1. Introduction
The study of differential equations and variational issues involving -growth conditions has received a majority of attention in recent years. The development of numerous significant models in electrorheological and thermorheological fluids, image processing, and other fields inspired a systematic study of partial differential equations with variable exponents; see [1,2,3]. The literature on the study of such operators is very large and rich, but we only list some newly published articles for interested readers, see, e.g., [4,5,6,7,8].
The study of elliptic equations with fractional operators is one of the most fascinating areas of nonlinear analysis. These issues have received much attention in both pure mathematics study and practical applications. In reality, this sort of operator often appears in a variety of settings. Few authors have also studied elliptic problems involving inequalities [9,10]. As far as we know, the fractional Sobolev spaces with variable exponents and the fractional -Laplacian were introduced firstly by U.Kaufmann, J.D.Rossi and R.Vidal in [11]. Here, the authors obtained the embedding result of fractional Sobolev spaces with variable exponents to variable-exponent Lebesgue spaces. In addition, they also discussed the existence result of a fractional -Laplacian problem.
After that, many mathematicians were concerned with equations involving the operator and studied it extensively, see [12,13,14,15,16,17]. In particular, this combination of fractional -Laplace operators and Kirchhoff functions is very interesting. For example, E. Azroul et al. [13] investigated a class of fractional -Kirchhoff type problems using the mountain pass lemma, direct variational method, Ekeland’s variational principle and concluded the existence of nontrivial weak solutions for the above problem in various cases of the competition between the growth rates of functions. In addition, we recommend that interested readers read the literature [18]. The basic Kirchhoff problem was first introduced by Kirchhoff [19] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic string. Kirchhoff’s model takes into account the changes in the length of the string produced by transverse vibrations. A detailed advancement in the Kirchhoff elliptic problem and its physical interpretation can be seen in [20].
On the other hand, elliptic, parabolic and hyperbolic equations with logarithmic nonlinearity have received extensive attention from many scholars, and many mathematicians have conducted extensive research; see [21,22,23,24,25,26,27]. In particular, we point out that Xiang et al. [26] investigated the existence of two local least energy solutions for fractional p-Kirchhoff problems involving logarithmic nonlinearity by means of the Nehari manifold approach. This method is used essentially because the functional corresponding to the equation is not bounded below in the whole workspace, so it is difficult to find the critical points in the whole workspace, and thus, we need to find the critical points on a smaller set. For more details on this approach, we recommend some very good papers for interested readers [28,29,30,31].
To our best knowledge, there are no results concerned with the Kirchhoff type problem driven by a -fractional Laplace operator with logarithmic nonlinearity. Motivated by the works discussed above, in this paper, we are interested in the existence of two nontrivial weak solutions for the following fractional -Kirchhoff type problems.
(1)
where is a smooth and bounded domain with for any for , is a positive parameter, for any and is a positive function, M is a Kirchhoff function model, is a -fractional Laplace operator, with , defined as follows: for each , along any , where p.v. is considered in the principal value sense.Let
A model of K proposed by Kirchhoff is of the form and if if . When for all , Kirchhoff problems are said to be nondegenerate and this happens, for example, if and in the model case (1), see for instance [20,32,33]. Otherwise, if and for all , the Kirchhoff problems are called degenerate and this occurs in the model case (1) when and , see also [34,35]. An interesting point regarding this problem is the involvement of comes from the fact that is sign changing and behaving at the origin similar to the power function for with a slow growth. In addition, the logarithmic function is not invariant by scaling, which does not hold for the power function. Furthermore, the presence of the variable exponent makes the problem more significant.
To study our main result, we need to make further assumptions.
-
(i). , is symmetric for all .
-
(ii). is a continuous function that satisfies the condition: there exists with such that , where , and .
An example that satisfies our hypothesis could be .
A function is a weak solution to the problem (1), if
for any , where .We are ready to state the main result of this paper.
Let . Assume that the assumptions (i) and (ii) hold. Then, there exists such that for any , problem (1) has at least two nontrivial weak solutions.
2. Functional Analytic Setup
In this section, first of all, we review some basic properties about the variable exponent Lebesgue spaces as well as the fractional Sobolev spaces with variable exponents.
Set
For any we define the variable exponent Lebesgue space as
and the Luxemburg norm defined on this space as,Clearly, is a separable reflexive Banach space, see [36] (Theorem and Corollaries and ).
Hölder’s inequality [14]: Let denote the conjugate space of , where and . If and then the following Hölder-type inequality holds:
A modular of the space is defined by
Assume that and . Then the following assertions hold (see [2]):
Let us set the fractional modular function as
Then the following assertions hold (see [2]):
Assume that and , then
-
(1). ,
-
(2). ,
-
(3). ,
-
(4). ,
-
(5). .
For any , the fractional Sobolev space with variable exponent, is denoted by
with the norm whereReaders may refer to [13,16] for more information related to this space. Define over as the space
and our solution space is defined as the space which is a convex, reflexive and separable Banach space (see [13]) with respect to the normWe will denote in all the upcoming results.
[14] Let denotes a smooth bounded domain and . Let be continuous variable exponents with for and for . Assume that is a continuous function such that , for . Then there exists a constant such that
Thus, the space is continuously embedded in for any . Furthermore, this embedding is compact and the result also holds for the space .
3. The Proof of Result
The functional corresponding to the problem (1) is defined as
which is well defined and of class on . Next, we show the necessity of considering the Nehari manifold.The functional I is not bounded below over .
Let .
Assumption (i) implies that, . Therefore, on passing the limit we conclude that the functional I is not bounded below over . □
Hence, we will seek weak solutions over the Nehari manifold. Define the Nehari manifold as . In particular, if and only if .
The functional I is coercive and bounded below over .
Since, so . This implies that
(2)
Now using above equation, we obtain
Now using assumption (ii) and the Theorem 2, we obtain
Since, and , hence we can conclude that the functional I is coercive and bounded below. □
Now we will divide the Nehari manifold into three sets
where,(3)
There exists such that for , the set is empty.
Let . We will prove the result by contradiction.
Since , so . Using this fact, Theorem 2 and Lemma 1, we obtain
This implies that
(4)
This further implies that
(5)
where, and .Again,
Using and assumption (ii), we obtain
Since, so
Since, coefficient of is negative as so using Theorem 2 and Proposition 1
Choosing small enough, say , so that we obtain a contradiction to (5) for . Hence, the set is empty. □
Since, , so by Lemma 4. Define and .
If , then we have
-
(i)
-
(ii)
.
(i) Let .
(6)
Since, so which implies that
This further implies
(7)
Furthermore,
(8)
Multiplying (8) by and adding to (7), we obtain
This implies that
(9)
Using (9) in (6), we obtain
Now using assumptions on and from (ii), we obtain
Since, from assumption (ii) so coefficient of is negative which along with using Theorem 2 further implies that
For say in the range of , we obtain and hence .
(ii) Let . Then, This implies that
This further implies that
(10)
Multiplying by and adding from (10) we obtain,
This implies
(11)
Using (11) and Proposition 1, we obtain
Thus, , and hence, . □
If , where then the functional I has a minimizer in and .
Since I is bounded below on and so on , there exists a minimizing sequence such that from Lemma 5. Furthermore, I is coercive so is bounded in from Lemma 3 and hence in up to a subsequence. By compact embedding, in for (Theorem 2). Since from assumption (i), so by compact embedding. Thus, as , we obtain and (refer [26]).
Now, we need to show that in . We will prove it by contradiction. Let in then
(12)
Furthermore, . Hence,
Since and from assumption (ii), we obtain
where, . Now taking limit infimum both sides and using (11) and Theorem 2, we obtainThis is a contradiction to for small enough. Hence, in and . Thus, is a minimizer for I on . □
If then the functional I has a minimizer in and .
Since I is bounded below on and so on , there exists a minimizing sequence such that from Lemma 5. Furthermore, I is coercive so is bounded in from Lemma 3, and hence, in up to a subsequence. By compact embedding, in . Furthermore, and =.
Moreover, there exists a constant such that . This can be verified as follows:
For, we have i.e.
Now,
Using from assumption (i) and from assumption (ii), we obtain
There arises two cases and . When , we obtain
Choosing small enough, say , we obtain .
Let , then
Choosing small enough, say , we obtain .
Hence, for .
Now, we will prove that in . Since in so in and in . Hence, . Therefore,
Furthermore, since and , by using continuity of the function K, we obtain
which is a contradiction to and hence . Furthermore, observe that the function attains its maximum at . Thus, we have which is absurd. Hence, in and therefore . Thus, is a minimizer for I on . □By Lemmas 6 and 7, we conclude that there exist and such that and . Hence, we obtain at least two distinct nontrivial weak solutions of the considered problem for , where .
4. Conclusions
In this article, we address the multiplicity of the solutions of an elliptic problem with variable exponents involving logarithmic nonlinearity and a nonlocal term using the analysis of the fibering map and Nehari manifold. The Nehari manifold technique via the fibering map applied for the variable exponents problem is interesting because of the non-homogeneity that arises from the variable exponents. It is likewise well worth citing that due to the presence of the variable exponents, most of the estimates are not maintained straight away, unlike inside the regular exponent set-up. Hence, to overcome this problem, some rigorous analysis has been performed.
Writing—original draft preparation, A.S. and J.Z.; validation, D.C. All authors have read and agreed to the published version of the manuscript.
This research is supported by Natural Science Foundational of Huaiyin Institute of Technology (Grant/Award number: 20HGZ002).
Not applicable.
Not applicable.
Not applicable.
We would like to thank A. Bahrouni (
The authors declare no conflict of interest.
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Abstract
In this paper, we investigate a fractional
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1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China;
2 School of Sciences, Christ (Deemed to be University), Delhi 201003, India
3 Department of Mathematics, NIT Rourkela, Rourkela 769008, India;