Existence of solutions for a boundary problem involving p(x)-biharmonic operator. Key Words: p(x)-biharmonic, Neumann problem, variational methods.

Transcription

1 Bol. Soc. Paran. Mat. (3s.) v. 3 (203): c SPM ISSN on line ISSN in press SPM: doi:0.5269/bspm.v3i.548 Existence of solutions for a boundary problem involving p(x)-biharmonic operator Abdel Rachid El Amrouss and Anass Ourraoui abstract: In this paper, we establish the existence of at least three solutions to a boundary problem involving the p(x)-biharmonic operator. Our technical approach is based on theorem obtained by B. Ricceri s variational principle and local mountain pass theorem without (Palais-Smale) condition. Key Words: p(x)-biharmonic, Neumann problem, variational methods. Contents Introduction 79 2 Preliminaries 8 2. Variable exponent space and Sobolev Spaces Ricceri s variational principle Proof of the main result 88. Introduction The study of various mathematical problems with variable exponent have received a lot of attention in recent years [,4]. Fourth order equations appears in many contexts. Some of these problems come from different areas of applied mathematics and physics such as Micro Electro-Mechanical systems, surface diffusion on solids, flow in Hele-Shaw cells (see [9]). In addition, this type of equation can describe the static from change of beam or the sport of rigid body, there are many authors have pointed out that type of non linearity furnishes a model to study traveling waves in suspension bridges (see [5,]). In this paper, we consider the following p(x) biharmonic problem with a boundary condition, (P) { 2 p(x) u + a(x) u p(x) 2 u = f(x,u) + λg(x,u) in, Bu = Tu = 0 on, here is a bounded open domain in R N with smooth boundary, 2 p(x) u = ( u p(x) 2 u) is the p(x) biharmonic with p C(), p(x) >, λ R, a L () such that inf x a(x) = a > 0. Bu = Tu = 0 denotes the following boundary conditions: () B = B,T = T, Navier boundary condition, i.e. B u = u = 0 and T u = 2000 Mathematics Subject Classification: 35J60, 35J48 79 Typeset by B S P M style. c Soc. Paran. de Mat.

2 80 Abdel Rachid El Amrouss and Anass Ourraoui u = 0 on. (2) B = B 2,T = T 2, Neumann boundary condition, i.e B 2 u = u ν = 0 and T 2u = ν ( u p(x) 2 u) = 0 on, where ν is the outward unit normal to. Denote by F(x,t) = t 0 f(x,s)ds, G(x,t) = t 0 g(x,s)ds, p := inf x p(x) and p + := sup x p(x). Throughout this paper, we suppose the following assumptions: (F) f,g C( R, R) such that f(x,t), g(x,t) C +C 2 t q(x) (.) (x,t) R, where q C(),C,C 2 > 0 and q(x) < p (x) x, with { p N p(x) (x) := N 2p(x) if p(x) < N 2, + if p(x) N 2. (F ) lim t [F(x,t) λ p(x) t p ] = uniformly for almost every x. (F 2 ) There exist x 0,r 0 ]0,[ and t 0 > with B(x 0,2r 0 ) such that F(x,t) 0 for x B(x 0,2r 0 ) and t ]0,t 0 ], F(x,t 0 ) C 0 for x B(x 0,r 0 ). Where and C 0 = [( 2 r 0 ) p+ (B(x 0,2r 0)) (2 N )+ a 2 N ] t 0 p+ (B(x 0,2r 0)) p (B(x 0,2r 0 )), p (B(x 0,2r 0 )) = (F 2) There exist ξ R such that F(x,ξ)dx > inf p(x), x B(x p+ (B(x 0,2r 0 )) = sup p(x). 0,2r 0) x B(x 0,2r 0) a(x) p(x) ξ p(x) dx. (F 3 ) There exist b 0 > 0,δ > 0 such that F(x,t) b 0 t q0(x), x, t < δ, where q 0 C() with p + < q 0 (x) < p (x) for x. (G ) There exist an open ball B(x,r ),β C(B(x,r ), R) with β(x) β + (B(x,r )) p (B(x,r )), b > 0 and γ > 0 such that G(x,t) b t β+ (B(x,r )) for all x (B(x,r )) and t < γ. With (G 2 ) limsup t 0 β + (B(x,r )) = sup β(x), β (B(x,r ) = inf β(x). x B(x,r ) x B(x,r ) inf x G(x,t) t p = +. In the case B = B and T = T, we claim the following theorem.

3 Existence of solutions 8 Theorem.. Suppose that assumptions (F ), (F 2 ), (F 3 ), (G ) and (F) hold. Then, there exists λ > 0 such that for any λ ]0,λ [, the problem (P) admits at least three weak solutions. The case B = B 2,T = T 2, we have the following result. Theorem.2. Under the assumptions (F ),(F 2),(F 3 ) (G 2 ) and (F). Then, there exists λ > 0 such that for any λ ]0,λ [, the problem (P) has at least three weak solutions. Many authors consider the existence of multiple nontrivial solutions for some fourth order problems [,6]. In particular, Li and Tang [0] consider the p- biharmonic equation. Using the modified three critical points theorem of B. Ricceri they get at least three solutions. The p(x) biharmonic operator possesses more complicated nonlinearities than p biharmonic, for example, it is inhomogeneous. Recently, in[4] A. Ayoujil and A. R. El Amrouss interested to the spectrum of a fourth order elliptic equation with variable exponent. They proved the existence of infinitely many eigenvalue sequences and supλ = +, where Λ is the set of all eigenvalues. Moreover, they present some sufficient conditions for infλ = 0. The technical approach is based on the Ricceri s variational principle and local mountain pass theorem[3], without Palais- Smale condition. One of the first result in this direction was obtained by Shao-Gao Deng [6] for the p(x) laplacien, here, we borrow some ideas from that work. The purpose of this work is to improve the results of [6] and extend them to the case of p(x)-biharmonic equation with Navier and Neumann condition. This article consists of three sections. In section 2, we start with some preliminary basic results on theory of Lesbegue-Sobolev spaces with variables exponent (we refer to the book of Musielak [3], Mihăilescu and Rădulescu [2]), we recall Ricceri s variational principle with some results which are needed later. In section 3, we give the proof of the main result. 2. Preliminaries 2.. Variable exponent space and Sobolev Spaces In order to deal with the problem (P), we need some theory of variable exponent Sobolev Space. For convenience, we only recall some basic facts which will be used later. Suppose that R N be a bounded domain with smooth boundary. Let C + () = {p C() such that inf x p(x) > }. For any p(x) C + (), denote by p k (x) = { N p(x) N kp(x) if kp(x) < N, + if kp(x) N. Define the variable exponent Lebesgue space L p(x) (), L p(x) () = {u L () : u p(x) dx < } then L p(x) () endowed with the norm u p(x) = inf{λ > 0 : u λ p(x) dx } becomes a Banach space separable and reflexive space.

4 82 Abdel Rachid El Amrouss and Anass Ourraoui Proposition 2.. Set, ρ(u) = u p(x) dx, if u L p(x) () we have : () u p(x) u p p(x) ρ(u) u p+ p(x). (2) u p(x) u p p(x) ρ(u) u p+ p(x). Define the variable exponent Sobolev space W k,p(x) () by W k,p(x) () = {u L p(x) () : D α u L p(x) (), α k}, where D α u = α α x... α N x N with α = (α,α 2,...,α N ) is a multi-index and α = Σ N i= α i. The space W k,p(x) () with the norm u = Σ α k D α u p(x) is a Banach separable and reflexive space. Proposition 2.2. ([2,7,3]). For p,r C + () such that r(x) p k (x) for all x, there is a continuous and compact embedding W k,p(x) () L r(x) (). We denote by W k,p(x) 0 () the closure of C 0 () in W k,p(x) (). Remark 2.. )(W 2,p(x) () W,p(x) 0 (),. ) is a Banach space separable and reflexive space. 2) Define u a = inf{λ > 0 : [ u λ p(x) +a(x) u λ p(x) ]dx }, for all u W 2,p(x) () or W 2,p(x) () W,p(x) 0 () then u a is a norm on W 2,p(x) () and W 2,p(x) () W,p(x) 0 (), equivalent the usual one. 3) By the above remark and proposition 2.2 there is a continuous and compact embedding of W 2,p(x) () W,p(x) 0 () into L r(x) (), where r(x) < p (x) for all x. Proposition 2.3. Set ρ a (u) = [ u p(x) +a(x) u p(x) ]dx. For u,u n W 2,p(x) () we have, () u a u p+ a ρ a (u) u p a. a. (2) u a u p+ a ρ a (u) u p (3) u n a 0 ρ a (u n ) 0. (4) u n a + ρ a (u n ) +. The proof is similar to proof in [7] [Theorem 3.]. Proposition 2.4. ([7]). For any u L p(x) (),v L q(x) (), we have uvdx ( p + q ) u p(x) v q(x), where p(x) + q(x) =. We denote by X = W 2,p(x) () W,p(x) 0 () when B = B,T = T and X = W 2,p(x) () if B = B 2 and T = T 2.

5 Existence of solutions 83 Definition 2.. Let u X, u is called weak solution of problem (P) if for all v X, u p(x) 2 u vdx+ a(x) u p(x 2) uvdx = f(x, u)vdx+λ g(x, u)vdx. We define, I(u) = p(x) [ u p(x) +a(x) u p(x) ]dx F(x,u)dx and J(u) = G(x,u)dx, where F(x,t) = t 0 f(x,s)ds, G(x,t) = t g(x,s)ds and ϕ = I + λj,λ R. 0 The following proposition will be used later, Proposition 2.5. () Let L(u) = p(x) [ u p(x) +a(x) u p(x) ]dx. Then the functional L : X R is sequentially weakly lower semi continuous, L C (X, R). (2) the mapping L : X X is a strictly monotone, bounded homeomorphism and is of type S +, namely, u n u and limsup n L (u n )(u n u) 0 implies that u n u, where and denote the strong and weak convergence respectively. Proof: : () Since p(x) u p(x) dx C (X, R) then L is well defined and L C (X, R). By the continuity and convexity of L, we deduce that L is sequentially weakly lower semi continuous. (2) Since L is Fréchet derivative of L then L is continuous and bounded. We set U p = {x : p(x) 2}, = {x : < p(x) < 2}. By the elementary inequalities, we have x,y R N x y γ 2 γ ( x γ 2 x y γ 2 y).(x y) if γ 2, x y 2 γ ( x + y )2 γ ( x γ 2 x y γ 2 y).(x y) if < γ < 2, where x.y denotes the usual inner product in R N. Then for all u,v X such that u v L (u) L (v),u v > 0, which means that L is strictly monotone. Let (u n ) n be a sequence of X such that and u n u in X lim sup L (u n ),u n u 0. n It suffices to show that ( u n u p(x) +a(x) u n u p(x) ) 0, (2.)

6 84 Abdel Rachid El Amrouss and Anass Ourraoui by the monotonicity of L, we claim that Since u n u in X then Put L (u n ) L (u),u n u 0. lim sup L(u n ) L(u),u n u = 0. (2.2) n χ n (x) = ( u p(x) 2 u n u p(x) 2 u).( u n u), ξ n (x) = ( u n p(x) 2 u n u p(x) 2 u).(u n u). By the compact embedding of X into L p(x) (), u n u in L p(x) (), u n p(x) 2 u n u p(x) 2 u in L q(x) (), where p(x) + q(x) = for all x. It follows that, ξ n (x)dx 0, (2.3) using (2.) and (3.3), then we have lim sup χ n (x)dx = 0. (2.4) n Thanks to the above inequalities, u n u k p(x) dx 2 p + U p χ n (x)dx, U p u n u k p(x) dx 2 p + U p ξ n (x)dx. U p It results that U p ( u n u ) p(x) +a(x) u n u p(x) dx 0 as n. (2.5) Besides, in, put δ n = u n + u, we have u n u p(x) dx p (χ n ) p(x) 2 (δn ) p(x) 2 (2 p(x)) dx. By Young s inequality, d u n u p(x) dx (dχ n ) p(x) 2 (δn ) p(x) 2 (2 p(x)) dx V p χ n (d) 2 p(x) dx + (δ n ) (x) dx. (2.6) p

7 Existence of solutions 85 From (2.4) and since χ n 0, one consider that 0 χ n dx <. If χ n dx = 0 then u n u p(x) dx = 0. Or else, we take d = ( χ n dx) 2 2 > and the fact that p(x) < 2, inequality (2.6) becomes u n u p(x) dx d ( χ n d 2 dx + V p (χ n dx) 2 ( + Note that, δp(x) n dx is bounded, which implies Similarly we can have Hence, it result that u n u p(x) dx 0 as n. u n u p(x) dx 0 as n. δ p(x) n dx), δ p(x) n dx). ( u n u p(x) +a(x) u n u p(x) )dx 0 as n. (2.7) Finally, (2.) is given by combining (2.5) and (2.7). It remains to show that L is is a homeomorphism. In view of strict monotonicity of L which implies the injectivity of L. Moreover, L is a coercive. Indeed, since p >, for each u X such that u we have L (u),u u = ρ a(u) u u p as n. Consequently, thanks to a Minty-Browder [5], L is surjective and admits an inverse mapping. It suffices to show the continuity of (L ). Let (f n ) n be a sequence of X such that f n f in X. Let u n and u in X such that (L ) (f n ) = u n and (L ) (f) = u. By the coercivity of L, one deducts that the sequence (u n ) is bounded in the reflexive space X. For a subsequence, we have u n û in X, which implies lim n L (u n ) L (u),u n û = lim n f n f,u n û = 0.

8 86 Abdel Rachid El Amrouss and Anass Ourraoui Since L is of (S + ) type and continuous, it follows that u n û in X and L (u n ) L (û) = L(u) in X. Moreover, since L is an injection, we deduce that u = û. Proposition 2.6. let σ(u) = G(x,u)dx, then σ is a C in L q(x) () and σ are weakly-strongly continuous, i.e u n u implies σ (u n ) σ (u). Proof: : by (.) we have, G(x,t) A(x) + B t q(x), where A L (),A(x) 0, B 0. Then Nemytskii operator properties implies that σ is a C in L q(x) (). By the continuous embedding of X into L q(x) (), we have σ is also C in X and for u,v X σ (u)v = g(x, u)vdx. Let (u n ) n X be a sequence such that u n u. Since there is a compact embedding of X into L q(x) (), there is a subsequence, noted also (u n ) n, such that u n u in L q(x) (). According to the Krasnoselki s theorem, the Nemytskii operator N g : L q(x) () L q(x) q(x) () u f(., u) is continuous. Hence, N g (u n ) N g (u) in L q(x) q(x) (). Using Holder s inequality and the continuous embedding of X into L q(x) (), we obtain σ (u n ) σ (u),v = (g(x,u n ) g(x,u))v(x)dx 2 N g (u n ) N g (u) q(x) v(x) q(x) q(x) C N g (u n ) N g (u) q(x) v a. q(x) Thus, σ (u n ) σ (u) Ricceri s variational principle Definition 2.2. Let G a bounded subset of X and ρ R. G is called a block of I with type ρ if I(u) < ρ x G and I(x) = ρ x G. Where G = G W \G and G W is the closure of G in X in the weak topology. Definition 2.3. Let D a bounded open subset of X and c < b is called Ricceri box of I with the type (c,d) if c = inf I < inf I = b. D D

9 Existence of solutions 87 Definition 2.4. Let Y be a Banach space, G 0 and G be two bounded open subset of Y with G 0 G and φ : Y R a functional. (G 0,G) is a valley box of φ if sup G 0 φ < inf G φ. Theorem 2.. (see [6,8]) Assume that I, J : X R are sequentially weakly lower semi continuous and G is a Ricceri block of I with type ρ. Let ρ I(x) λ = sup x G J(x) inf J G W then for each λ ]0,λ [, the restriction of I + λj to G W achieves its infimum at some x G, so x is a local minimizer of I + λj. Remark 2.2. i) let u X a strictly local minimizer of I, then for ǫ > 0 small enough, we have inf B(u,ǫ) I > I(u ) i.e B(u,ǫ) is a Recceri box of I. ii) In fact, by proposition 2.6) in [8], I, J : X R are sequentially weakly lower semi continuous. Proposition 2.7. [8] Suppose that G is a Ricceri box of I with type (c,b) and I : X R continuous. Then for every ρ ]c,b] we have I (],ρ[) G is a Ricceri block of I with type ρ. By Proposition 2.5, Remark 2.2 and Theorem 2. we obtain the following result, Proposition 2.8. [6] Suppose that I,J : X R are continuous. For some r > 0, u B(u 0,r), I(u 0 ) = inf B(u0,r) I = c 0 ; inf B(x0,r) I = b > c 0 and u is a strictly local minimizer of I and I(u ) = c > c 0. Then for ǫ > 0 small enough and ρ > c,ρ 0 ]c 0,min{b,c }[ and λ ]0,λ [, I + λj has at least two local minima u 0,u in B(u 0,r). Where u 0 I (],ρ 0 [) B(u 0,r),u 0 B(u,ǫ) and u I (],ρ [) B(u,r). Theorem 2.2. [6] Let Y be a reflexive Banach space. Assume that ) φ C (Y, R), the mapping φ : Y Y is of type S +. 2) (G 0,G) is a valley box of φ with G 0,G being connected and 0 G 0. 3) There exist e G 0 and r > 0 such that e > r, inf φ > max{φ(0),φ(e)}. B(0,r) Then, the functional φ has at least a critical point u 0 G with φ(u 0 ) = c, where c = inf γ Γ sup t [0,] φ(γ(t)) and Γ = {γ C([0,],G) : γ(0) = 0;γ() = e}. Corollary 2.. Under the same assumption as in previous theorem, furthermore, if J : Y R C and J : Y Y are weakly strongly continuous. Then, for each λ ]0,λ [, I + λj has still a mountain pass type critical point u 2 G.

10 88 Abdel Rachid El Amrouss and Anass Ourraoui 3. Proof of the main result The critical point of the integral functional ϕ = I+λJ is solution of the problem (P). Define p(x) λ = inf [ u p(x) +a(x) u p(x) ]dx u X\{0} p(x) u, p(x) dx Proof: [proof of Theorem.] The proof is divided into four steps, step(): We show that v 0 = 0 is strictly local minimizer of I. By (.) and assumption (F 3 ), we may find q C() with p + < q q (x) < p (x) such that F(x,t) c 3 t q(x), x, t R. (3.) We can assume that u a < is small enough, thus I(u) p(x) [ u p(x) +a(x) u p(x) dx] c 3 p + u + p a c 4 u q a. u q(x) Since q > p+ then there exists ǫ > 0 such that u B(0,ǫ) \ 0 we have I(u) > 0 = I(v 0 ). step(2): We show that the functional I has a global minimizer v 0. Set H(x,t) = F(x,t) λ p(x) t p. Then, from (F ) we conclude that, for every M > 0, there is R M > 0 such that H(x,t) M, t R M, almost every x. (3.2) We have I is coercive, or else there exist K R and (u) n X such that Putting v n = un u n a u n a and I(u n ) K. i.e v n a =. Then for subsequence, we may assume that for v X, we have v n v in X, v n v strongly in L p(x) (), v n (x) v(x) for almost every x. Now, using 3.2, we obtain K I(u n) = p(x) [ u p(x) +a(x) u n p(x) ]dx F(x, u n)dx. p(x) [ un p(x) +a(x) u n p(x) ]dx λ un p dx H(x, u n)dx p(x) un p p + a λ un p dx + M, (3.3) p(x) where M R. Dividing 3.3 by u n p a and passing to the limit, we conclude p + λ v p dx 0,

11 Existence of solutions 89 hence, v 0. Therefore > 0 such that = {x \ v 0 (x) 0}, then u n (x) + for almost every x. On the other hand, Where, λ [ u ] u p p(x) dx λ u p(x) dx [ u ] λ u p(x) dx p(x) [ u p(x) +a(x) u p(x) ]dx. (3.4) [ u ] = {x \ u } ; [ u < ] = {x \ u < }. It is clear that [ u n <] p(x) u n p(x) dx is bounded. From (F ) and the above inequalities 3.4 we deduce K p(x) [ u p(x) +a(x) u p(x) ]dx F(x,u n )dx = p(x) [ u n p(x) +a(x) u n p(x) u n p ]dx λ dx H(x,u n )dx p(x) = p(x) [ u n p(x) +a(x) u n p(x) ]dx λ u n p λ dx H(x,u n )dx p(x) λ [ u n <] [ u n <] u n p dx p(x) [ u n ] u n p dx p(x) H(x,u n )dx + H(x,u n )dx +, \ which is a contradiction. Hence I is coercive and has a global minimizer v. When the assumption (F 2 ) holds, taking ω C0 (B(x 0,2r 0 )) such that 0 ω t 0 for all x B(x 0,2r 0 ), ω (x) t 0 for x B(x 0,r 0 ) and ω (x) 2t0 r 0 then ω X. On the other hand, I(ω ) 2t0 dx+ a B(x 0,2r 0 )\B(x 0,r 0 ) r 0 B(x 0,2r 0 ) p(x) t0 p(x) dx F(x, ω )dx C 0 B(x 0, r 0) F(x, ω ).dx Since F(x,ω )dx > B(x 0,r 0) F(x,ω )dx C 0 B(x 0,r 0 ), we obtain I(ω ) < 0 then I(v ) < 0 = I(v 0 ). So v 0. step(3): We show that ϕ has two local minima. Since I is coercive there is r 0 > 0 large

12 90 Abdel Rachid El Amrouss and Anass Ourraoui enough such that v 0,v B(0,r 0 ) and inf B(0,r0) I > I(v 0 ) > I(v ). By proposition 2.8, given any ǫ > 0, ρ ]I(v ),0[ and ρ 2 > 0 then λ ]0,λ [, ϕ has at least two local minima u 0 B(0,ǫ) I (],ρ 2 [), u I (],ρ [) and u / B(0,ǫ). The minimizer u 0 0. In fact, when (G ) holds, taking ω C0 (B(x,r )) such that 0 ω and ω(x) for x B(x, r 2 ), then, it is easy to see that for λ ]0,λ [, when t > 0 is small enough, we get tω B(0,ǫ) I (],ρ 2 [) and I(u 0 ) + λj(u 0 ) I(tω) + λj(tω) < 0. In particular, u 0 0. step(4): ϕ has a mountain pass type critical point λ ]0,λ [. We take r > 0 such that B(0,r ) X and B(0,r ) I (],ρ [) B(0,ǫ). Since I is coercive, there exists r 2 > r such that inf I > sup I, B(0,r 2) B(0,r ) then (B(0,r ),B(0,r 2 )) is a valley box of I. Since I(v ) < 0 = I(v 0 ) and by step ), we have that for some ǫ 0 > 0 with ǫ 0 > v a, inf B(ǫ,0) I > 0. We apply the Corollary 3., then ϕ admits a mountain pass point u 2. Consequently, u 0, u and u 2 are at least three nontrivial solutions of the problem (P). Proof: [proof of Theorem.2] It was the same steps of the previous proof. step(): To show that v 0 = 0 is strictly local minimizer of I, we follow the same procedure as in step () in the previous proof. step(2): We show that the functional I has a global minimizer v 0. Similarly in step(2) in the last proof of Theorem.2, one shows the coercivity of I and then I has a global minimizer v. We use the condition (F 2), we obtain I(ξ) < 0 and then I(v ) < 0 = I(v 0 ). So v 0. step(3): We show that ϕ has two local minima. The same way as In step 3 in the last proof of Theorem.2, ϕ has at least two local minima u 0 and u 0. Moreover, the minimizer u 0 0. Indeed, by assumption (G 2 ), taking t n 0 such that inf x G(x,t n ) t n p +. Let w n = t n ( i.e w n (x) X). For all λ ]0,λ [, by 3., we get a(x) ϕ(w n ) t n p p(x) dx F(x,t n )dx λ G(x,t n )dx t n p K + c 3 t n q(x) G(x,t λ t n p n ) dx t n p t n p K + t n q K2 λ t n p G(x,t n ) t n p dx < 0, (3.5)

13 Existence of solutions 9 and Thus, w n B(0,ǫ) I (],ρ 2 [). ϕ(u 0 ) ϕ(w n ) < 0 = ϕ(0), so u 0 0 step(4): As in step(4) of the theorem., ϕ has a mountain pass type critical point u 2. Consequently, u 0, u and u 2 are at least three nontrivial solutions of the problem (P). References. E. Acerbi, G. Mingione, Regularity results for stationary electrorheological fluids, Arch. Ration. Mech.Anal.64 (2002) R.A. Adams, Sobolev Spaces, Academic Press, New York, A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and application, J.Funct.Anal.4 (973) A. Ayoujil, A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent. Nonlinear Anal. T.M.A. (2009) G.Bonanno, B.Di Bella, A boundary value problem for fourth-order elastic beam equations, J.Math.Anal.Appl. 343(2) (2008) S.G. Deng, A local mountain pass theorem and applications to a double perturbed p(x)- laplacien equations, J.Applied Mathematics and computation 2 (2009) X.L. Fan, D. Zhao, On the spaces L p(x) () and W m,p(x) (), J.Math.Anal.Appl. 263(200) X.L. Fan, Shao-Gao Deng, Remarks on Ricceri s variational principle and applications to the p(x) Laplacian equations, J.Nonlinear Anal 67 (2007) A. Ferrero, G. Warnault, On a solutions of second and fourth order elliptic with power type nonlinearities, Nonlinear Anal.T.M.A 70 (2009) C. Li, C-L Tang, Three solutions for a class of quasilinear elliptic systems involving the (p,q)-laplacian, Nonlinear Anal. 69 (2008) A.M. Micheletti, A. Pistoia, Multiplicity rsults for a fourth-order semilinear elliptic problem, Nonlinear Anal 3(7) (998) M. Mihăilescu, V. Rădulescu, A multiplicity result for nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc.R.Soc.Lond.Ser.A 462 (2006) J. Musielak, Orlicz Spaces and Modular Spaces, in:lecture Notes in Mathematics,vol.043, Springer, Berlin, M. R užička, Electrorheological Fluids: Modeling and Mthematical Theory,in:Lecture notes in Math, vol.748,springer-verlag,berlin, E. Zeidler, Non linear Fonction Analysis and its Applications, vol.ii/b:nonlinear Monotone Operators, Spring, New york, J. Zhang, S. Li, Multiple nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal(60) (2005)

14 92 Abdel Rachid El Amrouss and Anass Ourraoui Abdel Rachid El Amrouss University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco address: and Anass Ourraoui University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco address: