Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy-Sobolev exponent and singular nonlinearity.

In this paper, we study a class of critical elliptic problems of Kirchhoff type: [ a + b ( ∫ R 3 | ∇ u | 2 - μ u 2 | x | 2 d x ) 2 - α 2 ] ( - Δ u - μ u | x | 2 ) = | u | 2 ∗ ( α ) - 2 u | x | α + λ f ( x ) | u | q - 2 u | x | β , where a , b > 0 , μ ∈ [ 0 , 1 / 4 ) , α , β ∈ [ 0 , 2 ) , and q ∈ ( 1 , 2 ) are constants and 2 ∗ ( α ) = 6 - 2 α is the Hardy-Sobolev exponent in R 3 . For a suitable function f ( x ) , we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b > 0 as a parameter to obtain the convergence property of solutions for the given problem as b ↘ 0 + by the mountain pass theorem and Ekeland's variational principle.

Introduction and main results

In the present paper, we consider the following Schrödinger equation: where , , , and are constants and is the critical Hardy–Sobolev exponent.

We call (1.1) a Schrödinger equation of Kirchhoff type because of the appearance of the term which makes the study of (1.1) interesting. 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 a meaningful physical context. In fact, if we set and replace by a bounded domain , then we get the following Dirichlet problem: which is related to the stationary analogue of the equation proposed by Kirchhoff in [16] as an extension of the classical D’Alembert’s 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 J. L. Lions in his pioneer work [21] presented an abstract functional analysis framework to (1.2), this problem has been widely studied in extensive literature such as [8, 11, 12, 19, 20, 24, 25].

In their celebrated paper, Ambrosetti et al. [2] studied the following semilinear elliptic equation with concave-convex nonlinearities: where Ω is a bounded domain in , and with . By the variational method, they obtained the existence and multiplicity of positive solutions to the above problem. Subsequently, an increasing number of researchers have paid attention to semilinear elliptic equations with critical exponent and concave-convex nonlinearities; for example, see [1, 5, 13, 14, 27, 29] and the references therein.

Using the Nehari manifold and fibering maps, Chen et al. [6] extended the above analysis to the subcritical semilinear elliptic problem of Kirchhoff type: where M is the so-called Kirchhoff function depending on , Ω is a bounded domain with a smooth boundary in and the weight functions satisfy some specified conditions they proved the existence of multiple solutions of it. In the critical case, Lei et al. [19] considered the following Kirchhoff problem in three dimensions: where is a sufficiently small constant, and they employed the mountain pass theorem to show that the problem admits at least two different positive solutions. Some other related and important results can be found in [18, 23] and the references therein.

Before stating our main results, we introduce some function spaces. Throughout the paper, () is the usual Lebesgue space with the standard norm , and we consider the Hilbert space equipped with its usual inner product and norm By the well-known Hardy inequality [17] we derive that the induced inner product and norm are equivalent to the usual inner product and norm on for any . As a special case of [15, Lemma 2.3], for any and , we can define We also know that can be attained by a positive function satisfying

Motivated by all the works mentioned above, we are interested in the multiplicity and asymptotic behavior of solutions of (1.1) whose natural variational functional is Note that we can adopt the idea used in [28] 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 points of the functional . To this aim, we assume the following condition:

and there exists such that .

Since , using Hölder’s inequality and (1.3), we have For the convenience of narration, we set and where is given by Lemma 3.3 and only depends on .

Remark 1.1

It is easy to see that the constants for are independent of b, and then is also independent of b.

We are ready to state our first result.

Theorem 1.2

Assume (F), , , and , then for any problem (1.1) admits at least one positive solution for and two positive solutions for .

Remark 1.3

If the whole space is replaced by a bounded domain Ω and with , Theorem 1.2 can be seen as an improvement of the main results in [3, 7]. On the other hand, Theorem 1.2 extends the results of [6] to a more general case.

Inspired by the works in [8, 24, 25], we prefer to study the asymptotic behavior of multiple solutions to (1.1) because the solutions depend on the parameter b. By analyzing the convergence property, we establish the following result in this paper.

Theorem 1.4

Assume (F), , , and , then (1.1) has at least two positive solutions and for any . Moreover, let and be fixed constants, then there exist subsequences still denoted by themselves and such that in as for , where and are two nontrivial solutions of

Remark 1.5

A natural question is why we do not study the convergence of solutions obtained in Theorem 1.2. In fact, if we do this step by step, we can only prove that equation (1.6) has at least one nontrivial solution. The main reason for this phenomenon is that we cannot prove there exists independent of b such that (see Lemma 2.5 for details). To explain this in a little more detail, we assume there exists a sequence of solutions of (1.1) satisfying . By a standard method, we can show that there exists such that in as . Unfortunately, we fail to prove as , which yields .

The outline of this paper is as follows. In Sect. 2, we present some preliminary results. In Sect. 3, we obtain the existence of two local minimax solutions of (1.1). In Sect. 4, we prove the convergence property on the parameter .

Notations

Throughout this paper we shall denote by C and () various positive constants whose exact value may change from lines to lines but are not essential to the analysis of problem. We use “→” and “⇀” to denote the strong and weak convergence in the related function space, respectively. For any and any , denotes the ball of radius ρ centered at x, that is, .

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 that and as , where .

Nehari manifold and fibering map

In this section, we study the so-called Nehari manifold because the variational functional is not bounded from below on . Let us define and then any nontrivial solution of (1.1) belongs to . Obviously, if and only if

The following lemma tells us the behavior of on .

Lemma 2.1

The functional is coercive and bounded from below on .

Proof

For any , since and , we get which yields that is coercive and bounded from below on . □

The Nehari manifold is closely linked to the functions for any . As we all know, the above maps were introduced by Drábek and Pohozaev [9] and discussed in Brown and Zhang [4] (or Chen et al. [6]). For any , we have It is easy to see that for any and we obtain which gives that if and only if . In particular, if and only if . Arguing as Brown and Zhang [4], we split into three parts: Therefore, for any , we have

It is similar to the argument in Brown and Zhang [4, Theorem 2.3] that we can derive the following result.

Lemma 2.2

Suppose is a local minimizer for on and , then in .

Inspired by the above lemma, we will study when is established.

Lemma 2.3

If , then .

We argue it indirectly and assume that, for any , using (2.1) and (2.2) we have and by (1.5) which yields that

On the other hand, using (2.1) and (2.2) again we have and by (1.5) which yields that Combining (2.3) and (2.4), we obtain , which is a contradiction. Hence for any . The proof is complete. □

To find solutions of (1.1), it is necessary to consider whether are nonempty.

Lemma 2.4

Assume (F) and for any , then for any there exist and unique and with such that and

Compared with the results in [6], the proof is standard after some simple modifications and we omit it. □

From Lemma 2.3, we know that for any . Moreover, by Lemma 2.4 we have and by Lemma 2.1 we may define Then we have the following result.

Lemma 2.5

Under the assumptions of Theorem 1.2, we have

If , then ;

If , then there exists independent of b such that . In particular, we have .

(i) For any , by (2.1) we know which implies that Thus we obtain that .

(ii) To end the proof, we split it into the following two cases.

Case 1: .

Similar to (2.3), we can derive Then, for any and by (1.5), we have that Combining (2.5) and (2.6), we know that if , there exists independent of b such that .

Case 2: .

Similar to (2.4), we can derive Then, for any and by (1.5), we have that Combining (2.7) and (2.8), we know that if , there exists independent of b such that . The proof is complete. □

Proof of Theorem 1.2

In this section, we prove Theorem 1.2. Using Ekeland’s variational principle [10] and the argument in [6, Lemma 5.2], we have the following result.

Lemma 3.1

Under the assumptions of Theorem 1.2, we have

If , then has a sequence ;

If , then has a sequence .

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

Lemma 3.2

If , any sequence of contains a strongly convergent subsequence whenever , where and is a positive constant given by Lemma 3.3 below.

Let be a sequence of , and we conclude that is bounded in . In fact which yields that is bounded in since . Up to a subsequence if necessary, there exists such that in , in for and a.e. in . Next we prove that in .

By the concentration compactness principle [22], there exist a countable set Γ, a set of different points , nonnegative real numbers , for , and nonnegative real numbers , , and such that where is the Dirac mass at . Without loss of generality, we only consider the possibility of concentration at the singular point . To do it, for any , we let for all and choose to be a smooth cut-off function such that , when , when and . Then Since is bounded, using (3.2) we have In view of Sobolev inequality (1.3), that is, , we derive which gives that Therefore we have a contradiction! Hence we have which together with (1.5) implies Hence there holds which yields that in . The proof is complete. □

To apply in Lemma 3.2, we have the following result.

Lemma 3.3

Under the assumptions of Theorem 1.2, there holds for any . In particular, for any .

For given by (3.1), we have that Let us define As a consequence of (1.4), we have By some elementary calculations, we have which is equivalent to Since , we know that has a unique root, that is, Therefore we can conclude that which implies that

Since , there exists only depending on such that On the other hand, by (3.3) we have that which gives Finally, we can deduce that Since , by Lemma 2.4 there exists unique such that . Consequently, we have , which completes the proof. □

Now, we establish the existence of a local minimum for on .

Proposition 3.4

Assume (F), , , and , then for any there exists such that

is a positive solution of (1.1) and ;

as .

(i) In view of Proposition 3.1(i), any minimizing sequence of m can be chosen as a sequence of , that is, By Lemma 2.1, we know that is bounded in . Going to a subsequence if necessary, there exists such that in . It follows from the definitions of m and that . Hence in by Lemmas 3.2–3.3, then and . Since , we can derive is a nontrivial solution of (1.1) by Lemma 2.2. By the fact that is translation invariant, we know that and . By using Harnack’s inequality [26], it follows that in and then is a positive solution of (1.1). We now claim that . Indeed, we argue it indirectly and assume by Lemma 2.3. It follows from Lemma 2.4 that there exist unique and such that with . By the same idea used in [6, Lemma 4.2], we know that is strictly increasing on and hence a contradiction! So, we can obtain , which implies that . Consequently, the proof of (i) is complete.

(ii) Since , then similar to (2.3) and (2.4) we have which yields as . The proof is complete. □

Next, we establish the existence of a local minimum for on .

Proposition 3.5

Assume (F), , , and , then for any there exists such that

;

is a positive solution of (1.1).

It follows from Proposition 3.1(ii) that there exists a sequence of , Hence is bounded in by Lemma 2.1 and there exists such that in in the sense of a subsequence. Using Lemmas 3.2–3.3, we obtain in and then and . In view of Lemma 2.2 and Lemma 2.5(ii), we know that is a nontrivial solution of (1.1). Similar to Proposition 3.4, we have that is positive. The proof is complete. □

We are now in a position to complete the proof of Theorem 1.2.

Proof of Theorem 1.2

The part (i) is a corollary of Proposition 3.4. If , we can obtain two positive solutions and of (1.1) by Propositions 3.4–3.5. The definitions of give us , then we know that and are two different positive solutions of (1.1). □

Asymptotic behavior as

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 .

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

Lemma 4.1

The functional satisfies the mountain pass geometry around for any , that is,

there exist independent of b such that when ;

there exists with such that .

(i) It follows from (1.3) and (1.5) that where Therefore there exists such that when for any .

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

By Lemma 4.1 and the mountain pass theorem in [28], 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

Remark 4.2

By (4.1), we can conclude that for any . In fact, in view of the proof of Lemma 3.3, we obtain for any . As the proof of Lemma 4.1(ii), there exists sufficiently large such that . Hence let , then , which yields for any .

To obtain a solution with negative energy, we introduce the following lemma.

Lemma 4.3

(Ekeland’s variational principle [10], 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

Now, we establish the existence of multiple solutions of (1.1).

Proposition 4.4

Assume (F), , , and , then equation (1.1) has at least two positive solutions and satisfying

Let satisfy (4.2), by Lemma 3.2 and Remark 4.2 we derive there exists such that and .

On the other hand, for given by Lemma 4.1(i), we define clearly is a complete metric space with the distance . It is obvious that the functional is lower semicontinuous and bounded from below on . We claim that Indeed, choosing a nonnegative function , we have Therefore there exists a sufficiently small such that and , which imply that (4.3) holds. By Lemma 4.3, for any , there exists such that Then a standard procedure gives that is a bounded sequence of . Therefore, by Lemma 3.2 and (4.3), there exists such that and . It is similar to Proposition 3.4 that and are positive. □

For , we can obtain two sequences and of solutions of (1.1) by Proposition 4.4, that is, and The variational functional corresponding to (1.6) is given by which is of class of due to [28]. For any , we have where is independent of b.

Proof of Theorem 1.4

To end 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 4.1 is independent of any , then by (4.1) we have that . On the other hand, using (1.5) we have Let satisfy and since we can let such that , where is given by Lemma 4.1(ii). Therefore we can obtain So the proof of Step 1 is complete.

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

By (4.4) and (4.5), we know that () are sequences of the functionals . We claim that () are bounded. In fact, which yields that are bounded in since . With (4.4) and (4.5) in hand, we can see Lemma 3.2 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 as , which yields that are solutions of (1.6) for .

Step 3: .

Indeed, and

Summing the above three steps, we obtain that and are two nontrivial solutions of (1.6). The proof is complete. □

Conclusion

This paper is concerned with the qualitative analysis of solutions of a nonlocal problem with Sobolev–Hardy exponent of Kirchhoff type. Meanwhile, it seems that the study of Kirchhoff type equation involving Hardy term and singular nonlinearity via the Nehari manifold and fibering maps is new.