Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term

1 Introduction

In this article, we study the following Kirchhoff equation:

where V ∈ C ( R 3 , R ) , f ∈ C ( R , R ) , and a , b > 0 . This equation is a typical nonlocal equation, which arises from elasticity and population dynamics. Particularly in 1883, the Kirchhoff [11] proposed the following equation in the process of studying the classical D’Alembert wave equation of free vibration of telescopic rope

where L represents the rope length, h represents the cross-sectional area, E is the Young coefficient of the material, ρ is the mass density, and p 0 is the initial tension. For more details and physical background of equation (1.1), one can refer to [1,11] and references therein.

In the last two decades, the Kirchhoff equation has attracted much attention from the mathematicians and physicians, and there are plenty of results on the existence of positive solutions, multiple solutions, ground-state solutions, and semiclassical-sate solutions of equation (1.1) (see, for example, [2,4,9,10,13,16,21,23]). Particularly for the sign-changing solutions, Zhang and Perera [27] obtained the existence result of a Kirchhoff-type equation in bounded domain by using the minimax theorem and invariant sets of descent flow. Mao and Zhang [14] obtained multiple sign-changing solutions of the Kirchhoff-type problems without the Palais-Smale condition. Liu et al. [12] proved the existence and multiplicity of sign-changing solutions to the sub-cubic case by a novel perturbation approach and the method of invariant sets of descending flow. For more results about sign-changing solutions, one can refer to [19,20,22,26,27] and references therein.

Recently, by using the Nehari manifold method and gluing method, Deng et al. [5] obtained the existence and asymptotic behaviors of the radial nodal solutions with a prescribed number of nodes for Problem (1.1) by assuming the super-cubic condition: lim ∣ u ∣ → ∞ f ( u ) u 3 = + ∞ . Later, the authors in [7] extended this result to a general super-cubic case under a weaker monotonicity condition and differentiability of f , and to the cubic case f ( u ) = ∣ u ∣ 2 u by combining with the limit approach in [24]. However, when f ( u ) is an asymptotically cubic term satisfying lim ∣ u ∣ → ∞ f ( u ) u 3 = 1 , it is totally different from the super-cubic or the cubic term, because the nonlocal term b ∣ ∇ u ∣ L 2 ( R 3 ) 2 Δ u is 3-homogeneous in the sense that b ∣ ∇ t u ∣ L 2 ( R 3 ) 2 Δ t u = t 3 b ∣ ∇ u ∣ L 2 ( R 3 ) 2 Δ u , which competes complicatedly with the asymptotic cubic term f ( u ) . Now, a natural question arises: can we find such nodal solutions with the nodal characterization for equation (1.1) when f ( u ) is an asymptotically cubic term? Since the methods used in the aforementioned articles cannot be applied to the asymptotically cubic case directly, some new ideas are necessary. As is well known, equation (1.1) is not a point-wise identity, which also makes this question interesting and challenging. As far as we know, this problem is unsolved.

In order to achieve our goal, we make the following hypotheses:

1. V ( x ) = V ( ∣ x ∣ ) ∈ C ( [ 0 , + ∞ ) , R ) is bounded from below by a positive constant V 0 ;

2. f ∈ C ( R , R ) and f ( t ) = − f ( − t ) for any t ∈ R ;

3. lim t → 0 f ( t ) ∕ t = 0 ;

4. (asymptotically cubic case) lim t → ∞ f ( t ) ∕ t 3 = 1 and f ( t ) ∕ t 3 < 1 for all t ≠ 0 ;

5. the function t ↦ f ( t ) ∕ t 3 is strictly increasing on ( 0 , ∞ ) .

Before stating our main results, we briefly give some notations and useful lemmas. Let H r 1 ( R 3 ) be the radial Sobolev subspace of H 1 ( R 3 ) and

be endowed with norm ‖ u ‖ V = ∫ R 3 ( a ∣ ∇ u ∣ 2 + V ( ∣ x ∣ ) u 2 ) d x 1 ∕ 2 . Associated with (1.1), the corresponding energy functional I b : H V → R is

where F ( u ) = ∫ 0 u f ( s ) d s . Obviously, I b ∈ C 2 ( H V , R ) , and for any u , φ ∈ H V

As usual, let

be the Nehari manifold and

be the ground-state energy. It is proved in [6, Lemma 2.9] that m is attained by some U 0 ∈ N , namely,

For k ∈ N + and 0 ≕ r 0 < r 1 < ⋯ < r k < r k + 1 ≔ + ∞ , we denote by r k = ( r 1 , … , r k ) and

Clearly, B 1 r k is a ball, B 2 r k , … , B k r k are annuli, B k + 1 r k is the complement of a ball, and R 3 = ∪ i = 1 k + 1 B i r k ¯ . For u ∈ H V , we set u i = u in B i r k and u i = 0 in R 3 \ B i r k . Then, we define

the Nehari-type set

and the infimum level by

Our first existence result is stated as follows.

There are two main difficulties in the proof. First, because of the complicated competition between ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u and f ( u ) , it is not easy to prove that N k is nonempty. We shall prove it by using the Miranda theorem [15] and a construction method. Second, compared to [5], it is difficult for us to prove the existence of solutions for each partition in the process of using the gluing method. We shall solve it by classifying the domain partitions and making subtle analysis (see Lemmas 3.3 and 3.5). This case is totally different from the Schrödinger-Poisson equation, and the method used in [8] is not applicable. This is a novelty point of this article.

Our next result shows that the energy of the nodal solutions obtained in Theorem 1.2 increases as the number of nodes grows.

Obviously, U k obtained in Theorem 1.2 depends on b . Sometimes, we shall denote U k by U k b to emphasize this dependence. The following result shows the convergence properties of U k b as b → 0 + .

Different from the super-cubic case, the uniqueness of the projection is lost from H V onto the Nehari-type set N k . It will bring some difficulties to prove the energy of the solutions is the least one. We shall overcome it by using the Miranda theorem and some subtle analysis. This is another novelty point.

In Theorems 1.2–1.4, the oddness assumption on f ( u ) is actually unnecessary. Indeed, in this case, we only need to deal with f ( u ) as follows:

or

and define the corresponding I b ± ( u ) and c k ± = inf u ∈ N k I b ± ( u ) as those in [2] or [3]. Then, by similar arguments as Theorems 1.2–1.4, we can prove that c k ± can be attained by U k ± , which are the desired nodal solutions for the corresponding problem.

This article is organized as follows. In Section 2, we present some preliminary results and the variational framework. In Section 3, we prove the nonemptiness of Nehari-type set N k by a construction method and give some related properties of N k . In Section 4, we prove Theorem 1.1 by classifying the domain partitions and using the gluing method. In Section 5, we show the energy comparison and asymptotical behaviors of the nodal solutions by proving Theorems 1.3 and 1.4.

2 Preliminaries

In this section, we introduce some notations and preliminary results. For each fixed r k ∈ Γ k and thereby a family of annulus { B i r k } i = 1 k + 1 , we introduce a family of Hilbert spaces

endowed with the norm ‖ u ‖ i = ∫ B i r k ( a ∣ ∇ u ∣ 2 + V ( ∣ x ∣ ) u 2 ) d x 1 ∕ 2 , and a product space

We introduce a functional E b : ℋ k r k → R defined by:

where u i ∈ H i r k for i = 1 , … , k + 1 . It is obvious that

and thus, if ( u 1 , … , u k + 1 ) is a critical point E b , then each component u i satisfies

Note that

Then, for each r k ∈ Γ k , we define another Nehari manifold

and consider the infimum problem

Clearly, the nonemptiness of N k r k implies the nonemptiness of N k , which is defined in (1.5). It is worth pointing out that the nonemptiness of N k r k and N k is not obvious.

To prove this result, we introduce a C 2 functional E b , 4 : ℋ k r k → R defined by:

Then,

and thus, the corresponding Nehari manifold ℳ k , 4 r k is defined by:

3 Properties of the Nehari-type set

In this section, we are going to prove the nonemptiness and some properties of the Nehari-type sets N k and N k r k by using Lemmas 2.1 and (3.1), via a construction method and the Miranda theorem

We first define some continuous functions F i : ( R > 0 ) k + 1 → R by:

In view of Proposition 3.1, we let

Then, for any r k ∈ Γ k 1 , inf u ∈ N k r k I b ( u ) < + ∞ .

In the sequel, for each u ≔ ( u 1 , … , u k + 1 ) ∈ N k r k , we define continuous functions L i : ( R ≥ 0 ) k + 1 → R by: