Existence and multiplicity results for Dirichlet boundary value problems involving the (p 1 (x), p 2 (x))-laplace operator

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1 Note di atematica ISSN , e-issn Note at. 37 (207) no., doi:0.285/i v37np69 Existence and multiplicity results for Dirichlet boundary value problems involving the (p (x), p 2 (x))-laplace operator ostafa Allaoui FSTH, University of ohammed I, LANOL, Oujda, orocco allaoui9@hotmail.com Omar Darhouche Department of athematics, University of ohammed I, LANOL, Oujda, orocco omarda3@hotmail.com Received: ; accepted: Abstract. This paper is concerned with the existence and multiplicity of solutions for the following Dirichlet boundary value problems involving the (p (x), p 2(x))-Laplace operator of the form: div( u p (x) 2 u) div( u p 2(x) 2 u) = f(x, u) in, u = 0 on. By means of critical point theorems with Cerami condition and the theory of the variable exponent Sobolev spaces, we establish the existence and multiplicity of solutions. Keywords: variational methods, generalized Lebesgue-Sobolev spaces SC 2000 classification: 35A5, 46E35 Introduction The purpose of this article is to show the existence of solutions of the following Dirichlet problem involving the (p (x), p 2 (x))-laplace operator of the form div( u p(x) 2 u) div( u p2(x) 2 u) = f(x, u) in, () u = 0 on, where R N is a bounded domain with smooth boundary, p i C() such that < p i := inf x p i (x) p + i := sup x p i (x) < + for i =, 2, and f(x, u) : R R satisfies Carathéodory condition. The operator p(x) u := div( u p(x) 2 u) is called the p(x)-laplacian, and becomes p-laplacian when p(x) p (a constant). The p(x)-laplacian possesses more complicated properties than the p-laplacian; for example, it is inhomogeneous. The study of problems involving variable exponent growth conditions c 207 Università del Salento

2 70. Allaoui, O. Darhouche has a strong motivation due to the fact that they can model various phenomena which arise in the study of elastic mechanics [22], electrorheological fluids [] or image restoration [7]. The differential operator div( u p (x) 2 u) + div( u p 2(x) 2 u) is known as the (p (x), p 2 (x))-laplacian operator when p p 2. When p (x) and p 2 (x) are constant, this operator arises in problems of mathematical physics (see [3]) and in plasma physics and biophysics (see [8]). If p = p 2 = p, then we have a single operator the p(x)-laplacian and problem () becomes the p(x)-laplacian Dirichlet problem of the form p(x) u = div( u p(x) 2 u) = f(x, u) in, u = 0 on, (2) which have been studied sufficiently by several authors [4, 20] and the references therein. Define the family of functions { } F = G γ : G γ (x, t) = f(x, t)t γf (x, t), γ [2p m, 2p + ], where p m (x) = min{p (x), p 2 (x)}, p (x) = max{p (x), p 2 (x)} for all x, p m = inf x p m (x), p + = sup x p (x) and F (x, t) = t 0 f(x, s)ds. Noting that when p (x) p 2 (x) p is a constant, F = {f(x, t)t 2pF (x, t)} consists of only one element. Throughout this paper, we make the following assumptions on the function f: (f ) There exist C > 0 and q C + () with q(x) < p (x) for all x such that f(x, t) C( + t q(x) ) for each (x, t) R, where { } C + () = p(x) : p C(), p(x) > for all x and p is the critical exponent of p, i.e., Np (x) p N p (x) = (x) if p (x) < N, if p (x) N. (f 2 ) f(x, t) = o( t p+ ) as t 0 uniformly for a.e. x. (f 3 ) f(x, t) = f(x, t) for (x, t) R.

3 Existence and multiplicity results 7 (f 4 ) lim t f(x,t)t t p+ = +, uniformly for a.e. x. (g) There exists a constant δ such that for all γ, η [2p m, 2p + ] and (s, t) [0, ] R, the inequality δg γ (x, t) G η (x, st) holds for a.e. x. In [8] the authors consider problem () in the particular case f(x, u) = λ u q(x) 2 u, where λ > 0. Under the assumptions < p 2 (x) < q(x) < p (x) < N and max y q(y) < Np 2(x) N p 2 (x) for all x, they established the existence of two positive constants λ 0, λ with λ 0 λ such that any λ [λ, ) is an eigenvalue, while any λ (0, λ 0 ) is not an eigenvalue of the above problem. In [6] the authors consider problem () i.e., which is the well-known anisotropic p(.)-laplacian problem (see, e.g., [4] and references therein) in the case N =, 2, that is, N ) xi ( xi u pi(x) 2 xi u = f(x, u). i= Under proper growth condition and specially the well-known Ambrosetti- Rabinowitz type condition: ν > p +, > 0 such that x, t 0 νf (x, t) f(x, t)t, (AR) they obtained some existence and multiplicity results. The role of (AR) condition is to ensure the boundness of the Palais-Smale sequences of the Euler-Lagrange functional. This is very crucial in the applications of critical point theory. However, although (AR) is a quite natural condition, it is somewhat restrictive and eliminates many nonlinearities. Indeed, there are many superlinear functions which do not satisfy (AR) condition. For instance when p (x) = p 2 (x) 2 and δ = 2, the function below does not satisfy (AR), while it satisfies the aforementioned conditions. f(x, t) = 2t log( + t ). (3) But it is easy to see the above function (3) satisfies (f ) (f 4 ). As far as we are aware, elliptic problems like () involving the (p (x), p 2 (x))- Laplace operator without the (AR) type condition, have not yet been studied. That is why, at our best knowledge, the present paper is a first contribution in this direction. The purpose of this work is to improve the results of the above mentioned papers. Without assuming the Ambrosetti-Rabinowitz type conditions (AR), we prove the existence of solutions.

4 72. Allaoui, O. Darhouche Remark. If f(x, t) is increasing in t, then (AR) implies (g) when t is large enough, in fact, we can take δ = >, then p+ ν δg γ (x, t) G η (x, st) f(x, t)t f(x, st)st 0. But, in general, (AR) does not imply (g), see [20, Remark 3.4] when p (x) p 2 (x) p. Now we are ready to state our results. Theorem. Suppose that the conditions (f ), (f 2 ),(f 4 ) and (g) are satisfied. If q > p +, then the problem () has at least one nontrivial solution. Theorem 2. Assume that (f ), (f 3 ),(f 4 ) and (g) hold. If q > p +, then problem () has a sequence of weak solutions with unbounded energy. The present paper is divided into three sections, organized as follows: In section 2, we introduce some basic properties of the Lebesgue and Sobolev spaces with variable exponents and some min-max theorems like mountain pass theorem and fountain theorem with the Cerami condition that will be used later. In section 3, we prove our main results. Throughout the sequel, the letters c, c i, i =, 2,..., denotes positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. Preliminaries For the reader s convenience, we recall some necessary background knowledge and propositions concerning the generalized Lebesgue-Sobolev spaces. We refer the reader to [9, 0,, 2] for details. Let be a bounded domain of R N, denote p + = max{p(x) : x }, p = min{p(x) : x }, { } L p(x) () = u : u is a measurable real-valued function, u(x) p(x) dx <, with the norm u L p(x) () = u p(x) = inf { λ > 0 : u(x) } λ p(x) dx, becomes a Banach space [5]. We also define the space { } W,p(x) () = u L p(x) () : u L p(x) (),

5 Existence and multiplicity results 73 equipped with the norm u W,p(x) () = u L p(x) () + u L p(x) (). We denote by W,p(x) 0 () the closure of C0 () in W,p(x) (). Of course the,p(x) norm u = u L p(x) () is an equivalent norm in W0 (). Proposition ([9, 2]). (i) The conjugate space of L p(x) () is L p (x) (), where p(x) + p (x) =. For any u Lp(x) () and v L p (x) (), we have ( uv dx p + ) p u p(x) v p (x) 2 u p(x) v p (x). (ii) If p, p 2 C + () and p (x) p 2 (x) for all x, then L p 2(x) () L p (x) () and the embedding is continuous. Proposition 2 ([2]). Set ρ(u) = u(x) p(x) dx, then for u X and (u k ) X, we have () u < (respectively = ; > ) if and only if ρ(u) < (respectively = ; > ); (2) for u 0, u = λ if and only if ρ( u λ ) = ; (3) if u >, then u p ρ(u) u p+ ; (4) if u <, then u p+ ρ(u) u p ; (5) u k 0 (respectively ) if and only if ρ(u k ) 0 (respectively ). For x, let us define Np(x) p N p(x) if p(x) < N, (x) = if p(x) N. (i) W k,p(x) () and W k,p(x) 0 () are separable re- Proposition 3 ([2, 3]). flexive Banach spaces. (ii) If q C + () and q(x) p (x) (q(x) < p (x)) for x, then there is a continuous (compact) embedding X L q(x) (). (iii) There is constant C > 0 such that u p(x) C u p(x), u W k,p(x) 0 ().

6 74. Allaoui, O. Darhouche We recall the definition of the Cerami condition (C) introduced by G. Cerami (see [5]). Definition ([5]). Let (X,. ) be a Banach space and J C (X, R), given c R, we say that J satisfies the Cerami c condition (we denote condition (C c )), if (i) any bounded sequence {u n } X such that J(u n ) c and J (u n ) 0 has a convergent subsequence; (ii) there exist constants δ, R, β > 0 such that J (u) u β u J ([c δ, c + δ]) with u R. If J C (X, R) satisfies condition (C c ) for every c R, we say that J satisfies condition (C). Note that condition (C) is weaker than the (PS) condition. However, it was shown in [2] that from condition (C) it is possible to obtain a deformation lemma, which is fundamental in order to get some critical point theorems. ore precisely, let us recall the version of the mountain pass lemma with Cerami condition which is used in the sequel. Proposition 4 ([2]). Let (X,. ) a Banach space, J C (X, R), e X and r > 0, be such that e > r and b := If J satisfies the condition (C c ) with then c is a critical value of J. inf J(u) > J(0) J(e). u =r c := inf γ Γ max t [0,] J(γ(t)), Γ := {γ C([0, ], X) γ(0) = 0, γ() = e}, We also introduce the fountain theorem with the Condition (C) which is a variant of [9, 23]. Let X be a reflexive and separable Banach space. Then, from [2] there are {e i } X and {e i } X such that X = e i, i N, X = e i, i N, e i, e j = δ i,j, where δ i,j denotes the Kroneker symbol. For k N, put X k = Re k, We have the following lemma. Y k = k X i, i= Z k = X i. i=k

7 Existence and multiplicity results 75 Lemma. ([6]). For q C + () and q(x) < p (x) for all x, define Then lim k + β k = 0. β k = sup{ u q(x) : u =, u Z k }. Proposition 5 ([7]). Assume that (X,. ) is a separable Banach space, J C (X, R) is an even functional satisfying the Cerami condition. oreover, for each k =, 2,..., there exist ρ k > r k > 0 such that (A) inf {u Zk : u =r k } J(u) + as k ; (A2) max {u Yk : u =ρ k } J(u) 0. Then J has a sequence of critical values which tends to +. Consider the following functional Φ(u) = p (x) u p (x) dx+ p 2 (x) u p 2(x) dx, for all u X, where X := W,p (x) 0 () W,p 2(x) 0 () with the norm u = u p (x) + u p2 (x), x. It is obvious that (X,. ) is also a separable and reflexive Banach space. By using standard arguments, it can be proved that Φ C (X, R) (see [6]), and the (p (x), p 2 (x))-laplace operator is the derivative operator of Φ in the weak sense. Denote L = Φ : X X, then L(u), v = u p(x) 2 u v dx + u p2(x) 2 u v dx, for all u, v X, and.,. is the dual pair between X and its dual X. Proposition 6 ([6]). () L is a continuous, bounded homeomorphism and strictly monotone operator. (2) L is a mapping of type (S + ), namely: u n u and lim sup n + L(u n )(u n u) 0, imply u n u. Let us denote p (x) = max{p (x), p 2 (x)}, p m (x) = min{p (x), p 2 (x)}, x. It is easy to see that p (.), p m (.) C + (). For q(.) C + () such that q(x) < p (x) for any x we have X := W,p (x) 0 () W,p 2(x) 0 () = W,p (x) 0 () L q(x) () and the imbedding is continuous and compact.

8 76. Allaoui, O. Darhouche Definition 2. A function u X is said to be a weak solution of () if u p(x) 2 u v dx + u p2(x) 2 u v dx f(x, u)v dx = 0, for all v X. Define Φ(u) = p (x) u p (x) dx + p 2 (x) u p 2(x) dx, Ψ(u) = F (x, u)dx. The Euler-Lagrange functional associated to problem () is J(u) = Φ(u) Ψ(u). Under the hypothesis (f ), the functional J is well defined, of class C, and the Fréchet derivative is given by J (u), v = u p(x) 2 u v dx + u p2(x) 2 u v dx f(x, u)v dx for all u, v X. oreover a weak solution of problem () corresponds to a critical point of the functional J. Proof of main results First of all, we start with the following compactness result which plays the most important role. Lemma 2. Under assumptions (f ), (f 4 ) and (g), J satisfies the Cerami condition. Proof. For all c R, we show that J satisfies (i) of Cerami condition. Let {u n } X be bounded, J(u n ) c and J (u n ) 0. Without loss of generality, we assume that u n u, then J (u n )(u n u) 0. Thus we have J (u n )(u n u) = u n p(x) 2 u n ( u n u)dx + u n p2(x) 2 u n ( u n u)dx f(x, u n )(u n u)dx 0.

9 Existence and multiplicity results 77 From (f ), Propositions and 3, we can easily get that f(x, u n)(u n u)dx 0. Therefore, we have 2 i= Furthermore, since u n u in X, we have 2 i= From (4) and (5), we deduce that 2 i= u n p i(x) 2 u n ( u n u)dx 0. (4) u p i(x) 2 u( u n u)dx 0. (5) ( ) u n pi(x) 2 u n u pi(x) 2 u. ( u n u) dx 0. (6) Next, we apply the following well-known inequality ( ) ξ r 2 ξ ψ r 2 ψ. (ξ ψ) 2 r ξ ψ r, ξ, ψ R N, (7) valid for all r 2. From the relations (6) and (7), we infer that 2 i= u n u p i(x) dx 0, (8) and, consequently, u n u in X. Now, we check that J satisfies the assertion (ii) of Cerami condition. Arguing by contradiction, there exist c R and {u n } X satisfying: Let J(u n ) c, u n, J (u n ) u n 0. (9) p n = Choosing u n >, for n N, thus c = ( lim J(u n ) ) J (u n )(u n ) n + p n ( = lim n + ( ) u n p(x) + u n p 2(x) dx p (x) u n p(x) dx + p 2 (x) u n p2(x) dx. p n ) f(x, u n )u n dx F (x, u n )dx. (0)

10 78. Allaoui, O. Darhouche Denote w n = un u, then w n n =, so {w n } is bounded. Up to a subsequence, for some w X, we get w n w in X, w n w in L q(x) (), w n (x) w(x) a.e. in. If w 0, we can define a sequence {t n } R such that J(t n u n ) = max t [0,] J(tu n). ( ) For any B >, let b 2p + n = 2Bp + p w n, since b n 0 in L q(x) () and F (x, t) C( + t q(x) ), by the continuity of the Nemitskii operator, we see that F (., b n ) 0 in L () as n +, therefore Then for n large enough, J(t n u n ) J(b n ) b n p (x) = p (x) ( 2Bp + That is, p + lim n ( 2Bp + dx + ) p F (x, b n )dx = 0. () / u n (0, ) and b n p 2(x) p 2 (x) ) w n p (x) dx + p + 2 F (x, b n )dx 2B w n p(x) dx + 2B 2c B w n p + 2c2 B w n p 2 2c B + 2c 2 B F (x, b n )dx. dx Ψ(b n ) ( ) 2Bp + w n p2(x) dx w n p2(x) dx F (x, b n )dx F (x, b n )dx J(t n u n ) +. (2) From J(0) = 0 and J(u n ) c, we know that t n (0, ) and J d (t n u n ), t n u n = t n J(tu n ) = 0. dt t=tn

11 Existence and multiplicity results 79 Thus, from (2), we obtain that p tn Ψ (t n u n ), t n u n Ψ(t n u n ) = p tn Φ (t n u n ), t n u n Ψ(t n u n ) = J(t n u n ), (3) as n, where p tn = Φ (t nu n),t nu n Φ(t nu n). Let γ tnu n = p tn and γ un = p n, then γ tnu n, γ un [2p m, 2p + ]. Hence, G γ tnun, G γun p F. Using (g), (3) and the fact that inf tn n p n δ p n f(x, u n )u n dx > 0, we get F (x, u n )dx = G γun (x, u n )dx p n G γtnun (x, t n u n )dx p n δ ( p t n p n δ p tn Ψ (t n u n ), t n u n Ψ(t n u n ) ) +, which contradicts (0). If w 0, from (9), we write u n p(x) dx + u n p2(x) dx f(x, u n )u n dx = J (u n ), u n = o() u n, (4) that is, o() = = f(x, u n )u n u n p (x) dx + u n p 2(x) dx dx f(x, u n )u n dx u n p+ f(x, u n )u n w n p+ dx. u n p+ (5) Define the set Λ 0 = {x : w(x) = 0}. Then, for x Λ\Λ 0 = {x : w(x) 0} we have u n (x) + as n +. Hence by (f 4 ) we deduce f(x, u n (x))u n (x) w n (x) p+ + as n +. u n (x) p+

12 80. Allaoui, O. Darhouche In view of Λ\Λ 0 > 0, we deduce via the Fatou Lemma that f(x, u n )u n w n p+ dx + as n +. (6) Λ\Λ 0 u n p+ f(x,t)t t p+ On the other hand, from (f ) and (f 4 ), there exists d > such that d for t R and a.e. x. oreover, we have Λ 0 w n (x) p+ dx 0. Thus, there exists m > such that f(x, u n )u n w n p+ dx d w n p + dx m >. (7) Λ 0 u n p+ Λ 0 Combining (5), (6) and (7), there is a contradiction. This completes the proof of Lemma 2. Proof of Theorem. By Lemma 2, J satisfies conditions (C) in X. To apply Proposition 4, we will show that J possesses the mountain pass geometry. First, we claim that there exist µ, ν > 0 such that J(u) µ > 0, for all u X with u = ν. Let u. Then by Proposition 2, we have J(u) c c p + u p+ 2 + p + u p+ 2 F (x, u)dx 2 c p + u p+ F (x, u)dx, (8) where c = min{c, c 2 }. Since p + < q q(x) < p (x) for all x, we have the continuous embedding X L p+ () and X L q () and also there are two positive constants, c 3 and c 4 such that u p + c 3 u and u q c 4 u for all u X. (9) Let ɛ > 0 be small enough such that ɛc p+ 3 < c. By the assumptions (f 2p + ) and (f 2 ), we have F (x, t) ɛ t p+ + c ɛ t q(x), for all (x, t) R. Then, for u it follows that J(u) c p + u p+ ɛ u p+ dx cɛ u q(x) dx c p + u p+ p ɛc + 3 u p+ cɛ c 5 u q. (20)

13 Existence and multiplicity results 8 Therefore, there exist two positive real numbers µ and ν such that J(u) µ > 0, for all u X with u = ν. Next, we affirm that there exists e X\B ν (0) such that J(e) < 0. Let v 0 X \ {0}, by (f 4 ), we can choose a constant p A > v 0 p(x) dx + p v 0 p2(x) dx 2 v, 0 p+ dx such that F (x, t) A t p+ uniformly in x, t > C A, where C A > 0 is a constant depending on A. Let s > large enough, we have J(sv 0 ) = p (x) sv 0 p(x) dx + p 2 (x) sv 0 p2(x) dx F (x, sv 0 )dx sp+ p v 0 p(x) dx + sp+ 2 p v 0 p2(x) dx 2 F (x, sv 0 )dx F (x, sv 0 )dx, sv 0 >C A sv 0 C A sp+ p v 0 p(x) dx + sp+ 2 p v 0 p2(x) dx 2 As p+ v 0 p+ dx F (x, sv 0 )dx + A sv 0 sv p + dx 0 C A sv 0 C A sp+ p v 0 p(x) dx + sp+ 2 p v 0 p2(x) dx 2 As p+ v 0 p+ dx + c6, which implies that J(sv 0 ) as s +. Thus, there exist s 0 > and e = s 0 v 0 X\B r (0) such that J(e) < 0. Thereby, the Proposition 4 guarantees that problem () has a nontrivial weak solution. This completes the proof. Proof of Theorem 2. The proof is based on the Fountain Theorem. According to Lemma 2 and (f 3 ), J is an even functional and satisfies condition (C). We will prove that if k is large enough, then there exist ρ k > r k > 0 such that: (A) b k := inf{j(u) : u Z k, u = r k } + as k + ;

14 82. Allaoui, O. Darhouche (A2) a k := max{j(u) : u Y k, u = ρ k } 0 as k +. In what follows, we will use the mean value theorem in the following form: for every γ C + () and u L γ(x) (), there is ζ such that u γ(x) dx = u γ(ζ) γ(x). (2) Indeed, it is easy to see that = ( u u γ(x) ) γ(x) dx. On the other hand, by the mean value theorem for integrals, there exists a positive constant κ [γ, γ + ] depending on γ such that ( u u γ(x) ) γ(x) dx = ( u γ(x) ) κ u γ(x) dx. The continuity of γ ensures that there exists ζ such that γ(ζ) = κ. Combining all together, we get (2). (A): For any u Z k such that u = r k is big enough to ensure that u p (x) and u p2 (x) (r k will be specified below), by condition (f ) and (2) we have J(u) = p (x) u p (x) dx + p 2 (x) u p 2(x) dx F (x, u)dx ( ) p + u p p (x) + u p 2 p 2 (x) c 7 u q(x) dx c 8 c p + c p + c p + u p m c 7 u q(ζ) q(x) c 9, where ζ u p m c 7 c 9 if u q(x) u p m c 7 (β k u ) q+ c 9 if u q(x) > c p + u p m c 7 (β k u ) q+ c 0 ( ) = c u p m c β q+ k u q+ p + c 0. We fix r k as follows r k = ) (c q + β q+ p m k q+,

15 Existence and multiplicity results 83 then J(u) c ( c r p m k p + p + p k ) m (c 0 q + β q+ p m q+ c β q+ k (c q + β q+ ( ) q + c 0. q + k ) p m q+ ) c 0 From Lemma, we know that β k 0. Then since < p m p + < q+, it follows r k +, as k +. Consequently, J(u) + as u + with u Z k. The assertion (A) is valid. (A2): Since dimy k < and all norms are equivalent in the finite-dimensional space, there exists d k > 0, for all u Y k with u big enough to ensure that u p (x) and u p2 (x), we have Φ(u) p u p (x) dx + p 2 p u p+ p (x) + m p u p+ 2 p 2 (x) m c c p u p+ 2 + m p u p+ 2 m max(c, c 2 ) p u p+ m u p 2(x) dx dk u p+. (22) p + Now, from (f 4 ), there exists R k > 0 such that for all s R k, we have F (x, s) 2d k s p+. From (f ), there exists a positive constant k such that F (x, s) k for all (x, s) [ R k, R k ]. Then for all (x, s) R we have F (x, s) 2d k s p+ k. (23) Combining (22) and (23), for u Y k such that u = ρ k > r k, we infer that J(u) = Φ(u) F (x, u)dx d k u p+ + p + k max(c, c 2 ) p u p+ + k. m Therefore, for ρ k large enough (ρ k > r k ), we get from the above that a k := max{j(u) : u Y k, u = ρ k } 0.

16 84. Allaoui, O. Darhouche The assertion (A2) holds. Finally we apply the Fountain Theorem to achieve the proof of Theorem 2. Acknowledgements. The authors thank the referees for their careful reading of the manuscript and insightful comments. References [] E. Acerbi, G. ingione: Gradient estimate for the p(x)-laplacian system, J. Reine Angew. ath, 584 (2005), [2] P. Bartolo, V. Benci, D. Fortunato: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal, 7 (983), [3] V. Benci, P. D Avenia, D. Fortunato, L. Pisani: Solitons in several space dimensions: Derrick s problem and infinitely many solutions, Arch. Ration. ech. Anal, 54 (2000), [4]. Boureanua, P. Pucci, V. Radulescu: ultiplicity of solutions for a class of anisotropic elliptic equations with variable exponenty, Complex Variables and Elliptic Equations, 54 (20), [5] G. Cerami: An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 2 (978), [6] KC. Chang: Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, China, 986. [7] Y. Chen, S. Levine,. Rao: Variable exponent, linear growth functionals in image processing, SIA J. Appl. ath, 66 (2006), [8] L. Cherfils, Y. II yasov: On the stationary solutions of generalized reaction diffusion equations with p&q-laplacian, Commun. Pure Appl. Anal, 4 (2005), [9] D. Edmunds, J. Rákosník: Density of smooth functions in W k,p(x) (), Proc. R. Soc. A, 437 (992), [0] D. Edmunds, J. Rákosník: Sobolev embedding with variable exponent, Studia ath, 43 (2000), [] XL. Fan, J. Shen, D. Zhao: Sobolev embedding theorems for spaces W k,p(x) (), J. ath. Anal. Appl, 262 (200), [2] XL Fan, D. Zhao: On the spaces L p(x) and W m,p(x), J. ath. Anal. Appl, 263 (200), [3] XL. Fan, D. Zhao: On the generalized Orlicz-Sobolev space W k,p(x) (), J. Gansu Educ. College, 2, 998, 6. [4] XL. Fan, Q. Zhang: Existence of solutions for p(x)-laplacian Dirichlet problems, Nonlinear Anal, 52 (2003), [5] O. Kováčik, J. Rákosník: On the spaces L p(x) () and W,p(x) (), Czechoslovak ath. J, 4 (99),

17 Existence and multiplicity results 85 [6] D. Liu, X. Wang, J. Yao: On (p (x), p 2(x))-Laplace equations, (202), arxiv: v. [7] S. Liu, S. Li: Infinitely many solutions for a superlinear elliptic equation, Acta ath. Sinica, 46 (2003), [8]. ihailescu, V. Radulescu: Continuous spectrum for a class of nonhomogeneous differential operators, anuscripta athematica, 25 (2008), [9]. Willem: inimax theorems, Birkhäuser, Boston, 996. [20] A. Zang: p(x)-laplacian equations satisfying Cerami condition, J. ath. Anal. Appl, 337 (2008), [2] J. Zhao: Structure theory of Banach spaces, Wuhan Univ. Press, Wuhan, 99. [22] V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory, ath. USSR. Izv, 9 (987), [23] W. Zou: Variant fountain theorems and their applications, anuscripta athematica, 04 (200),

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