The Neumann problem for a class of generalized Kirchhoff-type potential systems

In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration-compactness principle for variable exponent under suitable assumptions on the nonlinearities.


INTRODUCTION
In this article, we are concerned with the existence and asymptotic behavior of nontrivial solutions for the following class of nonlocal quasilinear elliptic systems (x, u) in Ω, for i = 1, 2, . . . , n (n ∈ N * ), where Ω ⊂ R N (N ≥ 2) is a bounded domain with smooth boundary ∂ Ω, N i is the outward normal vector field on ∂ Ω, λ is a positive parameter, ∇F = (F u 1 , . . . , F u 2 ) is the gradient of a C 1 -function F : R N × R n → R, p i , q i , w i , s i ∈ C(Ω), and ℓ i ∈ C(∂ Ω) are such that and for all x ∈ Ω, where functions w i are given by condition (F 4 ) below, p − i := inf x∈Ω p i (x), p + i := sup x∈Ω p i (x), and analogously to is given by condition (A 2 ) below, and for all x ∈ Ω, and the function K : R + 0 → {0, 1} is defined by In addition, we consider both C 1 h i and C 2 h i as nonempty disjoint sets, which are respectively defined by The operators A j i : X i → R n for j = 1 or 2, and the operators B i : X i → R, are respectively defined by In what follows, we shall consider the functions a j i satisfying the following assumptions for all i ∈ {1, 2, . . . , n}, j ∈ {1, 2} : (A 1 ) a j i : R + → R + are of class C 1 .
(A 4 ) There exist positive constants β j i and γ i for all i ∈ {1, 2, . . . , n} , j ∈ {1, 2} such that , for all ξ ≥ 0. In particular, when M 0 1 = 0 and B 1 > 0, the Kirchhoff equation associated with M 1 is said to be degenerate. On the other hand, when M 0 1 > 0 and B 1 ≥ 0, the Kirchhoff equation associated with M 1 is said to be nondegenerate. In this case, when B 1 = 0, the Kirchhoff equation associated with M 1 (is a constant) reduces to a local quasilinear elliptic problem. The study of differential equations and variational problems driven by nonhomogeneous differential operators has received extensive attention and has been extensively investigated, see e.g., Papageorgiou et al. [42]. This is due to their ability to model many physical phenomena. It should be noted that the p(x)-Laplacian operator is a special case of the divergence form operator div B j i (∇u i ) . The natural functional framework for this operator is described by the Sobolev space with a variable exponent W 1,p(x) . In recent decades, there has been a particular focus on variable exponent Lebesgue and Sobolev spaces, L p(x) and W 1,p(x) , where p is a real function, e.g., Rǎdulescu and Repovš [45]. Traditional Lebesgue spaces L p and Sobolev spaces W 1,p with constant exponents have proven insufficient to tackle the complexities of nonlinear problems in the applied sciences and engineering. To address these limitations, the use of variable exponent Lebesgue and Sobolev spaces has been on the rise in recent years. This area of research reflects a new type of physical phenomena, such as electrorheological fluids, or "smart fluids," which can exhibit dramatic changes in mechanical properties in response to an electromagnetic field. These and other nonhomogeneous materials require the use of variable exponent Lebesgue and Sobolev spaces, where the exponent p is allowed to vary. Moreover, variable exponent Lebesgue and Sobolev spaces have found a wide range of applications, from image restoration and processing to fields such as thermorheological fluids, mathematical biology, flow in porous media, polycrystal plasticity, heterogeneous sand pile growth, and fluid dynamics. For a comprehensive overview of these and other applications, see e.g. Chen et al. [15], Diening et al. [20,21], Halsey [28], Rǎdulescu [44], Ružicka [46,47], and the references therein).
Furthermore, every single equation of system (1.1) is a generalization of the stationary problem of the first model introduced by Kirchhoff [33] in 1883 of the following form: where the parameters ρ, h , ρ 0 , t, L, E are constants with some physical meaning, which is an extension of the classical D'Alembert wave equation, by considering the effect of the change in the length of a vibrating string. Nearly a century later, in 1978, Jacques-Louis Lions [35] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as Later on, many interesting results have been obtained, see e.g., Caristi et al. [9], Dai and Hao [19], Ma [37] and the references therein. The main difficulty in studying these equations appears that they do not satisfy a pointwise identity any longer. It is generated by having the term containing M i in the equations, and it makes (1.1) a nonlocal problem. The nonlocal problem models arise in the description of biological systems and also various physical phenomena, where u describes a process that depends on the average of itself, such as population density. For more references on this subject, we refer the interested reader to Ambrosio et al. [3], Arosio and Pannizi [5], Cavalcanti et al. [10], Chipot and Lovat [17], He et al. [31], Corrêa and Nascimento [18], Yang and Zhou [50], Wang et al. [48] and the references therein. On the one hand, the differential equations with constant or variable critical exponents in bounded or unbounded domains have attracted increasing attention recently. They were first discussed in the seminal paper by Brezis and Nirenberg [8] in 1983, which treated Laplacian equations. Since then, there have been extensions of [8] in many directions. One of the main features of elliptic equations involving critical growth is the lack of compactness arising in connection with the variational approach. In order to overcome the lack of compactness, Lions [36] established the method using the so-called concentration compactness principle (CCP, for short) to show a minimizing sequence or a Palais-Smale ((PS), for short) sequence is precompact. Afterward, The variable exponent version of the Lions concentration-compactness principle for a bounded domain was independently obtained by Bonder et al. [6,7], Fu [26], and for an unbounded domain by Fu [27]. Since then, many authors have applied these results to study critical elliptic problems involving variable exponents, see e.g., Alves et al. [2,1], Chems Eddine et al. [14,12,13], Fang and Zhang [24], Hurtado et al. [32], Mingqi et al. [39], Zhang and Fu [51]. When M i satisfies conditions a 1 i ≡ 1 (with k 1 i = 1, and k 2 i > 0 and k 3 i = 0), and a 2 i ≡ 0, Chems Eddine [12] proved the existence of nontrivial weak solutions for the following class of Kirchhoff-type potential systems with Dirichlet boundary conditions for all x in Ω and the potential function F satisfies mixed and subcritical growth conditions. Following that, Chems Eddine [11] established the existence of an infinite of solutions for the following systems i (x)} nonempty and F ∈ C 1 (Ω × R n , R) satisfies some mixed and subcritical growth conditions. Next, Chems Eddine and Ragusa [14] dealt with cases when the above class of Kirchhoff-type potential systems under Neumann boundary conditions with two critical exponents, established the existence and multiplicity of solutions. Our objective in this article is to show the existence of nontrivial solutions for the nonlocal problem (1.1). As we shall see in this paper, there are some main difficulties in our situation, which can be summed up in three main problems. First, the assumption (M) provides only a positive lower bound for the Kirchhoff functions M i near zero, creating serious mathematical technical difficulties. To overcome these difficulties, we need to do a truncation of the Kirchhoff functions M i to obtain prior estimates of the boundedness from above, and thus obtain a new auxiliary problem, thereafter, we do another truncation to control the energy functional corresponding to the auxiliary problem. The second difficulty in solving problem (1.1) is the lack of compactness which can be illustrated by the fact that the embeddings W 1,h(x) (Ω) ֒→ L h * (x) (Ω) and W 1,h(x) (Ω) ֒→ L h ∂ (x) (∂ Ω) are no longer compact and, to overcome this difficulty, we use two versions of Lions's principle for the variable exponent extended by Bonder et al. [6,7]. Then by combining the variational method and Mountain Pass Theorem, we obtain the existence of at least one nontrivial solution to the auxiliary problem, see Theorem 3.1, and by truncating functions M i , we obtain the existence of at least one nontrivial solution to problem (1.1), see Theorem 1.2. Throughout this paper, we shall assume that F satisfies the following conditions: .
(F 4 ) There exists a positive constant c such that There are many potential functions F satisfying assumptions (F 1 ) and (F 2 ). For example, when n = 2, take F(x, u 1 , for all x ∈ Ω. By the standard calculus we can verify that F satisfies the assumption (F 1 ). Moreover, by using Young inequality we can check the assumption (F 2 ).
The main result of our paper is the following. The paper is organized as follows: In Section 2 we give some preliminary results of the variable exponent spaces. In Section 3 we introduce the auxiliary problem and obtain a nontrivial solution for the auxiliary problem. Section 4 is dedicated to proving the main results. Finally, in Section 5, we illustrate the degree of generality of the kind of problems we studied in this paper.

PRELIMINARIES AND BASIC NOTATIONS
In this section, we introduce some definitions and results which will be used in the next section. Throughout our work, let Ω be a bounded domain of R N (N ≥ 2) with a Lipschitz boundary ∂ Ω, and let us denote by D either Ω or its boundary ∂ Ω. Denote The functional ρ p,D is called the p(x)-modular of the L p(x) (D) space, it has played an important role in manipulating the generalized Lebesgue-Sobolev spaces. We endow the space L p(x) (D) with the Luxemburg norm  Furthermore, the following Hölder-type inequality If u ∈ L p(x) (D) and p < ∞, we have the following properties (see e.g., Fan and Zhao [23, Theorem 1.3, Theorem 1.4]): As a consequence, we have the equivalence of modular and norm convergence Proposition 2.2 (Edmunds and Rakosnik [22]). Let h and ℓ be variable exponents in e. x in Ω and p ∈ L ∞ (Ω). Then if u ∈ L ℓ(x) (Ω), u = 0, it follows that .
Now, let us pass to the Sobolev space with variable exponent, that is, where ∂ x i u = ∂ u ∂ x i represent the partial derivatives of u with respect to x i in the weak sense. This space has a corresponding modular given by which yields the norm . Both norms turn out to be equivalent but we use the first one for convenience. It is well known that W 1,p(x) (Ω) is a separable and reflexive Banach spaces (see, e.g., Kováčik and Rákosník [34, Theorem 3.1]). As usual, we define by p * (x) the critical Sobolev exponent and p ∂ (x) the critical Sobolev trace exponent, respectively by We recall the following crucial embeddings on W 1,p(x) (Ω) Proposition 2.3 (Diening et al. [21], Edmunds and Rakosnik [22]). Let p be Lipschitz continuous and satisfying for all x ∈ Ω, then this embedding is compact.
for all x ∈ Ω, then this embedding is compact.
For detailed properties of the variable exponent Lebesgue-Sobolev spaces, we refer the reader to Diening et al. [21], Kováčik and Rákosník [34]. As is well known, the use of critical point theory needs the well-known Palais-Smale condition ((PS) c for short) which plays a central role.
We say that {u m } is a Palais-Smale sequence with energy level c (or (u m ) is (PS) c for short). Moreover, if every (PS) c sequence for E has a strongly convergent subsequence in X, then we say that E satisfies the Palais-Smale condition at level c (or E is (PS) c short).
Our main tool is the following classical Mountain Pass Theorem.
Theorem 2.6 (Rabinowitz [43]). Let X be a real infinite dimensional Banach space and let E : X → R be of class C 1 and satisfying the (PS) c such that E(0 X ) = 0. Assume that In the sequel, we shall use the product space X : Definition 2.7. We say that u = (u 1 , u 2 , . . . , u n ) ∈ X is a weak solution of the system (1.1) if By standard calculus, one can see that under the above assumptions, the energy functional E λ : So the critical points of functional E λ are weak solutions of system (1.1).
To prove our existence result, since we have lost compactness in the inclusions . . , n}, we can no longer expect the Palais-Smale condition to hold. Nevertheless, we can prove a local Palais-Smale condition that will hold for the energy functional E λ below a certain value of energy, by using the principle of concentration compactness for the variable exponent Sobolev space W 1,h i (x) (Ω). For reader's convenience, we state this result in order to prove Theorem 1.2, see Bonder et al. [6,7] for its proof. Now, let O be a different subset of ∂ Ω, a closed set (possibly empty). Set where closure is taken with respect to v 1,h(x) . This is the subspace of functions vanishing on O. Evidently, if and only if the h i (x)-capacity of O equals zero, for more details we refer the interested readers to Harjulehto et al. [29]. The best Sobolev trace constant .

Theorem 2.9 (Bonder and Silva [6]). Let h i and s i be variable exponents
is the best constant in the Gagliardo-Nirenberg-Sobolev inequality for variable exponents Notations. Weak (resp. strong) convergence will be denoted by ⇀ (resp., →), C i , C i j , c j and c i j will denote positive constants which may vary from line to line and can be determined in concrete conditions. Here, X * denotes the dual space of X, δ x j is the Dirac mass at x j , for all ρ > 0, x ∈ Ω, where B(x, ρ) denotes the ball of radius ρ centered at x.

THE AUXILIARY PROBLEM AND VARIATIONAL FRAMEWORK
In order to prove Theorem 1.2, we shall introduce the auxiliary problem by defining the auxiliary functional E θ ,λ and showing that the energy functional E θ ,λ has the geometry of Mountain Pass Theorem 2.6. By assumption (M), we see that the functions M i are bounded only from below and do not give us enough information about the behavior of M i at infinity, which makes it difficult to prove that the functional E λ has the geometry of Mountain Pass Theorem and that the sequence of Palais-Smale is bounded in X. Hence, we truncate functions M i and study the associated truncated problem.
Take γ i as in assumption (F 3 ) and θ i ∈ R, for all i ∈ {1, . . . , N} such that M 0 we can introduce the following auxiliary problem and λ are as in Section 1. By assumption (M), we also know that Now, the next step is to prove that the auxiliary problem (3.1) has a nontrivial weak solution. We obtain the following result. For the proof of Theorem 3.1 we shall need some technical results. We observe that the auxiliary problem (3.1) has a variational structure, indeed it is the Euler-Lagrange equation of the functional E θ ,λ : X → R defined as follows Now we prove that the functional E θ ,λ has the geometric features required by Mountain Pass Theorem 2.6. Proof. For all u = (u 1 , . . . , u n ) ∈ X, we obtain under the assumptions (A 2 ) and (A 4 ), Hence, by using assumption (F 4 ), and relations (2.2)-(2.3), we have .

On the other hand, for all
Thus, u nm γ n ). Therefore, by using assumptions (A 4 ), (M) and (F 3 ), we can conclude that By using assumption (A 4 ), we can find positive constants C i1 and C i2 such that (3.12) To prove the assertion, we assume by contradiction that we obtain a contradiction. Hence, we can deduce that {u m } is bounded in X. When k 3 i > 0, we have three cases to analyze.
In the case (i), for m sufficiently large, we have u i m . Hence, by inequality (3.12), we get and this is a contradiction. In the case (ii), by using inequality (3.12), we conclude that c+o m (1) , Hence, we also get a contradiction when limit as m → +∞ because p − i > 1. In the case (iii), the proof is similar as in the case (ii) so we shall omit it. Finally, we can deduce that {u m } is bounded sequence in X.
Next, we shall prove that the auxiliary problem (3.1) possesses at least one nontrivial weak solution.
Proof of Theorem 3.1. By Lemmas 3.2 and 3.3, the functional E θ , λ satisfies the geometric structure required by Mountain Pass Theorem 2.6. Now, it remains to check the validity of the Palais-Smale condition. Let {u m = (u 1 m , u 2m , . . . , u n m )} m∈N be a Palais-Smale sequence at the level c θ , λ in X. Then Lemma 3.4 implies that there exists λ * such that where T ix j and S i are given respectively in relations (2.10) and (2.14) and So, there exists a subsequence strongly convergent in X. Indeed, applying Lemma 3.5, {u m } m∈N is bounded in X, passing to a subsequence, still denoted by {u m } m weakly convergent in X, so there exist positive bounded measures µ i , ν i ∈ Ω and ν i ∈ ∂ Ω such that  and suppψ ⊂ B(0, 1). We consider, for each j ∈ J 1 i and any ε > 0, the functions ψ j,ε := ψ 2ε). Therefore, we have and we can find (3.13) Next, we shall prove that Notice that, due to assumption (A 2 ), it suffices to show that First, by applying Hölder inequality, we obtain is also bounded, thus there exists a positive constant C i such that Moreover, the sequence {u i m } is bounded in W 1,p i (x) (B(x j , 2ε)), so there is a subsequence, again denoted by {u i m } , converging weakly to u i in L p i (x) (B(x j , 2ε)). Therefore, .

Note that
where ω N is the surface area of the N-dimensional unit sphere. Since we obtain that ∇ψ j,ε u i L p i (x) (Ω) → 0, which implies Analogously, we can verify relation (3.16). Hence, we have completed the proof of relation (3.14). Similarly, we can also obtain lim By applying Hölder inequality, assumption (F 2 ) and the fact that 0 ≤ ψ j,ε ≤ 1, we have .
This yields and the last term on the right-hand goes to zero, because ∑ n we have completed the proof of relation (3.19). On the other hand, we have lim ε→0 Ω ψ j,ε dµ i j = µ i j ψ(0) and lim ε→0 ∂ Ω ψ j,ε dν i j = ν i j ψ(0), and since C 1 The function ψ j,ε has compact support, so letting m → +∞ and ε → 0 in relation (3.13), we get from relations (3.14)- (3.19), (3.20) Note that, when k 3 i = 0, we have h i (x) = p i (x). Hence, by using relation (2.8), we have By applying the Lebesgue Dominated Convergence Theorem, we get lim ε→0 Ω |∇u i | p i (x) ψ j,ε dx = 0. Therefore, On the other hand, if k 3 i > 0, then h i (x) = q i (x) Therefore, it follows from relations (2.10) and (3.20) that By applying the Lebesgue Dominated Convergence Theorem again, we get lim ε→0 Ω |∇u i | q i (x) ψ j,ε dx = 0. Hence, By using relation (2.10), we obtain , for all j ∈ J 1 i . On the other hand, by using assumptions (M) and (F 3 ), we have (1).
. Since κ > 0 is arbitrary and ℓ i are continuous functions for all i ∈ {1, 2, . . . , n}, we get , the set ∪ n i=1 J 1 i is empty, which means that for . Next, consider J 2 i = / 0, by the same approach for the case J 1 i . We have On the other hand, we have . . , u n m ) → 0, i.e E ′ θ ,λ (u 1 m , . . . , u n m ) is a Cauchy sequence in X * . Furthermore, by using Hölder's inequality again, we find .
Similarly, we also have .

PROOF OF THE MAIN THEOREM
Now we are in position to prove Theorem 1.2.
Proof. Invoking Theorem 3.1, for all λ ≥ λ * let u λ = (u 1,λ , u 2,λ , . . . , u n,λ ) be a solution of system (3.1). We shall prove that there exists λ * ≥ λ * such that where τ 0 i is defined as at the beginning of Section 3. We argue by contradiction and suppose that there is a sequence {λ m } m∈N ⊂ R such that A i (u i,λ m ) ≥ τ 0 i , for all i ∈ {1, 2, . . . , n}. By assumption (A 1 ) and the fact .

SOME EXAMPLES
In the last section, we shall exhibit some examples which are interesting from the mathematical point of view and have a wide range of applications in physics and other scientific fields that fall within the general class of systems studied in this paper, under adequate assumptions on functions a i j .
Example 5.1. Taking a 1 i ≡ 1 and a 2 i ≡ 1, we see that a 1 i satisfies the assumptions (A 1 ), (A 2 ), and (A 3 ), with k 0 j i = k 1 j i = 1, k 2 j i > 0, and k 3 i = 0, for all i ∈ {1, 2, . . ., n} and j = 1 or 2. Hence, system (1.1) becomes for 1 ≤ i ≤ n(n ∈ N * ), where The operator ∆ p i (x) u i := div (|∇u i | p i (x)−2 ∇u i ) is so-called p i (x)-Laplacian, which coincides with the usual p i -Laplacian when p i (x) = p i , and with the Laplacian when p i (x) = 2.
Example 5.2. Taking a j i (ξ ) = 1 + ξ , we see that a j i satisfies the assumptions (A 1 ), (A 2 ), and (A 3 ), with k 0 j i = k 1 j i = k 2 j i = k 3 i = 1, for all i ∈ {1, 2, . . . , n} and j = 1 or 2. Hence, system (1.1) becomes the following p&q-Laplacian system for 1 ≤ i ≤ n(n ∈ N * ), where As explained in Cherfils and Il'yasov [16], the study of system (5.2) was motivated by the following more general reaction-diffusion system which has applications in biophysics (see, e.g., Fife [25], Murray [40]), plasma physics (see, e.g., Wilhelmsson [49]), and chemical reactions design (see, e.g., Aris [4]). In these applications, u represents a concentration, div[H(u)∇u] is the diffusion with diffusion coefficient, and the reaction term d(x, u) relates to source and loss processes. For further details we refer the interested reader to e.g., Mahshid and Razani [38], He and Li [30], and the references therein. We continue with other examples which are also interesting from the mathematical point of view.
Example 5.3. Taking a j i (ξ ) = 1 + ξ √ 1+ξ 2 and a 2 i ≡ 1, we see that a j i satisfies the assumptions (A 1 ), (A 2 ), and (A 3 ), with k 0 1 i = k 0 2 i = k 1 2 i = 1, k 1 1 i = 2, and k 3 i = 0, k 2 1 i > 0, and k 2 2 i > 0, for all i ∈ {1, 2, . . . , n}. Hence, system (1.1) becomes u) in Ω, The operator div 1 + |∇u| p(x) √ 1+|∇u| 2p(x) |∇u| p(x)−2 ∇u is said to be p i (x)-Laplacian like or is called a generalized capillary operator. The capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e. the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e. the attractive force between the molecules of the liquid. The study of capillary phenomenon has gained much attention. This increasing interest is motivated not only by the fascination in naturally occurring phenomena, such as motion of drops, bubbles and waves, but also by its importance in applied fields, raging from industrial and biomedical and pharmaceutical to microfluidic systems -for further details we refer the interested reader to e.g., Ni and Serrin [41], and the references therein.