MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC NEUMANN PROBLEMS IN ORLICZ-SOBOLEV SPACES

Here, Ω is a bounded domain with sufficiently smooth (e.g. Lipschitz) boundary ∂Ω and ∂/∂ν denotes the (outward) normal derivative on ∂Ω. We assume that the function φ :R→R, defined by φ(s)= α(|s|)s if s = 0 and 0 otherwise, is an increasing homeomorphism from R to R. Let Φ(s)= ∫ s 0 φ(t)dt, s∈R. Then Φ is a Young function. We denote by LΦ the Orlicz space associated withΦ and by ‖ · ‖Φ the usual Luxemburg norm on LΦ: ‖u‖Φ = inf { k > 0 : ∫


Introduction
In this paper, we consider the following elliptic problem with Neumann boundary condition, −div α ∇u(x) ∇u(x) = g(x,u) a.e. on Ω ∂u ∂ν = 0 a.e. on ∂Ω. (1.1) Here, Ω is a bounded domain with sufficiently smooth (e.g. Lipschitz) boundary ∂Ω and ∂/∂ν denotes the (outward) normal derivative on ∂Ω. We assume that the function φ : R → R, defined by φ(s) = α(|s|)s if s = 0 and 0 otherwise, is an increasing homeomorphism from R to R. Let Φ(s) = s 0 φ(t)dt, s ∈ R. Then Φ is a Young function. We denote by L Φ the Orlicz space associated with Φ and by · Φ the usual Luxemburg norm on L Φ : Also, W 1 L Φ is the corresponding Orlicz-Sobolev space with the norm u 1,Φ = u Φ + |∇u| Φ . The boundary value problem (1.1) has the following weak formulation in W 1 L Φ : (1.3) 300 Multiple solutions for Neumann problems First, let us recall the well known Palais-Smale (PS) condition. Let X be a Banach space and I : X → R. We say that I satisfies the (PS) condition if any sequence {u n } ⊆ X satisfying with ε n → 0, has a convergent subsequence.
Theorem 1.1 [1]. Let X be a Banach space with a direct sum decomposition with dimX 2 < ∞. Let J be a C 1 function on X with J(0) = 0, satisfying (PS) and, for some Assume also that J is bounded below and inf X J < 0. Then J has at least two nonzero critical points.
Note that our abstract main tool is the local linking theorem stated above. This method was first introduced by Liu and Li in [4] (see also [3]). It was generalized later by Silva in [6] and by Brézis and Nirenberg in [1]. The theorem stated above is a version of local linking theorems established in the last cited reference.

Existence result
First, let us state our assumptions on φ and g. Put It is easy to check that under hypothesis (H(φ)), both Φ and its Hölder conjugate satisfy the ∆ 2 condition. Let g : Ω × R → R be a Carathéodory function and let G be its anti-derivative: N. Halidias and V. K. Le 301 (H(g)) We suppose that g and G satisfy the following hypotheses.
It is easy to check that I is of class C 1 and the critical points of I are solutions of (1.3). Let and V = V ∩ X. It is clear that V (resp., V ) is the topological complement of R with respect to W 1,p 1 (Ω) (resp., with respect to X). From the Poincaré-Wirtinger inequality, we have the following estimates in V :

Lemma 2.1. If hypotheses (H(φ)) and (H(g)) hold, then the energy functional I satisfies the
for all n ∈ N, all φ ∈ X. We first show that {u n } is a bounded sequence in X. Suppose otherwise that the sequence is unbounded. By passing to a subsequence if necessary, we can assume that u n 1,Φ → ∞. Let y n (x) = u n (x)/ u n 1,Φ . Since {y n } is bounded in X, by passing once more to a subsequence, we can assume that y n y (weakly) in X and therefore y n −→ y (strongly) in L Φ (Ω).
(2.11) From (2.9), we have On the other hand, note that for all t > 0 and for ρ > 1. Simple calculations on these integrals give the above inequality. It follows from (2.13) that Dividing both sides of (2.12) by u n p 1 1,Φ > 1 and making use of (2.15), we obtain that is, y n → y (strongly) in X. Since y n 1,Φ = 1, we have y = 0. Furthermore, from the above arguments, y = c ∈ R with c = 0. From this we obtain that |u n (x)| → ∞. Choosing φ = u n in (2.10) and noting (2.9), we arrive at Ω p 1 G x,u n (x) − g x,u n (x) u n (x)dx + Ω φ ∇u n ∇u n − p 1 Φ ∇u n dx ≤ M + ε n u n 1,Φ .

(2.24)
From the definition of p 1 we have p 1 Φ(t) ≤ tφ(t). Using this fact and dividing the last inequality by u n 1,Φ , one gets From this we can see that Using Fatou's lemma and (H(g))(iv) we obtain a contradiction, which shows that the sequence {u n } is bounded. Passing to a subsequence, we can assume that u n u weakly in X and thus u n → u strongly in L a (Ω).
In order to show the strong convergence of {u n } in X, we get back to (2.10) and choose φ = u n − u. We obtain Ω α ∇u n ∇u n − α |∇u| ∇u ∇u n − ∇u dx ≤ Ω f x,u n u n − u dx + ε n u n − u 1,Φ − Ω α |∇u| ∇u ∇u n − ∇u dx.

(2.27)
Using again the compact imbedding X L a (Ω) and the fact that u n → u weakly in X we arrive at Ω a ∇u n ∇u n − a |∇u| ∇u ∇u n − ∇u dx −→ 0. (2.28) Using [2, Theorem 4] we obtain the strong convergence of {u n } in X.
In the next result, we verify that under the above assumptions, the functional I satisfies the saddle conditions in Brézis-Nirenberg's theorem.
Finally from (H(v)) we have that I is bounded from below and that inf X I < 0, thus we are allowed to use the multiplicity theorem of Brézis-Nirenberg and have the following result.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009