-Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

: In the present article, we study a class of Kirchhoff-type equations driven by the ( p ( x ) , q ( x )) Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not applicable. With the help of the Galerkin method and Brezis theorem, we obtain the existence of ﬁnite-dimensional approximate solutions and weak solutions. One of the main difﬁculties and innovations of the present article is that we consider competing ( p ( x ) , q ( x )) -Laplacian, convective terms, and logarithmic nonlinearity with variable exponents, another one is the weaker assumptions on nonlocal term M υ ( x ) and nonlinear term g .

From now on, we briefly state some major features of problem (1). One of the significant characteristics of the problem (1) is the presence of double non-local Kirchhoff terms, which were introduced in [1] as follows: where parameters ρ, p 0 , h, E, and L are real positive constants. Equation (4) is a nonlocal problem, which contains a nonlocal coefficient dx, and has a wide range of applications and research in physical systems, such as non-homogeneous Kirchhoff-type equations in R N [2], nonlocal Kirchhoff equations of elliptic type [3], Kirchhoff-Schrödinger type equations [4], p(x)-Laplacian Dirichlet problem [5,6], Kirchhoff-Choquard equations involving variable-order [7,8], fractional p(·)-Kirchhoff type problem in R N [9], Kirchhofftype equations involving the fractional p 1 (x)&p 2 (x)-Laplace operator [10], fractional p(x, ·)-Kirchhoff-type problems in R N [11], and fractional Sobolev space and applications to nonlocal variational problems [12]. For more Kirchhoff-type problems, we also mention that [13] studied a class of Kirchhoff nonlocal fractional equations and obtained the existence of three solutions, Ref. [14] discussed a class of p-Kirchhoff equations via the fountain theorem and dual fountain theorem, and Ref. [15] researched the existence of non-negative solutions for a Kirchhoff type problem driven by a non-local integro-differential operator.
Let M i : R + 0 → R + and p(x), q(x) : R N → (1, +∞) be continuous functions, which satisfy the following conditions: , for any t > ι. Another significant characteristic of the problem (1) is the presence of double operators, which comes from the following system where Dη = |∇η| p−2 + |∇η| q−2 and c(x, η) is a polynomial of η. System (5) had a wide range of applications in the field of physics and related sciences, for example, on the stationary solutions of generalized reaction diffusion equations [16], elliptic problems with critical growth in R N [17], nontrivial solutions to nonlinear elliptic equation in R N [18], and fractional Choquard problems with variable order [19]. The function η in (5) describes a concentration, and the first term corresponds to the diffusion with a (generally nonconstant) diffusion coefficient Dη, whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction term c(x, η) in (5) has a polynomial form with respect to the concentration η. When M υ(x) = 1(υ(x) ∈ {p(x), q(x)}) and µ = 1, Chung et al. in [20] devoted to the study of equations involving both p 1 (x)-Laplacian and p 2 (x)-Laplacian where p 1 , p 2 , q, r 1 , and r 2 are different continuous functions, while V 1 , V 2 are suitable weights. Equation (6) considered the local double Laplace operators, whose results differed from those of the single Laplace operator.
When M υ(x) = 1(υ(x) ∈ {p(x), q(x)}) and µ = −1, we mention that Motreanu in [21] considered Dirichlet problems with competing operators where Ω ⊂ R N is a bounded domain. Equation (7) includes the sum −∆ p + ∆ q of the negative p-Laplacian ∆ p and of the q-Laplacian ∆ q , due to competition between −∆ p and ∆ q , and the operator −∆ p + ∆ q has a different behavior in comparison to the operator −∆ p + ∆ q . Moreover, the ellipticity and monotonicity property of the operator −∆ p + ∆ q are lost.
The third significant characteristic of the problem (1) is the presence of convection term g(x, η, ∇η), depending on the function η and on its gradient ∇η, which makes the problem (1) non-variational, plays an important role in science and technology fields, and is widely used to describe physical phenomena. For example, due to convection and diffusion processes, particles or energy are converted and transferred inside physical systems. For the work related to this topic, we cite the interesting work [21][22][23][24] and their references.
The work in [25] focused on the p-Kirchhoff-type equations with gradient dependence in the reaction that is where Ω ⊂ R N is a bounded domain with a smooth boundary. The existence of solutions for the problem (8) was obtained by utilizing Galerkin's approach.
One more reference on convection is Vetro [26], which was devoted to the study of the following p(x)-Kirchhoff-type equation: The existence of weak solutions and generalized solutions for the problem (9) with gradient dependence was obtained via applying a topological method.
The nonlinearity g : Ω × R × R N → R is a Carathéodory function, satisfying H g 1 : There exist some constants c < 1, d > 0 and a function α ∈ [1, p − ) such that H g 2 : There exists a positive function φ(x) ∈ L p (x) (Ω) and some positive constants a and b such that The last significant characteristic of the problem (1) is the presence of logarithmic nonlinearity. The interest in studying problems with logarithmic nonlinearity is motivated not only by the purpose of describing mathematical and physical phenomena but also by their application in realistic models. For instance, in the biological population, we use the function η(x) to represent the density of the population, and the logarithmic nonlinear term |η| r(x)−2 η ln |η| to denote external influencing factors.
Many scholars make efforts to investigate logarithmic nonlinearity, and, indeed, some important results were obtained; for example, see [27][28][29][30]. Peculiarly, Xiang et al. in [31] considered the following equation: and h(x) is a sign-changing function. The existence of least energy solutions (10) was obtained by utilizing the Nehari manifold method.
Until now, there have been few papers to handle the equations involving logarithmic nonlinearity with variable exponents. Recently, Boudjeriou in [32] studied the following initial value problem: The weak solutions of Equation (11) were obtained under suitable conditions. Moreover, Zeng et al. in [33] were devoted to the study of equations with logarithmic nonlinearity and variable exponents by applying the logarithmic inequality.
Motivated by the previous and aforementioned cited works, there is no result for the Kirchhoff-type equations, which combine with variable exponents, competing (p(x), q(x))-Laplacian, logarithmic nonlinearity, and convection terms; therefore, we will investigate the existence of solutions for these kinds of equations, which are different from the work of [25,26,31,32]. Under weaker conditions on the nonlocal term M υ(x) and the nonlinearities g, we prove the existence of finite-dimensional approximate solutions by using the Galerkin method and obtain the existence of weak solutions with the help of the Brezis theorem. One of the main difficulties and innovations of the present article is that we consider competing (p(x), q(x))-Laplacian, convective term, and logarithmic nonlinearity with variable exponents; another one is the weaker assumptions on nonlocal term M υ(x) and nonlinear term g.
The present article is divided into six sections. Aside from Section 1, we have Section 2 given some preliminary notions and results about Lebesgue spaces and Sobolev spaces, and proved some technical lemmas. The finite-dimensional approximate solutions are obtained in Section 3. Section 4 discusses the existence of weak solutions by applying the Brezis theorem, and we give two examples of application of our theorems in Section 5 and present conclusions in Section 6.

Preliminary Results and Some Technical Lemmas
In this section, we briefly review some basic knowledge of generalized Lebesgue spaces and Sobolev spaces with variable exponents, and then give two technical lemmas.
For any real-valued function H defined on a domain Ω, we denote Letting ϑ(x) ∈ C + (Ω), we define the generalized Lebesgue spaces with variable exponents as ) is a separable and reflexive Banach spaces; see [34,35].
Let W 0 denote the closure of C ∞ 0 (Ω) in W with respect to the norm η W 0 , which is the subspace of W. Thus, the spaces (W 0 , · W 0 ) are also separable and reflexive Banach spaces.

Remark 1.
According to the Poincaré inequality, we know that ∇η ϑ(x) and η W 0 are equivalent norms in W 0 . From now on, we work on W 0 and replace η W 0 by ∇η ϑ(x) , that is,

Remark 2.
To simplify the presentation, we will denote the norm of W 0 by · instead of · W 0 . W * 0 denotes the dual space of W 0 .
Our technique of proof is based on Galerkin methods together with the fixed point theorem, whose proof may be found in Lions [37]. Lemma 3. Let W 0 be a finite dimensional space with the norm · and let G : W 0 → W * 0 be a continuous mapping. Assume that there is a constant R > 0 such that The following two Lemmas provide a useful growth estimate, related to logarithmic nonlinear terms, which play an important role during our proof process.

Lemma 4.
Assume that h(x) ∈ C + (Ω); then, we have the following estimate: Proof. Let h(x) ∈ C + (Ω), and we construct the following function: With respect to t, just by taking a simple derivative, we deduce and let f (t) = 0; then, t * = e h −1 (x) . It is obvious that t * is the unique maximum point of the function f (t), so f (t) ≤ f (t * ) = 0 for all t ∈ [1, +∞). Therefore, based on the above discussion, we can obtain the stated conclusion.
Lemma 5. Assume that, for all η ∈ W 0 and h(x), r(x) ∈ C + (Ω), then the following inequality holds: Since |η(x)| ≤ 1, there exist M r 1 > 0 and M r 2 > 0 such that |η| r(x) < M r 1 and ln |η| < M r 2 . By a simple calculation, we obtain where |Ω| denotes the Lebesgue measure of Ω and C Ω 1 > 0. Using in view of Lemma 2, and there exist some constants C h + +r + > 0 and C h − +r − > 0 such that It follows from (12) and (13) that This yields the stated conclusion.

Finite Dimensional Approximate Solutions
Since W 0 is a reflexive and separable Banach space, see [34], and there exists an orthonormal basis {e 1 , ..., e n , ...} in W 0 , such that Define X n = span{e 1 , ..., e n }, which means a sequence of vector X n subspaces of W 0 , satisfying dim(X n ) < ∞ for all n ≥ 1, X n ⊂ X n+1 for all n ≥ 1, and ∞ n=1 X n = W 0 .
It is known that X n and R N are isomorphic and, for η ∈ R N , we have a unique ξ ∈ X n by the identification Theorem 1. Assume that conditions H m , H pq , and H g 1 are satisfied; then, (1) admits a approximate solution for all µ < 0 and λ ≤ 0, • if 2q − > p + and q − > α + , problem (1) admits a approximate solution for all µ ≥ 0 and λ ≤ 0, • if 2p − > r + + h + and p − > α + , problem (1) admits a approximate solution for all µ ≥ 0 and λ > 0, • if 2p − > r + + h + , p − > q + and p − > α + , problem (1) admits a approximate solution for all µ < 0 and λ > 0, • if 2q − > r + + h + and q − > α + , problem (1) admits a approximate solution for all µ ≥ 0 and λ > 0, that is, for all n ≥ 1 and ϕ ∈ X n , there exists η n ∈ X n such that Proof. For all η ∈ X n , we consider the mapping G = (G 1 , G 2 , ..., The following work shows that, for each n ≥ 1, problem (1) has an approximate solution η n in X n , namely For η ∈ X n , we have From H g 1 and Lemma 5, we have the following estimate: According to Remark 1 and Lemma 2, there exist some positive constants C α + and C α − , such that If η > 1, then Combined with the above analysis, we deduce that Case 1: Utilizing that 2p − > p + and p − > α + with µ ≥ 0 and λ ≤ 0, there exists a positive constant R, provided at a sufficiently large size, such that for all η ∈ X n , with η = R. Case 2: Utilizing that 2p − > p + , p − > q + and p − > α + with µ < 0 and λ ≤ 0, there exists a positive constant R, provided at a sufficiently large size, such that for all η ∈ X n , with η = R. Case 3: Utilizing that 2q − > p + and q − > α + with µ ≥ 0 and λ ≤ 0, there exists a positive constant R, provided at a sufficiently large size, such that for all η ∈ X n , with η = R. Case 4: Utilizing that 2p − > r + + h + and p − > α + with µ ≥ 0 and λ > 0, there exists a positive constant R, provided at a sufficiently large size, such that for all η ∈ X n , with η = R. Case 5: Utilizing that 2p − > r + + h + , p − > q + and p − > α + with µ < 0 and λ > 0, there exists a positive constant R, provided at a sufficiently large size, such that for all η ∈ X n , with η = R. Case 6: Utilizing that 2q − > r + + h + and q − > α + with µ ≥ 0 and λ > 0, there exists a positive constant R, provided at a sufficiently large size, such that for all η ∈ X n , with η = R.
In the above six cases, G is continuous, so, in view of Lemma 3, problem (1) admits a approximate solution η n in X n ⊂ W 0 with η n ≤ R.

Corollary 1.
Assume that the conditions of Theorem 1 are satisfied, then the sequence {η n } n≥1 with η n ∈ X n constructed in Theorem 1 is bounded in W 0 .
Proof. If η n ≤ 1 for all n ∈ N, then the sequence {η n } n∈N is bounded in W 0 .
On the basis of condition H g 1 and Lemma 5, it gives
In the above six cases, we conclude that the sequence {η n } n≥1 is bounded in W 0 .

Existence of Weak Solutions
In this section, our interest is devoted to the existence of weak solutions for problem (1). The following are the main results of this section.

Corollary 2.
Assume that the conditions of Theorem 2 are satisfied; then, the sequence {η n } n≥1 with η n ∈ X n is bounded in W 0 .
Proof. The proof is similar to Corollary 1, which we omit.
To prove Theorems 2, we use the Brezis theorem for pseudomonotone operators in the separable reflexive space (see (Theorem 27.A [38]). Let us define the operator for all η, ϕ ∈ W 0 .

Lemma 6.
Assume that the conditions of Theorem 2 are satisfied; then, the operator T is bounded.

Lemma 7.
Assume that the conditions of Theorem 2 are satisfied; then, the operator T is demicontinuous.

Lemma 8.
Assume that the conditions of Theorem 2 are satisfied; then, for all µ > 0, the operator T is coercive.

Case 4: Utilizing that 2q
In the above four cases, we deduce the coerciveness of T from (35) as η → ∞.

Lemma 9.
Assume that the conditions of Theorem 2 are satisfied, then T is an (S + )-type operator.
First, note that Going if necessary up to a subsequence, we suppose there exists η ∈ W 0 such that η n η, weakly in W 0 , η n → η, a.e. in Ω.
Using the following Simon inequalities for all u 1 , u 2 ∈ R N , where c τ and c τ are positive constants depending only on τ, we obtain that is, if η n η in W 0 and lim sup n→∞ Tη n − Tη, η n − η ≤ 0, then η n → η in W 0 . This shows the (S + )-property of T.
Proof of Theorem 2. From Section 2, evidently, we know that W 0 is a real, separable, and reflexive Banach spaces. Moreover, it follows from Lemmas 6-9 that the operator T satisfies all conditions of the Brezis theorem. Hence, invoking the Brezis theorem, we obtain that Tη = 0 has at least one solution η in W 0 , i.e., problem (1) has at least one weak solution η.

Examples
Now, we give two easy examples of application of our theorems. The first is when M p(x) (t) = a p + b p t, for all t ≥ 0 with a p > 0, b p ≥ 0 and M q(x) (t) = a q + b q t, for all t ≥ 0 with a q > 0, b q ≥ 0. In this case, problem (1) reduces to the following form.
It is clear that M p(x) (t) ≥ a p > 0, for all t ≥ 0 and M q(x) (t) ≥ a q > 0, for all t ≥ 0. That is, the condition H m is satisfied. Thus, the results obtained in Theorems 1 and 2 stay true for problem (1). The problem and results are all new.

Conclusions
In this article, we study a kind of Kirchhoff-type elliptic problem, which combines with a variable exponent, competing (p(x), q(x))-Laplacian, logarithmic nonlinearity, and convection term. Due to the deficit of ellipticity, monotonicity, and variational structure, there are no available techniques to handle problem (1). A fundamental idea of the paper is to seek a solution to (1) as a limit of finite dimensional approximations. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions, respectively. Our study extends previous results, such as from the elliptic problem with logarithmic nonlinearity or the convection term to (p(x), q(x))-Kirchhoff-type equations both logarithmic nonlinearity with variable exponents and convection terms. Finally, we consider that it will be a new field to study such problems (1) in fractional Sobolev spaces with variable exponents and in Sobolev spaces with variable exponents and variable fractional order.