Existence results for a nonlinear elliptic transmission problem of p ( x )-Kirchhoff type

In this article, we establish the existence of weak solutions for a nonlinear transmission problem involving nonlocal coefficients of p(x)-Kirchhoff type in two different domains, which are connected by a nonlinear transmission condition at their interface. We get our results by means of the monotone operator theory and the (S+) mapping theory; the weak formulation takes place in suitable variable exponent Sobolev spaces.

We are concerned with the existence of solutions to the following nonlinear elliptic system A(x, ∇u 1 ) dx a(x, ∇u 1 ).
Transmission problems problems arise in several applications in physics and biology (see [36]).Some results are available for linear parabolic equations with linear and nonlinear conditions at interfaces, for biological models for the transfer of chemicals through semipermeable thin membranes (see [8,39,43]).There are cases where transmission conditions can allow to deal with models including chemical phenomena in materials with different porosity and diffusivity, and chemotaxis phenomena in regions with different substrate properties (see [30]).
Kirchhoff in 1883 [32] investigated an equation  [35], where a functional analysis framework for the problem was proposed; see e.g.[3,9,29] for some interesting results and further references.In recent years, various Kirchhoff-type problems have been discussed in many papers.The Kirchhoff model is an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, which takes into account the changes in length of the string produced by transverse vibrations; while purely longitudinal motions of a viscoelastic bar of uniform cross section and its generalizations can be found in [12,33,40,41].In particular, in a recent article, existence and multiplicity of nontrivial radial solutions are obtained via variational methods [34].The study of nonlocal elliptic problem has already been extended to the case involving the p-Laplacian (for details, see [11,19,20]) and p(x)-Laplacian (see [14,17,22,29]).More recently, Cabanillas L. et al. [7], have dealt with the p(x)-Kirchhoff type equation by topological methods.Our work is motivated by the ones of Feistauer et al. [28] and Cecik et al. [10].The aim of this article is to study the existence of a solution to the problem (1.1) in the Sobolev spaces with variable exponents; we use the well-known theorem named as Browder-Minty theorem and the degree theory of (S + ) type mappings to attack it.This paper is organized as follows.In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces.Section 3 is devoted to the proof of our general existence results.

Preliminaries
To discuss problem (1.1), we need some theory on W 1,p(x) (Ω) which is called variable exponent Sobolev space (for details, see [26]).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 with the norm and
In the sequel we shall assume that f ∈ L p (x) (Ω) 2 .
Let us define the Banach space E = W 1,p(x) (Ω 1 ) × W 1,p(x) (Ω 2 ) equipped with the norm It is obvious that Remark 2.7.From the assumptions on A, arguing as in [37], we get after some computations that

Existence of solutions
In this section, we shall state and prove the main result of the paper.For simplicity, we use c, c i , i = 1, 2, . . . to denote the general positive constants (the exact value may change from line to line).Let us define the forms We say that u = (u 1 , u 2 ) ∈ E is a weak solution of problem (1.1) if Here we recall how the theory of monotone operators is used to prove existence of solutions to (1.1) .For this, it will be useful to consider the differential operator as a mapping from E into its dual space, i.e., G : provided of course that for fixed u this indeed defines a bounded linear functional on E.
The following lemma states the rather obvious relation between the operator G and the differential equation (1.1) •) and L are linear and continuous on E .
Proof.The boundedness of the forms is an easy consequence of Hölder's inequality, Remark 2.7, Propositions 2.2-2.5 and monotonicity of M. Indeed, By Lemma 3.1 we can define the mapping G : E → E and the functional ϕ ∈ E by the identities for each u, v ∈ E. Now, it is clear that solving (1.1) is the same as finding u ∈ E such that Theorem 3.2.Assume that (M0), (H0) and (A1)-(A4) hold.In addition, suppose that (H1) h(x, 0) = 0 and h : Ω × R → R is a decreasing function with respect to the second variable, i.e. h(x, s 1 ) ≤ h(x, s 2 ) for a.e.x ∈ Ω and s 1 , If p + < α − , problem (1.1) has precisely one weak solution .
For the proof of our theorem we need to establish some lemmas.
Then there exists a positive constant c 2 = c 2 (r, s) such that Proof.Firstly, we prove that there exists c 2 > 0 such that for all u = (u 1 , u 2 ) ∈ E with u E = 1 .Let us assume that (3.2) is not valid.Then there exists a sequence {u ν } ⊂ E such that From Proposition 2.5 and b) it follows that Using (3.3) , the weak lower semicontinuity of the seminorm |u| E and c) we get therefore u = 0.This is a contradiction to a).Finally to prove (3.1) let u ∈ E, u = 0 and w = u u E . From (3.2) we have Multiplying this inequality by u ν β E the assertion (3.1) follows.

Lemma 3.4.
There exists a constant c 3 > 0 such that for any u ∈ E with u E ≥ 1 Proof.For any u = (u 1 , u 2 ) ∈ E we have using inequalities (3.5) and (3.6), it follows that , and noting that u for some c 2 > 0. For other cases, the proofs are similar and we omit them here.So we have This ends the proof of Lemma 3.4.
The proof of the next Lemma is done by adapting some arguments employed in the proof of Theorem 2.1 i) in [18].Lemma 3.5.The form a is strictly monotone: Noting that a(x, •) is monotone by assumption (A4), and following a similar procedure to that used in [18] , we get i.e. g is monotone.
If g(u, u − v) − g(v, u − v) = 0 then all four terms in the right-hand side of (3.7) are equal to zero.Hence, u i = v i = k i = const.a.e. in Ω i and u i = v i a.e. on Γ i , i = 1, 2. Therefore, k i = 0 and u = v a.e.Lemma 3.6.There exists a constant c 6 > 0 such that Proof.By definition of the form g, for all u, v, w ∈ E , we have Now in virtue of the Lipschitz condition satisfied by M and the elementary inequalities the equality (3.9) reduces to dx where L M i > 0, the Lipschitz constant of M, depends on max{ u E , w E }. Next, we use the degree theory of (S + ) type mappings to prove the second result of this paper.
Let us recall the definition of the mapping of type (S + ).Let X be a Banach space and D ⊂ X an open set."u ν u" and "u ν → u" denote respectively the weak convergence and the strong convergence in X.
A mapping A : D → X * is said to be of type (S + ) if for any sequence {u ν } ⊂ D for which u ν u in X and lim sup ν→∞ A(u ν ), u ν − u ≤ 0, u ν → u in X.For (S + ) mapping theory, including the degree theory and the surjection theorem, we refer the reader to [6,44,45].Theorem 3.7.Assume that (M0) and (H0) hold.If p + < α − and β + < p − , problem (1.1) has a weak solution .
Proof.By a simple adaptation of Theorem 2.1 ii) in [18] to our problem, we can prove that the mapping Define the mappings B, C, D and S : E → E * respectively by Then B(u) = ∑ 2 i=1 B 0 (u i ), L(u) = − ∑ 2 i=1 N h (u i ) and G(u) = B(u) + C(u) + D(u) + L(u) .It is clear that u ∈ E is a solution of (1.1) if an only if G(u) = T.
From the above analysis, it is obvious that B : E → E is continuous , bounded and of type (S + ).Moreover, using the compacity of embeddings W 1,p(x) (Ω i ) → L α(x) (Γ i ) and W 1,p(x) (Ω i ) → L β(x) (Ω i ) we deduce that the operators C, D, L are compact (cf.e.g.[5,26]).Noting that the sum of an (S + ) type mapping and a compact mapping is of type (S + ) , it follows that the mapping G = B + C + D + L is continuous, bounded, and of type (S + ).
Then, proceeding similarly as in the proof of Lemma 3.4, for u E large enough we have that

= +∞
So, the mapping G is coercive, and hence, by the surjection theorem for the pseudomonotone mappings (see [45,Theorem 27.A]), the mapping G is surjective.