EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR ANISOTROPIC ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT AND NONLINEAR ROBIN BOUNDARY CONDITIONS

Transcription

1 Electronic Journal of Differential Equations, Vol. 207 (207), No. 88, pp. 7. ISSN: URL: or EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR ANISOTROPIC ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT AND NONLINEAR ROBIN BOUNDARY CONDITIONS BRAHIM ELLAHYANI, ABDERRAHMANE EL HACHIMI Communicated by Jesus Ildefonso Diaz Abstract. This article presents sufficient conditions for the existence of solutions of the anisotropic quasilinear elliptic equation with variable exponent and nonlinear Robin boundary conditions, NX u p i (x)2 u NX u pi(x)2 u λ u m(x)2 u = γg(x, u) x i x i x i in, NX u p i (x)2 u υ i = µ u q(x)2 u on. x i x i Under appropriate assumptions on the data, we prove some existence and multiplicity results. The methods are based on Mountain Pass and Fountain theorems.. Introduction Many problems in physics and mechanics can be modeled with sufficient accuracy using classical Lebesgue and Sobolev spaces, L p () and W,p (), where p is a fixed constant and is an appropriate domain. But for the electrorheological fluids (Smart fluids), this is not adequate but rather, the exponent should be able to vary. This leads to study the problem in the frame-work of variable exponent Lebesgue and Sobolev spaces, L p( ) () and W,p( ) (), where p( ) is a real-valued function; see, e.g. [2, 3]. On the other hand, it has been experimentally shown that the above-mentioned fluids may have their viscosity undergoing a significant change; see, e.g. [3]. Consequently, the mathematical modelling of such fluids requires the introduction of the so-called anisotropic variable spaces.indeed, there is by now a large number of papers and increasing interest about anisotropic problems. With no hope of being complete, let us mention some pioneering works on anisotropic Sobolev spaces 200 Mathematics Subject Classification. 35J20, 35J25, 35J62. Key words and phrases. Anisotropic elliptic problems; Robin boundary conditions existence and multiplicity. c 207 Texas State University. Submitted May 24, 206. Published July 24, 207.

2 2 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 [20, 24] and some more recent regularity results for minimizers of anisotropic functionals [, 6, 2]. Therefore, in the recent years, the study of various mathematical problems modeled by quasilinear elliptic and parabolic equations with both anisotropic and variable exponent has received considerable attention. Let us mention many works in that direction by Antontsev and Shmarev; see, e.g. [2] and the references therein. Our paper is mainly devoted to the existence and multiplicity of solutions of quasilinear elliptic equations under nonlinear Robin boundary condition such as ( u 2 u ) x i x i x i u pi(x)2 u λ u m(x)2 u = γg(x, u) in, u 2 u υ i = µ u q(x)2 u x i x i on, (.) where R n is a bounded domain with n 2, with smooth boundary and υ i are the components of the outer normal unit vector and for i {,..., N}, p i, m C( ), q C( ). The functions p i and g are supposed to satisfy some conditions to be specified below, while λ, γ, and µ are real parameters, with γ, µ > 0. We shall give conditions under which problem (.) has infinitely many solutions. According to the behaviour of g and to the kind of results we want to prove, variational methods turn out to be more appropriate. When lim s 0 g(x, s)/ s σ = 0, σ to be made precise later, Mountain Pass theorem provides the existence of at least a solution of (.) and, on the other hand, when g is an odd function, Fountain s theorem yields the existence of infinitely many solutions. A host of publications exist for this type of problems when the boundary condition is replaced by u ν = 0 on and γ = 0 ; see, e.g. [6] and the references therein, where the authors obtained existence results by means of standard variational tools. The associated problem with Dirichlet boundary conditions has also been treated by many authors; see, e.g. [3, 5]. Furthermore, existence of positive solutions for nonlinear Robin problem involving the p(x)-laplacian have been studied by S. G. Deng; in [9], by using the sub-super solutions and variational methods. We consider here the case where µ is positive and g satisfies more hypotheses than in [7], to use the Mountain Passe and Fountain theorems. It turns out that the condition q > P plays an important role in the proofs of our main results. This article is divided into four sections. In the second section, we introduce some basic properties of the generalized Lebesgue-Sobolev space W,p(x) () and anisotropic Sobolev spaces W, p (x) (), and state the existence and multiplicity results concerning the problem (.). The third section is devoted to the proofs of the main results and finally, in the fourth section we deal with a generalized equation related to our problem (.). 2. Preliminaries and main results To study problem (.), we need to introduce the notions of Sobolev space W,p(x) () and anisotropic Sobolev spaces W, p (x) (), with variable exponent. For convenience, we only recall some basic facts which will be used later. Let

3 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 3 R N be a measurable subset with meas() > 0. We write C( ) = {u : u is a continuous function in }, C ( ) = {u C( ) : ess inf u }. Suppose that is a bounded domain of R N with a smooth boundary, and p C(, R) with p(x) >, for any x. Denote p = inf x p(x) and p = sup x p(x); then we have, p > and p <. Denote by M be either or. Define the variable exponent Lebesgue space L p(x) (M) = { u u : M R is measurable and u(x) p(x) dx < }, endowed with the Luxembourg norm u p(x) = u L p(x) (M) = inf Proposition 2. ([8]). Let ρ(u) = M, 2,... ), we have: M { τ > 0; u(x) M τ u(x) τ () u L p(x) (M) u p ρ(u) L p(x) u p (M) L p(x) (M). (2) u L p(x) (M) > u p ρ(u) L p(x) u p (M) L p(x) (M). (3) u k L p(x) (M) 0 ρ(u k ) 0. (4) u k L p(x) (M) ρ(u k ). We define the variable exponent Sobolev space } p(x) dx. p(x) dx. For u, u k L p(x) (M)(k = W,p(x) () = {u L p(x) () : u L p(x) ()}, endowed with the norm { ( u(x) u = inf τ > 0; p(x) u(x) p(x)) } dx. τ τ Proposition 2.2 (See [2]). Both (L p(x) (M)), p(x) ) and (W,p(x) (), ) are separable, reflexive and uniformly convex Banach spaces. Proposition 2.3 (See [2]). The Hölder inequality holds, namely uv dx 2 u p(x) v q(x) ; u L p(x) (M), v L q(x) (M), M where p(x) q(x) =. Proposition 2.4 (See [8]). Let ρ(u) = ( u(x) p(x) u(x) p(x) ) dx. For u, u k W,p(x) ()(k =, 2,... ), we have () u u p ρ(u) u p. (2) u > u p ρ(u) u p. (3) u k 0 ρ(u k ) 0. (4) u k ρ(u k ). Let p ( ) : R N be a vectorial function, p ( ) = (p ( ), p 2 ( ),..., p N ( )) such that 2 p i N and put p M (x) = max{p (x),..., p N (x)}.

4 4 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 The anisotropic Sobolev space with variable exponent is defined by W, p (x) () = {u L p M (x) () : endowed with the norm u p ( ) = For convenience, we denote: u x i L pi( ) (), i {,..., N}}, u x pi( ) N u pi( ). i P = inf{p, p 2,..., p N }, P = sup{p, p 2,..., p N } and write X = W, p (x) (). We know that X is reflexive if P >, (see e.g [22]). We define ( N J(u) = u N p i (x) x i p i (x) u pi(x)) dx, We have (J u, v) = G(x, u) = u 0 g(x, s)ds. ( N u 2 u v ) u pi(x)2 uv dx, x i x i x i for all v X. In all this paper C, C i (i = 0,, 2,... ) represents different positive real constants. We make the following assumptions on the functions q and g. (H0) q C( ) satisfies : q(x) (N)P for all x and q < P NP. (H) g : R R is a Caratheodory type function and there exist a constant C > 0 and a function α C( ) such that: < α(x) < NP, for all x NP and g(x, s) C( s α(x) ) for all (x, s) R. (H2) There exists M > 0 and θ λ m (resp θ λ m ) if λ 0 (resp λ < 0 ). such that for all s with s M and x, we have 0 < θ λ G(x, s) sg(x, s). (H3) g(x, s) = ( s P ) as s 0 and uniformly for x, (H4) g(x, s) = g(x, s), x, s R. We say that u X is a weak solution of (.) if ( N u 2 u v ) u pi(x)2 uv dx λ u m(x)2 uv dx x i x i x i = γg(x, u)v dx µ u q(x)2 uv dx, for all v X.

5 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 5 The energy functional associated with problem (.) is ( N Φ(u) = u N p i (x) x i p i (x) u pi(x)) dx λ m(x) u m(x) dx γ G(x, u) dx µ q(x) u q(x) dx. (2.) Proposition 2.5 (See [4, ]). () L J : X X is a continuous, bounded and strictly monotone operator; (2) L is a mapping of type (S ), i.e. if u n u in X, and lim n (L(u n ) L(u), u n u) 0, then u n u in X; (3) L : X X is a homeomorphism. The following are embedding results on anisotropic generalized Sobolev spaces and will be used later. Proposition 2.6 (See [2]). Suppose R N boundary. For any q C () satisfying q(x) < NP NP is continuous and compact. W, p (x) () L q(x) () is a bounded domain with smooth for all x, the embedding Proposition 2.7 (See [2]). Assume that the boundary of possesses the cone property and p i C(), 2 p i < N for all i {, 2,..., N}. If q C( ) satisfies the hypothesis < q(x) < (N)P NP is continuous and compact. for all x, then the embedding W, p (x) () L q(x) ( ) The main results of this article are as follows: Theorem 2.8. Suppose that (H0) (H2), (H4) hold with m < NP NP and P < min(α, m ). Then, for any λ R and µ, γ > 0, problem (.) has at least a nontrivial weak solution. Theorem 2.9. Suppose that (H0) (H2), (H5) hold with m < NP and P NP < min(α, m ). Then, for any λ R and µ, γ > 0, problem (.) has infinite many pairs of weak solutions. 3. Proofs of main results To prove Theorem 2.8, we shall use the Mountain Pass theorem [25]. We first start with the following lemmas. Lemma 3.. If (H0) (H2) hold, then for any λ R, the functional Φ satisfies the Palais Smale condition (PS). Proof. Suppose that (u n ) X is a Palais Smale sequence, ie, sup Φ(u n ) C, Φ (u n ) 0, as n. We shall prove that (u n ) has a convergent subsequence.

6 6 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 Let us show that (u n ) is bounded in X. Denote by m : m if λ > 0 and m : m if λ 0. Since Φ(u n ) is bounded, then by using (H), we have for large n, C C u n Φ(u n ) θ λ Φ (u n ) ( P ) N ( u un pi(x)) dx θ λ x i ( λ m ) u n θ λ m(x) dx γ (G(x, u n ) θ λ g(x, u n )u n ) dx ( Φ (u n ), u n µ ) u n q(x) dx θ λ θ λ q(x) ( P ) N u P θ λ x p (Φ (u n ), u n ) i i(x) θ λ ( µ ) θ λ q u n q(x) dx. Now, according to [4, page 6], we have u P p ( ) 2 P N P ( u ) P x p u P i i( ) p i( ) Then, ( C C u n 2 P N P P ( u x i u ) dx. θ λ ) u n P p ( ) C θ λ u n p ( ) C. Since µ > 0, then by using condition (H2) and the inequality above, we deduce that u n is bounded in X. The proof is complete. Lemma 3.2. There exist r, C > 0 such that Φ(u) C, for all u X such that u = r. Proof. Conditions (H0), (H) and (H2) ensure that, for any ɛ > 0, we have G(x, s) ɛ s P C(ɛ) s α(x), for all (x, s) R. For u small enough, we thus obtain Φ(u) ( u ) u P dx λ x i m(x) u m(x) dx (ɛ u P C(ɛ) u α(x) ) dx µ q u q(x) dx P 2 P N P u P p ( ) λ m u m(x) dx ɛ u P dx C(ɛ) u α(x) dx µ q u q(x) dx. (3.)

7 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 7 Since P < α α(x) < NP, for all x and q(x) < (N)P, for all x ; NP NP then, we have W, p (x) () L P () and W, p (x) () L q(x) ( ), with continuous and compact embeddings. Consequently, there exist two constants C > 0 and C 2 > 0 such that u L P () C 2 u, u L q(x) () C u, for all u X. (3.2) By using (3.2) for u small enough, we obtain from (3.) that λ Φ(u) P 2 P N P u P m max{ u m L m(x) (), u m ɛc 2 u P C(ɛ)C 3 u α µ q C u q. Since W, p () L m (), we have L m(x) () } λ C Φ(u) P 2 P N P u P m max{ u m, u m } ɛc 2 P u P C(ɛ)C 3 u α µ q C u q. Now, let ɛ > 0 be small enough so that: 0 < ɛc 2P 2P 2 P N P =: c 0. We have Φ(u) c 0 u P λ C, m max{ u m u m } C(ɛ) u α µc q u q ( u P c 0 λ C ) m max{ u m P, u m P } ( u P C(ɛ) u α P µc ) q u q P. Since P < min (α, m, q ), then there exist r > 0 and C > 0 such that Hence, the proof is complete. Φ(u) C > 0, for any u X. Proof of Theorem 2.8. To apply the Mountain Pass theorem ([25]), we have to prove that Φ(tu) as t, for some u X. From (H2), it follows that G(x, s) C s θ λ, x, s M. For u X and t >, we have Φ(tu) ( (tu) ) tu P dx λ x i G(x, tu) dx µ q tu q(x) dx m(x) tu m(x) dx

8 8 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 tp P Ct θ λ ( (tu) x i u θ λ dx µtq q ) u dx λt em u q(x) dx, m(x) u m(x) dx where again m = m if λ > 0 and m = m if λ 0. By (H0) and (H2), it follows that, for any λ R, Φ(tu) as (t ). Since Φ(0) = 0, it follows that Φ satisfies the condition of the Mountain Pass lemma, and so Φ admits at least one nontrivial critical point u 0 X; which is characterized by where τ = inf sup h Γ t [0,] Φ(h(t)), Γ = {h C([0, ], X); h(0) = 0 and h() = e}. Proof of Theorem 2.9. Let X be a reflexive and separable Banach space. It is well know (see, e.g. []) that there are {e j } j= X and {e j } j= X (where X is the topological dual of X) such that X = span{e j :, 2,... }, X = span{e j :, 2,... }, and e j, e i = { if i = j, 0 if i j. (3.3) For convenience, we write X j = span{e j }, Y k = k j= X j and Z k = j=k X j. Denote p (x) = { Np(x)/(N p(x)) if p(x) < N, if p(x) N. Lemma 3.3 (See [8, 0]). Let β(x) C ( ), with β(x) < p (x) for x and α k := sup{ u L β(x) (); u =, u Z k }. Then, we have lim k α k = 0. To prove Theorem 2.9, we shall use the Fountain theorem (see [25, Theorem 3.6] ). Obviously, Φ C (X, R) is an even functional. By using (H0) and (H) we first prove that if k is large enough, then there exist ρ k > ν k > 0 such that b k := inf{φ(u)/u Z k, u = ν k } as k ; (3.4) a k := max{φ(u)/u Y k, u = ρ k } 0 as k. (3.5)

9 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 9 Proof of (3.4): For any u Z k, u = r k >, we have Φ(u) ( u ) u P dx λ x i m C ( u α(x) ) dx µ u q(x) dx P q u P dx λ x i m C ( u α(x) ) dx µ q u m(x) dx u q(x) dx C λ P 2 P N P u p m u m(ξ) L m(x) u m(x) dx C u α(ξ) α(x) µ q u q(ξ) L q(x) ( ) C 2, for some ξ. (3.6) So, for the study of the previous inequality, we only need to consider either the case where m(x) α(x) or the case where m(x) < α(x) for all x. Let us assume that m(x) α(x) for all x. Then, we have L α(x) () L m(x) (). Thus, there is a positive constant C 3 > 0 such that So, for any ξ, we have u L m(x) () C 3 u L α(x) () for all u X. u m(ξ) L m(x) () Cm(ξ) u m(ξ) L α(x) (). Let us denote e := /(2 P N P ). Then, for any ξ, we have Φ(u) e u P Denote P e P u P C u m(ξ) L α(x) C u α(ξ) L α(x) () µ q u q(ξ) L q(x) ( ) C 2 C max{ u α(ξ) L α(x) (), u m(ξ) L α(x) () } µ q u q(ξ) L q(x) ( ) C 2. E = L α(x) () L q(x) ( ), A = {u E : u L α(x) (), u L q(x) ( ) }, B = {u E : u L α(x) () >, u L q(x) ( ) }, C = {u E; u L α(x) (), u L q(x) ( ) > }, D = {u E; u L α(x) (), u L q(x) ( ) > }. From (3.7), we have e u P P C if u A, e u P P Φ(u) C (α k u ) α C 2 if u B, e u P P µ q (β k u ) q C if u C, e u P C (α k u ) α µ q (β k u ) q C 2, if u D, P (3.7)

10 0 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 where β k = sup{ u L ( ); u =, u Z q(x) k }. It is obvious that Φ(u) as u in A. For u B C, we have Φ(u) e u P C2 ( α k u ) eα C 3, P By taking α k to be either α k or β k and α = α or q, we obtain ( Φ(u) e P Since α k 0 as k and α > P, then Consequently, we have If u D, then Φ(u) ) ( ) P C2 α e α α eα P eα k C 3. Φ(u) as u, u Z k. e P By assuming α q, we obtain Φ(u) e P e P e P ( e P ( C 2 e α α eα ) P k eα as k. u P C(α α k u α ) µ q (βq k u q ) C. u P C(α α k u q ) µ u P (Cα α k µ q βq q (βq k u q ) C k ) u q C u P C3 (α α k β q k ) u q C ) [ ] P q C 3 q (αk α β q k ) P q C. Since q > P, we then have [C 3 q (αk α β q k )] P q, as k. Consequently, we obtain Φ(u) as u, u Z k. Now, from condition (H2), we have G(x, s) C s θ λ C 2, for any (x, s) R. Then there exist constants C, C 3 > 0 such that Φ(u) C u λ dx P u m(x) dx C x i m 2 u θ λ C 3, where m = m if λ > 0 and m = m if λ 0. Hence, we obtain the inequality u ( P x C N u ) P L L p i (x) p () i x i, (x) () i Where C is a positive constant. In the case λ > 0, we obtain Φ(u) C P u P C 4 λ m u m C 2 u θ λ C 3.

11 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS But we have W, p (x) () L m(x) (), and W, p (x) () L q(x) ( ). Then, as θ λ > max(p, m ) and dim Y k = k, it is easy to see that For the case λ 0, we have Φ(u) as u for u Y k. Φ(u) C P u P C 2 u θ λ C 3. Now, as we have θ λ > P and dim Y k = k, it is also easy to see that Φ(u) as u for u Y k. 4. A generalized equation We shall now consider the generalized equation n u ( u 2 u ) n u pi(x)2 u x i x i x i = λg (x, u) νg 2 (x, u) in, n u 2 u υ i = µf(x, u) x i x i on, (4.) where λ, ν, µ > 0 are real numbers, p i (x) C( ) with 2 p i (x) N for all i {, 2,..., N}, and g, g 2 : R R are two functions of class C with respect to the -variable, and f : R R is of class C with respect to the -variable. We make the following assumptions on the functions q, g, g 2 and f. (H5) For i =, 2, q i C( ) satisfies < q i (x) < NP for all x. NP (H6) (i) For i =, 2, g i : R R satisfies the Caratheodory condition, and there exist some positive constant C i such that g i (x, s) C C 2 s qi(x) for (x, s) R. (ii) There exists M > 0, σ > P such that for all s M and x, 0 < σg 2 (x, s) g 2 (x, s)s. (H7) There exist δ > 0, C 3 > 0 and q 3 C( ) such that G (x, s) C 3 s q3(x), (x, s) (0, δ ], where max(q 3, q ) < P < P < q 2. (H8) g 2 (x, s) = ( s P ) as s 0 uniformly for x. (H9) (i) f : R R satisfies the Caratheodory condition and there exists a constant C > 0 such that f(x, s) C( s β(x) ), (x, s) R; where β(x) C( ) with < β β < P and β(x) < (N)P NP for all x. (ii) There exist R > 0, such that for all s R and x 0 < σf (x, s) f(x, s)s.

12 2 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 (H0) There exist δ 2 > 0, C 4 > 0 and q 4 (x) C( ) such that F (x, s) C 4 s q4(x), x, s δ 2, where < q 4 < (N)P and q NP 4 < P for all x. (H) For i =, 2, g i (x, s) = g i (x, s) for all (x, s) R, and f(x, s) = f(x, s) for all (x, s) R. We denote g(x, s) = λg (x, s) γg 2 (x, s), G i (x, s) = G(x, s) = s 0 g(x, t) dt, F (x, s) = and the associated functional ( u ) Φ(u) = u dx p i (x) x i s 0 s 0 g i (x, t) dt (i =, 2), f(x, t) dt; G(x, u) dx µ F (x, u) dx. Proposition 4.. If (H5), (H6) and (H9) hold, then for every λ, γ, µ 0 the functional Φ satisfies the Palais Small condition (PS). Proof. We use the following inequalities: For x, s R σg 2 (x, s) g 2 (x, s)s C 3, σg(x, s) g(x, s)s [σg (x, s) sg (x, s)] [σg 2 (x, s) sg 2 (x, s)] (C C 2 s q(x) ) C 3. Suppose that (u n ) X is a (PS) sequence ; i.e, sup Φ(u n ) C, Φ (u n ) 0, as n. Let us show that (u n ) is bounded in X. Since Φ(u n ) is bounded, then by using hypothesis (H6) and (H9), we have for n large enough C C u n σφ(u n ) Φ (u n ) ( P ) N ( u un pi(x)) dx σ x i (σg(x, u n ) g(x, u n )u n ) dx µ (σf(x, u n ) f(x, u n )u n ) dx ( 2 P N P p ) u n P p σ ( ) C u n q dx C (σf(x, u n ) f(x, u n )u n ) dx. Applying (H9) for u n large enough, we then get ( C C u n 2 P N P p ) u n P C u n q C σ 3. (4.2) Now, as W, p (x) () L q () is a continuous and compact embedding, from the inequality above, we deduce that u n is bounded in X. The proof is complete.

13 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 3 Remark 4.2. It follows from (H6) that G 2 (x, s) C 5 s /σ C 6, x, s R. The main results of this section are as follows: Proposition 4.3 ([4]). Assume that ψ : X R is weakly-strongly continuous and that ψ(0) = 0. Let ν > 0 be given. Set Then β k 0 as k. β k = β k (ν) = sup ψ(u). u Z k, u ν Theorem 4.4. Assume that (H5), (H6) and (H9) hold. () If in addition, (H0) holds, then for every γ, µ > 0, there exists r 0 (γ) > 0 such that when 0 λ, µ r 0 (γ), problem (4.) has a nontrivial solution u such that Φ(u ) > 0. (2) If in addition, (H7) and (H0) hold, then for every γ, µ > 0, there exists r 0 (γ) > 0 such that when 0 λ, µ r 0 (γ), problem (4.) has two nontrivial solutions u, v such that Φ(u ) > 0 and Φ(v ) < 0. (3) If in addition, (H7), (H0) and (H) hold, then for every λ, γ, µ > 0, problem (4.) has a sequence of solutions {±u k } such that Φ(±u k ) as k. Proof. () We denote ψ (u) = λ G (x, u(x)) dx, ψ 2 (u) = γ G 2 (x, u(x)) dx. When the assumptions in () hold, then for sufficiently small u, we get Then G 2 (x, u) ɛ u P C(ɛ) u q 2, (x, s) R, ψ 2 (u) γɛ u P dx γc(ɛ) u q2(x) dx. Since < q 2 < NP, for all x, then we have NP W, p (x) () L P (), and W, p (x) () L q2(x) (), with continuous and compact embeddings. This implies the existence of C, C 2 > 0 such that ψ 2 (u) γɛc u P γc(ɛ)c2 u q 2. Choose ɛ > 0 small enough so that 0 < γɛc 2 <. Then, we have 2 P N P ( u ) u dx ψ 2 (u) p i (x) x i 2 P N P u P γc(ɛ)c2 u q 2. Since q2 > P, there exist r > 0 and α > 0 such that ( u ) u dx ψ 2 (u) α > 0, for u = r. p i (x) x i

14 4 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 We can find r 0 (γ) > 0 such that when µ, λ r 0 (γ), we obtain ψ (u) α 2, u S r = {u X; u = r }. Therefore, λ, µ r 0 (γ). So, we obtain Φ(u) α 2 > 0, u S r. Let u X and t >, we have t pi(x) ( u ) Φ(tu) = u dx λ p i (x) x i γ G 2 (x, tu) dx µ F (x, tu) dx. From Remark 4.2, we obtain Φ(tu) t P ( u ) u dx λ p i (x) x i γc 5 t /σ u /σ dx µ F (x, tu) dx. Now, since G (x, tu) dx G (x, tu) dx (4.3) (4.4) G (x, tu) = o((t u ) q ), F (x, tu) = o((t u ) β ) when t, (because P q and β < P < σ ), we obtain Φ(tu), when t. Hence, It follows that there exist u 0 X such that u 0 > r and Φ(u 0 ) < 0. Therefore, By the Mountain Pass theorem, problem (4.) has a nontrivial solution u such that Φ(u ) > 0. (2) Under the assumpti9ns in (2) hold, (), we know that there exist r 0 (γ) > 0 such that when 0 λ, µ r 0 (γ), problem has a nontrivial solution u such that Φ(u ) > 0. For t (0, ) small enough, and v 0 C 0 () such that 0 v 0 (x) min{δ, δ 2 }, we have Φ(tv 0 ) = γ p i (x) ( (tv 0 ) x i G 2 (x, tv 0 ) dx µ tv0 pi(x)) dx λ F (x, tv 0 ) dx t P ( v 0 v0 pi(x)) dx λc 3 p i (x) x i γ G 2 (x, tv 0 ) dx µc 4 tv 0 q4(x) dx. For t (0, ) small enough, we obtain G (x, tv 0 ) dx tv 0 q3(x) dx (4.5) G 2 (x, tv 0 ) = o( tv 0 P ), as t

15 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 5 So, we have Φ(tv 0 ) t P ( tv 0 tv0 pi(x)) dx λc 3 t q p i (x) x i γmt P v 0 dx µc 4 t q 4 v 0 q4(x) dx. 3 v 0 q3(x) dx (4.6) Since min(q 3, q 4 ) < P, by factoring the right side of (4.6) by t q 3 if q 4 > q 3, and by t q 4 if q 3 > q 4, we obtain lim t 0 Φ(tv 0) < 0. Then there exist w X such that w r, and Φ(w) < 0. (3) Φ is an even functional. We denote ψ(u) = λ G (x, u) dx γ G 2 (x, u) dx µ F (x, u) dx As β k (ν) is defined in Proposition 2.6, for each positive integer, there exist a positive integer k 0 such that β k (n) for all k k 0 (n). We can assume k 0 (n) < k 0 (n ) for each n. We define {ν k : k =, 2,... } by { n if k 0 k < k 0 (n ), ν k = (4.7) if k < k 0. We see that ν k when k, then for u Z k with u = ν k, we obtain Consequently, Φ(u) = ( u ) u dx ψ(u) p i (x) x i P 2 P N (ν k) P P. inf Φ(u) as k. u Z k, u =ν k So the hypotheses (3.5) of Fountain theorem are satisfied. Indeed, by (H6), (H9) and Remark 4.2, for u we obtain Φ(u) C P u P C λ u q q C5γ u /σ (x) σ C 6 µ u β β(x) C 7. As the space Y k has finite dimension i.e all norms are equivalents, we then have Φ(u) C P u P C λ u q C 5 γ u /σ C µ u β C 7. Since min(q, q 2 ) < P < σ, we obtain Φ(u) as u, u Y k. Finally, the proof of (3) is complete. Acknowledgements. The authors would like to thank the anonymous referees for their helpful remarks.

16 6 B. ELLAHYANI, A. EL HACHIMI EJDE-207/88 References [] E. Acerbi, N. Fusco; partial regularity under anisotropic (p, q) growth conditions, J. Differential Equations. 07 (994), [2] S. Antontsev, S. Shmarev; Evolution PDEs with Nonstandard Growth Conditions. Atlantis Press, Amsterdam, 205. [3] M. Boureanu, F. Preda; Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions Nonlinear Differ. Equ. Appl. (202) 9: 235. [4] B. K. Bonzi, S. Ouaro, F. D. Y. Zongo; Entropy Solutions for Nonlinear, Int. J. Differ. Equ. Article (203), pp. 4. [5] J. Chabrowski, Y. Fu; Existence of solutions for p(x)-laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), [6] A. Cianchi; Local boundedness of minimizers of anisotropic functionals, Ann. Inst. H. Poincaré ANL 7 (2000), [7] S. G. Deng; A local mountain pass theorem and applications to a double perturbed p(x)- Laplacian equations, Appl. Math. Comput. 2 (2009), [8] S. G. Deng; Eigenvalues of the p(x)-laplacian Steklov problem, J. Math. Anal. Appl. 339 (2008), [9] S. G. Deng; Positive solutions for Robin problem involving the p(x)-laplacian, J. Math. Anal. Appl. 360(2009), [0] X. Fan; Solutions for p(x)-laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 32 (2005), [] X. L. Fan, Q. H. Zhang, D. Zhao; Eigenvalues of p(x)-laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), [2] X. L. Fan, D. Zhao; On the space L p(x) and W m,p(x), J. Math. Anal. Appl. 263 (200), [3] X. L. Fan, J. S. Shen, D. Zhao; Sobolev embedding theorems for space W k,p(x), J. Math. Anal. Appl. 262 (200), [4] X. L. Fan, Q. H. Zhang; Existence of solutions for p(x)-laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), [5] I. Fragalá, F. Gazzola, B. Kawohl; Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincar AN 2 (2004), [6] C. Ji; Nonlinear Analysis: Theory, Methods and Applications, 2009, Elsevier. [7] M. B. Gheami, S. Saiedinezhad, M. Eshaggi Gordji; The existence of weak solution for a calss of nonlinear p(x)-boundry value problem involving the priciple eignevalue, Trends in Applied Sciences Research 7(2), 60-67, 202. [8] J. P. Garcia Azorero, I. Peral Alonso; Hardy inequalities and some critical elliptic and parabolic problems. J. Differential Equations, 44 (998), [9] M. Giaquinta; Growth condition and regularity, a countreexample, Manuscripta Math. 59 (987), [20] S. N. Kruzhkov, I. M. Kolodii; On the theory of embedding of anisotropic Sobolev spaces, Russian Math. Surveys. 38 (983), [2] M. Mihǎilescu, P. Pucci, V. Rǎdulescu; Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, Comptes Rendus Mathematique. 345(2007), [22] S. M. Nikol skii; On imbedding, continuation and approximation theorems for differentiable functions of several variables, Russian Math. Surveys. 6 (96), [23] M. Růžička; Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Math 748, Springer-Verlag, Berlin, [24] J. Ràkosnik; Some remarks to anisotropic Sobolev space I, Beiträge zur Analysis 3(979) [25] M. Willem, Minimax Theorems, Birkhauser, Boston, 996. Brahim Ellahyani Department of Mathematics, Faculty of Sciences, Mohammed V University, P.B. 004, Rabat, Morocco address: ellahyani.brahim.99@gmail.com

17 EJDE-207/88 EXISTENCE AND MULTIPLICITY OF SOLUTIONS 7 Abderrahmane El Hachimi Department of Mathematics, Faculty of Sciences, Mohammed V University, P.B. 004, Rabat, Morocco address: aelhachi@yahoo.fr