LE Received 15 October 2004 and in revised form 21 January 2005 We investigate the existence of multiple solutions to quasilinear elliptic problems con-taining Laplace like operators φ-L
Trang 1NEUMANN PROBLEMS IN ORLICZ-SOBOLEV SPACES
NIKOLAOS HALIDIAS AND VY K LE
Received 15 October 2004 and in revised form 21 January 2005
We investigate the existence of multiple solutions to quasilinear elliptic problems con-taining 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 con-dition,
−div
α ∇ u(x) ∇ u(x)
= g(x, u) a.e onΩ
∂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 byφ(s) = α( | s |)s if s =0 and 0 otherwise, is an increasing homeomor-phism fromRtoR LetΦ(s) =s
0φ(t)dt, s ∈ R ThenΦ is a Young function We denote
byLΦthe Orlicz space associated withΦ and by · Φthe usual Luxemburg norm onLΦ:
u Φ=inf
k > 0 :
ΩΦu(x) k dx ≤1
Also,W1LΦis 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
W1LΦ:
u ∈ W1LΦ:
Ωα
|∇ u |∇ u · ∇ v dx =
Ωg( ·,u)v dx, ∀ v ∈ W1LΦ. (1.3) Our goal in this short note is to prove the existence of two nontrivial solutions to our problem under some suitable conditions ong 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
Trang 2First, let us recall the well known Palais-Smale (PS) condition Let X be a Banach
space andI : X → R We say thatI satisfies the (PS) condition if any sequence { u n } ⊆ X
satisfying
I
u n ≤ M I
u n
,φ ≤ ε n φ X, (1.4) withε n →0, has a convergent subsequence
Theorem 1.1 [1] LetX be a Banach space with a direct sum decomposition
with dimX2< ∞ Let J be a C1function 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 onφ and g Put
p1=inf
t>0
tφ(t) Φ(t), pΦ=lim inft→∞
tφ(t) Φ(t), p0=supt>0
tφ(t)
(H(φ)) We assume that
1< lim inf s→∞
sφ(s) Φ(s) ≤lim sups→∞
sφ(s)
It is easy to check that under hypothesis (H(φ)), bothΦ and its H¨older conjugate satisfy the∆2condition
Letg :Ω× R → Rbe a Carath´eodory function and letG be its anti-derivative:
G(x, u) =
u
Trang 3(H(g)) We suppose that g and G satisfy the following hypotheses.
(i) There exist nonnegative constantsa1,a2 such that| g(x, s) | ≤ a1+a2| s | a−1, for alls ∈ R, almost allx ∈ Ω, with p0< a < N p1/(N − p1)
(ii) We suppose that there existsδ > 0 such that G(x, u) ≥0, for a.e.x ∈Ω, all
u ∈[− δ, δ].
(iii) Assume that
lim
u→0
G(x, u)
| u | p0 =0, lim sup
u→∞
G(x, u)
uniformly forx ∈Ω
(iv) Suppose that
lim inf
|u|→∞
p1G(x, u) − g(x, u)u
withk ∈ L1(Ω), and such thatΩk(x)dx > 0.
(v) There exists somet ∗ ∈ Rsuch that
ΩG(x, t ∗)dx > 0 and G(x, u) ≤ j(x) for
| u | > M with M > 0 and j ∈ L1(Ω)
Our energy functional isI : W1LΦ→ Rwith
I(u) =
ΩΦ∇ u(x)dx −
ΩG
x, u(x)
It is easy to check thatI is of class C1and the critical points ofI are solutions of (1.3) Let
V =
u ∈ W1,p1(Ω) :
Ωu(x)dx =0
andV = V ∩ X It is clear that V (resp.,V ) is the topological complement ofRwith respect toW1,p1
(Ω) (resp., with respect to X) From the Poincar´e-Wirtinger inequality,
we have the following estimates inV :
u L p1(Ω)≤ C |∇ u |
L p1(Ω), ∀ u ∈ V , (2.8) (for some constantC > 0).
Lemma 2.1 If hypotheses (H( φ)) and (H(g)) hold, then the energy functional I satisfies the
(PS) condition.
Proof Let X = W1LΦ(Ω) Suppose that there exists a sequence{ u n } ⊆ X such that
I
I
u n
for alln ∈ N, allφ ∈ X We first show that { u n }is a bounded sequence inX 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 inX,
Trang 4by passing once more to a subsequence, we can assume thaty n y (weakly) in X and
therefore
From (2.9), we have
ΩΦ∇ u n(x)dx −
ΩG
x, u n(x)
On the other hand, note that
Φ(t) ≥ ρ p1Φt
ρ
Indeed, from the definition ofp1, we have thatΦ(t)p1≤ tφ(t) for t > 0 Thus,
t
t/ρ
p1
s ds ≤
t
t/ρ
φ(s)
for allt > 0 and for ρ > 1 Simple calculations on these integrals give the above inequality.
It follows from (2.13) that
ΩΦ∇ y n(x)dx ≤ 1
u n p1
1, Φ
ΩΦ∇ u n(x)dx. (2.15)
Dividing both sides of (2.12) by u n 1,p1Φ> 1 and making use of (2.15), we obtain
ΩΦ(∇ y n(x)dx ≤
Ω
G
x, u n(x)
u n p1
1,Φ
dx + u n M p1
1,Φ
Next, let us prove that
Ω
G
x, u n(x)
u n p1
1,Φ
In fact, from (H(g))(iii) we have that for every ε > 0 there exists M1> 0 such that for
| u | > M1we haveG(x, u)/ | u | p1
≤ ε for almost all x ∈Ω Thus,
Ω
G
x, u n(x)
u n p1
1,Φ
dx ≤
{x∈Ω:|u n(x)|≤M}
G
x, u n(x)
u n p1
1,Φ
dx +
{x∈Ω:|u n(x)|≥M} εy n(x)p1
dx.
(2.18)
Trang 5Becausep1≤ p0≤ a, we have W1LΦ L p1
(Ω) From this embedding, one obtains
Ω
G
x, u n(x)
u n p1
1,Φ
dx ≤
{x∈Ω:|u n(x)|≤M}
G
x, u n(x)
u n p1
1,Φ
dx + εc y n p1
1,Φ. (2.19)
Finally, noting that y n 1,Φ=1, we obtain (2.17)
From (2.16) and (2.17), we have
ΩΦ∇ y n(x)dx −→0, (2.20)
and thus∇ y n Φ→0 The lower semicontinuity of the norm · Φyields
(0≤)∇ y Φ≤lim inf
n→∞ ∇ y n
Hence,∇ y =0 a.e onΩ, that is, y ∈ R This also implies that
lim
n→∞ ∇ y n − y Φ=lim
n→∞ ∇ y n
From (2.11) and (2.22), we get
y n − y
1,Φ= y n − y
Φ+ ∇ y n − y Φ−→0 asn −→ ∞, (2.23)
that is,y n → y (strongly) in X Since y n 1, Φ=1, we havey =0 Furthermore, from the above arguments,y = c ∈ Rwithc =0 From this we obtain that| u n(x) | → ∞
Choosingφ = u nin (2.10) and noting (2.9), we arrive at
Ωp1G
x, u n(x)
− g
x, u n(x)
u n(x)dx
+
Ωφ ∇ u n ∇ u n − p1Φ∇ u ndx ≤ M + ε n u n
1, Φ.
(2.24)
From the definition ofp1we havep1Φ(t) ≤ tφ(t) Using this fact and dividing the last
inequality by u n 1, Φ, one gets
Ω
p1G
x, u n(x)
− g
x, u n(x)
u n(x)
u n(x) y n(x)dx ≤ M + ε n u n
1,Φ
u n
From this we can see that
lim inf
n→∞
Ω
p1G
x, u n(x)
− g
x, u n(x)
u n(x)
u n(x) y n(x)dx ≤0. (2.26)
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 thatu n u weakly
inX and thus u n → u strongly in L a(Ω)
Trang 6In order to show the strong convergence of{ u n }inX, we get back to (2.10) and choose
φ = u n − u We obtain
Ωα ∇ u n ∇ u n − α
|∇ u |∇ u
∇ u n − ∇ u
dx
≤
Ωf
x, u n
u n − u
dx + ε n u n − u
1, Φ−
Ωα
|∇ u |∇ u
∇ u n − ∇ u
dx.
(2.27)
Using again the compact imbeddingX 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
Using [2, Theorem 4] we obtain the strong convergence of{ u n }inX.
In the next result, we verify that under the above assumptions, the functionalI satisfies
the saddle conditions in Br´ezis-Nirenberg’s theorem
Lemma 2.2 If hypotheses (H( φ)) and (H(g)) hold, then there exists ρ > 0 such that for all
u ∈ V with u 1,Φ≤ ρ we have that I(u) ≥ 0 and I(e) ≤ 0 for all e ∈ R with | e | ≤ ρ Proof Choose u ∈ V with || u ||1,Φ= ρ, with ρ sufficiently small, to be specified later From (H(g))(iii) we have that for every ε > 0 there exists some δ > 0 for which
G(x, u) ≤ ε | u | p0
∀| u | ≤ δ and almost all x ∈ Ω. (2.29)
On the other hand, it follows from (H(g))(i) that there is ˜ a2> 0 such that
for allu ∈ Rand almost allx ∈ Ω Together with (H(g))(iii), this shows that there is γ > 0
such that
G(x, u) ≤ ε | u | p0
for allu ∈ R, almost allx ∈ Ω From the definition of p0we havep0/t ≥ φ(t)/ Φ(t)
Inte-grating this inequality in [t, t/ρ] with ρ < 1, t > 0 yields
Φ(t) ≥ ρ p0
Φ ρ t
Trang 7
Recall also that from the definition ofp1we can take fort ≥1
Φ(t) ≥ Φ(1)t p1
thus,LΦ L p1
(Ω) and there exists k0> 0 such that
for allu ∈ LΦ( · p1is the usual Lebesgue norm onL p1
(Ω))
Because u 1, Φ≤1 we have also∇ u Φ≤1 Then, we have the estimate
ΩΦ|∇ u |dx ≥ |∇ u | p0
Φ ≥ C |∇ u | p0
noting that
ΩΦ(|∇ u | / ∇ u Φ)=1 (see [5, Proposition 6, page 77])
Using now the Poincar´e-Wirtinger inequality, we arrive at
ΩΦ|∇ u |dx ≥ C u 1,p0p1. (2.36) Also,
ΩG(x, u)dx ≤ ε u p p00+γ1 u a
1,p1≤ εc1 u 1,p0p1+γ1 u a
1,p1. (2.37) Choosing small enoughε we arrive at I(u) ≥ C u 1,p0p1− γ1 u a
1,p1 Therefore, we choose small enoughρ to obtain I(u) ≥0 for u 1,Φ≤ ρ.
Fort ∈ Rwe haveI(t) = −ΩG(x, t)dx But from (H(g))(ii) we have that G(x, t) ≥0 for small enought ∈ R Thus, for such at ∈ Rwe obtainI(t) ≤0 Finally from (H(v)) we have thatI is bounded from below and that inf X I < 0, thus we
are allowed to use the multiplicity theorem of Br´ezis-Nirenberg and have the following result
Theorem 2.3 Under hypotheses (H( φ)) and (H(g)) hold, the boundary value problem ( 1.3 ) has at least two nontrivial solutions.
We conclude with a simple example to illustrate the above conditions and arguments
Example 2.4 Let α and g be defined by
α(s) =ln
e + s2
g(u) =
4u3 if| u | ≤ √1
5,
u − u3 if| u | > √1
5.
(2.39)
It can be easily checked thatΦ(s) =1/2(e + s2)[ln(e + s2)−1](s ∈ R) and thuspΦ= p1=
2 andp0≈2.6 Because G(u) = u4for| u |small andG(u) ≈ u2/2 − u4/4 for | u |large, we see that the conditions in (H(φ)) and (H(g)) are satisfied.
Trang 8[1] H Br´ezis and L Nirenberg, Remarks on finding critical points, Comm Pure Appl Math 44
(1991), no 8-9, 939–963.
[2] V K Le, A global bifurcation result for quasilinear elliptic equations in Orlicz-Sobolev spaces,
Topol Methods Nonlinear Anal 15 (2000), no 2, 301–327.
[3] S J Li, Some advances in Morse theory and minimax theory, Morse Theory, Minimax Theory
and Their Applications to Nonlinear Differential Equations (H Br´ezis, S J Li, J Q Liu, and
P H Rabinowitz, eds.), New Stud Adv Math., vol 1, International Press, Massachusetts,
2003, pp 91–115, Proceedings of the Workshop and International Academic Symposium Held in Beijing, April 1–September 30, 1999.
[4] J Q Liu and S J Li, An existence theorem for multiple critical points and its application, Kexue
Tongbao (Chinese) 29 (1984), no 17, 1025–1027.
[5] M M Rao and Z D Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and
Applied Mathematics, vol 146, Marcel Dekker, New York, 1991.
[6] E A B Silva, Critical point theorems and applications to differential equations, Ph.D thesis,
University of Wisconsin-Madison, Wisconsin, 1988.
Nikolaos Halidias: Department of Statistics and Actuarial Science, University of the Aegean, Karlovassi 83200, Samos, Greece
E-mail address:nick@aegean.gr
Vy K Le: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO
65401, USA
E-mail address:vy@umr.edu
... Since{ y n }is bounded in< i>X, Trang 4by passing...
t/ρ
φ(s)
for allt > and for ρ > Simple calculations on these integrals give the above inequality.
It follows from (2.13) that
...
ρ
Indeed, from the definition ofp1, we have thatΦ(t)p1≤ tφ(t) for t > Thus,
t