Existence and multiplicity of a-harmonic solutions for a Steklov ...

(3s.) v. 36 2 (2018): 125–136. ISSN-00378712 in press doi:10.5269/bspm.v36i2.31071

Bol. Soc. Paran. Mat. c

SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm

Existence and multiplicity of a-harmonic solutions for a Steklov problem with variable exponents B. Karim, A. Zerouali and O. Chakrone.

abstract: Using variational methods, we prove in a different cases the existence and multiplicity of a-harmonic solutions for the following elliptic problem: div(a(x, ∇u)) = 0 a(x, ∇u).ν = f (x, u)

in Ω, on ∂Ω,

where Ω ⊂ RN (N ≥ 2) is a bounded domain of smooth boundary ∂Ω and ν is the outward unit normal vector on ∂Ω. f : ∂Ω × R → R, a : Ω × RN → RN , are fulfilling appropriate conditions.

Key Words: Variable exponents; Elliptic problem; Nonlinear boundary condition; a-harmonic solutions; Recceri’s variational principle, mountain pass theorem. Contents 1 Introduction and main results

125

2 Preliminaries

128

3 Proof of Theorem 1.1

129

4 Proof of Theorem 1.2

132

1. Introduction and main results N

Let Ω ⊂ R (N ≥ 2) be a bounded domain with smooth boundary ∂Ω and consider the elliptic Steklov problem with variable exponents div(a(x, ∇u)) = 0 in Ω,

(1.1)

a(x, ∇u).ν = f (x, u) on ∂Ω,

where ν is the outward unit normal vector on ∂Ω and f : ∂Ω × R → R is a continuous function which will be specified later. Let p ∈ C(Ω) be a variable exponent. Throughout this paper, we denote p− = min p(x); p+ = max p(x); x∈Ω

p∂ (x) =

x∈Ω

(N − 1)p(x)/[N − p(x)] ∞

if p(x) < N, if p(x) ≥ N,

2010 Mathematics Subject Classification: 35J65, 35J60, 47J30, 58E05. Submitted February 23, 2016. Published July 05, 2016

125

Typeset by BSP style. M c Soc. Paran. de Mat.

126

B. Karim, A. Zerouali and O. Chakrone

and C+ (Ω) = {p ∈ C(Ω) : 1 < p− < p+ < ∞}. Our exponent p fulfills p ∈ C+ (Ω) and for this p we introduce a characterization of the Carathéodory function a : Ω × RN 7→ RN . (H0 ) a(x, −s) = −a(x, s) for a.e. x ∈ Ω and all s ∈ RN . (H1 ) There exists a Carathéodory function A : Ω × RN 7→ R continuously differentiable with respect to its second argument, such that a(x, s) = ∇s A(x, s) all s ∈ RN and a.e. x ∈ Ω. (H2 ) A(x, 0) = 0 for a.e. x ∈ Ω. (H3 ) There exists c > 0 such that a satisfies the growth condition |a(x, s)| ≤ c(1 + |s|p(x)−1 ) for a.e. x ∈ Ω and all s ∈ RN , where |.| denotes the Euclidean norm. (H4 ) The monotonicity condition 0 ≤ [a(x, s1 ) − a(x, s2 )](s1 − s2 ) holds for a.e. x ∈ Ω and all s1 , s2 ∈ RN , with equality if and only if s1 = s2 . (H5 ) The inequalities |s|p(x) ≤ a(x, s)s ≤ p(x)A(x, s) hold for a.e. x ∈ Ω and all s ∈ RN . A first remark is that hypothesis (H0 ) is only needed to obtain the multiplicity of solutions. As in [9], we have decided to use this kind of function a satisfying (H0 )-(H5 ) because we want to assure a high degree of generality to our work. Here we invoke the fact that, with appropriate choices of a, we can obtain many types of operators. We give, in the following, two examples of well known operators which are present in lots of papers. Examples: 1 |s|p(x) . 1. If a(x, s) = |s|p(x)−2 s, we have A(x, s) = p(x) (H0 ) − (H5 ) are verified, and we arrive to the p(x)-Laplace operator div(a(x, ∇u)) = div(|∇u|p(x)−2 ∇u) = △p(x) u. 1 2. If a(x, s) = (1 + |s|2 )(p(x)−2)/2 s, we have A(x, s) = p(x) [(1 + |s|2 )p(x)/2 − 1]. (H0 ) − (H5 ) are verified, and we find a generalized mean curvature operator div(a(x, ∇u)) = div((1 + |∇u|2 )(p(x)−2)/2 ∇u).

The above operator appears in [16] and it is used in the study of two antiplane frictional contact problems of elastic cylinders. Functions fulfilling conditions related to (H0 )–(H5 ) are used not only in the framework of the spaces with variable exponents [5], but also in the framework of the classical Lebesgue-Sobolev spaces [21] and the anisotropic spaces with variable exponents. In the present paper, we study problem 1.1 in the particular case f (x, t) = λ |t|q(x)−2 t − |t|r(x)−2 t − |t|p(x)−2 t,

Existence and multiplicity of a-harmonic solutions for a Steklov problem. . . 127

where λ ≥ 0 is a real number and p, q, r ∈ C+ (Ω). The energy functional corresponding to problem 1.1 is defined on W 1,p(x) (Ω) as Z Z Z q(x) |u|p(x) |u| |u|r(x) A(x, ∇u)dx + Φλ (u) = dσ, (1.2) dσ − λ − q(x) r(x) ∂Ω p(x) Ω ∂Ω where dσ is the N − 1 dimensional Hausdorff measure. Let us recall that a weak solution of 1.1 is any u ∈ W 1,p(x) (Ω) such that Z Z a(x, ∇u)∇vdx + |u|p(x)−2 uvdσ Ω ∂Ω Z =λ |u|q(x)−2 uv − |u|r(x)−2 uv dσ for all v ∈ W 1,p(x) (Ω). ∂Ω

The study of differential and partial differential equation with variable exponent has been received considerable attention in recent years. This importance reflects directly into a various range of applications. There are applications concerning elastic materials [22], image restoration [11], thermorheological and electrorheological fluids [4,19] and mathematical biology [13]. In the case when p(x) = p is a constant and a(x, s)) = |s|p−2 s, the authors in [1] are considered the following Steklov problem △p u = 0 in Ω, p−2 = λm|u| u on ∂Ω. |∇u|p−2 ∂u ∂ν They are interested to the existence of p-harmonic solutions ( when ∆p u = 0). Motivated by the recent works [5,6], we will study the existence and multiplicity of a-harmonic solutions (when div(a(x, ∇u)) = 0) for the problem 1.1 with variable exponents. These solutions becomes p(x)-harmonic when a(x, s) = |s|p(x)−2 s. This is a generalization of the classical p-harmonic solutions obtained in the case when p is a positive constant. Our main results in this paper are the proofs of the following theorems, which are based on the Ricceri Theorem and the Mountain Pass Theorem. ¯ such that N < p− and Theorem 1.1. Assume (H0 )–(H5 ) and let p, q, r ∈ C+ (Ω), − + − + − 1 ≤ r ≤ r < q ≤ q(x) ≤ q < p , for all x ∈ Ω. Then there exist an open interval ∧ ⊂ (0, ∞) and a positive constant ρ > 0 such that for any λ ∈ ∧, problem 1.1 has at least three weak solutions whose norms are less than ρ. ¯ such that r+ ≤ p+ < Theorem 1.2. Assume (H0 )–(H5 ) and let p, q, r ∈ C+ (Ω), ∂ − + ∂ q ≤ q < p (x) for all x ∈ Ω, where p (x) is defined above. Then for any λ > 0 problem 1.1 possesses a non trivial weak solutions. This present work extends some of the results known with Neuman or Dirichlet boundary conditions on bounded domain(see [16,18]), and generalize some results knouwn in the Steklov problems (see [2,3]). This paper consists of four sections. Section 1 contains an introduction and the main results. In section 2, which has a preliminary character, we state some

128


elementary properties concerning the generalized Lebesgue-Sobolev spaces and an embedding results. The proofs of our main theorems are given in Section 3 and Section 4. 2. Preliminaries We first recall some basic facts about the variable exponent LebesgueSobolev. ¯ we introduce the variable exponent Lebesgue space For p ∈ C+ (Ω), Z p(x) p(x) L (Ω) := u; u : Ω → R is a measurable and |u| dx < +∞ , Ω

endowed with the Luxemburg norm ( |u|Lp(x) (Ω) := inf

) Z u(x) p(x) α > 0; dx ≤ 1 , α Ω

which is separable and reflexive Banach space (see [15]). Let us define the space

W 1,p(x) (Ω) := {u ∈ Lp(x) (Ω); |∇u| ∈ Lp(x) (Ω)}, equipped with the norm ) ( Z Z u(x) p(x) ∇u(x) p(x) dx + dx ≤ 1 ; ∀u ∈ W 1,p(x) (Ω). kukΩ = inf α > 0; α α Ω Ω

Proposition 2.1. [10] For any u ∈ W 1,p(x) (Ω). Let ||u|| := |∇u|Lp(x) (Ω) + |u|Lp(x) (∂Ω) . Then the norm ||u|| is a norm on W 1,p(x) (Ω) which is equivalent to ||u||Ω . Proposition 2.2. [12,14] (1) W 1,p(x) (Ω) is separable reflexive Banach space; ¯ and s(x) < p∂ (x) for any x ∈ Ω, ¯ then the embedding from (2) If s ∈ C+ (Ω) 1,p(x) s(x) W (Ω) to L (∂Ω) is compact and continuous.

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping ρ defined by Z Z p(x) ρ(u) := |∇u| dx + |u|p(x) dσ, ∀u ∈ W 1,p(x) (Ω). Ω

∂Ω

Proposition 2.3. [10] For u, uk ∈ W 1,p(x) (Ω); k = 1, 2, ..., we have −

+

(1) kuk ≥ 1 implies kukp ≤ ρ(u) ≤ kukp ;

Existence and multiplicity of a-harmonic solutions for a Steklov problem. . . 129 +

−

(2) kuk ≤ 1 implies kukp ≥ ρ(u) ≥ kukp ; (3) kuk k → 0 if and only if ρ(uk ) → 0; (4) kuk k → ∞ if and only if ρ(uk ) → ∞. Remark 2.4. If N < p− ≤ p(x) for any x ∈ Ω, by Theorem 2.2 in [15] and remark 1 in [18], we have W 1,p(x) (Ω) is compactly embedded in C(Ω). Defining ||u||∞ = sup |u(x)|, we find that there exists a positive constant c > 0 such that x∈Ω

||u||∞ ≤ c||u|| for all u ∈ W 1,p(x) (Ω). 3. Proof of Theorem 1.1 The key argument in the proof of Theorem 1.1 is the following version of Ricceri theorem (see Theorem 1 in [8]). Theorem 3.1. [8] Let X be a separable and reflexive real Banach space; Φ : X → R a continuously Gˆ ateaux differentiable and sequentially weakly lower. semicontinuous functional whose Gˆ ateaux derivative admits a continuous inverse on X ∗ ; Ψ : X → R a continuously Gˆ ateaux differentiable functional whose Gˆ ateaux derivative is compact. Assume that (i) lim (Φ(u) + λΨ(u)) = +∞ for all λ > 0; and that are r ∈ R and u0 , u1 ∈ X ||u||→∞

such that (ii) Φ(u0 ) < r < Φ(u1 ); (iii) inf Ψ(u) > −1 u∈Φ

((−∞,r])

(Φ(u1 )−r)Ψ(u0 )+(r−Φ(u0 ))Ψ(u1 ) . Φ(u1 )−Φ(u0 )

Then there exist an open interval ∧ ⊂ (0, +∞) and a positive real number ρ0 such that for each λ ∈ ∧ the equation Φ′ (u) + λΨ′ (u) = 0 has at least three solutions in X whose norme are less than ρ0 . Let X denote the generalized Sobolev space W 1,p(x) (Ω). In order to apply Ricceri’s result we define the functionals Φ, Ψ : X → R by Z Z 1 Φ(u) = A(x, ∇u)dx + |u|p(x) dσ, (3.1) Ω ∂Ω p(x) Z q(x) |u| |u|r(x) dσ, (3.2) Ψ(u) = − − q(x) r(x) ∂Ω Its clear that from (H1 ), the Fréchet derivative of Φ is the operator Φ′ : X → X ′ defined as Z Z hΦ′ (u), vi = a(x, ∇u)∇vdx + |u|p(x)−2 uvdσ for any u, v ∈ X. Ω

∂Ω

On the other hand the Fréchet derivative of Ψ is Ψ′ defined as Z |u|q(x)−2 uv − |u|r(x)−2 uv dσ, for any u, v ∈ X. hΨ′ (u), vi = − ∂Ω

130


Thus we deduce that u ∈ X is a weak solution of problem 1.1 if there exist λ > 0 such that u is a critical point of the operator Φ + λΨ. We start by proving some properties of the operator Φ′ . Theorem 3.2. Suppose that the mapping a satisfies (H0 )-(H5 ). Then the following statements holds. (1) Φ′ is continuous, bounded and strictly monotone; (2) Φ′ is of (S+ ) type; (3) Φ′ is an homeomorphism. Proof. The same approach in proofRof Theorem 1.1 in [3], by taking λ = 0 and R 1 asp(x) 1 |u| dx by ∂Ω p(x) |u|p(x) dσ in the expression of energy replacing the term Ω p(x) ✷ functional φλ,0 defined in [3]. Now we can give the proof of our main result. Proof of Theorem 1.1. Set Φ and Ψ as 3.1, 3.2 . So, for each u, v ∈ X, one has Z Z |u|p(x)−2 uvdσ, a(x, ∇u)∇vdx + hΦ′ (u), vi = ∂Ω Ω Z ′ |u|q(x)−2 u − |u|r(x)−2 u v dx hΨ (u), vi = − Ω

From Theorem 3.2, the functional Φ is a continuous Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X ′ , moreover, Ψ is continuously Gâteaux differentiable functional whose Gˆ ateaux derivative is compact. Obviously, Φ is bounded on each bounded subset of X under our assumptions. From (H5 ) and using Proposition2.3, if kuk ≥ 1 then Z Z 1 A(x, ∇u)dx + Φ(u) = |u|p(x) dσ ∂Ω p(x) Ω Z Z 1 1 p(x) |∇u| dx + |u|p(x) dσ ≥ p(x) p(x) ∂Ω Ω 1 ≥ + ρ(u) p − 1 ≥ + kukp , p Meanwhile, for each λ ∈ Λ, λΨ(u) = −λ

Z

∂Ω

|u|q(x) |u|r(x) − q(x) r(x) −

+

≥ −λ(c1 ||u||q + c2 ||u||q )

dσ


for any u ∈ X, where c1 and c2 are positive constants. Combining the two inequalities above, we obtain Φ(u) + λΨ(u) ≥

− − + 1 kukp − λ(c1 ||u||q + c2 ||u||q ), p+

since q + < p− , it follows that lim

(Φ(u) + λΨ(u)) = +∞ ∀u ∈ X,

kuk→+∞

λ ∈ [0, +∞).

Then assumption (i) of Theorem 3.1 is satisfied. Next, we will prove that assumption (ii) is also satisfied. In order to do that we define the function q(x) r(x) G : Ω × [0, ∞[→ R by G(x, t) = tq(x) − tr(x) , ∀x ∈ Ω and t ∈ (0, ∞). It is clear that G is of class C 1 with respect to t, uniformly when x ∈ Ω. Define also the function Gt (x, t) = tr(x)−1 (tq(x)−r(x) − 1), ∀x ∈ Ω and t ∈ (0, ∞). Thus Gt (x, t) ≥ 0 for all t ≥ 1 and all x ∈ Ω; Gt (x, t) ≤ 0 for all t ≤ 1 and all x ∈ Ω. Consequently G(x, t) is increasing when t ∈ (1, ∞) and decreasing when t ∈ (0, 1), uniformly with respect to x. Furthermore, lim G(x, t) = +∞ uniformly t→+∞

which respect to x ∈ Ω. On the other hand G(x, t) = 0 imply that t = t0 = 0 or 1 q(x) q(x)−r(x) . So we have G(x, t) ≤ 0 for all 0 ≤ t ≤ tx and G(x, t) > 0 t = tx = r(x)

such that0 < a < min(1, c), for all t > tx and all x ∈ Ω. Let a, b two real +numbers 1 1 q 1 − −r+ q , ( |∂Ω| ) p− . with c given in Remark 2.4 and b > max ( r− )

Consider u0 , u1 ∈ X, u0 (x) = 0, u1 (x) = b, for any x ∈ Ω. Consequently by Remark 2.4 we have u0 (x) = 0 and u1 (x) = b, for any x ∈ Ω. Thus we have Z Z G(x, b)dσ. sup G(x, t)dσ ≤ 0 < ∂Ω 0≤t≤a

∂Ω

p+ We also define r = p1+ ac , we have r ∈ (0, 1) and Φ(u0 ) = −Ψ(u0 ) = 0. R p+ R 1 1 p− p(x) Φ(u1 ) = ∂Ω p(x) b dx ≥ p+ b |∂Ω| > p1+ . ac = r, Ψ(u1 ) = − ∂Ω G(x, b)dσ. Thus we deduce that Φ(u0 ) < r < Φ(u1 ), so (ii) in Theorem 3.1 is verified. On the other hand we have R G(x, b)dσ Ψ(u1 ) (Φ(u1 ) − r) Ψ(u0 ) + (r − Φ(u0 )) Ψ(u1 ) > 0. = −r = r R ∂Ω 1 p(x) − Φ(u1 ) − Φ(u0 ) Φ(u1 ) b dσ ∂Ω p(x) Let u ∈ X with Φ(u) ≤ r < 1. Then by Proposition 2.3, we have

+ 1 1 1 a p p+ ||u|| ≤ ρ(u) ≤ Φ(u) ≤ r = < 1. p+ p+ p+ c

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Using Remark 2.4, we deduce that for any u ∈ X with Φ(u) ≤ r, we have 1

|u(x)| ≤ c.||u|| ≤ c.(p+ .r) p+ = a, ∀x ∈ Ω. The above inequality shows that −

inf

u∈Φ−1 ((−∞,r])

Thus

Ψ(u) =

sup u∈Φ−1 ((−∞,r])

−Ψ(u) ≤

Z

sup G(x, t)dσ ≤ 0

∂Ω 0≤t≤a

R G(x, b)dσ − inf Ψ(u) < r R ∂Ω 1 p(x) , −1 u∈Φ ((−∞,r]) b dσ ∂Ω p(x)

i.e. inf

u∈Φ−1 ((−∞,r])

Ψ(u) >

(Φ(u1 ) − r) Ψ(u0 ) + (r − Φ(u0 )) Ψ(u1 ) , Φ(u1 ) − Φ(u0 )

consequently the condition (iii) in Theorem 3.1 is verified. We proved that all assumptions of Theorem 1.2 are verified. We conclude that there exists an open interval ∧ ⊂ (0, ∞) and a positive constant ρ0 > 0 such that for any λ ∈ ∧ the equation Φ′ (u) + λΨ′ (u) = 0 has at least three solution in X whose norms are less ✷ than ρ0 . The proof of Theorem 1.1 is complete. 4. Proof of Theorem 1.2 The proof of Theorem1.2 relies on the following version of the mountain pass theorem. Theorem 4.1 ( [20]). Let X endowed with the norm k.kX , be a Banach space. Assume that φ ∈ C 1 (X; R) satisfies the Palais-Smale condition. Also, assume that φ has a mountain pass geometry, that is, (i) there exists two constants η > 0 and ρ ∈ R such that φ(u) ≥ ρ if kukX = η; (ii) φ(0) < ρ and there exists e ∈ X such that kekX > η and φ(e) < ρ. Then φ has a critical point u0 ∈ X such that u0 6= 0 and u0 6= e with critical value φ(u0 ) = inf sup φ(u) ≥ ρ > 0. γ∈P u∈γ

Where P denotes the class of the paths γ ∈ C([0, 1]; X) joining 0 to e. The energy functional corresponding to problem 1.1 is defined as Z q(x) Z Z |u| |u|r(x) |u|p(x) dσ. dσ − λ − A(x, ∇u)dx + Φλ (u) = q(x) r(x) ∂Ω ∂Ω p(x) Ω Where dσ is the N − 1 dimensional Hausdorff measure. Standard arguments imply that Φλ ∈ C 1 (X; R)


¯ such that r+ ≤ p+ < Lemma 4.2. Assume (H0 )–(H5 ) and let p, q, r ∈ C+ (Ω), − + ∂ q ≤ q < p (x) for all x ∈ Ω. Then there exist η, b > 0 such that Φλ (u) ≥ b for u ∈ W 1,p(x) (Ω) with ||u|| = η. ¯ by Proposition 2.2 and (H5 ), we have the Proof. Since q + < p∂ (x) for all x ∈ Ω, following inequality Φλ (u) ≥

+ − + 1 λ if kuk ≤ 1. kukp − − C1 kukq + C2 kukq + p q

Thus Φλ (u) ≥ kuk

p+

1 λ q+ −p+ q− −p+ if kuk ≤ 1. + C2 kuk − − C1 kuk p+ q

As p+ < q − ≤ q + , the functional h : [0, 1] → R defined by h(t) =

λC1 + + λC2 − + 1 − − tq −p − − tq −p + p q q

is positive on neighborhood of the origin. So the Lemma 4.2 is proved.

✷

¯ such that r+ ≤ p+ < Lemma 4.3. Assume (H0 )–(H5 ) and let p, q, r ∈ C+ (Ω), − + ∂ q ≤ q < p (x) for all x ∈ Ω. Then there exists e ∈ W 1,p(x) (Ω) with kek > η such that Φλ (e) < 0; where η is given in Lemma 4.2. ¯ ϕ ≥ 0 and ϕ 6≡ 0, on ∂Ω. For t > 1, and using Proof. Choose ϕ ∈ C0∞ (Ω), (H2 ), (H3 ) we have +

−

λtq c 1 tp |∇ϕ|dx + − ρ(ϕ) − + Φλ (tϕ) ≤ tc p q Ω Z

Z

∂Ω

+

|ϕ|

q(x)

tr dσ + λ − r

Z

|ϕ|r(x) dσ.

∂Ω

Since r+ ≤ p+ < q − , we deduce that lim Φλ (tϕ) = −∞. Therefore for all ε > 0 t→+∞

there exists α > 0 such that |t| > α, Φλ (tϕ) < −ε < 0. This completes the proof. ✷ ¯ such that r+ ≤ p+ < Lemma 4.4. Assume (H0 )–(H5 ) and let p, q, r ∈ C+ (Ω), − + ∂ q ≤ q < p (x) for all x ∈ Ω. Then the functional Φλ satisfies the Palais-Smale (PS) condition. Proof. Let (uk ) ⊂ W 1,p(x) (Ω) be a sequence such that C = sup Φλ (uk ) and k∈N∗

Φ′λ (uk ) → 0. Suppose by contradiction that kuk k → ∞, there exists k0 ∈ N∗

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such that kuk k > 1 for any k ≥ k0 , using (H5 ) Then we have 1 C + kuk k ≥ Φλ (uk ) − − hΦ′λ (uk ), uk i q Z Z Z 1 1 1 p(x) q(x) r(x) A(x, ∇uk )dx + ≥ dσ |uk | dσ − λ |uk | − |uk | q(x) r(x) ∂Ω p(x) Ω Z ∂Ω Z Z λ 1 a(x, ∇uk )∇uk dx + |uk |p(x) dσ + − |uk |q(x) − |uk |r(x) dσ − − q q Ω ∂Ω ∂Ω Z 1 1 1 1 1 1 q(x) r(x) ≥ dσ |uk | + |uk | − − ρ(uk ) + λ − − p+ q q− q(x) r(x) q − ∂Ω 1 1 − − ρ(uk ) ≥ p+ q − 1 1 ≥ − − kuk kp . p+ q Since p+ < q − , this contradicts the fact that p− > 1. So, the sequence (uk ) is bounded in W 1,p(x) (Ω). As W 1,p(x) (Ω) is reflexive (Proposition 2.2), for a subsequence still denoted (uk ), we have uk ⇀ u in W 1,p(x) (Ω), uk → u in Lp(x) (∂Ω), uk → u in Lq(x) (∂Ω), u in Lr(x) (∂Ω) (see Proposition 2.2). Therefore R R uk → p(x)−2 ′ uk (uk −u)dσ → 0, ∂Ω |uk |q(x)−2 uk (uk −u)dσ → hΦλ (uk ), uk −ui → 0, ∂Ω |uk | R R 0 and ∂Ω |uk |r(x)−2 uk (uk −u)dσ → 0. Thus lim sup Ω a(x, ∇uk )(∇uk −∇u)dx ≤ 0. k→+∞

The following theorem assure that uk → u strongly in W 1,p(x) (Ω) as k → +∞. ✷

Theorem 4.5. ( [17] Theorem 4.1) The Carathéodory function a : Ω × Rn → Rn described by (H0 )–(H5 ) is an operator R of type S+ that is, if un ⇀ u weakly in W 1,p(x) (Ω) as n → +∞ and lim sup Ω a(x, ∇un )(∇un − ∇u)dx ≤ 0, then un → u n→+∞

strongly in W 1,p(x) (Ω) as n → +∞.

Proof of Theorem 1.2. Using the Lemmas 4.2 and 4.3, we obtain max (Φλ (0), Φλ (e)) = Φλ (0) < inf Φλ (u) =: β. ||u||=µ

By Lemma 4.4 and Theorem 4.1, we deduce the existence of critical points of Φλ associated of the critical value given by inf sup Φλ (γ(t)) ≥ β,

γ∈Γ t∈[0,1]

where Γ = {γ ∈ C([0, 1], W 1,p(x) (Ω)); γ(0) = 0 and γ(1) = e}. This completes the proof.

✷

Acknowledgment The autors would like to thank the anonymous referee for valuable suggestions.


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B. Karim, Facult´ e des Sciences et T´ echniques, Errachidia, Maroc E-mail address: [email protected] and ´ A. Zerouali, Centre R´ egional des M´ etiers de l’Education et de la Formation, F` es, Maroc E-mail address: [email protected] and O. Chakrone,Facult´ e des Sciences Oujda, Maroc E-mail address: [email protected]