# Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz-Sobolev Spaces

Authors: Liu Yang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz-Sobolev space.


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## Body

## 1. Introduction

Consider the following nonhomogeneous Neumann problem with a perturbed term:(Pλ,μ)-div(α(|∇u|)∇u)+α(|u|)u=λf(x,u)+μg(x,u),inΩ,∂u∂ν=0,on∂Ω,
where Ω is a bounded domain in ℝN(N≥3) with smooth boundary ∂Ω, ν is the outer normal to ∂Ω, f,g:Ω×ℝ→ℝ are two Carathéodory functions, λ>0,μ≥0 are two parameters, and the function α:(0,∞)→ℝ is such that φ(t):ℝ→ℝ defined by
(1.1)φ(t)={α(|t|)t,t≠0,0,t=0,
is an odd, strictly increasing homeomorphism from ℝ to ℝ.It is well known that these kinds of problems are important in applications in many fields, such as elasticity, fluid dynamics,  and image processing (see [1–4]). Since the operator in the divergence form is nonhomogeneous, we introduce Orlicz-Sobolev space which is an appropriate setting for these problems. Such space originated with Nakano [5] and was developed by Musielak and Orlicz [6]. Many properties of Sobolev spaces have been extended to Orlicz-Sobolev space (see [7–10]). Several authors have widely studied the existence of solutions for the relevant problem by means of variational techniques, monotone operator methods, fixed point, and degree theory (see [11–15]). To the best of our knowledge, for the perturbed nonhomogeneous Neumann problem, there has so far been few papers concerning its multiple solutions. Motivated by the above facts, in this paper, we establish some new sufficient conditions under which such a problem possesses three weak solutions in Orlicz-Sobolev space.This paper is organized as follows. In Section2, some preliminaries are presented. In Section 3, we discuss the existence of three weak solutions for problem (Pλ,μ).


## 2. Preliminaries

We start by recalling some basic facts about Orlicz-Sobolev space. Letφ be as in Introduction and Φ(t):ℝ→ℝ,
(2.1)Φ(t)=∫0t‍φ(s)ds.
We observe that Φ is, a Young function, that is Φ(0)=0, Φ is convex and limt→∞Φ(t)=+∞. Furthermore, since Φ(t)=0 if and only if t=0, limt→0(Φ(t)/t)=0, and limt→∞(Φ(t)/t)=+∞, then Φ is called an N-function. The function Φ* is called the complementary function of Φ and it satisfies
(2.2)Φ*(t)=sup{st-Φ(s);s≥0},∀t≥0.
Assume that Φ satisfies the following structural hypotheses (Φ1)1<liminft→∞(tφ(t)/Φ(t))≤p0:=supt>0(tφ(t)/Φ(t))<∞;(Φ2)N<p0:=inft>0(tφ(t)/Φ(t))<liminft→∞(log(Φ(t))/log(t)).Further, we also assume that the function(Φ3)[0,∞)∋t→Φ(t) is convex.The Orlicz spaceLΦ(Ω) defined by Φ is the space of measurable functions u:Ω→ℝ such that
(2.3)∥u∥LΦ:=sup{∫Ω‍u(x)v(x)dx;∫Ω‍Φ*(|v(x)|)dx≤1}<∞.
Then (LΦ(Ω),∥·∥LΦ) is a Banach space whose norm is equivalent to the Luxemburg norm
(2.4)∥u∥Φ:=inf{k>0;∫Ω‍Φ(|u(x)|k)dx≤1}.
We denote by W1LΦ(Ω) the Orlicz-Sobolev space, defined by
(2.5)W1LΦ(Ω)={u∈LΦ;∂u∂xi∈LΦ,i=1,2,…,N}.
This is a Banach space with respect to the norm
(2.6)∥u∥1,Φ=∥|∇u|∥Φ+∥u∥Φ.Lemma 2.1 (see [13]).
OnW1LΦ(Ω) the norms
(2.7)∥u∥1,Φ=∥|∇u|∥Φ+∥u∥Φ,∥u∥2,Φ=max{∥|∇u|∥Φ,∥u∥Φ},∥u∥:=inf{μ>0;∫Ω‍[Φ(|u(x)|μ)+Φ(|∇u(x)|μ)]dx≤1},
are equivalent. Moreover, for every u∈W1LΦ(Ω), one has
(2.8)∥u∥≤2∥u∥2,Φ≤2∥u∥1,Φ≤4∥u∥.Lemma 2.2.
Letu∈W1LΦ(Ω), then
(2.9)∥u∥p0≤∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx≤∥u∥p0,if∥u∥>1,∥u∥p0≤∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx≤∥u∥p0,if∥u∥<1.Proof.
For the proof of(2.10)∥u∥p0≤∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx,if∥u∥>1,∥u∥p0≤∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx,if∥u∥<1,
we can see Lemma 2.2 of the paper [13]. Since p0≥(tφ(t))/Φ(t) for all t≥0, it follows that letting σ>1, we have
(2.11)log(Φ(σt))-log(Φ(t))=∫tσt‍φ(s)Φ(s)ds≤∫tσt‍p0sds=log(σp0).
Thus, one has
(2.12)Φ(σt)≤σp0Φ(t),t>0,σ>1.
Moreover, by the definition of the norm, we remark that
(2.13)∫Ω‍[Φ(|u(x)|∥u∥)+Φ(|∇u(x)|∥u∥)]dx≤1.
Therefore, we have
(2.14)∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx=∫Ω‍[Φ(∥u∥|u(x)|∥u∥)+Φ(∥u∥|∇u(x)|∥u∥)]dx≤∥u∥p0∫Ω‍[Φ(|u(x)|∥u∥)+Φ(|∇u(x)|∥u∥)]dx≤∥u∥p0,
for all ∥u∥>1.
Similar techniques as those used in the proof of (2.12), we have
(2.15)Φ(t)≤τp0Φ(tτ),t>0,0<τ<1.
Therefore, we can obtain
(2.16)∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx≤∥u∥p0∫Ω‍[Φ(|u(x)|∥u∥)+Φ(|∇u(x)|∥u∥)]dx≤∥u∥p0,
for all ∥u∥<1.Lemma 2.3 (see [13]).
Letu∈W1LΦ(Ω) and assume that
(2.17)∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx≤r
for some 0<r<1, then one has ∥u∥<1.Lemma 2.4 (see [13]).
Ifp0>N, then W1LΦ(Ω) is compactly embedded in C0(Ω-) and there exists a constant c>0 such that
(2.18)∥u∥∞≤c∥u∥1,Φ,∀u∈W1LΦ(Ω),
where ∥u∥∞∶=supx∈Ω-|u(x)|.Now,one recall, a three critical theorem of B. Ricceri. If X is a real Banach space, denote by 𝒲X (see [16]) the class of all functionals Φ:X→ℝ possessing the following property: if {un} is a sequence in X converging weakly to u and liminfn→∞Φ(un)≤Φ(u), then {un} has a subsequence converging strongly to u. For example, if X is uniformly convex and g:[0,+∞)→ℝ is a continuous, strictly increasing function, then, by a classical results, the functional u→g(∥u∥) belongs to the class 𝒲X.Lemma 2.5 (see [16]).
LetX be a separable and reflexive real Banach space; let I:X→ℝ be a coercive, sequentially weakly lower semicontinuous C1 functional, belonging to 𝒲X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X*; J:X→ℝ a C1 functional with compact derivative. Assume that I has a strict local minimum u0 with I(u0)=J(u0)=0. Finally, setting
(2.19)α′=max{0,limsup∥u∥→+∞J(u)I(u),limsupu→u0J(u)I(u)},β′=supu∈I-1(0,+∞)J(u)I(u),
assume that α′<β′. Then for each compact interval [a,b]⊂(1/β′,1/α′) (with the conventions (1/0)=+∞,(1/+∞)=0), there exists B>0 with the following property: for every λ∈[a,b] and every C1 functional Ψ:X→ℝ with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation
(2.20)I'(x)=λJ'(x)+μΨ'(x)
has at least three solutions in X whose norms are less than B.Lemma 2.6 (see [17]).
LetX be a reflexive real Banach space; S⊂ℝ an interval, let I:X→ℝ be a sequentially weakly lower semicontinuous C1 functional, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X*; J:X→ℝ a C1 functional with compact derivative. Assume that
(2.21)lim∥u∥→∞[I(u)-λJ(u)]=+∞
for all λ∈S, and that there exists ρ∈ℝ such that
(2.22)supλ∈Sinfu∈X[I(u)+λ(ρ-J(u))]<infu∈Xsupλ∈S[I(u)+λ(ρ-J(u))].
Then there exist a nonempty open set A⊂S and a positive number B, with the following property: for every λ∈A and every C1 functional Ψ:X→ℝ with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation
(2.23)I′(x)=λJ′(x)+μΨ′(x)
has at least three solutions in X whose norms are less than B.Lemma 2.7 (see [18]).
LetX be a nonempty set and I,J two real functions on X. Assume that there are r>0 and x0,x1∈X such that
(2.24)I(x0)=J(x0)=0,I(x1)>r,supx∈I-1(]-∞,r])J(x)<rJ(x1)I(x1).
Then for each ρ satisfying
(2.25)supx∈I-1(]-∞,r])J(x)<ρ<rJ(x1)I(x1),
one has
(2.26)supλ≥0infu∈X[I(u)+λ(ρ-J(u))]<infu∈Xsupλ≥0[I(u)+λ(ρ-J(u))].


## 3. Proof of the Main Results

Setγ=inf{(∫Ω‍(Φ(|∇u(x)|)+Φ(|u(x)|))dx/∫Ω‍F(x,u(x))dx):u∈X,∫Ω‍F(x,u(x))dx>0}.Theorem 3.1.
LetΦ be a function satisfying the structural hypotheses (Φ1–Φ3) and the following conditions hold(H1)
max{limsupξ→0(supx∈ΩF(x,ξ)/|ξ|p0),limsup|ξ|→∞(supx∈ΩF(x,ξ)/|ξ|p0)}≤0,(H2)
supu∈X∫Ω‍F(x,u(x))dx>0.Then, for each compact interval [a,b]⊂(γ,∞), there exists B>0 with the following property: for every λ∈[a,b] and g, there exists δ>0 such that, for each μ∈[0,δ], the problem (Pλ,μ) has at least three weak solutions whose norms in X are less than B.Proof of Theorem3.1.
In order to apply Lemma2.5, we let
(3.1)I(u)=∫Ω‍[Φ(|∇u(x)|)+Φ(|u(x)|)]dx,J(u)=∫Ω‍F(x,u(x))dx,Ψ(u)=∫Ω‍G(x,u(x))dx.
We divide our proof into two steps as follows.
Step 1. We show that some fundamental assumptions are satisfied.
X
:
=
W
1
L
Φ
(
Ω
). Obviously, X is a separable and reflexive real Banach space (see [13]). By Lemma 2.2, it is easy to see that I(u) is a coercive, bounded on each bounded subset of X. On the other hand, I,J,Ψ∈C1(X,ℝ) with the derivatives given by
(3.2)〈I′(u),v〉=∫Ω‍[α(|∇u(x)|)∇u(x)·∇v(x)+α(|u(x)|)u(x)v(x)]dx,〈J′(u),v〉=∫Ω‍f(x,u(x))v(x)dx,〈Ψ′(u),v〉=∫Ω‍g(x,u(x))v(x)dx,
for any u,v∈X. Hence, the critical points of the functional I-λJ-μΨ are exactly the weak solutions for problem (Pλ,μ). Moreover, owing that Φ is convex, it follows that I is convex. Hence, one has that I is sequentially weakly lower semicontinuous. The fact X is compactly embedded into C0(Ω¯) implies that operators J',Ψ' is compact. As the proof of Lemma 3.2 in [15], we know that I' has a continuous inverse.
Moreover, if{un} is a sequence in X converging weakly to u and liminfn→∞I(un)≤I(u), remark that I is sequentially weakly lower semicontinuous, one has
(3.3)I(u)≤liminfn→∞I(un)≤I(u).
Then, up to a subsequence, we deduce that I(un)→I(u)=d. Taking into account that {(un+u)/2} converges weakly to u and I is sequentially weakly lower semicontinuous, we have
(3.4)d=I(u)≤liminfn→∞I(un+u2).
We assume by contradiction that un does not converge to u in X. Hence, there exist ε0>0 and a subsequence {unm} of (un) such that
(3.5)∥unm-u2∥>ε0,∀m.
Then there exists ε1>0 such that
(3.6)I(unm-u2)>ε1,∀m.
On the other hand,
(3.7)12I(u)+12I(unm)-I(unm+u2)≥I(unm-u2)>ε1,(see [19]). Letting m→∞ in the above inequality we obtain
(3.8)d-ε1≥limsupm→∞I(unm+u2)
and that is a contradiction with (3.4). It follows that un converges strongly to u and I∈𝒲X. In addition, I(0)=J(0)=0.
Step 2. We show that α′=0,β′>0.
In view of(H1), for all ε>0, there exists τ1>0 such that
(3.9)F(x,ξ)≤ε|ξ|p0,
for any |ξ|∈[0,τ1]. For ∥u∥<min{1,(τ1/2c)}, we have
(3.10)|u(x)|≤∥u∥∞≤c∥u∥1,Φ≤2c∥u∥≤τ1,limsupu→0J(u)I(u)≤limsupu→0ε∫Ω‍|u|p0dx∥u∥p0≤limsupu→0ε|Ω|(2c∥u∥)p0∥u∥p0≤0.
By (H1), for all ε>0, there exists 0<τ1<τ2 such that
(3.11)F(x,ξ)≤ε|ξ|p0,
for any |ξ|>τ2. Further, for each ∥u∥>1, we have
(3.12)J(u)I(u)≤∫Ω(|u|≤τ2)‍F(x,u(x))dx∥u∥p0+∫Ω(|u|>τ2)‍F(x,u(x))dx∥u∥p0≤∫Ω(|u|≤τ2)‍F(x,u(x))dx∥u∥p0+ε|Ω|(2c∥u∥)p0∥u∥p0.
So we get
(3.13)limsup∥u∥→∞J(u)I(u)≤0.
Then, with the notation of Lemma 2.5, we have α'=0. By assumption (H2), we have β'>0. Thus, all the hypotheses of Lemma 2.5 are satisfied. Clearly, γ=1/β′. Finally, by Lemma 2.5, we can obtain the Theorem 3.1.Example 3.2.
Letp>N+1. Define
(3.14)φ(t)=|t|p-2tlog(1+|t|),t≠0,
and φ(0)=0. By [13], one has
(3.15)p0=p-1<p0=p.
Let F(x,t)=|t|p+1-|t|p+2. Since F≤0 if |t| is large enough and F>0 if |t| is small enough, moreover, it is easy to see limsupξ→0(supx∈ΩF(x,ξ)/|ξ|p0)=0, the conditions of Theorem 3.1 can be satisfied.Remark 3.3.
SinceF in [13] is p0-sublinear, the results of [13] do not fit to the problem treated in the previous Example 3.2 even if μ=0, that is, there is no perturbed nonlinear term. In addition, for nonhomogeneous Neumann problem with a perturbed term, we can have the following result when F is p0-sublinear, which extends the results of [13].Theorem 3.4.
LetΦ be a function satisfying the structural hypotheses (Φ1–Φ3) and the following conditions.
(
H
3
) There exist two constants γ,δ with γ<2c such that Φ(δ)>(γp0/(2c)p0|Ω|) and
(3.16)∫Ω‍max|ξ|≤γF(x,ξ)dxγp0<∫Ω‍F(x,δ)dx(2c)p0|Ω|Φ(δ),
where |Ω| denotes the Lebesgue measure of the set Ω.
(
H
4
) There exist h(x),k(x)∈L1(Ω;ℝ+) and 0<s<p0 such that |F(x,t)|≤h(x)+k(x)|t|s for every (x,t)∈Ω×ℝ.
Then, there exist a nonempty open setA⊂[0,∞) and a positive number B, for each λ∈A and for every g, there exists δ>0 such that, for each μ∈[0,δ], the problem (Pλ,μ) has at least three weak solutions whose norms in X are less than B.Proof of Theorem3.4.
Let us considerI,J,Ψ as the proof of Theorem 3.1. For any λ>0,u∈X, by (H4) we have
(3.17)I(u)-λJ(u)≥∫Ω‍[Φ(|u(x)|)+Φ(|∇u(x)|)]dx-λ∥u∥∞s∫Ω‍k(x)dx-λ∫Ω‍h(x)dx≥∥u∥p0-λ(2c)s∥u∥s∫Ω‍k(x)dx-λ∫Ω‍h(x)dx,
for ∥u∥>1. Since 0<s<p0, one has lim∥u∥→∞[I(u)-λJ(u)]=+∞ for all λ≥0.
Letr∶=(γp0/(2c)p0),   x1=δ. For r>0,I(u)≤r, we have
(3.18)|u(x)|≤∥u∥∞≤c∥u∥1,Φ≤2c∥u∥≤γ,∀x∈Ω.
Hence, one has
(3.19)supu∈I-1(]-∞,r])J(u)r≤(2c)p0∫Ω‍max|ξ|≤γF(x,ξ)dxγp0.
From (H3), it follows that
(3.20)supI(u)≤rJ(u)r<J(δ)I(δ).
Since all the assumptions of Lemmas 2.7 and 2.6 are satisfied, then, there is a nonempty open set A⊂[0,∞) and a positive number B, for each λ∈A and for every g there exists δ>0 such that, for each μ∈[0,δ], the problem has at least three weak solutions whose norms in X are less than B.



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