Multiplicity and asymptotic behavior of solutions to a class of Kirchhoff-type equations involving the fractional p-Laplacian.

The present study is concerned with the following fractional p-Laplacian equation involving a critical Sobolev exponent of Kirchhoff type: [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are constants, and [Formula: see text] is the fractional p-Laplacian operator with [Formula: see text] and [Formula: see text]. For suitable [Formula: see text], the above equation possesses at least two nontrivial solutions by variational method for any [Formula: see text]. Moreover, we regard [Formula: see text] and [Formula: see text] as parameters to obtain convergent properties of solutions for the given problem as [Formula: see text] and [Formula: see text], respectively.

Introduction and main results

In this paper, we consider the following fractional p-Laplacian equation involving critical Sobolev exponent of Kirchhoff type: where and are constants, is the critical Sobolev exponent, and is the fractional p-Laplacian operator with and which, up to normalization factors, works on the Riesz potential as where is the complement set in of . As for some recent results on the p-Laplacian, we refer to [1-6] and the references therein.

We call Eq. (1.1) a Kirchhoff-type p-fractional Schrödinger equation because of the appearance of the term . Indeed, if we choose , , and let , then (1.1) transforms to the following classical Kirchhoff-type equation: which is degenerate if and non-degenerate otherwise. Equation (1.2) arises in an interesting physical context. In fact, if we set and replace by a bounded domain in (1.2), then we get the following Kirchhoff Dirichlet problem: which is related to the stationary analog of the equation proposed by Kirchhoff in [7] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. This model takes the changes in length of the string produced by transverse vibrations into account. After Lions in his pioneering work [8] presented an abstract functional analysis framework to use for (1.2), this problem has been widely studied in extensive literature such as [9-13]. In view of the above facts, it is reasonable to consider the p-fractional Kirchhoff equation.

When , , and let , then (1.1) can be reduced to the following fractional Schrödinger equation: which was used to study the standing wave solutions for the equation where ħ is the Planck constant, is an external potential and k is a suitable nonlinearity. Since the fractional Schrödinger equation appears in problems involving nonlinear optics, plasma physics and condensed matter physics, it is one of the main objects of fractional quantum mechanics. To learn more, the reader can refer to [14-23] and the references therein.

Very recently, great attention has been paid to the study of fractional p-Laplacian problems. For example, Pucci–Xiang–Zhang [2] were concerned with the nonhomogeneous Schrödinger equations involving the fractional p-Laplacian of Kirchhoff type where M is the so-called Kirchhoff function, satisfies the subcritical growth. They employed the mountain-pass theorem and Ekeland’s variational principle to prove that the existence of at least two solutions for (1.4). In [24], Xiang–Zhang–Zhang studied problem (1.1) with and they obtained infinitely many solutions when and for different . They also proved the existence of multiple solutions for suitable . Subsequently, if , Wang–Zhang [25] established the existence of infinitely many solutions which tend to zero for suitable positive parameters ξ and τ by the Kajikiya version of the symmetric mountain-pass theorem. Some other important and meaningful results on the p-fractional Schrödinger equation of Kirchhoff type can be found in [26-30] and the references therein.

Before stating our main results, we introduce some useful notations and definitions. Let denote the completion of with respect to the norm and . We write for the closure of with respect to the norm . Since a finite measure on is a continuous linear functional on , for a measure μ we write Throughout this paper we shall denote C and () for various positive constants whose exact value may change from line to line but are not essential to the analysis of the problem. is the usual Lebesgue space with the standard norm . We use “→” and “⇀” to denote the strong and weak convergence in the related function spaces, respectively. Let be a Banach space with its dual space , and Ψ be its functional on X. The Palais–Smale sequence at level ( sequence in short) corresponding to Ψ satisfies and as , where .

Motivated by all the work mentioned above, we are interested in the multiplicity and asymptotic behavior of solutions for problem (1.1) whose natural variational functional is given by Note that we can employ the idea used in [31] (or [2]) to prove that is well defined on and of class . Furthermore, any solution of (1.1) is a critical point of . Hence we obtain the solutions of it by finding the critical point of the functional . To this aim, we assume the following condition:

with and .

Definition 1.1

We say that is a (weak) solution of (1.1) if for all .

Our first result is as follows.

Theorem 1.2

Assume (F) and , then for any there exists a constant such that Eq. (1.1) has at least two nontrivial solutions, and , satisfying

Remark 1.3

We point out here that if in (1.1), the results in Theorem 1.2 can be seen as a part of [24]. Although the generalization in this sense is trivial, the main interest of this paper is not here, but more attention is paid to the relation between the solutions obtained in Theorem 1.2 and the parameters and , and the convergent properties (see Theorems 1.5 and 1.7 below) of the solutions are given. Also, our results extend the results of [32] to fractional Kirchhoff type. Briefly speaking, if , and in (1.1), the results in Theorem 1.2 can be found in [32].

Remark 1.4

When the nontrivial solutions of (1.1) are obtained, we can prove that the existence of ground state solutions of it. In fact, with Theorem 1.2 in hand, we know that and are well defined. Hence any minimizing sequence of m is bounded, then by Lemmas 2.6–2.7 below we derive that m is attained by some function and it is a ground state solution.

It is worth mentioning that the idea of proving the asymptotic behavior of solutions to (1.1) comes from [12, 33]. Since the solutions and obtained in Theorem 1.2 depend on the parameter b, we next denote and by and to emphasize this dependence, respectively. By analyzing the convergence property of and as , we establish one of the following main results in this paper.

Theorem 1.5

Assume (F) and , let and be fixed constants, if and are nontrivial solutions of (1.1) obtained in Theorem 1.2, there exist subsequences still denoted by themselves and such that in as for , where and are two nontrivial solutions of

Remark 1.6

If the whole space is replaced with a bounded domain Ω and assume suffciently small, Lei–Liu–Guo [13] proved that problem (1.1) admits at least two nontrivial solutions when , , and . In a more general case, Theorem 1.5 tells us that the solutions of problem (1.1) are actually the solutions of problem (1.5) if the positive parameter b is small enough.

Inspired by Theorem 1.5, the solutions of problem (1.1) also depend on the parameter and then we have the following result.

Theorem 1.7

Assume (F) and , then there exists such that the problem (1.1) admits at least two nontrivial solutions. Furthermore if we let and be fixed constants and denote and are nontrivial solutions of (1.1) obtained above, then there exist subsequences still denoted by themselves and such that in as for , where and are two nontrivial solutions of

Remark 1.8

In this paper, we only consider the convergence of the solutions with and as the parameters, respectively. It is natural to raise the following two open problems: (i) Do our results still remain valid when and ? (ii) If we take as the parameter and let the positive constants a and b be fixed, does the convergent property of the solutions still exist when ? Xiang–Zhang–Zhang [24] studied the existence of solutions for problem (1.1) with and , but from our point of view, it seems to be different when it comes to taking as a parameter.

We note that, to the best of our knowledge, there is no result on asymptotic behavior of solutions of critical Kirchhoff-type equations involving the fractional p-Laplacian. We now sketch our proofs of Theorems 1.2, 1.5 and 1.7 based on variational method. What makes the proof of Theorem 1.2 more complicated is not only the lack of compactness imbedding of into , but also how to estimate the critical value. To deal with the difficulties mentioned above, some arguments are in order. Using the idea of the well-known Brézis–Nirenberg argument [34], we obtain the threshold value by solving a quadratic algebra equation with one unknown, where is the best Sobolev constant, that is, After pulling the mountain-pass energy level down below the critical value, we use the celebrated concentration–compactness principle developed by Lions [35] and extended to the fractional Sobolev space at some level by Xiang–Zhang–Zhang [24] to show that any (PS) sequence of contains a strongly convergent subsequence. As to the proof of Theorem 1.5, Although most difficult, the lack of compactness imbedding of into has been solved, we cannot draw the conclusion that the two sequences of solutions of (1.1) converge to some functions which are nontrivial solutions of (1.5). To overcome it, we have to further estimate the mountain-pass value and local minimum carefully; see (4.3) below for example. Compared with the proof of Theorem 1.5, there are some necessary modifications. For example, Lemma 2.5 below which plays a vital role in the proof Theorem 1.2 can never take positive effect when we take as a parameter. Therefore, we can successfully prove Theorems 1.2, 1.5 and 1.7 step by step.

The outline of this paper is as follows. In Sect. 2, we present some preliminary results for Theorem 1.2. In Sect. 3, we obtain the existence of two nontrivial solutions of problem (1.1). In Sects. 4 and 5, we prove the convergent properties on the parameters and , respectively.

Some preliminaries

In this section, we first recall the concentration–compactness principle in the setting of the fractional p-Laplacian and then investigate the mountain-pass geometry and the behavior of the sequence. The following definition can be found in [31].

Definition 2.1

Let denote the finite nonnegative Borel measure space on . For any , . We say that weakly ∗ in , if holds for all as .

The proofs of the Propositions 2.2–2.4 can be found in [24].

Proposition 2.2

Let with upper bound for all and Then where J is at most countable, are positive constants, is the Dirac mass centered at , μ̅ is a non-atomic measure, is given by (1.8) and

Proposition 2.3

Let be a bounded sequence such that and for any we define Then the quantities and are well defined and satisfy Moreover,

Proposition 2.4

Assume that is the sequence given by Proposition 2.2, let be fixed and ϕ be a smooth cut-off function such that , when , when and . For any , we set for any , then

Now we will verify that the functional J exhibits the mountain-pass geometry.

Lemma 2.5

There exists such that the functional satisfies the mountain-pass geometry around for any , that is,

there exist such that when and ;

there exists with such that .

Proof

(i) It follows from (1.8) and Hölder’s inequality that Therefore if we set then there exists such that when for any .

(ii) Choosing , then since and is nonnegative one has Hence letting with sufficiently large, we have and . The proof is complete. □

By Lemma 2.5, and the mountain-pass theorem in [31], a sequence of the functional at the level can be constructed, where the set of paths is defined as In other words, there exists a sequence such that

As the existence of the critical Sobolev exponent in (1.1), we have to estimate the mountain-pass value given by (2.2) carefully. Thanks to the results in [36], there exists a positive function satisfying and .

Lemma 2.6

There exists such that the mountain-pass value satisfies for any , and S are given by (1.7) and (1.8), respectively.

It is obvious that there exists independent of b such that

We then claim that Indeed, let us define where By some elementary calculations, we have which is equivalent to Since , we know that has a unique root, that is, where we use the fact that . Therefore we can conclude that which together with the fact is nonnegative gives (2.4).

Since , there exists such that On the other hand, the facts and (2.5) show that Taking then we have Finally, choosing we can deduce that which yields the proof of this lemma. □

The following lemma provides the interval where the condition holds for .

Lemma 2.7

If , any sequence satisfying (2.2) contains a strongly convergent subsequence whenever , where is given by (1.7).

Let be a sequence verifying (2.3) and we conclude that is bounded in . Recalling that , then we have which shows that is bounded in since . Up to a subsequence if necessary, there exists such that in , in for and a.e. in . Obviously, the conclusions in Proposition 2.2 are true in the sense of a subsequence. Now we prove that in .

To do it, we first claim that the set J given by Proposition 2.2 is an empty set. Arguing it by contradiction, for some and for any choosing to be a smooth cut-off function such that , when , when and . It follows from Proposition 2.2 that and where we have used Proposition 2.4. We also know that and Since is bounded, we have , that is, It is easy to see that Coming the above six formulas, we have . In view of (2.1) and , we obtain which gives that Using Proposition 2.2 and (2.3) again, we derive a contradiction. Hence we have .

We then claim that the quantities and given by Proposition 2.3 satisfy . For any , let to be a smooth function such that , when , when and . Now repeating the same process of proving the above claim, we can obtain .

Finally, based on the above two claims and [24, Lemma 4.5], we have in . Therefore and . The proof is complete. □

The proof of Theorem 1.2

In this section, we will prove Theorem 1.2 in detail.

Existence of a first solution for (1.1)

Proof

Let be given as in Lemma 2.6, then for any there exists a sequence verifying (2.3) by Lemma 2.5. In view of the proof of Lemma 2.7, we know that there exists a critical point of J such that . Hence is a nontrivial solution of (1.1). □

Existence of a second solution for (1.1)

Before we obtain the second solution, we introduce the following proposition.

Proposition 3.1

(Ekeland’s variational principle [37], Theorem 1.1)

Let V be a complete metric space and be lower semicontinuous, bounded from below. Then, for any , there exists some point with

We are in a position to show the existence of a second positive solution for (1.1).

For given by Lemma 2.5(i), we define and clearly is a complete metric space with the distance . It is obvious that the functional J is lower semicontinuous and bounded from below on (see [31]).

We claim first that Indeed, choosing a nonnegative function and then we have Therefore there exists a sufficiently small such that and , which imply that (3.1) holds. By Proposition 3.1, for any there exists such that Then we claim that for sufficiently large. In fact, we will argue it by contradiction and just suppose that for infinitely many n, without loss of generality, we may assume that for any . It follows from Lemma 2.5 that and by (3.2) we have which is a contradiction to (3.1). Next, we will show that in . Indeed, set where small enough such that for fixed n large, then which imply that . So it follows from (3.2) that that is, Letting , then we have for any fixed n large. Similarly, choosing and small enough, and repeating the process above we have for any fixed n large. Therefore the conclusion as for any implies that in .

Hence, we know that is a sequence for the functional with . Therefore, Lemma 2.7 implies that there exists a function such that and . Hence is a nontrivial solution of (1.1). □

Asymptotic behavior as

In this section, we prove Theorem 1.5. In the following, we regard as a parameter in problem (1.1) and analyze the convergence property of as for . The variational functional corresponding to (1.5) is given by which is of class of due to [31] (or [2]). For any , we have where M is independent of b. Let () be the solutions of (1.1) obtained in Theorem 1.2, that is, and where

Proof of Theorem 1.5

To present the proof clearly, we will split it into several steps:

Step 1: there exist four constants independent of such that In fact, the constant given by Lemma 2.5 is independent of any , then by (2.2) we have . On the other hand, using (4.2) we have where is given by Lemma 2.6. We choose a nonnegative function to satisfy . Since we can let such that , where is given by Lemma 2.5(ii). Therefore we can obtain So the proof of Step 1 is complete.

Step 2: the sequences () contain strongly convergent subsequences.

By (4.1) and (4.2), we know that () are sequences of the functionals . We claim that () are bounded. In fact, which shows that are bounded in since . With (4.1) and (4.2) at hand, we can see Lemma 2.7 as a special case to show that the sequences () contain strongly convergent subsequences with . Hence there exist subsequences still denoted by themselves and such that in as for . Therefore, we have which shows that are solutions of (1.5) for .

Step 3: .

Indeed, and

Summing the above three steps, we see that and are two nontrivial solutions of (1.5). The proof is complete. □

In this section, we regard as a parameter in problem (1.1) and analyze the convergence property. To do it, we have to prove that problem (1.1) admits at least two nontrivial solutions again. We introduce the following variational functional: to emphasize the independence of . In order to eliminate the influence of parameter , we have the following lemma which is different from Lemma 2.5.

Lemma 5.1

There exists such that the functional satisfies the mountain-pass geometry around for any , that is:

there exist such that when and ;

there exists with such that .

(i) It follows from (1.8) and Hölder’s inequality that Therefore if we set then there exists such that when for any .

(ii) Choosing , then since and is nonnegative one has Hence letting with sufficiently large, we have and . The proof is complete. □

By Lemma 5.1, and the mountain-pass theorem in [31], a sequence of the functional at the level can be constructed, where the set of paths is defined as In other words, there exists a sequence such that

The following two lemmas are very similar to Lemmas 2.6 and 2.7, respectively.

Lemma 5.2

There exists such that the mountain-pass value satisfies for any , and S are given by (1.7) and (1.8), respectively.

Lemma 5.3

If , any sequence satisfying (2.2) contains a strongly convergent subsequence whenever , where is given by (1.7).

Remark 5.4

Since , always where is independent of a. Consequently, in addition to the proper adjustment of , the proof of Lemma 5.2 is exactly the same as that of Lemma 2.6. The above formula is applied to Eq. (2.6) to get a contradiction, hence we can prove Lemma 5.3.

In view of Sect. 3.2 and using Lemmas 5.1–5.3, we have the following proposition.

Proposition 5.5

Assume (F), then for any there exists a constant such that Eq. (1.1) has at least two nontrivial solutions, and , satisfying

Now let and be fixed; we have the following.

Proposition 5.6

Let and be nontrivial solutions of (1.1) obtained in Proposition 5.5, then there exist subsequences still denoted by themselves and such that in as for , where and are two nontrivial solutions of (1.6).

For any , there exists independent of a such that Recalling Steps 1–3 in the proof of Theorem 1.5, we have the following facts.

Fact 1: there exist four constants independent of such that where .

Fact 2: the sequences () contain strongly convergent subsequences with . Hence there exist subsequences still denoted by themselves and such that in as for . Therefore, we have which shows that are solutions of (1.6) for .

Fact 3: , where

Therefore we know that and are two nontrivial solutions of (1.6). The proof is complete. □

Proof of Theorem 1.7

In view of Sect. 3.2 and using Lemmas 5.1–5.3, we know that for any there exists a constant such that (1.1) has at least two nontrivial solutions, and , satisfying Now we use Proposition 5.6 to obtain the desired result directly. The proof is complete. □

Conclusion

This paper is concerned with the qualitative analysis of solutions of a nonlocal problem driven by the fractional p-Laplace operator. A key feature of this paper is the presence of the critical Sobolev exponent of Kirchhoff-type. We are interested both in the existence of solutions and in the multiplicity properties of the solutions. We also establish the convergence of solutions as the positive parameters converge to zero. There are obtained several very nice results and the variational arguments play a central role in the arguments developed in this paper. Finally, we obtain the threshold value by solving a quadratic algebra equation with one unknown which does not seem to have appeared in previous literature.