# Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces

Authors: Bilal Cekic; Rabil A. Mashiyev
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

In this paper, by means of adequate variational techniques and the theory of the variable
exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations ofp(x)-Kirchhoff type with nonstandard growth condition.


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## Body

## 1. Introduction

LetΩ be smooth bounded domain of ℝn,n≥2, and let Ω1⊂Ω be a subdomain with smooth boundary Σ satisfying Ω¯1⊂Ω. Writing Γ=∂Ω and Ω2=Ω∖Ω¯1 we have Ω=Ω¯1∪Ω2 and ∂Ω2=Σ∪Γ (Figure 1).Figure 1We are concerned with the existence of positive solutions to the following system of nonlinear elliptic equations:(P)-M(∫Ω11p(x)|∇u|p(x)dx)div⁡(|∇u|p(x)-2∇u)=f(x,u),inΩ1,-N(∫Ω21p(x)|∇υ|p(x)dx)div⁡(|∇υ|p(x)-2∇υ)=g(x,υ),inΩ2,υ=0,onΓ,
with the transmission conditions(1.1)M(∫Ω1|∇u|p(x)dx)∂u∂η=N(∫Ω2|∇υ|p(x)dx)∂υ∂η,onΣu=υ,onΣ,
where M  and N are positive continuous functions, η is outward normal to Ω2 and is inward to Ω1, and p(x)∈C(Ω¯). By means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations of p(x)-Kirchhoff type with nonstandard growth condition. We investigate the problem (P) when f(x,u)=λ1|u|q(x)-2u, g(x,υ)=λ2|υ|q(x)-2υ, where λ1,λ2>0 and p(x), q(x)∈C(Ω¯) such that 1<q(x)<p*(x) for all x∈Ω¯, where p*(x)=np(x)/n-p(x) if p(x)<n or p*(x)=+∞ if p(x)≥n.The operator −div(|∇u|p(x)-2∇u) is called the p(x)-Laplacian which is a natural generalization of the p-Laplacian when p(x)≡p>1 (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1, 2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [3].Transmission problems arise in several applications in physics and biology. For instance, one of the important problems of the electrodynamics of solid media is the electromagnetic process research in ferromagnetic media with different dielectric constants. These problems appear as well as in solid mechanics if a body consists of composite materials. The existence and regularity results for linear transmission problems are well known, and a complete study can be found in [4]. We refer the reader to [5] for nonlinear elliptic transmission problems, to [6] for a nonlinear nonlocal elliptic transmission problem. Furthermore, uniqueness and regularity of the solutions to the thermoelastic transmission problem were investigated in [7].We note that problem (P) with the transmission condition is a generalization of the stationary problem of two wave equations of Kirchhoff type,(1.2)utt-M(∫Ω1|∇u|2dx)Δ2u=f(x,u)inΩ1,υtt-N(∫Ω2|∇υ|2dx)Δ2υ=g(x,υ)inΩ2,
which models the transverse vibrations of the membrane composed by two different materials in Ω1 and Ω2. Controllability and stabilization of transmission problems for the wave equations can be found in [8, 9]. We refer the reader to [10] for the stationary problems of Kirchhoff type, to [11] for elliptic equation p-Kirchhoff type, and to [12, 13] for p(x)-Kirchhoff type equation.


## 2. Auxiliary Results

We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spacesLp(x)(Ω), W1,p(x)(Ω), and W01,p(x)(Ω). In that context we refer to [14, 15] for the fundamental properties of these spaces.ByΩ we always denote a nonempty open subset of ℝn. Set(2.1)C+(Ω̅)={h;h∈C(Ω̅),h(x)>1∀x∈Ω̅}.
For any h∈C+(Ω̅) we define(2.2)h1:=inf⁡x∈Ωh(x),h2:=sup⁡x∈Ωh(x).
We define the variable exponent Lebesgue spaceLp(·)(Ω) to consist of all measurable functions u:Ω→ℝ for which the modular(2.3)ρp(x),Ω(u)=∫Ω|u(x)|p(x)dx
is finite. We define the Luxemburg normon this space by the formula(2.4)‖u‖p(x),Ω=inf⁡{δ>0:ρp(x),Ω(uδ)≤1}.
Equipped with this norm, Lp(·)(Ω) is a separable and reflexive Banach space. Define the variable exponent Sobolev spaceW1,p(x)(Ω) by(2.5)W1,p(x)(Ω)={u∈Lp(x)(Ω);|∇u|∈Lp(x)(Ω)},
and the norm(2.6)‖u‖1,p(x),Ω=‖u‖p(x),Ω+‖∇u‖p(x),Ω,∀u∈W1,p(x)(Ω)
makes W1,p(x)(Ω) a separable and reflexive Banach space. The space W01,p(x)(Ω) is denoted by the closure of C0∞(Ω) in W1,p(x)(Ω). W01,p(x)(Ω) is a separable and reflexive Banach space.Proposition 2.1 (see [14, 15]).
Letp∈C+(Ω). Then conjugate space of Lp(x)(Ω) is  Lq(x)(Ω), where 1/p(x)+1/q(x)=1. For any u∈Lp(x)(Ω) and υ∈Lq(x)(Ω), one has
(2.7)|∫Ωuυdx|≤2‖u‖p(x),Ω‖υ‖q(x),Ω.The next proposition illuminates the close relation between the∥·∥p(x),Ω and the convex modular ρp(x),Ω.Proposition 2.2 (see [14, 15]).
Ifu∈Lp(x)(Ω), then one has(i)
∥u∥p(x),Ω<1(=1;>1)⇔ρp(x),Ω(u)<1(=1;>1),(ii)
∥u∥p(x),Ω>1⇒∥u∥p(x),Ωp1≤ρp(x),Ω(u)≤∥u∥p(x),Ωp2,(iii)
∥u∥p(x),Ω<1⇒∥u∥p(x),Ωp2≤ρp(x),Ω(u)≤∥u∥p(x),Ωp1,(iv)
∥u∥p(x),Ω=a>0⇔ρp(x),Ω(u/a)=1.Proposition 2.3 (see [14, 15]).
Ifu,un∈Lp(x)(Ω), n=1,2,…, then the following statements are equivalent to each other:(i)
limn→∞∥un-u∥p(x),Ω=0;(ii)
limn→∞ρp(x),Ω(un-u)=0;(iii)
un→u in measure in Ω and limn→∞ρp(x),Ω(un)=ρp(x),Ω(u).Proposition 2.4 (see [14]).
InW01,p(x)(Ω) the Poincaré inequality holds, that is, there exists a positive constant C such that
(2.8)‖u‖p(x),Ω≤C‖∇u‖p(x),Ω,∀u∈W01,p(x)(Ω).
Consequently,∥u∥1,p(x),Ω=∥|∇u|∥p(x),Ω are equivalent norms on W01,p(x)(Ω).  In what follows, W01,p(x)(Ω), with p∈C+(Ω¯), will be considered as endowed with the norm ∥u∥1,p(x).  We will use ∥u∥1,p(x),Ω=∥∇u∥p(x),Ω for u∈W01,p(x)(Ω) in the following discussions.Proposition 2.5 (see [14, 16]).
Assume thatΩ is bounded, the boundary of Ω possesses the cone property and p∈C+(Ω¯). If q∈C+(Ω¯) and q(x)<p*(x) for allx∈Ω¯ then the embedding W1,p(x)(Ω)↪Lq(x)(Ω) is the compact and continuous.Lemma 2.6 (see [17]).
Assume thatΩ is bounded and has a Lipschitz boundary with the cone property and p∈C+(Ω¯). Then there is a continuous boundary trace embedding W1,p(·)(Ω)↪Lp(·)(∂Ω).Lemma 2.7 (see [17]).
For anyu∈W1,p(x)(Ω), let
(2.9)‖u‖∂:=‖∇u‖p(x),Ω+‖u‖p(x),∂Ω.
Then, ∥u∥∂ is a norm on W1,p(x)(Ω), equivalent to the standard norm of W1,p(x)(Ω).Our analysis is based on the Sobolev space(2.10)E={(u,υ)∈W1,p(x)(Ω1)×WΓ1,p(x)(Ω2):u=υonΣ},
where(2.11)WΓ1,p(x)(Ω2)={υ∈W1,p(x)(Ω2):υ=0onΓ}.
Then, we have following lemma that will permit the variational setting of the problem (P).Lemma 2.8.
E is a closed subspace of W1,p(x)(Ω1)×W1,p(x)(Ω2), and
(2.12)‖(u,υ)‖E=‖∇u‖p(x),Ω1+‖∇υ‖p(x),Ω2
defines a norm in E, equivalent to the standard norm of W1,p(x)(Ω1)×W1,p(x)(Ω2).Proof.
It is clear that (2.12) defines a seminorm. Then suppose that ∥(u,υ)∥E=0. Since ∥∇υ∥p(x),Ω2 defines a norm on W1,p(x)(Ω2) and Poincaré inequality holds in W1,p(x)(Ω2), thus υ=0. From transmission condition, u=υ on Σ, and therefore u∈W01,p(x)(Ω1). This shows that (u,υ)=0, and hence (2.12) defines a norm in E. Now, applying the trace theorem in W1,p(x)(Ω2), there exists c>0 such that for all υ∈WΓ1,p(x)(Ω2)(2.13)‖υ‖p(x),Σ≤C‖∇υ‖p(x),Ω2.
But it is known that ∥∇u∥p(x),Ω1+∥u∥p(x),Σ defines an equivalent norm in W1,p(·)(Ω1) from Lemma 2.7. Then combining these two remarks the result follows.


## 3. Main Results

Let us precisely describe our assumptions in order to establish the main result. The energy functional corresponding to problem (P) is defined as J:E→ℝ,(3.1)J(u,υ)=M̂(∫Ω1|∇u|p(x)p(x)dx)+N̂(∫Ω2|∇υ|p(x)p(x)dx)-∫Ω1F(x,u)dx-∫Ω2G(x,υ)dx,
where M̂(t)=∫0tM(ξ)dξ, and F(x,u)=∫0uf(x,ξ)dξ. It is not difficult to show that J∈C1(E,ℝ), and as a matter of fact, J is of class C1 and weakly lower semicontinuous. In particular we have, for all (u,v), (φ,ψ)∈E,(3.2)〈J′(u,υ),(φ,ψ)〉=M(∫Ω1|∇u|p(x)p(x)dx)∫Ω1|∇u|p(x)-2∇u∇φdx+N(∫Ω2|∇υ|p(x)p(x)dx)∫Ω2|∇υ|p(x)-2∇υ∇ψdx-∫Ω1f(x,u)φdx-∫Ω2g(x,υ)ψdx.We assume the following hypotheses forM and N.There are positive constantsM1, M2, N1, N2, and α such that(E1):
M1tα-1≤M(t)≤M2tα-1,(E2):
N1sα-1≤N(s)≤N2sα-1for allt,s>0 and α≥1.Now we state our main result.Theorem 3.1.
Let us assume that(E1) and (E2) hold. If q2<αp1, then the problem (P) with the transmission condition has at least one nonnegative solution.Lemma 3.2.
There existsλ*>0 such that for any λ1+λ2∈(0,λ*) there exist ρ>0 and r>0 such that
(3.3)Jλ(u,υ)≥r,∀(u,υ)∈E,with‖(u,υ)‖E=ρ.Proof.
Sinceq(x)<p*(x) for all x∈Ω¯ it follows that (Proposition 2.5)
(3.4)‖u‖q(x),Ω1≤C1‖u‖1,p(x),Ω1,∀u∈W1,p(x)(Ω1),‖υ‖q(x),Ω2≤C2‖υ‖1,p(x),Ω2,∀υ∈W1,p(x)(Ω2).
By Lemma 2.8 we have
(3.5)‖u‖q(x),Ω1+‖υ‖q(x),Ω2≤C1‖u‖1,p(x),Ω1+C2‖υ‖1,p(x),Ω2≤C3‖(u,υ)‖E.
We fix ρ∈(0,1) such that ρ<1/C3. Then the above relation implies
(3.6)‖u‖q(x),Ω1+‖υ‖q(x),Ω2<1,∀(u,υ)∈E.
By Proposition 2.2(ii) and 2.5 and Lemma 2.8, we deduce that
(3.7)∫Ω1|u|q(x)dx+∫Ω2|v|q(x)dx≤‖u‖q(x),Ω1q1+‖v‖q(x),Ω2q1≤‖u‖q(x),Ω1+‖v‖q(x),Ω2≤C4(‖u‖1,p(x),Ω1+‖v‖1,p(x),Ω2)(3.8)≤C5‖(u,v)‖E.
Using (E1), (E2), Proposition 2.2(iii), and (3.8), we obtain that for any (u,υ)∈E with ∥(u,υ)∥E<1 the following inequalities hold true:
(3.9)Jλ(u,υ)=M̂(∫Ω1|∇u|p(x)p(x)dx)+N̂(∫Ω2|∇υ|p(x)p(x)dx)∫Ω2|∇υ|p(x)dx-λ1∫Ω1|u|q(x)q(x)dx-λ2∫Ω2|υ|q(x)q(x)dx≥M1∫0(1/p2)ρp(x),Ω1(∇u)tα-1dt+N1∫0(1/p2)ρp(x),Ω2(∇u)ηα-1dη-λ1q1∫Ω1|u|q(x)dx-λ2q1∫Ω2|υ|q(x)dx≥M1α(1p2∫Ω1|∇u|p(x)dx)α+N1α(1p2∫Ω2|∇υ|p(x)dx)α-λ1q1∫Ω1|u|q(x)dx-λ2q1∫Ω2|υ|q(x)dx≥min⁡{M1,N1}α(p2)α(‖∇u‖p(x),Ω1αp2+‖∇υ‖p(x),Ω2αp2)-C5(λ1+λ2)q1‖(u,υ)‖E≥21-αp2min⁡{M1,N1}α(p2)α(‖∇u‖p(x),Ω1+‖∇υ‖p(x),Ω2)αp2-C5(λ1+λ2)q1‖(u,υ)‖E≥21-αp2min⁡{M1,N1}α(p2)α‖(u,υ)‖Eαp2-C5(λ1+λ2)q1‖(u,υ)‖E.
By the above inequality if we define
(3.10)λ*=21-αp2ραp2-1q1min⁡{M1,N1}C5α(p2)α,
then for any λ1+λ2∈(0,λ*) and (u,υ)∈E there exists r>0 such that Jλ(u,υ)≥r>0.
The proof of Lemma3.2 is complete.Lemma 3.3.
There existsφ,ϕ∈E such that φ,ϕ≥0 and φ,ϕ≠0 and Jλ(tφ,tϕ)<0 for t>0 small enough.Proof.
Let|Ω1|>0 and |Ω2|>0. Moreover, let choose φ,ϕ∈C0∞(Ω) and 0≤φ(x)≤1 in Ω1,  0≤ϕ(x)≤1 in Ω2. Then, for any t∈(0,1), by (E1) and (E2) it follows
(3.11)Jλ(tφ,tϕ)=M̂(∫Ω1|t∇φ|p(x)p(x)dx)+N̂(∫Ω2|t∇ϕ|p(x)p(x)dx)-λ1∫Ω1|tφ|q(x)q(x)dx-λ2∫Ω2|tϕ|q(x)q(x)dx≤M2∫0(1/p1)ρp(x),Ω1(t∇φ)sα-1ds+N2∫0(1/p1)ρp(x),Ω2(t∇ϕ)ηα-1dη-λ1∫Ω1|tφ|q(x)q(x)dx-λ2∫Ω2|tϕ|q(x)q(x)dx≤M2tαp1α(p1)αρp(x),Ω1α(∇φ)+N2tαp1α(p1)αρp(x),Ω2α(∇ϕ)-λ1tq2q2ρq(x),Ω1(φ)-λ2tq2q2ρq(x),Ω2(ϕ).
Let
(3.12)Rα=M2ρp(x),Ω1α(∇φ)+N2ρp(x),Ω2α(∇ϕ),Sλ1,λ2=λ1ρq(x),Ω1(φ)+λ2ρq(x),Ω2(ϕ).
Then
(3.13)Jλ(tφ,tϕ)≤tαp1α(p1)αRα-tq2q2Sλ1,λ2.
Therefore, we conclude
(3.14)Jλ(tφ,tϕ)<0,
for 0<t<σ1/αp1-q2 providing that
(3.15)0<σ<min{1,α(p1)αSλ1,λ2q2Rα}.
The proof of Lemma 3.3 is complete.Proof of Theorem3.1.
From Lemma3.3, we infer that there exists a ball centered at the origin Bρ(0)¯⊂E, such that
(3.16)inf⁡∂Bρ(0)Jλ>0.
Furthermore, by Lemma 3.3, we know that there exists (φ,ω)∈E such that Jλ(tφ,tω)<0 for t>0 small enough. Therefore, considering also inequality (3.14), we obtain that
(3.17)-∞<c:=inf⁡Bρ(0)Jλ<0.
Let us choose ɛ>0. Then, it follows that
(3.18)0<ɛ≤inf⁡∂Bρ(0)Jλ-inf⁡Bρ(0)¯Jλ.
Now, if we apply the Ekeland's variational principle [18] to the functional Jλ:Bρ(0)¯→ℝ, it follows that there exists (uɛ,υɛ)∈Bρ(0)¯ such that
(3.19)Jλ(uɛ,υɛ)<inf⁡Bρ(0)¯Jλ+ɛ,Jλ(uɛ,υɛ)<Jλ(u,υ)+ɛ‖(u-uɛ,υ-υɛ)‖E,uɛ≠u,υ≠υɛ.
By the fact that
(3.20)Jλ(uɛ,υɛ)<inf⁡Bρ(0)¯Jλ+ɛ<inf⁡Bρ(0)Jλ+ɛ<inf⁡∂Bρ(0)Jλ,
we can infer that (uɛ,υɛ)∈Bρ(0).
Now, let us defineΦλ:Bρ(0)¯→ℝ by Φλ(u,υ)=Jλ(u,υ)+ɛ∥(u-uɛ,υ-υɛ)∥E. It is not difficult to see that (uɛ,υɛ) is a minimum point of Φλ, and thus
(3.21)Φλ(uɛ+t⋅τ,υɛ+t⋅σ)-Φλ(uɛ,υɛ)t≥0,
for t>0 small enough and any (τ,σ)∈B1(0). By the above expression, we have
(3.22)Jλ(uɛ+t⋅τ,υɛ+t⋅σ)-Jλ(uɛ,υɛ)t+ɛ‖(τ,σ)‖E≥0.
Letting t→0, we have
(3.23)〈Jλ′(uɛ,υɛ),(τ,σ)〉+ɛ‖(τ,σ)‖E>0,
and this implies that ∥Jλ′(uɛ,υɛ)∥≤ɛ. So, we infer that there exists a sequence {ωn,νn}⊂Bρ(0) such that
(3.24)Jλ(ωn,νn)⟶c=inf⁡Bρ(0)¯Jλ<0,Jλ′(ωn,νn)⟶0.
It is obvious that{ωn,νn} is bounded in E. Therefore, there exists (ω,ν)∈E such that, up to a subsequence, {ωn,νn} converges weakly to (ω,ν) in E and converges strongly to ωn→ω in Lq(x)(Ω1) and νn→ν in Lq(x)(Ω2) (Proposition 2.5). Thus 〈Jλ′(un,υn),(un-u,υn-υ)〉→0 and
(3.25)〈Jλ′(ωn,νn),(ωn-ω,νn-ν)〉=M(∫Ω1|∇ωn|p(x)p(x)dx)∫Ω1|∇ωn|p(x)-2∇ωn(∇ωn-∇ω)dx+N(∫Ω2|∇νn|p(x)p(x)dx)∫Ω2|∇νn|p(x)-2∇νn(∇νn-∇ν)dx-λ1∫Ω1|ωn|q(x)-2ωn(ωn-ω)dx-λ2∫Ω2|νn|q(x)-2νn(νn-ν)dx⟶0.
By Proposition 2.1, it follows that
(3.26)|∫Ω1|ωn|q(x)-2ωn(ωn-ω)dx|≤‖|ωn|q(x)-1‖q′(x),Ω1‖ωn-ω‖q(x),Ω1,|∫Ω2|νn|q(x)-2νn(νn-ν)dx|≤‖|νn|q(x)-1‖q′(x),Ω2‖|νn-ν|‖q(x),Ω2.
Since {ωn} converges strongly to ω in Lq(x)(Ω1), that is, ∥ωn-ω∥q(x),Ω1→0 as n→∞, we get
(3.27)∫Ω1|ωn|q(x)-2ωn(ωn-ω)dx⟶0
and similarly
(3.28)∫Ω2|vn|q(x)-2νn(νn-ν)dx⟶0.
Hence,
(3.29)M(∫Ω1|∇ωn|p(x)p(x)dx)∫Ω1|∇ωn|p(x)-2∇ωn(∇ωn-∇ω)dx⟶0,N(∫Ω2|∇νn|p(x)p(x)dx)∫Ω2|∇νn|p(x)-2∇νn(∇νn-∇ν)dx⟶0.
From (E1) and (E2), it follows that
(3.30)∫Ω1|∇ωn|p(x)-2∇ωn(∇ωn-∇ω)dx⟶0,∫Ω2|∇νn|p(x)-2∇νn(∇νn-∇ν)dx⟶0.
Eventually, by [19, Theorem  3.1], we get that {ωn,νn} converges strongly to (ω,ν) in E, so we conclude that (ω,ν) is a nontrivial weak solution for problem (P). The proof of Theorem 3.1 is complete.



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