Multiple solutions for quasilinear elliptic Neumann problems in Orlicz-Sobolev spaces
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We investigate the existence of multiple solutions to quasilinear elliptic problems containing Laplace like operators (φ-Laplacians). We are interested in Neumann boundary value problems and our main tool is Br´ezis-Nirenberg’s local linking theorem. 1. 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 (1.1)
∂u = 0 a.e. on ∂Ω. ∂ν
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 homeomors phism from R to R. Let Φ(s) = 0 φ(t)dt, s ∈ R. Then Φ is a Young function. We denote by LΦ the Orlicz space associated with Φ and by · Φ the usual Luxemburg norm on LΦ : u(x) uΦ = inf k > 0 : Φ dx ≤ 1 .
(1.2)
k
Ω
Also, W 1 LΦ is the corresponding Orlicz-Sobolev space with the norm u1,Φ = uΦ + |∇u|Φ . The boundary value problem (1.1) has the following weak formulation in W 1 LΦ :
1
u ∈ W LΦ :
Ω
α |∇u| ∇u · ∇v dx =
Ω
g(·,u)v dx,
∀v ∈ W 1 L Φ .
(1.3)
Our goal in this short note is to prove the existence of two nontrivial solutions to our problem under some suitable conditions on g. The main tool that we are going to use is an abstract existence result of Br´ezis and Nirenberg [1], which is stated here for the sake of completeness. Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 299–306 DOI: 10.1155/BVP.2005.299
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 {un } ⊆ X satisfying I u n ≤ M
I un ,φ ≤ εn φX ,
(1.4)
with εn → 0, has a convergent subsequence. Theorem 1.1 [1]. Let X be a Banach space with a direct sum decomposition X = X1 ⊕ X2
(1.5)
with dimX2 < ∞. Let J be a C 1 function on X with J(0) = 0, satisfying (PS) and, for some R > 0, J(u) ≥ 0,
for u ∈ X1 ,
u ≤ R,
J(u) ≤ 0,
for u ∈ X2 ,
u ≤ R.
(1.6)
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 Br´ezis and Nirenberg in [1]. The theorem stated above is a version of local linking theorems established in the last cited reference. 2. Existence result First, let us state our assumptions on φ and g. Put p1 = inf t>0
tφ(t) , Φ(t)
pΦ = liminf t →∞
tφ(t) , Φ(t)
p0 = sup t>0
tφ(t) . Φ(t)
(2.1)
(H(φ)) We assume that
1 < liminf s→∞
sφ(s) sφ(s) ≤ limsup < +∞. Φ(s) s→∞ Φ(s)
(2.2)
It is easy to check that under hypothesis (H(φ)), both Φ and its H¨older conjugate satisfy the ∆2 condition. Let g : Ω × R → R be a Carath´eodory function and let G be its anti-derivativ
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