

3

!

≥c₂.

Multiplying this inequality byku^νk^β_E the assertion (3.1) follows.

Lemma 3.4. There exists a constant c3 >₀such that for any u∈ E withkuk_E ≥1

g(u,u)≥c₃kuk_E^p−. (3.4) Proof. For anyu= (u₁,u2)∈ Ewe have

g(u,u)≥

∑

M^Z

Ωi

A(x,∇u_i)dx^Z

Ωi

a(x,∇u_i)∇u_idx

+k

∑

Z

Γi

|u_i|^α(x)dS+k Z

Γ3

|u₂−u₁|^α(x)dS−

∑

Z

Ωi

h(x,u_i)u_idx

≥m0

∑

minn

|∇u_i|^p_p⁻_(x),Ω

i,|∇u_i|^p_p⁺_(x),Ω

i

o +k

∑

minn

|u_i|^α_p⁻_(x),Γ

i,|u_i|^α_p⁺_(x),Γ

i

o

(3.5) +kminn

|u₁−u2|^α_α⁻_(x),Γ

3,|u₁−u2|^α_α⁺_(x),Γ

3

o . Now, if

minn

|∇u_i|^p_p⁻_(x),Ω

i,|∇u_i|^p_p⁺_(x),Ω

i

o

=|∇u_i|^p_p⁻_(x),Ω

i

minn

|u_i|^α_p⁻_(x),Γ

i,|u_i|^α_p⁺_(x),Γ

i

o

=|u_i|^α_p⁻_(x),Γ

i (3.6)

minn

|u₁−u2|^α_α⁻_(x),Γ

3,|u₁−u2|^α_α⁺_(x),Γ

3

o

=|u₁−u2|^α_α⁻_(x),Γ

3, using inequalities (3.5) and (3.6), it follows that

g(u,u)≥ m₀c₄|u|_E^p−+k

∑

|u_i|^α_p⁻_(x),Γ

i+k|u₁−u₂|^α−

α(x)_,Γ3.

Provided thatkuk_E >1 , putting β= p⁻, r = s= α⁻ in (3.1) , and noting thatkuk^p_E^−−α− ≤ 1 we obtain

g(u,u)≥m₀c₄|u|^p_E⁻ +k

∑

|u_i|^α_p⁻_(x),Γ

i+k|u₁−u₂|^α−

α(x)_,Γ3 ≥c₂⁰kuk^p_E⁻

for somec₂⁰ >0. For other cases, the proofs are similar and we omit them here. So we have g(u,u)≥c₃minn

kuk^p_E⁻,kuk^p_E^+o= c₃kuk^p_E⁻. This ends the proof of Lemma3.4.

The proof of the next Lemma is done by adapting some arguments employed in the proof of Theorem 2.1 i) in [18].

Lemma 3.5. The form a is strictly monotone:

g(_u,u−v)−g(_v,u−v)≥0, for all u,v∈E,u 6=v.

Proof. Denoteρ_1,p(x)(u_i) =R

Ωi

1

p(x)|∇u_i|^p(x)dx, for allu_i ∈W^1,p(x)(_Ωi), i=1, 2.

So, for any u_i,v_i ∈ W^1,p(x)(_Ωi)with u_i 6= v_i we may assume, without loss of generality , that ρ_1,p(x)(u_i)>ρ_1,p(x)(v_i). By virtue of monotonicity of M we have

M

Υ2

Z

Ωi

1

p(x)|∇u_i|^p(x)dx

≥ M

Υ1

Z

Ωi

1

p(x)|∇v_i|^p(x)dx

Noting thata(x,·)is monotone by assumption(A4), and following a similar procedure to that used in [18] , we get

g(u,u−v)−g(v,u−v)

≥ ^m0 2

∑

Z

Ωi

[a(x,∇u_i)−a(x,∇v_i)] (∇u_i− ∇v_i)dx

+ ^k 2

∑

Z

Γi

(|u_i|^α(x)−2− |v_i|^α(x)−2)(|u_i|²− |v_i|²)dS + ^k

2 Z

Γ3

(|u₁−u₂|^α(x)−2− |v₁−v₂|^α(x)−2)(|u₁−u₂|²− |v₁−v₂|²)dS

−

∑

Z

Ωi

(h(x,u_i)−h(x,v_i)(u_i−v_i)dx ≥0,

(3.7)

i.e.g is monotone.

Ifg(u,u−v)−g(v,u−v) =0 then all four terms in the right-hand side of (3.7) are equal to zero. Hence, u_i = v_i = k_i = const. a.e. inΩi andu_i = v_i a.e. onΓi, i = 1, 2. Therefore, k_i =0 andu=va.e.

Lemma 3.6. There exists a constant c₆>0such that

|g(u,v)−g(w,v)|

≤c₆h

max{kuk_E^p+,kuk_E^p−,kuk^α_E⁺⁻²,kuk^α_E⁻⁻²}

+_max{kwk_E^p+_,kwk_E^p−_,kwk^α_E⁺⁻²_,kwk^α_E⁻⁻²}ku−wk_E +

∑

|N_h(u_i)−N_h(w_i)| β(x) β(x)−1,Ωi

+M

Υ2

p⁻max{|∇u_i|^p_p⁻_(x),Ω

i,|∇u_i|^p_p⁺_(x),Ω

i}

× |a(x,∇u_i)−a(x,∇w_i)|_p0(x),Ωi

ikvk_E

(3.8)

Proof. By definition of the form g, for allu,v,w∈E, we have g(u,v)−g(w,v)

=

∑

nh M^Z

Ωi

A(x,∇u_i)dx

−M^Z

Ωi

A(x,∇w_i)dxi^Z

Ωi

a(x,∇u_i)∇v_idx

+M^Z

Ωi

A(x,∇w_i)dx^Z

Ωi

(a(x,∇u_i)−a(x,∇w_i))∇v_idxo

+k

∑

Z

Γi

(|u_i|^α(x)−2u_i− |w_i|^α(x)−2w_i)v_idx +k

Z

Γ3

|u₂−u₁|^α(x)−2(u₂−u₁)− |w₂−w₁|^α(x)−2(w₂−w₁)(v₂−v₁)dx

−

∑

Z

Ωi

(h(x,u_i)−h(x,w_i))v_idx

(3.9)

Now in virtue of the Lipschitz condition satisfied byM and the elementary inequalities a)|z|^α−2z− |y|^α−2y

≤C|z−y|^α−1 for all y,z∈_Rⁿ, if 1<α≤2 b)|z|^α−2z− |y|^α−2y

≤C|z−y|(|z|+|y|)^α−2 for all y,z∈_Rⁿ, if 2≤α< _∞, the equality (3.9) reduces to

|g(u,v)−g(w,v)|

≤c

∑

h^Z

Ωi

L_Mi|^|∇ui|^p(x)−|∇wi|^p(x)|

p(x) dx

Z

Ωi

|∇u_i|^p(x)−1|∇v_i|dx

+M

Υ2

p⁻ maxn

|∇u_i|^p_p⁻_(x),Ω

i,|∇u_i|^p_p⁺_(x),Ω

i

o|a(x,∇u_i)−a(x,∇w_i)|_p0(x)_,Ωi|∇v_i|_p(x),Ωiⁱ

+kcα

∑

Z

Γi

|u_i−w_i|(|u_i|^α(x)−2+|w_i|^α(x)−2)|v_i|dx +kc_α

Z

Γ3

|(u₂−u₁)−(w₂−w₁)||u₂−u₁|^α(x)−2+|w₂−w₁|^α(x)−2|v₂−v₁|dx +c₇

∑

|N_h(u_i)−N_h(w_i)|_β0(x)_,Ωi|v_i|_β(x),Ωi,

where L_Mi > 0, the Lipschitz constant of M, depends on max{kuk_E,kwk_E}. Therefore, by Propositions2.2–2.6, after some calculations, we arrive at the estimate (3.8).

Proof of Theorem3.2. First, we note that using hypothesis(A2)and Proposition 2.2 in [24] we get

|a(x,∇u_i)−a(x,∇w_i)|_p0(x)_,Ωi →0 ifu→vin E

From Lemmas3.4–3.6 the operatora is bounded, coercive, strictly monotone and continuous (hence hemicontinuous) in E. Therefore, by the Browder–Minty theorem [4, Theorem 7.3.2], problem (1.1) admits a unique weak solution.

Next, we use the degree theory of (S+)type mappings to prove the second result of this paper.