Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions

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1 Nonlinear Differ. Equ. Appl. 9 (202), c 20 Springer Basel AG /2/ published online July 3, 20 DOI 0.007/s Nonlinear Differential Equations and Applications NoDEA Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions Maria-Magdalena Boureanu and Felician Preda Abstract. We analyze a class of quasilinear elliptic problems involving a p( )-Laplace-type operator on a bounded domain R N, N 2, and we deal with nonlinear conditions on the boundary. Working on the variable exponent Lebesgue Sobolev spaces, we follow the steps described by the fountain theorem and we establish the existence of a sequence of weak solutions. Mathematics Subject Classification (200). 35J25, 35J62, 35D30, 46E35, 35J20. Keywords. Quasilinear elliptic equations, Multiple weak solutions, Variable exponent Lebesgue Sobolev spaces, Fountain theorem.. Introduction and preliminaries We consider the problem { div(a(x, u)) + b(x) u p(x) 2 u = f(x, u) for x, () a(x, u) ν(x) =g(x, u) for x, where R N,N 2 is a bounded domain with smooth boundary. We assume that a : R N R N, a = a(x, η), is a Carathéodory function such that it is the continuous derivative with respect to η of a function A : R N R, A = A(x, η). More exactly, a(x, η) = η A(x, η). The mappings a and A verify the following. (A) The equality A(x, 0) = 0 holds for all x. (A2) There exists a constant c 0 > 0 such that a(x, η) c 0 ( + η p(x) ), for all x and all η R N.

2 236 M.-M. Boureanu and F. Preda NoDEA (A3) The inequality 0 [a(x, η ) a(x, η 2 )] (η η 2 ) holds for all x and all η,η 2 R N, with equality if and only if η = η 2. (A4) The inequalities η p(x) a(x, η) η p(x) A(x, η) hold for all x and all η R N. (A5) The mapping A is even with respect to its second variable, that is, A(x, η) =A(x, η) for all x and all η R N. The above type of assumptions can be found in other papers too. For example, in [6] and [7] we meet the assumptions (A) (A4). In [6] the approach relies on the sub and supersolution method, so the author extends the sub and supersolutions concept to the Sobolev space with variable exponents. He establishes the existence of the solution between a number of sub and supersolutions and presents some of its properties. Although the author does not discuss a problem similar to (), he treats a variational inequality that can be reduced to a variational equality and can be linked to a problem with Neumann or Robin boundary conditions. In [7] the authors establish the existence of a solution for an elliptic problem with Dirichlet boundary conditions. Their strategy relies on a variant of the mountain pass theorem of Ambrosetti and Rabinowitz combined with adequate variational methods. Our strategy is to apply a theorem related to the mountain pass theorem, namely the fountain theorem, which will be reminded in Sect. 2. This theorem requires the symmetry condition (A5) and allows us to obtain a sequence of solutions. We note that, in [6] and [7], as well as in our paper, the study is conducted in the framework of the variable exponent Lebesgue Sobolev spaces. However, nonstandard operators closely related to the ones described with the aid of (A) (A5) can appear in different situations, such is the case of classical Lebesgue Sobolev spaces (see [4]), or the case of anisotropic Lebesgue Sobolev spaces with variable exponent (see [3]). There is a good reason for working under hypotheses like (A) (A5), here and elsewhere. This set of hypotheses is only natural when we think at the operators that can be obtained from it by making the suitable choices. Indeed, for A(x, η) = p(x) η p(x) we deduce that a(x, η) = η p(x) 2 η and for η = u we find the p( )-Laplace operator div(a(x, u)) = div ( ) u p(x) 2 u. A second well-known example of operator arises when we choose A(x, η) = p(x) [( + η 2 ) p(x)/2 ], thus a(x, η) =(+ η 2 ) (p(x) 2)/2 η and for η = u we

3 Vol. 9 (202) Infinitely many solutions 237 find the generalized mean curvature operator ( ) div(a(x, u)) = div ( + u 2 ) (p(x) 2)/2 u. We notice that in [7] it is considered a convexity-type condition in addition to (A) (A4). This condition permits to consider the above presented examples of operators only for the case when p(x) 2. We, on the other hand, can manage without it, thus in our examples p>. Another remark is that the first example enables us to connect () tothe problem presented in [29], { Δp u + b(x) u p 2 u = f(x, u) for x, p 2 u u ν = g(x, u) for x, (2) where R N is a bounded domain with smooth boundary. In (2) the exponent p is constant, hence they are dealing with the classical p-laplace operator Δ p u =div( u p 2 u). A similar problem is considered also in [27], where p is not a constant, but a continuous function. More exactly, they consider the problem { Δp(x) u + u p(x) 2 u = λ f(x, u) for x, p(x) 2 u u ν = λ (3) 2g(x, u) for x. The p( )-Laplace operator Δ p( ) is an extension of the p-laplace operator Δ p and in the present paper we generalize it to the even more general operator described by (A) (A5). Thus we are entitled to say that our problem extends (2) and (3), since the constants λ,λ 2 are positive and the functions b, f, g fulfill conditions resembling to those used in our paper and described in what follows. For the function b : R we assume that (B) b L () and there exists b 0 > 0 such that b(x) b 0 for all x. In what concerns the rest of the functions involved in problem (), in order to present the general hypotheses more easily, we introduce some notation. We set { C + () = h C():< min x and for all h C + () we denote h + =suph(x), x In addition, we denote and h (x) = h (x) = } h(x) < max h(x) < x h =inf x h(x). { Nh(x)/[N h(x)] if h(x) <N, if h(x) N, { (N )h(x)/[n h(x)] if h(x) <N, if h(x) N.

4 238 M.-M. Boureanu and F. Preda NoDEA We consider p C + () and we suppose that f, g are Carathéodory functions verifying the next conditions. (F) For q C + () with q(x) <p (x) for all x, there exists c > 0 such that f(x, t) c t q(x) for all x and all t R. (F2) There exists α >p + such that 0 <α F (x, t) tf(x, t) for all x and all t R, where F (x, t) = t 0 f(x, s)ds. (F3) The function f is odd with respect to its second variable, that is, f(x, t) = f(x, t) for all x and all t R. (G) For r C + () with r(x) <p (x) for all x, there exists c 2 > 0 such that g(x, t) c 2 t r(x) for all x and all t R. (G2) There exists α 2 >p + such that 0 <α 2 G(x, t) tg(x, t) for all x and all t R, where G(x, t) = t 0 g(x, s)ds. (G3) The function g is odd with respect to its second variable, that is, g(x, t) = g(x, t) for all x and all t R. We point out that by our main theorem we will establish the existence of a sequence of weak solutions for () without adding further conditions on p, q and r. On the other hand, the corresponding result from [27] is provided under the additional assumptions that q,r >p +. As previously said, the nonlinear elliptic problem () will be studied in the framework of the variable exponent Lebesgue Sobolev spaces which are briefly described in Sect. 2. The interest for such problems is based on the multiple possibilities to apply them. There are applications in elastic mechanics [30], in the mathematical modeling of non-newtonian fluids [6,2,8 23,26] and in image restoration [5]. To be more specific, we recall that some of the first applications are concerning the electrorheological fluids, shortly, the ER fluids. These fluids are suspensions of extremely fine non-conducting particles (up to 50 μm diameter) in an electrically insulating fluid. The ER fluids have a special property: when disposed to an electromagnetic field, their apparent

5 Vol. 9 (202) Infinitely many solutions 239 viscosity undergoes a significant change which is reversible. For example, a typical ER fluid can go from the consistency of a liquid to that of a gel, and back, with response times on the order of milliseconds. After he made this discovery, Winslow obtained an US patent [25] on the effect who now carries his name and wrote an article published in 949 [26]. For more details it is enough to perform a simple search on the free encyclopedia site, Wikipedia. This is a rapid way to learn that the electrorheological fluids have applications in fast acting hydraulic valves and clutches [22], in ER brakes [2] and shock absorbers [23]. Recently, there are appearing novel uses, including use in the US army s planned future force warrior project or use in accurate abrasive polishing and tactile displays [5]. ER fluids have also been proposed to have potential applications in flexible electronics and Motorola filed a patent application for mobile device applications in Further information on properties, modelling and the applications of variable exponent spaces to these fluids can be found in [,2,4,6,8,0,2,8 20]. These are all very interesting matters that sustain our determination to investigate problems with p( )-growth conditions but, as usually, there is a downside. Indeed, if we take the mathematical point of view, when dealing with variable exponents it is obvious that we are in fact dealing with more mathematical difficulties. Moreover, when we pass from the classical p-laplace operator to the variable p( )-Laplace operator we get into more trouble, since the homogeneity is lost. However, in our case, we are manipulating even more general operators, as we saw above. And, as this would not be difficult enough, we also have nonlinear conditions on the boundary. Therefore, in order to overcome all this, we need some good tools, some solid auxiliary results. We provide them in the next section. 2. Notation and auxiliary results For the bounded domain with smooth boundary from the introductory section, R N, N 2, we denote by M one of the following two sets: or. We define the variable exponent Lebesgue space by L p( ) (M)= { u : u is a measurable real valued function such that endowed with the Luxemburg norm { u L p( ) (M) =inf μ>0 : M u(x) μ p(x) }. M } u p(x) < The space ( L p( ) (M), L p( ) (M)) has many qualities. It is a separable and reflexive Banach space ([3, Theorem 2.5, Corollary 2.7]) and the inclusion between spaces generalizes naturally: if 0 < M < and p, p 2 C + (M) are such that p p 2 in M, then the embedding L p2( ) (M) L p( ) (M) is continuous ([3, Theorem 2.8]). Furthermore, for all u L p( ) (M) andv L p ( ) (M)

6 240 M.-M. Boureanu and F. Preda NoDEA we have the following Hölder-type inequality (see [3, Theorem 2.]) uv dx 2 u L (M) v p( ) L p ( ) (M) (4) M where we denoted by L p ( ) (M) the conjugate space of L p( ) (M), obtained by conjugating the exponent pointwise, that is, /p(x) +/p (x) =([3, Corollary 2.7]). We denote by ρ M,p( ) : L p( ) (M) R the mapping defined by ρ M,p( ) (u) = u p(x), which is called the p( )-modular of the L p( ) (M) space. This is very helpful when working on the generalized Lebesgue spaces. We present some of its properties ([, Theorem.3, Theorem.4]). If u L p( ) (M) andp< then, u L p( ) (M) < (=;> ) ρ M,p( ) (u) < (=;> ) M u L p( ) (M) > u p L p( ) (M) ρ M,p( )(u) u p+ L p( ) (M) (5) u L p( ) (M) < u p+ L p( ) (M) ρ M,p( )(u) u p L p( ) (M) (6) u L p( ) (M) 0( ) ρ M,p( ) (u) 0( ). (7) If, in addition, (u n ) n L p( ) (M), then lim u n u L p( ) (M) =0 lim ρ M,p( )(u n u) =0 n n (u n ) n converges to u in measure and lim ρ M,p( )(u n )=ρ M,p( ) (u). n For previously introduced we define the variable exponent Sobolev space W,p( ) (), { } W,p( ) () = u L p( ) () : u L p( ) () endowed with the norm u = u L p( ) () + u L (), p( ) where by u L p( ) () we understand u L p( ) (). Thespace(W,p( ) (), ) is a separable and reflexive Banach space ([3, Theorem.3]). Moreover, we have the following embedding result and trace theorem. Theorem. ([8, Proposition 2.4]) Let R N, N 2 be a bounded domain with smooth boundary. If p, q C() satisfy the condition q(x) <p (x), x, then there is a compact embedding W,p( ) () L q( ) (). (8)

7 Vol. 9 (202) Infinitely many solutions 24 Theorem 2. ([7, Corollary 2.4]) Let R N, N 2 be a bounded domain with smooth boundary. Suppose that p C + () and r C() satisfy the condition r(x) <p (x), x. Then there is a compact boundary trace embedding W,p( ) () L r( ) ( ). We refer to [7] for further details concerning the extension of the classical trace to Lebesgue Sobolev spaces with variable exponent. For simplicity, when we refer to the trace of u we will write u instead of u or γu. The third useful theorem concerns the application a. Theorem 3. ([6, Theorem 4.]) The mapping a is an operator of type (S + ), that is, if u n u(weakly) in W,p( ) () and lim sup a(x, u n ) ( u n u) dx 0, n then u n u (strongly) in W,p( ) (). The next theorem that needs to be recalled here is the fountain theorem. But first we have to present the general context. It is known ([28, Sect. 7]) that for a separable and reflexive Banach space there exist {e n } n= X and {f n } n= X such that { if n = m, f n (e m )=δ n,m = 0 if n m, X = span{e n : n =, 2,...,} and X = span{f n : n =, 2,...,}. (Note that, since [28] is written in Chinese, the above assertions are based on the affirmations made by Fan and Han in [9], or by Yao in [27].) For k =, 2,... we denote X k = span{e k }, Y k = k X j and Z k = X j. (9) j= For a separable reflexive Banach space X and for X k, Y k, Z k previously defined we give the statement of the fountain theorem (see for example [24]) that we are going to use in Sect. 3. Theorem 4. Assume that I C (X, R) and that for each k =, 2,..., there exist ρ k >γ k > 0 such that (H) inf u Zk, u X =γ k I(u) as k. (H2) max u Yk, u X =ρ k I(u) 0. (H3) I satisfies the (PS) c condition for every c>0(that is, any sequence (u n ) n X such that I(u n ) c and I (u n ) 0 in X as n contains a subsequence converging to a critical point of I). Then I has a sequence of critical values tending to +. j=k

8 242 M.-M. Boureanu and F. Preda NoDEA In the present paper we choose X = W,p( ) () which, as said above, is a separable reflexive Banach space. To apply Theorem 4 we will operate with energetic functionals and rely on the critical point theory. This is why we provide properties of some of the functionals that will be involved in our future calculus. We start with Λ : W,p( ) () R defined by Λ(u) = A(x, u) dx. Proposition. ([7, Lemma ]) (i) The functional Λ is well-defined on W,p( ) (). (ii) The functional Λ is of class C (W,p( ) (), R) and Λ (u),v = a(x, u) vdx, for all u, v W,p( ) (). Remark. The study from [7] is conducted for Λ : W,p( ) 0 () R, where W,p( ) 0 () represents the Sobolev space with zero boundary values defined as the closure of C 0 () with respect to the norm W,p( ) 0 () = u L p( ) (). Since the calculus is almost identical, it is omitted for brevity. In addition, we denote by J : W,p( ) () R the functional defined by [ J(u) = u p(x) + u p(x)] dx. Proposition 2. ([9, Proposition 2.3]) If u W,p( ) () then u > u p J(u) u p+ u < u p+ J(u) u p. Other two functionals that have interesting properties are ψ, ϕ: W,p( ) () R defined by ψ(u) = q(x) u q(x) dx, ϕ(u) = r(x) u r(x) ds. (0) Proposition 3. The functionals ψ, ϕ : W,p( ) () R are weakly-strongly continuous, that is, u n u (weakly) implies ψ(u n ) ψ(u), respectively, ϕ(u n ) ϕ(u), asn. Proof. Let (u n ) n W,p( ) () be such that u n u (weakly) in W,p( ) (). By Theorem and Theorem 2, wehave and W,p( ) () L q( ) () compactly W,p( ) () L r( ) ( ) compactly.

9 Vol. 9 (202) Infinitely many solutions 243 Therefore, passing eventually to a subsequence, we get the strong convergence of (u n ) n to u in L q( ) () and L r( ) ( ). This only means that u n u L q( ) () 0, respectively, u n u L r( ) ( ) 0 as n. Using (8) wehavearrivedto lim,q( )(u n )=ρ,q( ) (u), n respectively, lim,r( )(u n )=ρ,r( ) (u). n Since ψ(u n ) ψ(u) q ρ,q( ) (u n ) ρ,q( ) (u) and ϕ(u n ) ϕ(u) ρ,r( ) r (u n ) ρ,r( ) (u), our proof is complete. We remind now the following. Proposition 4. ([9, Lemma 3.3]) For a separable reflexive Banach space X and for X k, Y k, Z k defined by (9) assume that Φ:X R is weakly-strongly continuous, Φ(0) = 0 and γ>0 is a given positive number. Then sup Φ(u) 0 u Z k, u X γ as k. For X = W,p( ) () with X k, Y k, Z k defined by (9) andψ, ϕ defined by (0), we set β k = sup u Z k, u ψ(u), θ k = sup u Z k, u ϕ(u). () By Proposition 3 and Proposition 4 we deduce our final auxiliary result. Proposition 5. If k, then β k 0 and θ k 0. Now we have the necessary tools to obtain the desired multiplicity result that will be stated and proved in the third section. 3. Main result Everywhere below we work under the hypotheses described in the previous sections. Since we are preoccupied with the existence of multiple weak solutions of problem (), we begin by giving the definition of such solutions. Definition. Afunctionu W,p( ) () which verifies a(x, u) vdx+ b(x) u p(x) 2 uv dx f(x, u)vdx for all v W,p( ) () is called a weak solution of (). g(x, u)vds=0

10 244 M.-M. Boureanu and F. Preda NoDEA We associate to problem () the energetic functional I : W,p( ) () R defined by b(x) I(u) = A(x, u) dx + p(x) u p(x) dx F (x, u) dx G(x, u) ds. Due to Proposition, by a standard calculus it can be shown that I is well defined and I C (W,p( ) (); R) with I (u),v = a(x, u) vdx+ b(x) u p(x) 2 uv dx f(x, u)vdx g(x, u)vds for all u, v W,p( ) (). Hence any critical point u W,p( ) () of I is a weak solution to problem (). Our main result is given by the following theorem. Theorem 5. Problem () admits a sequence (u n ) n of weak solutions such that I(u n ) when n. The idea of the proof of Theorem 5 is to show that (H) (H3) are fulfilled. To this end we will prove three corresponding lemmas. Lemma. For every k N, there exists γ k > 0 such that inf u Zk, u =γ k I(u),whenk. Proof. By (F), (F3) and (G), (G3) we deduce that there exist c,c 2 > 0 such that F (x, u) c q(x) u q(x) x (2) and G(x, u) c 2 r(x) u r(x) x, (3) where q, r C + () with q(x) <p (x) for all x andr(x) <p (x) for all x. On the other hand, by (A4), Λ(u) p + u p(x) dx. (4) Clearly, by (B) and (2) (4), p + c 2 u p(x) dx + b 0 p + q(x) u q(x) dx r(x) u r(x) ds. (5) u p(x) dx c Next, by (5), Proposition 2, (0) and (), p + min{; b 0} u p c u q+ β k c 2 u r+ θ k, (6)

11 Vol. 9 (202) Infinitely many solutions 245 for u Z k with u >. Depending on the order relation between p and q +, respectively, p and r +, we distinguish several cases to treat separately. Case : p <q + and p <r +. We consider the following two equations: 4p + min{; b 0}t p c t q+ β k =0, (7) Then and 4p + min{; b 0}t p c 2 t r+ θ k =0. (8) Let ξ k 0 be the solution of (7) andζ k 0 be the solution of (8). ( ξ k = ( ζ k = 4p + c β k min{; b 0 } 4p + c 2 θ k min{; b 0 } ) q + p ) r + p. Using Proposition 5 we obtain that lim ξ k = and lim ζ k =. k k We choose γ k = min{ξ k,ζ k },foru Z k with u = γ k and by (6) (8) we have 2p + min{; b 0}γ p k c as k, where c is a positive constant. With that, the proof of the first case is complete. Case 2: p >q + and p <r +. By Proposition 5 we deduce that, for sufficiently large k, β k < 2p + c min{; b 0 },thus(6) yields 2p + min{; b 0} u p c 2 θ k u r+, for u Z k with u >. We choose γ k = ζ k, that is, the nonzero solution of (8), and we discover that, for u Z k with u = γ k, 4p + min{; b 0}γ p k as k, (9) hence the proof of our second case is complete. Case 3: p <q + and p >r +. Proposition 5 implies that, for sufficiently large k, θ k < 2p + c 2 min{; b 0 }, so we notice that, using (6), 2p + min{; b 0} u p c β k u q+, for u Z k with u >. We choose γ k to be the nonzero solution of (7), that is, ξ k.foru Z k with u = γ k we have arrived at relation (9) and the proof of our third case is complete.

12 246 M.-M. Boureanu and F. Preda NoDEA Case 4: p >q + and p >r +. Again, by Proposition 5, for sufficiently large k, β k < 4p + c min{; b 0 } and θ k < 4p + c 2 min{; b 0 }. Therefore by (6), 2p + min{; b 0} u p. By choosing (γ k ) k such that γ k as k,foru Z k with u = γ k, we have that I(u) and the proof of our last case is complete. Remark 2. The proof of the fourth case can be done without relying on Proposition 5. Indeed, using (2) (4), (B) and Proposition 2 for u W,p( ) () with u > we get that p + min{; b 0} u p c q Relations (5) and (6) imply that u q(x) dx c 2 r u r(x) ds. (20) u q(x) dx u q+ L q( ) () + u q L q( ) (). Taking into account Theorem we can establish the existence of a constant c 3 > 0 such that u q(x) dx c 3 u q+ for u >. (2) In the same manner, using Theorem 2 instead of Theorem, we can find a constant c 4 > 0 such that u r(x) ds c 4 u r+ for u >. (22) We introduce (2), (22) into(20) and, for u Z k with u = γ k >, we have p + min{; b 0}γ p k c c 3 + q γq k c 2c 4 + r γr k. We choose (γ k ) k such that γ k as k and we infer that I(u) as k because q + <p and r + <p. Let us pass now to the proof of our second lemma, corresponding to hypothesis (H2) from the fountain theorem. Lemma 2. For every k N there exists ρ k >γ k (γ k given by Lemma ) such that the following inequality max I(u) 0 u Y k, u =ρ k takes place. Proof. The assumptions (A) and (A2) yield that, for all η R N and all x, ( A(x, η) = a(x, tη) ηdt c 0 η + ) p(x) η p(x). 0

13 Vol. 9 (202) Infinitely many solutions 247 Therefore, by (B), I(u) c 0 u L ()+ max{c 0; b L ()} p J(u) F (x, u) dx G(x, u) ds. (23) Since <p(x) we have the continuous embedding L p( ) () L () hence there exists c 5 > 0 such that u L () c 5 u L p( ) () c 5 u. (24) From (F2) and (G2) it follows that there exist M,M 2 > 0 such that M u α F (x, u) x (25) and M 2 u α2 G(x, u) x. (26) Due to Proposition 2 and relations (23) (26), for u Y k with u > we obtain I(u) c 0 c 5 u + max{c 0; b L ()} p u p + M u α L α () M 2 u α2 L α 2 ( ). But <p + <α i, i {, 2}, andy k is a finite dimensional space, thus all the norms are equivalent and we can conclude that, for u Y k with u = ρ k >γ k big enough, I(u) 0. Finally, we give the third lemma, corresponding to hypothesis (H3). Lemma 3. The functional I satisfies the (PS) c condition for every c>0. Proof. Let c>0 and (u n ) n W,p( ) () be such that ( I(u n ) c and I (u n ) 0in W ()),p( ) as n. (27) To show that (u n ) n is strongly convergent in W,p( ) () we start by showing that (u n ) n is bounded. Arguing by contradiction, we assume that, passing eventually to a subsequence still denoted by (u n ) n, u n as n. We denote α = min{α,α 2 } andwechooseτ (p +,α). Then, for sufficiently large n, by(27) and (A4) we deduce that c ++ u n I(u n ) τ I (u n ),u n ( p + ) ( u n p(x) + b(x) u n p(x)) dx τ (F (x, u n ) τ ) f(x, u n)u n dx (G(x, u n ) τ ) g(x, u n)u n ds. (28)

14 248 M.-M. Boureanu and F. Preda NoDEA Taking into consideration (F2) and (G2), the inequality (28) reduces to ( c ++ u n p + ) ( u n p(x) + b(x) u n p(x)) dx. (29) τ Furthermore, due to (B) and Proposition 2, (29) becomes ( c ++ u n p + ) min{,b 0 } u n p, τ for sufficiently large n. Weletn go to and we divide by u n p in the above relation. This way, we have obtained a contradiction, therefore (u n ) n is bounded in W,p( ) (). However, W,p( ) () is a reflexive Banach space, thus, up to a subsequence, (u n ) n is weakly convergent to some u 0 in W,p( ) (). Using (27) we infer that lim n I (u n ),u n u 0 =0, more precisely, [ lim a(x, u n ) ( u n u 0 ) dx + b(x) u n p(x) 2 u n (u n u 0 ) dx n ] f(x, u n )(u n u 0 ) dx g(x, u n )(u n u 0 ) ds =0. (30) At the same time, by (B) and the Hölder-type inequality (4), b(x) u n p(x) 2 u n (u n u 0 ) dx 2 b L () u n p(x) p( ) u n u 0 L (). (3) p( ) L () Since p(x) <p (x), for all x, then, by Theorem, the space W,p( ) () is compactly embedded in L p( ) (). Hence u n u 0 in L p( ) (). (32) By (3), (7) and (32) we deduce that b(x) u n p(x) 2 u n (u n u 0 ) dx =0. (33) lim n In a similar manner, using (F), (4), Theorem and (7), we get f(x, u n )(u n u 0 ) dx =0, (34) lim n while using (G), (4), Theorem 2 and (7), we get g(x, u n )(u n u 0 ) ds =0. (35) lim n Combining (30) with (33) (35) we arrive at a(x, u n ) ( u n u 0 ) dx =0. lim n

15 Vol. 9 (202) Infinitely many solutions 249 This means that, since a is a (S + )-type operator, by Theorem 3, (u n ) n is strongly convergent to u 0 in W,p( ) (). From (27) it follows that u 0 is a critical point for I and the proof is complete. ProofofTheorem5. The proof follows immediately from Lemma, Lemma 2, Lemma 3 and Theorem 4. Acknowledgment. The first author was supported by Grant CNCSIS PCEE 08/200. The study was completed while M.-M. Boureanu was visiting Milan Bicocca University. Also, the authors would like to thank the referees for their valuable suggestions. References [] Acerbi, E., Mingione, G.: Gradient estimates for the p(x)-laplacean system. J. Reine Angew. Math. 584, 7 48 (2005) [2] Alves, C.O., and Souto, M.A.: Existence of solutions for a class of problems in R N involving the p(x)-laplacian, contributions to nonlinear analysis, a tribute to D.G. de Figueiredo on the occasion of his 70th birthday. In: Cazenave, T., Costa, D., Lopes, O., Manàsevich, R., Rabinowitz, P., Ruf, B., Tomei, C. (eds.) Series: progress in nonlinear differential equations and their applications, vol. 66, pp Birkhäuser, Basel (2006) [3] Boureanu, M.-M.: Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent. Taiwanese J. Math. (accepted) [4] Chabrowski, J., Fu, Y.: Existence of solutions for p(x)-laplacian problems on a bounded domain. J. Math. Anal. Appl. 306, (2005) [5] Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math. 66, (2006) [6] Diening, L.: Theoretical and Numerical Results for Electrorheological Fluids. Ph.D. thesis, University of Frieburg, Germany (2002) [7] Fan, X.: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl. 339, (2008) [8] Fan, X.: Solutions for p(x)-laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 32, (2005) [9] Fan, X., Han, X.: Existence and multiplicity of solutions for p(x)-laplacian equations in R N. Nonlinear Anal. 59, (2004) [0] Fan, X., Zhang, Q., Zhao, D.: Eigenvalues of p(x)-laplacian Dirichlet problem. J. Math. Anal. Appl. 302, (2005) [] Fan, X., Zhao, D.: On the spaces L p(x) () and W m,p(x) (). J. Math. Anal. Appl. 263, (200)

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17 Vol. 9 (202) Infinitely many solutions 25 [29] Zhao, J.-H., Zhao, P.-H.: Infinitely many weak solutions for a p-laplacian equation with nonlinear boundary conditions. Electron. J. Differen. Equ. 2007, 4 (2007) [30] Zhikov, V.V.: Averaging of functionals in the calculus of variations and elasticity. Math. USSR Izv. 29, (987) Maria-Magdalena Boureanu Department of Mathematics University of Craiova Craiova Romania mmboureanu@yahoo.com Felician Preda Institute of Mathematical Statistics and Applied Mathematics 0507 Bucharest Romania felicianpreda@yahoo.com Received: 6 February 20. Accepted: 7 June 20.