The Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity
In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a -linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that . The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter .
1. Introduction and the Main Results
In this paper, we are devoted to investigating the existence of the least energy sign-changing solutions for the following p-Kirchhoff-type problem with a (2p-1)-linear growth nonlinearity:where is a bounded domain in , , , is the first eigenvalue of the following problem:
In fact, the related problems have been studied extensively, especially on the existence of the positive solutions, multiple solutions, ground state solutions, and least energy sign-changing solutions. In , Li and Sun studied the existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities; the mountain pass theorem was used in the paper. Guo, Ma, and Zhang  studied a class of autonomous Kirchhoff-type equation. By a simple transformation, they found that the solutions of autonomous Kirchhoff-type equation or system could be obtained by using the known solutions of the corresponding local equation or system, which is very interesting. In , Ying Li and Lin Li considered the existence and multiplicity of solutions to a class of p(x)-Laplacian-like equations. They introduced a revised Ambrosetti-Rabinowitz condition and obtained that the problem had a nontrivial solution and infinitely many solutions, respectively. Meanwhile, in , Luca Vilasi proved an eigenvalue theorem for a stationary p(x)-Kirchhoff problem by using variational techniques, and the author also provided an estimate for the range of such eigenvalues. For more details, we refer the reader to [5–30].
In [31, 32], the authors studied the following Kirchhoff-type problems in bounded domains:under different assumptions on , the authors mainly use the quantitative deformation lemma and the degree theory to get the existence of the least energy sign-changing solution and its corresponding convergence property as the parameter From the assumptions on , we can easily find that both in [31, 32] satisfies 3-superlinear growth at infinity and superlinear growth at zero.
Motivated by the above works, a natural question is that if there exists a ground state sign-changing solution for problem (1). However, up to now, no paper has appeared in the literature which discusses the existence and convergence property of the solution for the p-Kirchhoff-type problem with a -linear growth nonlinearity. This paper attempts to fill this gap in the literature.
Throughout this paper, we will make full use of the following notations. Let be the usual Sobolev space equipped with the following norm: denotes the usual Lebesgue space norm. is the best Sobolev constant for the embedding of in ; that is,
From the above definition, we give the energy functional corresponding to problem (1) byClearly, is well defined on and is of class. For each , by a simple calculation, we haveObviously, the critical points of are corresponding to the weak solutions of problem (1). If is a sign-changing solution of problem (1), then
(i) is a solution of problem (1), that is, is a critical point of ;
(ii) , where .
When , problem (1) reduces to the following problem:
The corresponding energy functional is defined by Also, we can compute that
For , problem (1) is called a nonlocal problem since the appearance of the nonlocal term . The differences posed by the nonlocal term make the method in solving problem (11) cannot be applied to solve problem (1), which makes the study of our paper very interesting and meaningful.
To get the ground state solutions, we define the following sets:and consider the following minimization problem:
Since , we can immediately get . The main results of the paper are described as follows.
Theorem 1. For and , problem (1) has at least one ground state sign-changing solution, which precisely has two nodal domains. Moreover, .
Theorem 2. For each , for any sequence small enough with as , there exists a subsequence still denoted by , such that convergent to strongly in , where is a ground state sign-changing solution of problem (11), which changes sign only once.
2. Some Critical Preliminaries
The following several lemmas are crucial to prove our main results.
Lemma 3. If , , satisfies andthen there exists a unique pair of positive numbers such that
Proof. (i) If , then from (7), (9), and (10), we haveandLet and , the above equations correspond to the following system:Obviously, if we can prove that system (20) has the unique solution , then is the unique solution for (18) and (19).
LetFor , we have . Since , then Similarly, we haveFrom (21)-(23), we have , and is the unique solution for system (20). Accordingly, is the unique positive solution for (18) and (19). Thus, (i) is proved.
(ii) Next, we give the proof of (ii).
From (6), we haveBy a simple computation, we haveandFrom , we haveandWe consider the Hessian matrix of ; then from (17), we have The above deduction implies that is a maximal point of for . Since we cannot get the maximal point of on the boundary of , is the unique maximal point; that is,
Lemma 4. Assume that and , then (17) holds.
Proof. For , we haveSince , we have . Thus, we have
Lemma 5. Assume that , with and , there exists a unique pair such that .
Proof. If with and , we haveSince , thenFrom Lemma 3, there is a unique pair of positive numbers such that , which implies that is the solution of system (20). Then, we haveTherefore, we have . Similarly, we have . Thus, there exists a unique pair such that .
Lemma 6. If for any with , there exists a unique such that . Moreover, for all and .
Proof. If and satisfies , implies thatThus, there exists a unique satisfying (35). From (6), we haveBy a simple deduction, we haveThus, attains its maximal point at . In other words, we have for all and .
Lemma 7. Assume ; we have that
(i) if , is attained by some and is a constant sign critical point of , where is given by (5);
(ii) if , is attained by some and is a sign-changing critical point of .
Proof. (i) Firstly, we will show that for all , there exists such that , which implies . From (5), we know that there exists such that For , we haveThus, we have .
For each , it follows from and (5) thatThen, . Thus, we havethat is, and is coercive and bounded below on for and .
Let be a minimizing sequence for . From and , we assume that in for all . Since is coercive and bounded below on , the sequence is bounded in , so that, up to subsequences, in and . Next, we will prove that strongly in . We suppose by contradiction that . Therefore, we haveIf , the above inequality makes a contradiction. Thus, we have in . From the fact that , we have . By Lemma 6, there exists a unique such that and for all Thus, we havewhich leads to a contradiction. Therefore, we have , strongly in and . Then, by a standard argument, which is similar to the discussion in , we can deduce that is a constant sign critical point of .
(ii) From a similar deduction as (i), we know that for , there exists such thatObviously, if such that satisfies (43), then also satisfies (43). Therefore, we assume that a.e. in . We let and define for all , where and . Then, from (43), we haveLet ; we can obtain that and andthat is,Similarly, we also haveBy Lemma 3, we know that for .
Assume that is a minimizing sequence for , such that . Since is coercive on , the sequence is bounded in ; going if necessary to a subsequence, still denoted by , we can assume that there exists a such that for sufficiently large,From , we have ; that is,Therefore,In the same way, we have and Passing to the limit, we havewhich implies that andFrom , Lemmas 3 and 5, there exists a unique pair such thatFrom the definition of , we haveThus, , , and , is the required minimizer.
Next, we will prove that is indeed a sign-changing solution; that is, . We mainly use the quantitative deformation lemma  to prove the results.
If , there exists and , such thatLet , , and . It follows from Lemma 3 thatLet and ; there exists a deformation such that
(i) if ;
From (56), Lemma 3 and (ii), we can easily getWe prove that , which contradicts the definition of . We define and