Abstract
In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form where , , is a positive parameter, , , is a potential function, and is a Riesz potential of order . Here, with is the Sobolev critical exponent, and is the biharmonic operator. Under certain assumptions on and , we prove that the equation has ground state solutions by variational methods.
1. Introduction
In this article, we study the following fourth-order elliptic Kirchhoff-type equation involving the critical growth of the form: where , , is a positive parameter, , is a potential function, and is a Riesz potential of order defined by . Here, with is the Sobolev critical exponent, and is the biharmonic operator, that is, . Besides, is a potential function satisfying
(V1) and
(V2) , where meas denotes the Lebesegue measure in and and are positive constants.
Furthermore, we suppose that the function satisfies
() as
()
() is increasing on and decreasing on
(f4) is increasing on
On the one hand, in 2012, by the variational methods, Wang and An [1] studied the following fourth-order equation of Kirchhoff type: and obtained the existence and multiplicity of solutions. Later, Cabada and Figueiredo [2] considered a class of generalized extensible beam equations with critical growth in as follows: where is continuous, , , and is a parameter. With the help of the minimax theorem and the truncation technique, the existence of nontrivial solutions of equation (3) is proved for sufficiently large. Recently, Song and Shi [3] proved the multiplicity of solutions for the following fourth-order elliptic equation with critical exponent where is an open bounded domain with smooth boundary, is continuous, , , and is a positive parameter. Soon after that, Liang et al. [4] obtained multiplicity of solutions to the following generalized extensible beam equation with critical growth:
In fact, in the earlier time, Ma has already applied the variational methods to study the existence and multiplicity of solutions for the following fourth-order boundary value problem of Kirchhoff type:
For more details, readers can refer to [5, 6] and the references therein.
Actually, for the special case of problem (2) with , then problem (2) is reduced to the following fourth-order elliptic equations of Kirchhoff type:
This problem is related to the stationary analog of the evolution equation of Kirchhoff type:
Dimensions one and two are relevant from the point of view of physics, engineering, and other sciences, because in those situations model (8) is considered a good approximation for describing nonlinear vibrations of beams or plates (see [7, 8]). Different approaches have been taken to attack this problem under various hypotheses on the nonlinearity. For example, very recently, Wang et al. [9] concentrated on the following Navier BVPs: where is a smooth bounded domain and , , . Applying mountain pass techniques and the truncation method, they obtained the existence of nontrivial solution to equation (9) for small enough when satisfies some superlinear assumptions. For whole space , Song and Chen [10] studied the class of Schrödinger-Kirchhoff-type biharmonic problems: where satisfies the Ambrosetti-Rabinowitz type conditions. Under appropriate assumptions on and , the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.
On the other hand, in the past decades, many scholars have studied the following problem: which is called nonlinear Choquard type equation. For the physical background, we refer to [11–13] and the references therein. Mathematically, the existence and qualitative properties of solutions to equation (5) have been studied for decades by variational methods. See [11, 14, 15] for earlier results and [16–26] for a recent work.
Motivated by the works we mentioned above, especially by [4, 10], we consider the combination of equations (7) and (11) and extend to the general convolution case in . In our paper, we get the ground state solution of problem (1).
Our main results are as follows:
Theorem 1. If satisfies (V1)-(V2) and verifies (f1)-(f4), then problem (1) has a ground state solution.
For the convenience of expression, hereafter, we use the following notations:
equipped with scalar product
equipped with scalar product , therefore, (i) denotes the Lebesgue space with the norm (ii)For any is denoted as
(iii) denote positive constants possibly different in different lines
Remark 2. By the assumptions of , it is obvious that the embedding is continuous. Furthermore, the embedding is continuous for , and compact for , where if and if or 2. Thus, for each , there is a such that for all .
2. Preliminaries
In this section, we will give some very important lemmas.
Lemma 3. Assume (f1)-(f4) hold, then we have the following: (1)For all , there is such that and (2)For all , there is a such that for every , and (3)For any and
Proof. One can easily obtain the results by elementary calculation.
Lemma 4 (Hardy-Littlewood-Sobolev inequality [27]). Let and be such that (1)For any and one has (2)For any one has
Remark 5. By Lemma 3(1), Lemma 4(1) and Sobolev imbedding theorem, we can get
3. Variational Formulation
The associated energy function of problem (1) is given by
i.e. the critical points of the functional are weak solutions of problem (1). Under the assumptions, , and for all , it holds that
Thus,
In this section, we prove the following results.
Lemma 6. The functional possesses the mountain-pass geometry, i.e., (1)There exists such that for all (2)There exists such that and
Proof. (1) By Lemma 3(1) and Lemma 4, we have
Thus, there exists such that for all small enough.
(2) For any ,
as , since ; thus, we see for large. Note that
Taking , with large, we have and .
Hence, we define the mountain-pass level of : where .
Lemma 7. If is a sequence of , then is bounded in .
Proof. Let be sequence, i.e., and ; then, we have Consequently, is bounded in .
Remark 8. By Lemma 7, we can assume that there exists a such that Then, by the similar method as Lemma 3.3 in [4], we can obtain =.
Lemma 9. satisfies condition.
Proof. Let be sequence, i.e., and ; by Lemma 7, is bounded in . Hence, up to a subsequence, we may assume that there exists a such that Then, using the lower semicontinuity of the norm, Brezis-Lieb Lemma [28] and Remark 8, we have Thus, we can have , which implies that strongly converges to in . This completes the proof of Lemma 9.
Remark 10. Now, we recall the Nehari manifold Let then for any , we have Hence, is well defined. Moreover, by the similar argument as Chapter 4 [29], we have the following characterization:
4. Ground State Solution for Problem (1)
In this section, we prove the main theorem.
Proof of Theorem 11. From Lemmas 6 and 7, we know that there exists a bounded sequence , that is, , . Next, let . We claim . On the contrary, by similar argument as Lions’ concentration compactness principle, we can proof in for . By Lemma 3(2), for any , there exists a constant such that
Note that is arbitrary; we get
Combining with and Remark 8, we can get
which implies that
Then, we have , which implies in . We deduce that , which contradicts to the fact that . Hence, , and there exists such that . We set ; then, and . Thus, there exists a such that
Then, for any , we have , which means is a solution of equation (1).
On the other hand, combining with the Fatou Lemma, we can obtain
At the same time, we know by the definition of . Then, we can deduce that is a ground state solution of equation (1). Thus, we complete the proof of Theorem 11.
Data Availability
Data is not available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
LZ curated the data and is responsible for the writing of the original draft. CZ acquired funding and is assigned to the validation. LZ and CZ did the writing, review, and editing.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771198 and 11901276) and Science and Technology Project of Education Department of Jiangxi Province (Grant No. GJJ218406).