# Solutions of Nonlocalp1x,p2x-Laplacian Equations

Authors: Mustafa Avci; Rabil Ayazoglu (Mashiyev)
Journal: International Journal of Partial Differential Equations
 (2013)

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## Abstract

In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involvingp1x,p2x-Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.


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## Body

## 1. Introduction

We study the existence and multiplicity of solutions of the nonlocal equation(P)-M1(∫Ω|∇u|p1(x)dxp1(x))div(|∇u|p1(x)-2∇u)-M2(∫Ω|∇u|p2(x)dxp2(x))div(|∇u|p2(x)-2∇u)=f(x,u)inΩ,u=0on∂Ω,
where Ω⊂ℝN (N≥3) is a smooth bounded domain, pi∈C(Ω¯) such that 2≤pi(x)<N for any x∈Ω¯, and i=1,2.The problem(P) is related to the stationary version of a model, the so-called Kirchhoff equation, introduced by [1]. To be more precise, Kirchhoff established a model given by the equation
(1)ρ∂2u∂t2-(P0h+E2L∫0L|∂u∂x|2dx)∂2u∂x2=0,
where ρ, P0, h, E, and L are constants, which extends the classical D’Alambert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. A distinguish feature of the Kirchhoff equation (1) is that the equation contains a nonlocal coefficient P0/h+(E/2L)∫0L‍|∂u/∂x|2dx which depends on the average (E/2L)∫0L‍|∂u/∂x|2dx of the kinetic energy (1/2)|∂u/∂x|2 on [0,L], and hence the equation is no longer a pointwise identity. For Kirchhoff-type equations involving the p(x)-Laplacian operator, see, for example, [2–4].Thep(x)-Laplacian operator -Δp(x)u:=div(|∇u|p(x)-2∇u) is a natural generalization of the p-Laplacian operator -Δpu:=-div(|∇u|p-2∇u) where p>1 is a real constant. The main difference between them is that p-Laplacian operator is (p-1) homogenous, that is, Δp(μu)=μp-1Δpu for every μ>0, but the p(x)-Laplacian operator, when p(x) is not constant, is not homogeneous. This causes many problems; some classical theories and methods, such as the Lagrange multiplier theorem and the theory of Sobolev spaces, are not applicable. For p(x)-Laplacian operator, we refer the readers to [5–9] and references there in. Moreover, the nonlinear problems involving the p(x)-Laplacian operator are extremely attractive because they can be used to model dynamical phenomenons which arise from the study of electrorheological fluids or elastic mechanics. Problems with variable exponent growth conditions also appear in the modelling of stationary thermorheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes of filtration of an ideal barotropic gas through a porous medium. The detailed application backgrounds of the p(x)-Laplacian can be found in [10–14] and the references therein.In the present paper, by considering the joint effects of different (p1(x),p2(x))-Laplace operator -Δp1(x)u-Δp2(x)u, we study the existence and multiplicity of solutions for a nonlocal problem, that is, problem (P) via Mountain-Pass theorem and Fountain theorem. As far as we know, there is no paper that deals with a nonlocal problem involving (p1(x),p2(x))-Laplace operator except [15] in which the authors consider problem (P) for the case M1≡1 and M2≡1. Therefore, our paper deals with more general results than those obtained in [15]. Moreover, if we choose the functions M1≡M2≡1 in problem (P), we get the equation
(2)-[div(|∇u|p1(x)-2∇u)+div(|∇u|p2(x)-2∇u)]=f(x,u),
which is the well-known anisotropic p→(·)-Laplacian problem (see, e.g., [16] and references therein) in the case N=1,2, that is,
(3)-∑i=1N∂xi(|∂xiu|pi(x)-2∂xiu)=f(x,u).
As mentioned above, the p(x)-Laplacian can be applied to describe the physical phenomenon with pointwise different properties which earliest arose from the nonlinear elasticity theory. In that context, the systems involving the (p1(x),p2(x))-Laplacian (or (p1(x),…,pN(x))-Laplacian) can be good candidates for modeling phenomena which ask for distinct behavior of partial differential derivatives in various directions. For a mathematical model of a real physical phenomenon, one can consider the mean curvature operator
(4)-∑i=12∂xi[(1+|∂xiu|2)(pi(x)-2)/2∂xiu]=f(x,u).
It is obvious that problem (P) is a degenerate version of (4) when M1≡M2≡1.


## 2. Preliminaries

We state some basic properties of the variable exponent Lebesgue-Sobolev spacesLp(x)(Ω) and W1,p(x)(Ω), where Ω⊂ℝN is a bounded domain (for details, see, e.g., [17–19]).Set(5)C+(Ω¯)={p:p∈C(Ω¯),p(x)>1foranyx∈Ω¯}.
For any p∈C+(Ω¯), denote
(6)p-:=infx∈Ω¯p(x),p+:=supx∈Ω¯p(x)<∞,
and define the variable exponent Lebesgue space by
(7)Lp(x)(Ω)={∫Ω|u(x)|p(x)dx<∞uisameasurablerealfunctiononΩ:∫Ω|u(x)|p(x)dx<∞}.
We define a norm, the so-called Luxemburg norm, on Lp(x)(Ω) by the formula
(8)|u|p(x)=inf{λ>0:∫Ω|u(x)λ|p(x)dx≤1},
and then (Lp(x)(Ω),|·|p(x)) becomes a Banach space.Define the variable exponent Sobolev space by(9)W1,p(x)(Ω)={u∈Lp(x)(Ω);|∇u|∈Lp(x)(Ω)},
then it can be equipped with the norm
(10)∥u∥1,p(x)=|u|p(x)+|∇u|p(x),∀u∈W1,p(x)(Ω).
The space W01,p(x)(Ω) is defined as the closure of C0∞(Ω) in W1,p(x)(Ω) with respect to the norm ∥u∥1,p(x). For u∈W01,p(x)(Ω), we can define an equivalent norm
(11)∥u∥=|∇u|p(x),
since Poincaré inequality holds; that is, there exists a positive constant C>0 such that
(12)|u|p(x)≤C|∇u|p(x),
for all u∈W01,p(x)(Ω).Proposition 1 (see [18, 19]).
The conjugate space ofLp(x)(Ω) is Lp′(x)(Ω), where (1/p′(x))+(1/p(x))=1. For any u∈Lp(x)(Ω) and v∈Lp′(x)(Ω), we have
(13)|∫Ωuvdx|≤(1p-+1(p-)′)|u|p(x)|v|p′(x).Proposition 2 (see [18, 19]).
Denote
ρ
(
u
)
=
∫
Ω
|
u
(
x
)
|
p
(
x
)
d
x
, for all u,un∈Lp(x)(Ω)(n=1,2,…); one has(i)
|u|p(x)>1(≤1)⇒|u|p(x)p-≤ρ(u)≤|u|p(x)p+(|u|p(x)p+≤ρ(u)≤|u|p(x)p-);(ii)
limn→∞|un|p(x)=0⇔limn→∞ρ(un)=0;limn→∞|un|p(x)→∞⇔limn→∞ρ(un)→∞.Proposition 3 (see [18, 19]).
If
u
,
u
n
∈
L
p
(
x
)
(
Ω
)
(
n
=
1,2
,
…
), then the following statements are equivalent:(i)
limn→∞|un-u|p(x)=0;(ii)
limn→∞ρ(un-u)=0;(iii)
un→u in measure Ω and limn→∞ρ(un)=ρ(u).Proposition 4 (see [18, 19]).
(
i
) If 1<p-≤p+<∞, then spaces Lp(x)(Ω), W1,p(x)(Ω), and W01,p(x)(Ω) are separable and reflexive Banach spaces.
(
ii
) If q∈C+(Ω¯) and q(x)<p*(x) for any x∈Ω¯(p*(x)=Np(x)/N-p(x) if p(x)<N and p*(x)=+∞ if p(x)≥N), then the embedding W1,p(x)(Ω)↪Lq(x)(Ω) is compact and continuous.Definition 5.
LetX be a Banach space and J:X→ℝ a C1 functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short), if any sequence {un} in X such that {J(un)} is bounded and J′(un)→0 as n→∞ admits a convergent subsequence.


## 3. Main Results and Proofs

Let us consider the functional(14)J(u)=∫Ω|∇u|p1(x)p1(x)dx+∫Ω|∇u|p2(x)p2(x)dx,∀u∈X,
where X:=W01,p1(x)(Ω)∩W01,p2(x)(Ω) with its norm given by ∥u∥:=∥u∥1,p(x)+∥u∥2,p(x), for all u∈X. It is obvious that (X,∥·∥) is also a separable and reflexive Banach space.By using standard arguments, it can be proved thatJ∈C1(X,ℝ) (see [20]), and the (p1(x),p2(x))-Laplace operator is the derivative operator of J in the weak sense. Denote L:=J′:X→X*; then
(15)〈L(u),φ〉=∫Ω|∇u|p1(x)-2∇u∇φdx+∫Ω|∇u|p2(x)-2∇u∇φdx,∀u,φ∈X,
where 〈·,·〉 is the dual pair between X and its dual X*.Let us denote(16)pM(x)=max{p1(x),p2(x)},pm(x)=min{p1(x),p2(x)},∀x∈Ω¯.
By the definition, it is not difficult to see that pM(x), pm(x)∈C+(Ω¯). For q(x)∈C+(Ω¯) such that q(x)<pM*(x) for any x∈Ω¯, we have X:=W01,p1(x)(Ω)∩W01,p2(x)(Ω)=W01,pM(x)(Ω)↪Lq(x)(Ω), and the imbedding is continuous and compact.We say thatu∈X is a weak solution of (P) if
(17)M1(∫Ω|∇u|p1(x)p1(x)dx)∫Ω|∇u|p1(x)-2∇u∇φdx+M2(∫Ω|∇u|p2(x)p2(x)dx)∫Ω|∇u|p2(x)-2∇u∇φdx=∫Ωf(x,u)φdx,
for any φ∈X.We associate to the problem(P) the energy functional, defined as I:X→ℝ:
(18)I(u)=M^1(∫Ω|∇u|p1(x)p1(x)dx)+M^2(∫Ω|∇u|p2(x)p2(x)dx)-∫ΩF(x,u)dx,
where M^i(t)=∫0tMi(ξ)dξ (i=1,2.) and F(x,u)=∫0uf(x,s)ds. We know that from (M0) and (f0) (see below) I is well defined and in a standard way we can prove that I∈C1(X,ℝ) and that the critical points of I are solutions of (P).Moreover, the derivative ofI is given by
(19)〈I′(u),φ〉=M1(∫Ω|∇u|p1(x)p1(x)dx)∫Ω|∇u|p1(x)-2∇u∇φdx+M2(∫Ω|∇u|p2(x)p2(x)dx)∫Ω|∇u|p2(x)-2∇u∇φdx-∫Ωf(x,u)φdx,
for any u,φ∈X.Now, we are ready to set and prove the first main result of the present paper.Theorem 6.
Assume that the following assumptions hold:(M0)
M1,M2:ℝ+→ℝ+ are continuous functions and satisfy the conditions
(20)C1tα-1≤M1(t),C2tα-1≤M2(t),for allt>0, where C1 and C2 are positive constants and α>1;(f0)
f:Ω¯×ℝ→ℝ is a Carathéodory function and satisfies the growth condition
(21)|f(x,t)|≤C3+C4|t|q(x)-1,∀(x,t)∈Ω¯×ℝ,whereC3 and C4 are positive constants and p1,p2,q∈C+(Ω¯) such that q+<αpm-<pM*(x), for all x∈Ω¯.
Then problem(P) has a weak solution.Proof.
Let∥u∥>1. By the assumptions (M0) and (f0), we have
(22)I(u)=M^1(∫Ω|∇u|p1(x)p1(x)dx)+M^2(∫Ω|∇u|p2(x)p2(x)dx)-∫ΩF(x,u)dx≥C1∫0(1/p1+)ρ1(∇u)tα-1dt+C2∫0(1/p2+)ρ2(∇u)tα-1dt-C3∫Ω|u|q(x)dx-C4=C1α(p1+)α(∫Ω|∇u|p1(x)dx)α+C2α(p2+)α(∫Ω|∇u|p2(x)dx)α-C3∫Ω|u|q(x)dx-C4≥C1α(p1+)α∥u∥αp1-+C2α(p2+)α∥u∥αp2--C3∥u∥q+-C4≥cα(pM+)α∥u∥αpm--C3∥u∥q+-C4⟶+∞(∥u∥⟶∞),
where ρ1(∇u)=∫Ω|∇u|p1(x)dx, ρ2(∇u)=∫Ω|∇u|p2(x)dx and c=min{C1,C2}. So, I is coercive. Since I is sequentially weakly lower semicontinuous, I has a minimum point u in X and u is a weak solution of (P).Theorem 7.
Assume that the following assumptions hold:(M1)
M1,M2:ℝ+→ℝ+ are continuous functions and satisfy the conditions
(23)C5tα-1≤M1(t)≤C6tα-1,C7tα-1≤M2(t)≤C8tα-1,for allt>0, where C5, C6, C7, C8, and α are positive constants such that C5≤C6≤C7≤C8 and α>1;(f1)
f:Ω¯×ℝ→ℝ is a Carathéodory function and satisfies the growth condition
(24)|f(x,t)|≤C9+C10|t|β(x)-1,∀(x,t)∈Ω¯×ℝ;(f2)
f(x,t)=o(|t|αpM+-1), t→0  uniformly  for  x∈Ω¯,whereC9 and C10 are positive constants and β∈C+(Ω¯) such that αpM+<β-≤β+<pM*(x), for all x∈Ω¯;(f3)
f(x,-t)=-f(x,t),∀(x,t)∈Ω¯×ℝ;(AR)
∃K>0, θ>(pM+)αsuch that(25)0<θF(x,t)≤f(x,t)t,|t|≥Ka.e.x∈Ω¯.Then problem(P) has at least one nontrivial weak solution.To obtain the result of Theorem7, we need to show Lemmas 8 and 9 hold.Lemma 8.
Suppose(M1), (f1), and (AR) hold. Then I satisfies (PS) condition.Proof.
Let us assume that there exists a sequence{un} in X such that
(26)|I(un)|≤c,I′(un)⟶0.
Then by the assumptions (26), (M1), and (AR), we get
(27)c+∥un∥≥I(un)-1θ〈I′(un),un〉≥(C5α(p1+)α-C6θα(p1-)α-1)(∫Ω|∇un|p1(x)dx)α+(C7α(p2+)α-C8θα(p2-)α-1)(∫Ω|∇un|p2(x)dx)α+∫Ω(1θf(x,un)un-F(x,un))dx≥λ∥un∥αpm-+c,
where λ=((C5/α(p1+)α)-(C6/θα(p1-)α-1))+((C7/α(p2+)α)-(C8/θα(p2-)α-1)). Since θ>(pM+)α, we have λ>0 for θ large enough. Therefore, {∥un∥} is bounded in X. Passing to a subsequence, if necessary, there exists u∈X such that un⇀u. Therefore, we have the embeddings
(28)un⇀uinX,un⟶uinLβ(x)(Ω),un⟶ua.e.inΩ.
By (26), we have 〈I′(un),un-u〉→0. Thus
(29)〈I′(un),un-u〉=M1(∫Ω|∇un|p1(x)p1(x)dx)×∫Ω|∇un|p1(x)-2∇un(∇un-∇u)dx+M2(∫Ω|∇un|p2(x)p2(x)dx)×∫Ω|∇un|p2(x)-2∇un(∇un-∇u)dx-∫Ωf(x,un)(un-u)dx.
From (f1) and Proposition 1, it follows that
(30)|∫Ωf(x,un)(un-u)dx|≤C10|∫Ω|un|β(x)-2un(un-u)dx|+C9|∫Ω(un-u)dx|≤C10||un|β(x)-1|β(x)/(β(x)-1)×|un-u|β(x)+C9∫Ω|un-u|dx.
If we consider the relations given in (28), we get
(31)∫Ωf(x,un)(un-u)dx⟶0.
Hence,
(32)M1(∫Ω|∇un|p1(x)p1(x)dx)×∫Ω|∇un|p1(x)-2∇un(∇un-∇u)dx+M2(∫Ω|∇un|p2(x)p2(x)dx)×∫Ω|∇un|p2(x)-2∇un(∇un-∇u)dx⟶0.
From (M1), it follows that
(33)∑i=12∫Ω|∇un|pi(x)-2∇un(∇un-∇u)dx⟶0.
Furthermore, since un⇀u in X, we have
(34)∑i=12∫Ω|∇u|pi(x)-2∇u(∇un-∇u)dx⟶0.
From (33) and (34), we deduce that
(35)∑i=12∫Ω(|∇un|pi(x)-2∇un-|∇u|pi(x)-2∇u)×(∇un-∇u)dx⟶0.
Next, we apply the following well-known inequality
(36)(|ξi|ri-2ξi-|ψi|ri-2ψi)·(ξi-ψi)≥2-ri|ξi-ψi|ri,ξi,ψi∈ℝN,
valid for all ri≥2 (i=1,2). From the relations (35) and (36), we infer that
(37)∑i=12∫Ω|∇un-∇u|pi(x)dx⟶0,
and, consequently, un→u in X. We are done.Lemma 9.
Suppose(M1),(f1),(f2), and (AR) hold. Then the following statements hold: (i)
there exist two positive real numbersγ and a such that I(u)≥a>0, u∈X with ∥u∥=γ;(ii)
there existsu∈X such that ∥u∥>γ, I(u)<0.Proof.
(i) Let∥u∥<1. Then by (M1) and Proposition 2, we have
(38)I(u)≥C5α(p1+)α∥u∥αp1++C7α(p2+)α∥u∥αp2+-∫ΩF(x,u)dx≥c*α(pM+)α∥u∥αpM+-∫ΩF(x,u)dx,
where c*=min{C5,C7}. Since αpM+<β-≤β+<pM*(x) for all x∈Ω¯, we have the continuous embeddings X↪LαpM+(Ω) and X↪Lβ+(Ω)↪Lβ-(Ω), and also there are positive constants C11,C12 and C13 such that
(39)|u|αpM+≤C11∥u∥,|u|β-≤C12∥u∥,|u|β+≤C13∥u∥,∀u∈X.
Let ε>0 be small enough such that εC11αpM+<(c*/2α(pM+)α). By the assumptions (f1) and (f2), we have F(x,t)≤ε|t|αpM++Cε|t|β(x), for all (x,t)∈Ω¯×ℝ.
Then, for∥u∥≤1 it follows that
(40)I(u)≥c*α(pM+)α∥u∥αpM+-ε∫Ω|u|αpM+dx-Cε∫Ω|u|β(x)dx≥c*α(pM+)α∥u∥αpM+-εC11αpM+∥u∥αpM+-CεC12β-∥u∥β-.
Therefore, there exists two positive real numbers γ and a such that I(u)≥a>0, for all u∈X  with  ∥u∥=γ.
(ii) From(AR) it follows that F(x,t)≥c|t|θ, for all x∈Ω¯ and |t|≥K. In the other hand, when |t|≥K from (M1) we obtain that
(41)M^1(t)≤C6αtα≤C6αtαpM+,M^2(t)≤C8αtα≤C8αtαpM+.
Hence, for any fixed ω∈X∖{0} and t>1 we have
(42)I(tω)≤C6α(∫Ω|∇tω|p1(x)p1(x)dx)αpM++C8α(∫Ω|∇tω|p2(x)p2(x)dx)αpM+-∫ΩF(x,tω)dx≤C6αtαpM+(∫Ω|∇ω|p1(x)p1(x)dx)αpM++C8αtαpM+(∫Ω|∇ω|p2(x)p2(x)dx)αpM+-tθ∫ΩF(x,ω)dx,
which implies I(tω)→-∞(t→+∞).Proof of Theorem7.
From Lemmas8 and 9 and the fact that I(0)=0, I satisfies the Mountain-Pass theorem (see [20, 21]). Therefore, I has at least one nontrivial critical point; that is, (P) has a nontrivial weak solution. The proof is complete.In the following, we will prove the second main result of the present paper.Theorem 10.
Suppose(M1),(AR), (f1), (f2), and (f3) hold. Then I has a sequence of critical points {un} such that I(un)→+∞ and (P) has infinite many pairs of solutions.SinceX is a reflexive and separable Banach space, then there are {ei}⊂X and {ei*}⊂X* such that
(43)X=span{ei:i=1,2,…}¯,X*=span{ei*:i=1,2,…}¯,〈ei,ej*〉={1,i=j,0,i≠j.
For convenience, we write Xi=span{ei}, Yk=⨁i=1kXi, and Zk=⨁i=k∞Xi.Lemma 11.
Ifβ∈C+(Ω¯) such that β(x)<pM*(x) for any x∈Ω¯, denote
(44)δk:=sup{|u|β(x):∥u∥=1,u∈Zk}.
Then limk→∞δk=0.Since the proof of Lemma11 is similar to that of Lemma 4.9 in [7], we omit it.Proof of Theorem10.
By the assumptions(M1), (AR), and (f1), I satisfies (PS) condition and from (f3) it is also an even functional. In the sequel, we will show that if k is large enough, then there exist ρk>rk>0 such that(i)
bk:=infu∈Zk,∥u∥=rkI(u)→+∞(k→∞);(ii)
ak:=maxu∈Yk,∥u∥=ρkI(u)≤0.
Therefore, to obtain the results of Theorem10 it is enough to apply Fountain theorem (see [21]).(i)
For anyu∈Zk with ∥u∥ big enough, we have
(45)I(u)≥C5α(p1+)α∥u∥αp1-+C7α(p2+)α∥u∥αp2--C14∫Ω|u|β(x)dx-c1≥c*α(pM+)α∥u∥αpm--C14|u|β(x)β(ξ)-c2,whereξ∈Ω.I(u)≥c*α(pM+)α∥u∥αpm--C14-c2,if|u|β(x)≤1,I(u)≥c*α(pM+)α∥u∥αpm--C14δkβ+∥u∥β+-c2,if|u|β(x)>1,I(u)≥c*α(pM+)α∥u∥αpm--C14δkβ+∥u∥β+-c3=c*α(1(pM+)α∥u∥αpm--C15δkβ+∥u∥β+)-c3.Set ∥u∥=rk=(C15β+δkβ+)1/(αpm--β+). Because δk→0 and αpm-≤αpM+<β+, we have
(46)I(u)≥c*α(1(pM+)α∥u∥αpm--C15δkβ+∥u∥β+)-c3=c*α(1(pM+)α(C15β+δkβ+)αpm-/(αpm--β+)-C15δkβ+(C15β+δkβ+)β+/(αpm--β+)1(pM+)α(C15β+δkβ+)αpm-/(αpm--β+))-c3=c*α(1(pM+)α-1(β+)α)×(C15β+δkβ+)αpm-/(αpm--β+)-c3⟶+∞(k⟶∞).(ii)
From(AR), we have F(x,t)≥C16|t|θ-C17. Because θ>(pM+)α and dimYk=k, it is obvious that I(u)→-∞ as ∥u∥→∞ for u∈Yk.



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