Nontrivial Solutions for a Class of p-Kirchhoff Dirichlet Problem

Abstract

This paper is devoted to the following p-Kirchhoff type of problems with the Dirichlet boundary value. We show that the p-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.

1. Introduction and Main Results

Consider the following p-Kirchhoff type of problems with the Dirichlet boundary value:where is a smooth bounded domain in , are real constants, denotes the p-Laplacian operator by , and is continuous on .

We look for the weak solutions of (1) which are the same as the critical points of the functional defined bywhere and is the Sobolev space endowed with the norm

is of and with derivatives given by

Problem (1) began to attract the attention of researchers after the work of Lions [1], where a functional analysis approach was proposed to attack it. Since then, much attention has been paid to the existence of nontrivial solutions, sign-changing solutions, ground state solutions, multiplicity of solutions, and concentration of solutions.

Many researchers studied the existence of weak solution of (1) in -dimensional whole spaces (see [2–7] and references therein). For example, Wu [2] showed that problem has a nontrivial solution and a sequence of high-energy solutions by using the mountain-pass theorem and the symmetric mountain-pass theorem. Similar consideration can be found in Nie and Wu [3], where radial potentials were considered. Jia and Li [6] studied multiplicity and concentration behaviour of positive solutions for Schrdinger-Kirchhoff type equations involving the -Laplacian. Recently, Li and Niu [7] have obtained the existence of nontrivial solutions for the -Kirchhoff type equations with critical exponents by the Ekeland variational principle and mountain-pass lemma.

Kirchhoff-type problem setting on a bounded domain also attracts a lot of attention, see [8–22] and the references therein. For example, we refer to some recent works [8, 9] in which some interesting results on the problems with Dirichlet or Neumann boundary conditions have been obtained. Chung [10] studied Kirchhoff type problems with Robin boundary conditions and indefinite weights. Chen et al. [11] treated problem (1) when , using the Nehari manifold and fibering maps, and they established the existence of multiple positive solutions for (1). In [14–19], the authors studied the existence of solutions of the -Kirchhoff problem in the following form or -Kirchhoff problem such as (1). In [16, 17], the authors obtained the existence and multiplicity of solutions for the -Kirchhoff equation by using the genus theory. In [18, 19], using variational methods, the researchers studied fractional -Kirchhoff equations and received the multiplicity and centration of solutions. In [20, 21], the authors used the fountain theorem and concentration-compactness principle to consider multiplicity of solutions for -Kirchhoff equations. By applying a variant of the mountain-pass theorem and the Ekeland variational principle, Cheng et al. [22] obtained the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity. In [23, 24], the authors studied the equations with nonlinearities which are superlinear in one direction and linear in the other. They use the Ambrosetti-–Rabinowitz (AR) condition to express the superlinear growth at . The AR condition has been used in a technical but crucial way not only in establishing the mountain-pass geometry of the functional but also in obtaining the boundedness of Palais–Smale (PS) sequences. Since the publication of [25], the AR condition has been used extensively throughout the literature (see [26, 27]). However, this condition is very restrictive, eliminating many nonlinearities. There are always many functions that satisfy the AR condition. For example,

Under the motivation of Pei and Zhang [28], the aim of this paper is to consider existence of nontrivial solutions for problem (1) with We shall assume that the nonlinearity term does not satisfy the AR condition and it is asymmetric as approaches and .

Nevertheless, to ensure the global compactness, one needs to impose the subcritical growth condition on the nonlinearity : there exists a position constant such thatwhere . However, under the motivation of Lan and Tang [29], we consider a class of elliptic partial differential equations with a more general growth condition. We assume that the following condition holds:

It is weaker than subcritical growth condition.

We next state the other hypotheses on . Suppose , and this satisfies:  uniformly a.e. , where   uniformly a.e. , where   uniformly a.e.   is increasing when , for a.e.

Theorem 1. Assume satisfies (F) and . If , then problem (1) has at least one nontrivial solution when is not any of the eigenvalues of on .

Theorem 2. Let and assume satisfies (F) and . If and uniformly for a.e. , then problem (1) has at least one nontrivial solution.

Theorem 3. Assume satisfies (F) and . If and , then problem (1) has at least one nontrivial solution.

Here, is the first eigenvalue of and there is a corresponding eigenfunction in . For any is the usual Lebesgue space endowed with the norm

The paper is organized as follows. In Section 1, we introduced the main purpose of this paper and get some conclusions. The proofs of main results will be given in Section 2. We denote various positive constants as or for convenience.

2. Proofs of the Main Results

Proof of Theorem 1.  Step 1 (the condition). Let be a sequence, then we have First, we prove that is bounded. Assume for contradiction that, up to a sequence, as . Since , for every , we can write , namely, Define Obviously, with Then, there exists a subsequence (denoted also by ) such that where and . Dividing both sides of (10) by , we obtain Passing to the limit, we get Let us now prove that for a.e. . To verify this, we chose in (13), and we have where . Moreover, using and by , we have Therefore, if , by Fatou’s lemma, we will obtain that which contradicts (10). Thus, and the claim () is proved. Clearly, . By contradiction , we know that , where . Therefore, we obtain , which is a contradiction. When , we have . If this is the case, by , we first obtain Dividing both sides of (17) by , we obtain Passing to the limit, we get Then, Using Lebesgue’s dominated convergence theorem and for the left hand side of (20), we have Then, (20) can be written as follows: This means that is an eigenvalue, which contradicts our assumption. So, is bounded. Now, we prove that has a convergent subsequence. By the continuity of embedding, we have for all . Going if necessary to a subsequence, we have By (F), for every , we can find a constant such that Let means , we have Hence, is equiabsolutely continuous. Moreover, is equi-integrable and bounded in , and in measure. From the Vitali convergence theorem, it follows that By , for every , we can find a constant such that where for all and are positive constants with the assumption of . Form Hlder’s inequality, for every , we have Let , we have Hence, is equiabsolutely continuous. Moreover, is equi-integrable and bounded in , and in measure. From the Vitali convergence theorem, it follows that Since it follows that Thus, we have , which means that satisfies the condition. Step 2 (mountain-pass geometric structure). has mountain-pass geometry, i.e., there exist and such that Let be a -eigenfunction with , and assume and hold. If there exist so small such that for all , and has mountain-pass geometry. By and , we have Choose such that . By (35), using Poincaré inequality and Sobolev inequality, we get Then, (33) is proved if we choose small enough. On the other hand, by , we have Take such that , and let such that . Using (37), we have Thus, it shows and . Therefore, has mountain-pass geometry. Step 3 (a critical value of ). For in Step 2, we define It turns out that the mountain-pass theorem holds. Then, is a critical value of .

Proof of Theorem 2.  Step 1 (the condition). Let be a sequence, then we have Similar to the proof of Theorem 1, we have and By maximum principle, is an eigenfunction of . When , we have . By our assumption, uniformly in , which implies that On the other hand, which implies that It contradicts (43). Hence, is bounded. According to Step 1, the proof of Theorem 1, we have , which means that satisfies . Step 2 (mountain-pass geometric structure). Let be a -eigenfunction with , and assume and hold. If , similar to the proof of Theorem 1, (33) holds. We only need to show (34) holds. Let be such that . By , we have By (43), we get By our assumption , we haveThus, (34) is proved. Therefore, has mountain-pass geometry.

Proof of Theorem 3.  Step 1 (the condition). Let be a sequence. We can prove that is bounded in . Assume . Similar to the proof of Theorem 1, we have , and when . Let Since is bounded in , we have a subsequence denoted also by ) such that . From the proof of Theorem 1, we know , so , and . By , we have where is a large enough constant. Since , then we get Therefore, we have So, for a.e. . However, if , then . Hence, On the other hand, by as , for a suitable subsequence, we may assume that From (54), we have Then, for any , We set Then, By , hence So, for all . Similarly, we set Then, Hence, So, for all . Therefore, On the other hand, from (54), we have Then, That is, Thus, we have for all . By , we have (defined by (49)) as . Thus, we get Obviously, it contradicts (53). So, is bounded in . According to the proof of Theorem 1, we have which means that satisfies . Step 2 (mountain-pass geometric structure). Let be a -eigenfunction with , and assume and hold. If , for any , there exists so small such that for all , and has mountain-pass geometry. Similar to the proof of Theorem 1, (33) holds. We only need to show that (34) holds. By , we have Let such that . Using (69), we have Thus, it shows and . Therefore, has mountain-pass geometry.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11671331) and National Foundation Training Program of Jimei University (ZP2020057).