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

In the present article, we deal with a class of Kirchhoff-type equations involving a logarithmic nonlinearity and a convection term. Due to the lack of a variational structure, the well-known variational methods are not applicable. Using a topological approach based on the Galerkin method together with fixed point theorem, we obtain the existence of the finite-dimensional approximate solutions, generalized solutions, and strong generalized solutions. One of the main difficulties and innovations of present article is that we consider both convective term and logarithmic nonlinearity with variable exponents, another one is the weak assumptions on nonlocal term Mp(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{p(x)}$$\end{document} and nonlinear term g, and finally, we discuss the existence of solutions for discontinuous Kirchhoff-type equations.


Introduction
In recent decades, Kirchhoff-type equations or systems have been some of the classical topics in the qualitative analysis of partial differential equations, and the existence of solutions to these problems has been obtained under various assumptions on nonlinearity and Kirchhoff functions.
Kirchhoff in [12] introduced the following model, which came to be known as the Kirchhof-type equation: where parameters ρ, p 0 , h, E, and L are real positive constants. Equation (1.1) is nonlocal problems which contains a nonlocal coefficient p0 where λ is a parameter, and Ω is an open bounded domain in R N with smooth boundary.
Here, Kirchhoff function M p(x) is the following form: and Δ p(x) is a p(x)-Laplace operator, which can be defined by for all x ∈ Ω and η ∈ C ∞ 0 (R N ). Hence, Δ P: The condition that we impose on the continuous function p(x) is as follows: For the framework of variable exponential, there have been some papers on this topic, for instance, see [1,8,15,16,27,28,30,32]. Most of these papers deal with problems for power-type nonlinearities; however, few papers consider the existence of solutions for both a logarithmic nonlinearity and a convection term.
One of the main features of the problem (1.2) is the presence of convection term g(x, η, ∇η), depending on the function η and on its gradient ∇η, which 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 a physical system. For the work related to this topic, we cite the interesting work [7,17,20,34] and their references. The work in [2] focuses on the p-Kirchhoff-type equations with gradient dependence in the reaction, that is (1.6) where Ω ⊂ R N is a bounded domain with a smooth boundary. The existence of solutions for the problem (1.6) was obtained using Galerkin's approach. One more reference on convection is Vetro [6], which was devoted to the study of the following p(x)-Kirchhoff-type equations: The existence of weak solutions and generalized solutions for the problem (1.7) with gradient dependence was gotten via applying a topological method. The nonlinearity g : Ω × R × R N → R is a Carathéodory function, satisfying G 1 : There exists a positive function φ(x) ∈ L (p * (x)) (Ω) and some positive constants a and b, such that Remark 1.1 Notice that the convection term g(x, η, ∇η) makes the problem (1.2) non-variational. To overcome this difficulty and tricky challenge, our approach is the Galerkin method combined with Brouwer's fixed-point theorem.
Another significant characteristic of the problem (1.2) 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 practical 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| p(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 [3,18,25,36]. Peculiarly, Xiang et al. in [21] considered the following equations: studied the following initial value problem: (1.9) The weak solutions of Eq. (1.9) were obtained under suitable conditions and an appropriate space of functions. Moreover, Zeng et al. in [11] were devoted to the study of equations with logarithmic nonlinearity and variable exponent by applying the logarithmic inequality.
Remark 1.2 Since the logarithmic nonlinearity does not satisfy the monotonicity condition or Ambrosetti-Rabinowitz condition, the problem (1.2) becomes extremely complex. Therefore, we do need more careful analysis.
To the best of my knowledge, there is no result for the Kirchhoff-type equations, which combines with variable exponent, logarithmic nonlinearity, and convection term. Therefore, motivated by the previous and above-cited works, we will investigate the existence of solutions for this kind of equation, which are different from the work of [2,6,21,29]. Under weak conditions on the nonlocal term M p(x) and the nonlinearities g, we prove the existence of solutions with the help of the Galerkin method. Our study extends previous results, such as from the elliptic problem with logarithmic nonlinearity or convection term to p(x)-Kirchhoff-type equations both logarithmic nonlinearity with variable exponent and convection term.
The paper is divided into eight sections. Aside from Sect. 1, we have Sect. 2 given some preliminary notions and results about Lebesgue spaces and Sobolev spaces and Sect. 3 proved some technical lemmas. The finitedimensional approximate solutions, generalized solutions, strong generalized solutions are obtained in Sects. 4, 5, 6, respectively. Section 7 discuss the discontinuous case of kirchhoff functions and we make a conclusion in Sect. 8.

Preliminary Results
In this subsection, we briefly review some basic knowledge, lemmas, and propositions of generalized Lebesgue spaces and Sobolev spaces with variable exponents.
For any real-valued function H defined on a domain Ω, we denote Let ϑ(x) ∈ C + (Ω), we define the generalized Lebesgue spaces with variable exponents as provided with the Luxemburg norm: ) is a separable and reflexive Banach spaces, see [24,35].
Suppose that η n , η ∈ L ϑ(x) (Ω), then the following properties hold Now, we consider the following generalized Sobolev spaces with variable exponents: endowed with the norm: then (W, · W ) is a separable and reflexive Banach spaces, see [35]. [35]) Assume that q(x) ∈ C + (Ω) fulfill Then, there exists C q = C q (N, ϑ, q, Ω) > 0 such that Let W 0 denote the closure of C ∞ 0 (Ω) in W with respect to the norm η W0 , which is the subspace of W . Thus, the spaces (W 0 , · W0 ) is also a uniformly convex and reflexive Banach spaces.
Remark 2.1 According to the Poincaré inequality, we know that ∇η ϑ(x) and η W0 are equivalent norms in W 0 . From now on, we work on W 0 and replace η W0 by ∇η ϑ(x) , that is, Remark 2.2 To simplify the presentation, we will denote the norm of W 0 by · instead of · W0 . 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 [9].

Some Technical Lemmas
The following Lemma 3.1 provides an useful growth estimate, related to the reaction term g(x, η(x), ∇η(x)), and is proved using the Hölder inequality. Lemma 3.1. Suppose that condition G 1 holds, then the following inequality holds for all η, ξ ∈ W 0 , and C > 0.
Proof. From hypothesis G 1 and Lemma 2.1, we have . MJOM Kirchhoff-type problems involving Page 7 of 22 77 Since p(x) < p * (x) then using the continuous embedding where C is some positive constant.
Notice that the problem (1.2) contains logarithmic nonlinear terms, we need to establish the following two estimates, which play an important role during our proof process.

Lemma 3.2.
Suppose that h(x) ∈ C + (Ω), then we have the following estimate Proof. Suppose that h(x) ∈ C + (Ω), we construct the following function With respect to t, just by taking a simple derivative, we deduce 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 3.3.
Suppose that for all η n , η ∈ W 0 and h(x), p(x) ∈ C + (Ω), then the following properties hold where |Ω| denotes the Lebesgue measure of Ω and C Ω1 > 0. Using in view of Lemma 2.2, there exist some constants C h + +p + > 0 and It follow from (3.1) and (3.2) that This yields the stated conclusion. (ii) Going if necessary up to a subsequence, we suppose there exists η ∈ W 0 , such that In fact, by a simple calculation for the logarithmic nonlinear term, we deduce Sincep + < p * (x) then using the continuous embedding L p * (x) (Ω) → L p + (Ω) and combining Lemma 3.2, we deduce where C Ω2 > 0. From Lemma 2.1, we obtain . (3.5)

MJOM
Kirchhoff-type problems involving Page 9 of 22 77 Notice that the relation (3.4) implies that where C p + p + −1 > 0. Therefore, it follows from (3.3), (3.5) and (3.6) that This yields the stated conclusion. (iii) To prove the third result, we are entitled to use all the formulas in (ii).
Hence, from (3.3), it yields This yields the stated conclusion.

Finite Dimensional Approximate Solutions
Since W 0 is a reflexive and separable Banach space, 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 It is known that X n and R N are isomorphic and for η ∈ R N , we have an unique ξ ∈ X n by the identification: that is, for all n ≥ 1 and ϕ ∈ X n , there exists η n ∈ X n , such that Proof. Inspired by the literature [2,6,7], Theorem 4.1 was proved using Galerkin's method. For all η ∈ X n , we consider the mapping G = (G 1 , G 2 , ..., The following work shows that, for each n ≥ 1, the problem (1.2) has an approximate solution η n in X n , namely For η ∈ X n , we have From G 2 and Lemma 3.3, we have the following estimate According to Remark 2.1 and Lemma 2.2, 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: If 2p − > p + and p − > α + with λ ≤ 0, there exists some positive constant R, provided sufficiently large, such that G, η ≥ 0, for all η ∈ X n , with η = R.
In both cases, G is continuous, so, in view of Lemma 2.3, the problem (1.2) admits a approximate solution η n in X n ⊂ W 0 with η n ≤ R. Proof. If η n ≤ 1 for all n ∈ N, then the sequence {η n } n∈N is bounded in W 0 . If η n > 1 for all n ∈ N, with η n in place of ϕ in (4.1), we have On the basis of condition G 2 and Lemma 3.3, it gives

Existence of Generalized Solutions
Before stating our theorem, we make the following definition of the generalized solution.  for some η in W 0 . Define the Nemytskii operator N g : W 0 → W * 0 with respect to the function g : Ω × R × R N → R, given by Lemma 3.1 shows that the Nemytskii operator N g is well-defined. Moreover, there exists some positive constant C such that N g (η n ) W * 0 ≤ C φ (p * s (x)) + aM 1 + bM 2 , for all η n ∈ W 0 . Hence, the Nemytskii operator N g is bounded. In combination with (5.1), we obtain Define another Nemytskii operator N f : W 0 → W * 0 with respect to the function f : Ω × R → R given as The Nemytskii operator N f is also well-defined, and the following estimate holds for all η n ∈ W 0 and C f , C h + +p + −1 , C h − +p − −1 > 0. Thus, the Nemytskii operator N f is also bounded. In combination with (5.1), we deduce Since, the sequence η n constructed in Theorem 4.1 is bounded in W 0 and p(x) η n − λN f (η n ) − N g (η n ) ζ, in W * 0 as n → ∞, (5.5) for some ζ in W 0 . Let ϕ ∈ ∪ ∞ n=1 X n . There is an integer m ≥ 1, such that ϕ ∈ X m . Applying Theorem 4.1, we see that equality (4.1) holds true for all n ≥ m. Letting n → ∞ in (4.1) entails ζ, ψ = 0, for all ϕ ∈ ∪ n≥1 X n .
Since ∪ n≥1 X n is dense in W * 0 , we obtian that ζ = 0. As a result, the expression (5.5) becomes 0, as n → ∞. As a result, it follows from (5.1), (5.6) and (5.9) that there exists a generalized solution to problem (1.2) in the sense of Definition 5.1.