Electronic Journal of Qualitative Theory of Differential Equations
2011, No. 17, 1-10;http://www.math.u-szeged.hu/ejqtde/
Existence and multiplicity of solutions for a Neumann-type
p
(
x
)
-Laplacian equation with nonsmooth potential
Bin Ge
1,∗Qingmei Zhou
21 Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, P. R. China
2Library, Northeast Forestry University, Harbin, 150040, P. R. China
Abstract: In this paper we study Neumann-type p(x)-Laplacian equation with nonsmooth potential. Firstly, applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem inW1,p(x)(Ω). Finally, we obtain the existence
of at least two nontrivial solutions, whenα−> p+.
Key words: p(x)-Laplacian, Differential inclusion problem, Three critical points theo-rem, Neumann-type problem.
§
1
Introduction
The study of differential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1, 2]). It also has wide applications in different research fields, such as image processing model (see e.g. [3, 4]), stationary thermorheological viscous flows (see [5]) and the mathematical description of the processes filtration of an idea barotropic gas through a porous medium (see [6]).
The study on variable exponent problems attracts more and more interest in recent years, many results have been obtained on this kind of problems, for example [7-14].
In this paper, we investigate the following Neumann-type differential equation with p(x )-Laplacian and a nonsmooth potential:
−div(|∇u|p(x)−2∇u) +|u|p(x)−2u∈λ∂j(x, u), in Ω,
∂u
∂γ = 0, on∂Ω,
(P)
where Ω is a bounded domain ofRNwith smooth boundary,λ >0 is a real number,p(x)∈C(Ω) with 1 < p− := min
x∈Ω
p(x) ≤p+ := max x∈Ω
p(x) < +∞, ∂j(x, u) is the Clarke subdifferential of
j(x,·),γis the outward unit normal to the boundary∂Ω.
∗Corresponding author: gebin04523080261@163.com
In [7], Dai studied the particular case p(x) ∈ C(Ω) with N < p−. He established the existence of three solutions by using the non-smooth critical -points theorem [15]. In this paper we will study problem (P) in the case when 1< p(x)<+∞for anyx∈Ω. We will prove that there also exist three weak solutions for problem (P), and existence of at least two nontrivial solutions, whenα−> p+.
This paper is organized as follows. We will first introduce some basic preliminary results and lemma. In Section 2, including the variable exponent Lebesgue, Sobolev spaces, generalized gradient of locally Lipschitz function and non-smooth three-critical-points theorem. In section 3, we give the main results and their proof.
§
2
Preliminaries
In this part, we introduce some definitions and results which will be used in the next section. Firstly, we introduce some theories of Lebesgue-Sobolev space with variable exponent. The detailed can be found in [8-13].
Assume thatp∈C(Ω) andp(x)>1, for allx∈Ω. SetC+(Ω) ={h∈C(Ω) :h(x)>1 for
anyx∈Ω}. Define
h−= min x∈Ω
h(x), h+= max x∈Ω
h(x) for anyh∈C+(Ω).
Forp(x)∈C+(Ω), we define the variable exponent Lebesgue space:
Lp(x)(Ω) ={u:uis a measurable real value function R
Ω|u(x)|p
(x)dx <+∞},
with the norm|u|Lp(x)(Ω)=|u|p(x)=inf{λ >0 :R
Ω| u(x)
λ |p
(x)dx≤1},
and define the variable exponent Sobolev space
W1,p(x)(Ω) ={u∈Lp(x)(Ω) :|∇u| ∈Lp(x)(Ω)},
with the normkuk=kukW1,p(x)(Ω)=|u|p(x)+|∇u|p(x).
We remember that spaces Lp(x)(Ω) and W1,p(x)(Ω) are separable and reflexive Banach
spaces. Denoting by Lq(x)(Ω) the conjugate space of Lp(x)(Ω) with 1 p(x) +
1
q(x) = 1, then
the H¨older type inequality
Z
Ω
|uv|dx≤( 1
p− +
1
q−)|u|Lp(x)(Ω)|v|Lq(x)(Ω), u∈Lp(x)(Ω), v∈Lq(x)(Ω) (1)
holds. Furthermore, define mappingρ:W1,p(x)→Rby
ρ(u) =
Z
Ω
(|∇u|p(x)+|
u|p(x))
dx,
then the following relations hold
kuk<1(= 1, >1)⇔ρ(u)<1(= 1, >1), (2)
kuk>1⇒ kukp−
≤ρ(u)≤ kukp+, (3)
kuk<1⇒ kukp+≤ρ(u)≤ kukp−
. (4)
Hereafter, letp∗(x) =
N p(x)
N−p(x), p(x)< N,
+∞, p(x)≥N.
Remark 2.1. If h ∈ C+(Ω) and h(x) ≤ p∗(x) for any x ∈ Ω, by Theorem 2.3 in [11],
we deduce that W1,p(x)(Ω) is continuously embedded in Lh(x)(Ω). When h(x) < p∗(x), the
Let X be a Banach space andX∗ be its topological dual space and we denote <·,· >as the duality bracket for pair (X∗, X). A functionϕ:X 7→Ris said to be locally lipschitz, if for everyx∈X,we can find a neighbourhoodU ofxand a constantk >0(depending onU), such that|ϕ(y)−ϕ(z)| ≤kky−zk,∀y, z∈U.
The generalized directional derivative ofϕat the pointu∈X in the directionh∈X is
ϕ0(u;h) = lim sup u′→u;λ↓0
ϕ(u′+λh)−ϕ(u′)
λ .
The generalized subdifferential ofϕat the pointu∈X is defined by
∂ϕ(u) ={u∗∈X∗;< u∗, h >≤ϕ0(u;h), ∀h∈X},
which is a nonempty, convex andw∗−compact set ofX. We say thatu∈X is a critical point ofϕ, if 0∈∂ϕ(x). For further details, we refer the reader to [16].
Finally, for proving our results in the next section, we introduce the following lemma:
Lemma 2.1(see [15]). Let X be a separable and reflexive real Banach space, and let Φ,Ψ : X →R be two locally Lipschitz functions. Assume that there existu0 ∈X such that
Φ(u0) = Ψ(u0) = 0 and Φ(u)≥0 for everyu∈X and that there exists u1∈X andr >0 such
that:
(1)r <Φ(u1);
(2) sup
Φ(u)<r
Ψ(u)< rΨ(u1)
Φ(u1),and further, we assume that function Φ−λΨ is sequentially lower
semicontinuous, satisfies the (PS)-condition, and (3) lim
kuk→∞(Φ(u)−λΨ(u)) = +∞
for everyλ∈[0, a], where
a= hr
rΨ(u1 )
Φ(u1 )−Φ(supu)<rΨ(u)
,withh >1.
Then, there exits an open interval Λ1 ⊆[0, a] and a positive real numberσ such that, for
everyλ∈Λ1, the function Φ(u)−λΨ(u) admits at least three critical points whose norms are
less thanσ.
§
3
Existence theorems
In this section, we will prove that there also exist three weak solutions for problem (P). Our hypotheses on nonsmooth potentialj(x, t) as follows.
H(j) : j : Ω×R→Ris a function such that j(x,0) = 0 a.e. on Ω and satisfies the following facts:
(i) for all t∈R,x7→j(x, t) is measurable;
(ii) for almost allx∈Ω,t7→j(x, t) is locally Lipschitz;
(iii) there existα∈C+(Ω) withα+< p− and positive constantsc1, c2, such that
|w| ≤c1+c2|t|α(x)−1
for everyt∈R, almost allx∈Ω and allw∈∂j(x, t);
(iv) there exists at0∈R+, such thatj(x, t0)>0 for allx∈Ω;
(v) there existq∈C(Ω) such thatp+< q−≤q(x)< p∗(x) and lim
|t|→0
j(x, t)
Remark 3.1. It is easy to give examples satisfying all conditions inH(j). For example, the following nonsmooth locally Lipschitz functionj: Ω×R→R, satisfies hypothesesH(j):
j(x, t) = convex, hence locally Lipschitz on W1,p(x)(Ω). On the other hand, because of hypotheses
H(j)(i),(ii),(iii), Ψ is locally Lipschitz (see Clarke [16], p.83)). Therefore ϕ(u) is locally Lips-chitz. We state below our main results
Theorem 3.1. If hypothesesH(j) hold, Then there are an open interval Λ⊆[0.+∞) and a numberσsuch that, for eachλ∈Λ the problem (P) possesses at least three weak solutions inW1,p(x)(Ω) whose norms are less thanσ.
Proof: The proof is divided into the following three Steps.
Step 1. We will show thatϕis coercive in the step.
Firstly, for almost allx ∈ Ω, byt 7→j(x, t) is differentiable almost everywhere on Rand
Lα(x)(Ω)(compact embedding). Furthermore, there exists acsuch that |u|
α(x)≤ckukfor any
Step 2. We show that (PS)-condition holds.
Suppose {un}n≥1 ⊆W1,p(x)(Ω) such that|ϕ(un)| ≤c and m(un) → 0 as n→ +∞. Let
u∗n ∈∂ϕ(un) be such thatm(un) =ku∗nk(W1,p(x)(Ω))∗, n≥1, then we know that
where the nonlinear operator Φ′ :W1,p(x)(Ω)→(W1,p(x)(Ω))∗ defined as
+)(see [14, Theorem 3.1]). Thus we obtain
Fixr0 such thatr0<p1+min{ku1kp
, we can analogously get
sup
Thus, Φ and Ψ satisfy all the assumptions of Lemma 2.1, and the proof is complect. @
Thus far the results involved potential functions exhibitingp(x)-sublinear. The next theorem concerns problems where the potential function is p(x)-superlinear. The hypotheses on the nonsmooth potential are the following:
(vi) For almost allx∈Ω, allt∈Rand allw∈∂j(x, t), we have
j(x, t)≤ν(x) withν ∈Lβ(x)(Ω),1≤β(x)< p−.
Remark 3.2. It is easy to give examples satisfying all conditions inH(j)1. For example, the following nonsmooth locally Lipschitz functionj: Ω×R→R, satisfies hypothesesH(j)1:
j(x, t) =
Theorem 3.2. If hypotheses H(j)1 hold, then there exists a λ0 > 0 such that for each
λ > λ0, the problem (P) has at least two nontrivial solutions.
Proof: The proof is divided into the following five Steps.
Step 1. We will show thatϕis coercive in the step.
ByH(j)1(vi), for allu∈W1,p(x)(Ω),kuk>1, we have
Step 2. We will show that theϕis weakly lower semi-continuous.
Letun⇀ uweakly inW1,p(x)(Ω), by Remark 2.1, we obtain the following results:
Hence, by The Weierstrass Theorem, we deduce that there exists a global minimizer u0∈
and
We will show that for each λ > λ0, the problem (P) has two nontrivial solutions. In order
to do this, fort∈[t1, t2], let us define
Step 4. We will check the C-condition in the following.
condition.
Step 5. We will show that there exists another nontrivial weak solution of problem (P). From Lebourg Mean Value Theorem, we obtain
j(x, t)−j(x,0) =hw, ti
for somew∈∂j(x, ϑt) and 0< ϑ <1. Thus, fromH(j)1(iv), there existsβ ∈(0,1) such that
|j(x, t)| ≤ |hw, ti| ≤µ(x)|t|γ(x), ∀|t|< βand a.e.x∈Ω. (10) On the other hand, by the conditionH(j)1(iii), we have
j(x, t)≤c1|t|+c2|t|α(x)
≤c1|
t β|
α(x)−1|t|+c 2|t|α(x)
=c1|
1
β|
α+−1|t|α(x)+c 2|t|α(x)
=c5|t|α(x)
(11)
for a.e. x∈Ω, all|t| ≥β withc5>0.
Combining (10) and (11), it follows that
|j(x, t)| ≤µ(x)|t|γ(x)+c 5|t|α(x)
for a.e. x∈Ω and allt∈R.
Thus, For allλ > λ0,kuk<1,|u|γ(x)<1 and|u|α(x)<1, we have
ϕ(u) =
Z
Ω
1
p(x)|∇u|
p(x)dx−λZ Ω
j(x, u(x))dx
≥ 1
p+kuk p+
−λ Z
Ω
µ(x)|u|γ(x)dx−λc5
Z
Ω
|u|α(x)dx
≥ 1
p+kuk p+−λc
6kukγ
−
−λc7kukα
−
.
So, forρ >0 small enough, there exists aν >0 such that
ϕ(u)> ν, forkuk=ρ
and ku0k > ρ. So by the Nonsmooth Mountain Pass Theorem, we can getu1 ∈ W1,p(x)(Ω)
satisfies
ϕ(u1) =c >0 andm(u1) = 0.
Therefore,u1is second nontrivial critical point of ϕ. @
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