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International Journal of Advances in Applied Mathematics and Mechanics
Existence of solutions for a class of p(x)-Kirchhoff type equation via topological methods
Research Article
Eugenio Cabanillas L.∗, Adrian G. Aliaga LL., Willy Barahona M., Gabriel Rodriguez V.
Instituto de Investigación, Facultad de Ciencias Matemáticas-UNMSM, Lima-Perú
Received 08 April 2015; accepted (in revised version) 09 May 2015
Abstract: The aim of this work is to obtain weak solutions for a class of p(x)-Kirchhoff type problem under no-flux boundary conditions. Our result is obtained using a Fredholm-type result for a couple of nonlinear operators and the theory of the variable exponent Sobolev spaces.
MSC: 47H05 • 46E35 • 35B38.
Keywords:p-Kirchhoff type equations • Variational methods • No-flux boundary condition.
© 2015 The Author(s). This is an open access article under the CC BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
In this paper we discuss the existence of weak solutions for the following Kirchhoff type problem
−M(
Z
Ω(A(x, ∇u) + 1
p(x)|u|p(x)) d x)[div(a(x, ∇u)) − |u|p(x)−2u] = f (x,u)|u|t (x)s(x) inΩ, u = constant on ∂Ω,
Z
∂Ωa(x, ∇u).νdΓ = 0.
(1)
whereΩ is a bounded domain in Rnwith a smooth boundary∂Ω, and n ≥ 1, p,s,t ∈ C(Ω) for any x ∈ Ω; M : R+→ R+ is a continuous function, f is a Caratheodory function and div¡a(x,∇u)¢ is a p(x)-Laplacian type operator. The study of differential and partial differential equations with variable exponent has been received considerable attention in recent years .This importance reflects directly into various range of applications.There are applications concerning elastic mechanics [1], thermorheological and electrorheological fluids [2,3], image restoration [4] and mathematical biology [5]. Eq. (1) is called a nonlocal problem because of the term M , which implies that the equation in (1) is no longer a pointwise equation. This causes some mathematical difficulties which make the study of such a problem particularly interesting. Nonlocal differential equations are also called Kirchhoff-type equations because Kirchhoff [6]
investigated an equation of the form
ρ∂2u
∂t2−
³P0 h + E
2L ZL
0
¯
¯
∂u
∂x
¯
¯
2d x´∂2u
∂x2= 0, (2)
which extends the classical D’Alembert’s wave equation, by considering the effect of the changing in the length of the string during the vibration. A distinct feature is that the (2) contains a nonlocal coefficientP0
h + E 2L
Z L
0 |∂u
∂x|2d x which
∗ Corresponding author.
E-mail address:[email protected](Eugenio Cabanillas L.)
depends on the average 1 2L
Z L 0
¯
¯
∂u
∂x
¯
¯
2d x, and hence the equation is no longer a pointwise equation. The parameters in (2) have the following meanings: L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material,ρ is the mass density and P0is the initial tension. Lions [38] has proposed an abstract framework for the Kirchhoff-type equations. After the work by Lions [38], various equations of Kirchhoff-type have been studied extensively, see e.g. [7,8] and [9]-[17].The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see [9,10,12,15,17,18]) and p(x)-Laplacian (see [11,13,14,19,20]).
In [21] , Fan has discussed the nonlocal p(x) -Laplacian Dirichlet problem with non-variational form
−A (u) div(|∇u|p(x)−2∇u) = B(u) f (x, u) inΩ, u = 0 on ∂Ω,
and nonlocal p(x)-Laplacian Dirichlet problem with variational form
−a
³Z
Ω
1
p(x)|∇u|p(x)d x´
div(|∇u|p(x)−2∇u) = b
³Z
ΩF (x, u)d x´
f (x, u) inΩ, u = 0 on ∂Ω,
respectively and has obtained the existence of solutions , whereA ,B are two functional defined on W01,p(x), and F (x, t ) =
Z t 0
f (x, s)d s.
More recently, Yucedag et al.[22] , have dealt with Kirchhoff type problem
−M³Z
ΩA(x, ∇u)´
div¡a(x,∇u)¢ = m(x)|u|q(x)−2u inΩ, u = 0 on ∂Ω,
by variational methods.
The nonlocal boundary condition in (1) have been studied by Berestycki and Brezis [23], Ortega [24] , Amster et al. [25], Zhao et al.[26], Boureanou et al.[27], Cabanillas L. et al. al.[28] and the references therein.They arise from certain models in plasma physics:specifically,a model describing the equilibrium of a plasma confined in a toroidal cavity, called a Tokamak machine. A detailed description of this model can be found in the Appendix of [29].
Motivated by the above papers and the results in [22], we consider (1) to study the existence of weak solutions.We note that our problem has no variational structure , so the most usual variational techniques can not used to study it.
To attack it we will employ a Fredholm type theorem for a couple of nonlinear operators due to Dinca [16].
This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable expo- nent Sobolev spaces.In Sections 3, we give some existence results of weak solutions of problem (1) and their proofs.
2. Preliminaries
To discuss problem (1), we need some theory on W1,p(x)(Ω) which is called variable exponent Sobolev space (for details, see [30]). Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. Write
C+(Ω) = {h : h ∈ C(Ω),h(x) > 1 for any x ∈ Ω}, h−:= min
Ω h(x), h+:= max
Ω h(x) for every h ∈ C+(Ω).
Define
Lp(x)(Ω) = {u ∈ S(Ω) :Z
Ω|u(x)|p(x)d x < +∞ for p ∈ C+(Ω)}
with the norm
|u|Lp(x)(Ω)= |u|p(x)= inf{λ > 0 : Z
Ω|u(x)
λ |p(x)d x ≤ 1}, and
W1,p(x)(Ω) = {u ∈ Lp(x)(Ω) : |∇u| ∈ Lp(x)(Ω)}
with the norm
kuk ≡ kukW1,p(x)(Ω)= |u|Lp(x)(Ω)+ |∇u|Lp(x)(Ω).
Proposition 2.1 ([30]).
The spaces Lp(x)(Ω) and W1,p(x)(Ω) are separable and reflexive Banach spaces.
Proposition 2.2 ([30]).
Setρ(u) = Z
Ω|u(x)|p(x)d x. For any u ∈ Lp(x)(Ω), then (1) for u 6= 0, |u|p(x)= λ if and only if ρ(u
λ) = 1;
(2) |u|p(x)< 1 (= 1; > 1) if and only if ρ(u) < 1 (= 1; > 1);
(3) if |u|p(x)> 1, then |u|pp(x)− ≤ ρ(u) ≤ |u|pp(x)+ ;
(4) if |u|p(x)< 1, then |u|pp(x)+ ≤ ρ(u) ≤ |u|pp(x)− ; (5) lim
k→+∞|uk|p(x)= 0 if and only if lim
k→+∞ρ(uk) = 0;
(6) lim
k→+∞|uk|p(x)= +∞ if and only if lim
k→+∞ρ(uk) = +∞.
Proposition 2.3 ([30,31]).
If q ∈ C+(Ω) and q(x) ≤ p∗(x) (q(x) < p∗(x)) for x ∈ Ω, then there is a continuous (compact) embedding W1,p(x)(Ω) ,→
Lq(x)(Ω), where
p∗(x) = (N p(x)
N −p(x) if p(x) < N , +∞ if p(x) ≥ N .
Proposition 2.4 ([30,32]).
The conjugate space of Lp(x)(Ω) is Lq(x)(Ω), where 1 q(x)+ 1
p(x)= 1 holds a.e. in Ω. For any u ∈ Lp(x)(Ω) and v ∈ Lq(x)(Ω), we have the following Hölder-type inequality
¯
¯ Z
Ωuv d x¯
¯≤ ( 1 p−+ 1
q−)|u|p(x)|v|q(x).
Theorem 2.1 ([16]).
Let X and Y be real Banach spaces and two nonlinear operators T, S : X → Y such that 1. T is bijective and T−1is continuous.
2. S is compact.
3. Letλ 6= 0 be a real number such that: k(λT − S)(x)k → +∞ as kxk → +∞;
4. There is a constant R > 0 such that
k(λT − S)(x)k > 0 if kxk ≥ R, dLS(I − T−1(S
λ), B (θ,R),0) 6= 0.
ThenλI − S is suryective from X onto Y .
Here dLS(G, B, 0) denotes the Leray-Schauder degree.
Throughout this paper, let
V = {u ∈ W1,p(x)(Ω) : u|∂Ω= constant}.
The space V is a closed subspace of the separable and reflexive Banach space W1,p(x)(Ω) (See [33]) ,so V is also sepa- rable and reflexive Banach space with the usual norm of W1,p(x)(Ω).The space V is the space where we will try to find weak solutions for problem (1).
Definition 2.1.
A function u ∈ V is said to be a weak solution of (1) if M³Z
Ω(A(x, ∇u) + 1
p(x)|u|p(x)) d x´·Z
Ωa(x, ∇u)∇v d x + Z
Ω|u|p(x)−2uv d x
¸
= Z
Ωf (x, u)|u|t (x)s(x)v d x , for all v ∈ V .
We assume that a(x,ξ) : Ω × RN→ RNis the continuous derivative with respect toξ of the mapping A :Ω × RN→ R, A = A(x, ξ), i.e. a(x, ξ) = ∇ξA(x,ξ). Suppose that a and A satisfy the following assumptions:
(A1) a satisfies the growth condition |a(x,ξ)| ≤ c0(a0(x) + |ξ|p(x)−1) for all x ∈ Ω, ξ ∈ RN, for some constant c0> 0;
a0∈ Lp0(x)is a nonnegative function.
(A2) A(x, 0) = 0 for all x ∈ Ω;
(A3) |ξ|p(x)≤ a(x, ξ)ξ ≤ p(x)A(x, ξ) for all x ∈ Ω, ξ ∈ RN.
(A4) The monotonicity condition 0 ≤£a(x,η1) − a(x,η2)(η1− η2)¤ , for all x ∈ Ω and all η1,η2∈ RN,with equality if and only ifη1= η2.
(F1) f :Ω × R → R is a Caratheodory function satisfying the following conditions
| f (x, s)| ≤ c1+ c2|s|α(x)−1), ∀x ∈ Ω, s ∈ R,
for someα ∈ C+(Ω) such that 1 < α(x) < p∗(x) for x ∈Ω and c1, c2are positive constants.
(M0) M : [0, +∞[→]m0, +∞[ is a continuous and nondecreasing function with m0> 0.
3. Existence of solutions
In this section we will discuss the existence of weak solutions of (1). Our main result is as follows.
Theorem 3.1.
Assume that hypotheses (A1)–(A4), (F1) and (M0) hold. Then (1) has a weak solution in V .
Proof. In order to apply theorem (2.1), we take Y = V0and the operators T, S : V → V0in the following way
〈Tu, v〉 = M³Z
Ω(A(x, ∇u) + 1
p(x)|u|p(x)) d x´·Z
Ωa(x, ∇u)∇v d x + Z
Ω|u|p(x)−2uv d x
¸
〈Su, v〉 = Z
Ωf (x, u)|u|t (x)s(x)v d x
for all u, v ∈ V Then u ∈ V is a solution of (1) if and only if Tu = Su in V0.
Next, we split the proof in several steps.
Step1.We prove that T is an injection.
First we observe that Φ(u) =Mc³Z
Ω(A(x, ∇u) + 1
p(x)|u|p(x)) d x´
is a continuously Gâteaux differentiable function whose Gâteaux derivative at the point u ∈ V is the functional Φ0(u) ∈ V0given by
〈Φ0(u), v〉 = 〈T (u), v〉 for all u ∈ V.
On the other hand, we consider the functional L : V → R defined by
L(u) = Z
Ω(A(x, ∇u) + 1
p(x)|u|p(x)) d x for all u ∈ V then L ∈ C1(V,R and
〈L0(u), v〉 = Z
Ωa(x, ∇u)∇v d x + Z
Ω|u|p(x)−2uv d x for all u, v ∈ V.
Using (A4) and taking into account the inequality [37, (2.2)]
〈|x|p−2x − |y|p−2y, x − y〉 ≥(Cp|x − y|p if p ≥ 2 Cp |x−y|2
(|x|+|y|)p−2, (x, y) 6= (0,0) if 1 < p < 2, (3)
for some for Cp> 0 and for all x, y ∈ RN, we obtain for all u, v ∈ V with u 6= v
〈L0(u) − L0(v), u − v〉 > 0
which means that L0is strictly monotone. So, by [34, Prop. 25.10], L is strictly convex. Moreover, since M is nonde- creasing,M is convex in [0, +∞[. Thus, for every u, v ∈ X with u 6= v, and every s, t ∈ (0,1) with s + t = 1, one hasc
M (L(su + t v)) < cc M (sL(u) + tL(v)) ≤ s cM (L(u)) + t cM (L(v)).
This shows thatΦ is strictly convex, and as Φ0(u) = T (u) in V0 we infer that T is strictly monotone in V , then T is an injection.
Step2. We prove that the inverse T−1: V0→ V of T is continuous.
For any u ∈ V with kuk > 1 , one has
〈T (u), u〉
kuk =
M³ R
Ω(A(x, ∇u) +p(x)1 |u|p(x)) d x
´£R
Ωa(x, ∇u)∇u d x +R
Ω|u|p(x)d x¤
kuk ≥ c0kukp−−1,
from which we have the coercivity of T .
Since T is the Fréchet derivative ofΦ , T is continuous.Thus in view of the well known Minty Browder theorem T is a surjection and so T−1: V0→ V and it is bounded.
Now we prove the continuity of T−1.
First, we verify that T is of type (S+).In fact, if uν* u in V (so there exists R > 0 such that kuνk ≤ R ) and the strict monotonicity of T we have
0 = limsup
ν→∞ 〈Tuν− Tu, uν− u〉 = limν→∞〈Tuν− Tu, uν− u〉
Then
ν→∞lim〈Tuν, uν− u〉 = 0 That is
ν→∞limM³Z
Ω(A(x, ∇uν) + 1
p(x)|uν|p(x)) d x´·Z
Ωa(x, ∇uν)∇(uν− u) d x + Z
Ω|uν|p(x)−2uν(uν− u) d x
¸
= 0 (4)
Now, for any x ∈ Ω and ζ ∈ Rnwe have
A(x,ζ) =Z 1 0
d
d tA(x, tζ)dt =Z 1 0
a(x, tζ).ζdt
Then using (A1) , we get
A(x,ζ) ≤ c1|ζ| + c2|ζ|p(x) for all x ∈ Ω, ζ ∈ Rn The above inequality implies
A(x, ∇uν) ≤ c1|∇uν| + c2|∇uν|p(x) (5)
By (5), propositions(2.2), (2.4) and the continuous embedding W1,p(x)(Ω) ,→ Lp(x)we infer that³Z
Ω(A(x, ∇uν) d x + 1
p(x)|uν|p(x)) d x
´
ν≥1is bounded .
Then , since M is continuous, up to a subsequence there is t0≥ 0 such that M³Z
Ω(A(x, ∇uν)d x + 1
p(x)|uν|p(x)) d x´
→ M(t0) ≥ m0 asν → ∞
This and (4) imply
ν→∞lim hZ
Ωa(x, ∇uν)∇(uν− u) d x + Z
Ω|uν|p(x)−2uν(uν− u) d x i
= 0
Using, again, the compact embedding W1,p(x)(Ω) ,→ Lp(x), we have
ν→∞lim Z
Ω|uν|p(x)−2uν(uν− u) d x = 0
Thus
ν→∞lim Z
Ωa(x, ∇uν)∇(uν− u) d x = 0.
By [35, theorem 4.1] we get that
uν→ u strongly in W1,p(x)(Ω) as ν → ∞
Since (uν) ⊆ V and V is a closed subspace of W1,p(x)(Ω) , we have u ∈ V , so uν→ u in V .
Let (gν)ν≥1be a sequence of V0such that gν→ g in V0.Let uν= T−1gν, u = T−1g , then Tuν= gν, Tu = g .
By the coercivity of T , we deduce that (uν)ν≥1is bounded in V ;up to a subsequence , we can assume that uν* u in V . Since gn→ g ,
n→+∞lim 〈Tun− Tu, un− u〉 = limn→+∞〈gn− g , un− u〉 = 0.
Since T is of type (S+), un→ u, so T−1is continuous.
Step3 We prove that S is a compact operator.
1.- S is well defined.Indeed,using ( f1) and t ∈ C (Ω) ,for all u, v in V we have
|〈Su, v〉| ≤ Z
Ω| f (x, u)||u|t (x)s(x)|v| d x
≤ C | f (x, u)| α(x)
α(x)−1|v|α(x)≤ C | f (x, u)| α(x)
α(x)−1kvk < ∞ 2.- S is continuous on V .
Let uν→ u in V . Then proposition (2.3) implies that uν→ u in Ls(x)(Ω) and Lα(x)(Ω) So, up to a subsequence we deduce
uν→ u a.e. in Ω (6)
|uν(x)|α(x)≤ k(x) a.e. x ∈ Ω for some k ∈ L1(Ω) (7)
Since t ∈ C (Ω)
|uν|t (x)s(x)→ |u|t (x)s(x) a.e. x ∈ Ω.
Furthermore
f (x, uν) → f (x,u) a.e. x ∈ Ω, Thus , we have
f (x, uν)|uν|t (x)s(x)→ f (x, u)|u|t (x)s(x) a.e. x ∈ Ω.
But, it follows from (f1) and (7) that
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯
α0(x)
≤ C 2(α0)+ h
| f (x, uν)|(α0)++ | f (x, u)|(α0)+ i
≤ C (1 + k(x) + |u|α(x))
Note that C (1 + k(x) + |u|α(.)) ∈ L1(Ω) . Then, applying the Dominated Convergence Theorem , we obtain
ν→∞lim Z
Ω
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯
α0(x)
d x = 0
This implies that
ν→∞lim
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯α0(x)= 0 (8)
Hence by direct computations we get
|〈Suν, v〉 − 〈Su, v〉| ≤ Z
Ω
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯|v | d x
≤ C
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯α0(x)kvk
therefore,from (8)
|Suν− Su| ≤ C
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯α0(x)→ 0 (9)
So, Suν→ Su in V0.
3.- Every bounded sequence (uν)νhas a subsequence (still denoted by (uν)ν) for which (Suν)νconverges.
Let (uν)νbe a bounded sequence of V , there exists a subsequence again denoted by (uν)νand u in V , such that uν* u weakly in W1,p(x)(Ω)
and by the compact embedding W1,p(x)(Ω) ,→ Lα(x)(Ω) , we have uν→ u in Lα(x)(Ω).
Hence,similarly to the proof of (9)we get
|Suν− Su| ≤ C
¯
¯
¯ f (x , uν)|uν|t (x)s(x)− f (x, u)|u|t (x)s(x)
¯
¯
¯α0(x)→ 0 So Suν→ Su.
Step4
k(T − S)(u)k → ∞ as kuk → ∞ for u ∈ V.
In fact, after some computations we get
kTuk ≥ C0kukp−−1 for all u ∈ V with kuk > 1 and
kSuk ≤ C1kukθ+C2 for all u ∈ V, for some θ ∈ [α−− 1, α+− 1]
Combining the above inequalities,we obtain
k(T − S)(u)k ≥ kTuk − kSuk ≥ C0kukp−−1−C01kukα+−1−C2 (10)
Since
t →∞lim(C0tp−−1−C10tα+−1−C2) = ∞
and from (10) we conclude that k(T − S)(u)k → ∞ as kuk → ∞.
Moreover, there exists r0> 1 such that k(T − S)(u)k > 1 for all u ∈ V , with kuk > r0. Step5 Set
W = {u ∈ V : ∃t ∈ [0,1] such that u = tT−1(Su)}
Next,we prove that W is bounded in V .
For u ∈ W \ 0, i.e. u = tT−1(Su) for some t ∈ [0,1] we have kT (u
t)k = kSuk ≤ C1kukθ+C2with t > 0 (11)
Then there exist two constants a, b > 0 such that m0kukp+−1≤ akukα−−1+ b if 0 < kuk < t, m0kukp−−1≤ akukα−−1+ b if t ≤ kuk ≤ 1, m0kukp−−1≤ akukα+−1+ b if 1 < kuk Let g1, g2: [0, 1] → R and g3:]1, ∞[→ R be defined by
g1(t ) = m0tp+−1− atα−−1− b, g2(t ) = m0tp−−1− atα−−1− b, g3(t ) = m0tp−−1− atα+−1− b.
The sets {t ∈ [0,1] : g1(t ) ≤ 0},{t ∈ [0,1] : g2(t ) ≤ 0} and {t ∈]1,∞[: g3(t ) ≤ 0} are bounded in R.
From the above inequalities and (11) we infer that W is bounded in V , so W ⊆ B(0,r1) for some r1> 0
Now, taking r = max{r0, r1} , it follows from [36, theorem 1.8] that dLS(I − tT−1(S), B (0, r ), 0) = 1 for all t ∈ [0,1].
In particular
dLS(I − T−1(S), B (0, r ), 0) = 1
Thus, the couple of nonlinear operators (T, S) satisfies the hypotheses of theorem (2.1) forλ = 1.Then T −S : V → V0is surjective.Therefore, there exists u ∈ V such that
(T − S)u = 0 in V0 This completes the proof.
Acknowledgements
This article is part of the doctoral thesis of the second author and thanks his advisor Dr. E. Cabanillas L. for their constant support
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