Multiple nodal solutions of the Kirchhoff-type problem with a cubic term

Tao Wang; Yanling Yang; Hui Guo

doi:10.1515/anona-2022-0225

Open Access Published by De Gruyter March 9, 2022

Multiple nodal solutions of the Kirchhoff-type problem with a cubic term

Tao Wang , Yanling Yang and Hui Guo

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2022-0225

Abstract

In this article, we are interested in the following Kirchhoff-type problem

(0.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) ,

where a , b > 0 , N = 2 or 3, the potential function V is radial and bounded from below by a positive number. Because the nonlocal b ∣ ∇ u ∣ L 2 ( R N ) 2 Δ u is 3-homogeneous which is in complicated competition with the nonlinear term ∣ u ∣ 2 u . This causes that not all function in H 1 ( R N ) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer k , equation (0.1) admits a radial nodal solution U k , 4 b having exactly k nodes. Moreover, we show that the energy of U k , 4 b is strictly increasing in k and for any sequence { b n } with b n → 0 + , up to a subsequence, U k , 4 b n converges to U k , 4 0 in H 1 ( R N ) , which is a radial nodal solution with exactly k nodes of the classical Schrödinger equation

− a Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) .

Our results extend the existence result from the super-cubic case to the cubic case.

Keywords: Kirchhoff-type equations; nodal solutions; Miranda theorem; Gersgorin disk theorem

MSC 2010: 35A15; 35J20; 35J50

1 Introduction

In this article, we are interested in the following Kirchhoff-type problem

(1.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( x ) u = f ( x , u ) x ∈ R N , u ∈ H 1 ( R N )

where a , b are positive constants, V is bounded from below by a positive number. As is well known, problem (1.1) is related to the stationary analogue of the equation

ρ ∂ 2 u ∂ t 2 − p 0 h + E 2 L ∫ 0 L ∂ u ∂ x 2 d x ∂ 2 u ∂ t 2 = 0 ,

which was proposed in 1883 by Kirchhoff [1] in the process of studying the classical D’Alembert wave equation of free vibration of retractable rope. Here, L denotes the rope length, h denotes the cross-sectional area, E denotes the Young coefficient of material, ρ denotes the mass density and p 0 denotes the initial tension. For more details and physical background of problem (1.1), one can refer to [1,2, 3,4] and references therein.

In the last two decades, via the variational methods, the existence of positive solutions, multiple solutions, ground state solutions and semi-classical state solutions of problem (1.1) have been widely studied (see [5,6,7, 8,9,10] for example). Particularly for the sign-changing solutions, Zhang and Perera [11] obtained the existence result of a Kirchhoff-type equation in a bounded domain by using the minimax theorem and invariant sets of descent flow in 2006. Later, Mao and Zhang [12] obtained multiple sign-changing solutions of the Kirchhoff-type problems without the P.S. condition. For more results about sign-changing solutions, one can refer to [11,12,13, 14,15,16, 17,18] and references therein.

Recently, Deng et al. [19] and Guo et al. [20] obtained the existence and asymptotic behaviors of nodal solutions with a prescribed number of nodes for problem (1.1) by assuming the following super-cubic conditions:

For some p ∈ ( 4 , 6 ) , lim u → ∞ f ( r , u ) u p − 1 = 0 uniformly in r > 0 ;
lim u → ∞ F ( r , u ) u 4 = + ∞ , where F ( r , u ) = ∫ 0 u f ( r , t ) d t .

Corresponding to the classical pure power nonlinearity model f ( ∣ x ∣ , u ) = ∣ u ∣ q − 2 u , their main results in [19,20] only solve

(1.2) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ q − 2 u in R N

for the case q ∈ ( 4 , 6 ) . For q ≤ 2 or q ≥ 6 , equation (1.2) admits only trivial solutions due to the Pohoazaev identity [8, Lemma 2.1]. Very recently, Liu et al. [21] proved that (1.2) admits infinitely many sign-changing solutions for any q ∈ ( 2 , 6 ) by the minimax theorem and invariant sets of descent flow. But unfortunately, their result does not give any information of nodes of these sign-changing solutions. Then a natural problem arises that

Can one find a nodal solution with a prescribed number of nodes for problem (1.2) when 2 < q ≤ 4 ?

To the best of our knowledge, this problem still remains unsolved.

In this article, we are devoted to the existence and asymptotic behaviors of nodal solutions to the cubic case q = 4 in (1.2), namely, the Kirchhoff-type equation with a cubic term,

(1.3) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u x ∈ R N , u ∈ H 1 ( R N ) ,

where N = 2 or 3, a , b are positive constants and V satisfies

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

As is well known, there is a sophistically competition between the nonlocal term ∫ R N ∣ ∇ u ∣ 2 d x Δ u and the cubic term ∣ u ∣ 2 u , which causes that all the techniques of the aforementioned papers concerning the super-cubic case do not work. Hence, some new ideas are necessary. Besides, equation (1.3) is no longer a point-wise identity due to the appearance of the nonlocal term ∫ R N ∣ ∇ u ∣ 2 d x Δ u , which makes this problem more interesting and challenging.

Before we present our main results, we first give several definitions and notations. Let H r 1 ( R N ) be the radial Sobolev subspace of H 1 ( R N ) and

H V ≔ u ∈ H r 1 ( R N ) : ∫ R N ( a ∣ ∇ u ∣ 2 + V ( ∣ x ∣ ) u 2 ) d x < + ∞

be endowed with norm ‖ u ‖ V = ∫ R N ( a ∣ ∇ u ∣ 2 + V ( ∣ x ∣ ) u 2 ) d x 1 / 2 . Associated with (1.3), the energy functional I b , 4 : H V → R is

(1.4) I b , 4 ( u ) ≔ 1 2 ‖ u ‖ V 2 + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 − 1 4 ∫ R N ∣ u ∣ 4 .

Obviously, I b , 4 ∈ C 2 ( H V , R ) and

⟨ I b , 4 ′ ( u ) , u ⟩ = ‖ u ‖ V 2 + b ∫ R N ∣ ∇ u ∣ 2 d x 2 − ∫ R N ∣ u ∣ 4 .

The usual Nehari manifold N is

(1.5) N = { u ∈ H V \ { 0 } : ⟨ I b , 4 ′ ( u ) , u ⟩ = 0 } ,

and the ground state energy is defined as

(1.6) m ≔ inf N I b , 4 ( u ) .

In [8, Theorem 1.1], it is proved that m is attained by some U 0 ∈ N , namely,

(1.7) m = I b , 4 ( U 0 ) > 0 .

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

B 1 r k ≔ { x ∈ R N : 0 ≤ ∣ x ∣ < r 1 } , B i r k ≔ { x ∈ R N : r i − 1 < ∣ x ∣ < r i } , i = 2 , … , k + 1 .

Clearly, B 1 r k is a ball, B 2 r k , … , B k r k are annulus and B k + 1 r k is the complement of a ball. Then we define the Nehari-type set

(1.8) N k , 4 = { u ∈ H V : there exists r k s.t. u i ≠ 0 in B i r k , ⟨ I b , 4 ′ ( u ) , u i ⟩ = 0 , i = 1 , … , k + 1 } ,

where u i = u in B i r k and u i = 0 on ∂ B i r k . By considering the infimum level

(1.9) c k , 4 = inf u ∈ N k , 4 I b , 4 ( u ) ,

we shall obtain the following existence result via a limit approach.

Theorem 1.1

For each k ∈ N + , problem (1.3) admits a radial nodal solution U k , 4 ∈ N k , 4 having exactly k nodes such that I b , 4 ( U k , 4 ) = c k , 4 .

Different from [19], the nonempty of N k , 4 is not obvious, because of the complicated competition between ∫ R N ∣ ∇ u ∣ 2 d x Δ u and ∣ u ∣ 2 u . Even if the nonempty is proved, it is still difficult to show Theorem 1.1 by the Nehari manifold method and gluing method, which depends heavily on the super-cubic conditions (F1) and (F2) as stated in [19]. Hence, some new ideas are necessary. We shall prove it by the limit approach and Gersgorin disk theorem.

Our next result yields that the energy of those solutions obtained in Theorem 1.1 increases as the number of nodes grows.

Theorem 1.2

Under the assumptions of Theorem 1.1, the energy of U k , 4 is strictly increasing in k , i.e.,

I b , 4 ( U k + 1 , 4 ) > I b , 4 ( U k , 4 ) , for any k ∈ N + .

Moreover, I b , 4 ( U k + 1 , 4 ) > ( k + 2 ) I b , 4 ( U 0 ) , where U 0 is the ground state solution that appeared in (1.7).

Obviously, U k , 4 obtained in Theorem 1.1 depends on b . So we denote U k , 4 by U k , 4 b to emphasize this dependence. The following result shows the convergence properties of U k , 4 b as b → 0 + .

Theorem 1.3

Under the assumptions of Theorem 1.1, for any sequence { b n } with b n → 0 + as n → ∞ , there exists a subsequence, still denoted by { b n } , such that U k , 4 b n converges to U k , 4 0 strongly in H V as n → ∞ , where U k , 4 0 is a least energy radial nodal solution among all the nodal solutions having exactly k nodes to the following equation:

(1.10) − a Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u .

Remark 1.4

As we see, the number q = 4 is a key number for the existence of nodal solutions with any prescribed number of nodal domains to equation (1.3). We stress here that the existence problem is still left open for the sub-cubic case q ∈ ( 2 , 4 ) .

The article is organized as follows. In Section 2, we present some preliminary results and the variational framework of equation (1.3). In Section 3, we prove the nonempty of Nehari-type set N k , 4 by construction method and Miranda theorem. In Section 4, by using the Gersgorin disk theorem and the limit approach, we prove Theorem 1.1. In Section 5, we investigate the energy comparison and asymptotic behaviors of the nodal solutions of (1.3).

2 Preliminaries

In this section, we give some notations and preliminary results. For each k ∈ N + , we define

Γ k = { r k = ( r 1 , … , r k ) ∈ ( 0 , ∞ ) k : 0 ≕ r 0 < r 1 < ⋯ < r k < r k + 1 ≔ ∞ } .

For fixed r k ∈ Γ k and thereby a family of annulus { B i r k } i = 1 k + 1 , we introduce an Hilbert space

H i r k ≔ { u ∈ H 0 1 ( B i r k ) : u ( x ) = u ( ∣ x ∣ ) , u ( x ) = 0 for x ∈ ∂ B i r k }

endowed with norm ‖ u ‖ i = ∫ B i r k ( a ∣ ∇ u ∣ 2 + V ( ∣ x ∣ ) u 2 ) d x 1 / 2 . Let

ℋ k r k = H 1 r k × ⋯ × H k + 1 r k

and define a functional E b , 4 : ℋ k r k → R by

(2.1) E b , 4 ( u 1 , … , u k + 1 ) ≔ 1 2 ∑ i = 1 k + 1 ‖ u i ‖ i 2 + b 4 ∑ i , j = 1 k + 1 ∫ B i r k ∣ ∇ u i ∣ 2 ∫ B j r k ∣ ∇ u j ∣ 2 − 1 4 ∑ i = 1 k + 1 ∫ B i r k ∣ u i ∣ 4 .

It is obvious that

(2.2) E b , 4 ( u 1 , … , u k + 1 ) = I b , 4 ∑ i = 1 k + 1 u i .

If ( u 1 , … , u k + 1 ) is a critical point E b , 4 , then each component u i satisfies

− a + b ∑ j = 1 k + 1 ∫ B j r k ∣ ∇ u j ∣ 2 Δ u i + V ( ∣ x ∣ ) u i = ∣ u i ∣ 2 u i x ∈ B i r k , u i = 0 x ∉ B i r k .

Note that

⟨ E b , 4 ′ ( u 1 , … , u k + 1 ) , u i ⟩ = ‖ u i ‖ i 2 + b ∑ j = 1 k + 1 ∫ B j r k ∣ ∇ u j ∣ 2 d x ∫ B i r k ∣ ∇ u i ∣ 2 d x − ∫ B i r k ∣ u i ∣ 4 d x .

For each r k ∈ Γ k , we define another Nehari-type set

(2.3) ℳ k , 4 r k ≔ { ( u 1 , … , u k + 1 ) ∈ ℋ k r k : u i ≠ 0 , ⟨ E b , 4 ′ ( u 1 , … , u k + 1 ) , u i ⟩ = 0 , i = 1 , … , k + 1 } .

However, the nonempty of ℳ k , 4 r k is not obvious. In order to prove it, we introduce the following important lemmas.

Lemma 2.1

([22] Miranda theorem) Suppose that

D = { x = ( x 1 , … , x m ) ∈ R m : ∣ x i ∣ < L , ∀ 1 ≤ i ≤ m }

and H = ( h 1 , … , h m ) ∈ C ( D ¯ ; R m ) satisfies H ( x ) ≠ ( 0 , … , 0 ) for any x ∈ ∂ D and

h i ( x 1 , … , x i − 1 , − L , x i + 1 , … , x m ) ≥ 0 for 1 ≤ i ≤ m ,
h i ( x 1 , … , x i − 1 , L , x i + 1 , … , x m ) ≤ 0 for 1 ≤ i ≤ m .

Then there exists x 0 ∈ D such that H ( x 0 ) = ( 0 , … , 0 ) .

The following result is a corollary of the Gersgorin disk theorem [23].

Lemma 2.2

([14], Lemma 2.3) For any a i j = a j i > 0 with i ≠ j and s i > 0 with i = 1 , … , m , if the matrix B ≔ ( b i j ) m × m is defined by

b i j = − ∑ l ≠ i s l a i l s i i = j , a i j > 0 i ≠ j ,

then ( b i j ) m × m is a non-positive definite symmetric matrix.

Lemma 2.3

If f ∈ C 2 ( R m , R ) is a strictly concave function and has a critical point s ¯ ≔ ( s ¯ 1 , … , s ¯ m ) in R m , then s ¯ is the unique critical point of f in R m .

Proof

We prove it by contradiction. Suppose on the contrary that there is another critical point s of f in R m such that s ≠ s ¯ . We define g : [ 0 , 1 ] → R by

g ( t ) = f ( s ¯ + t ( s − s ¯ ) ) .

Then g ( 0 ) = f ( s ¯ ) , g ( 1 ) = f ( s ) , g ′ ( 0 ) = g ′ ( 1 ) = 0 and the strict concavity of f gives that g ″ ( t ) < 0 for any t ∈ [ 0 , 1 ] . By the Taylor expansion, we see that

(2.4) f ( s ) − f ( s ¯ ) = g ( 1 ) − g ( 0 ) = g ′ ( 0 ) + g ″ ( θ ) 2 < 0 ,

where θ ∈ [ 0 , 1 ] .

By applying similar argument to g ¯ ( t ) ≔ f ( s + t ( s ¯ − s ) ) , we have f ( s ) > f ( s ¯ ) , which contradicts with (2.4). Hence, the conclusion follows.□

3 Properties of the Nehari-type set

In this section, we are going to prove the nonempty and some properties of the Nehari-type set ℳ k , 4 r k and N k , 4 via the construction method and Miranda theorem.

Lemma 3.1

There exists r k ∈ Γ k such that

ℳ k , 4 r k ≠ ∅ ,

where ℳ k , 4 r k is defined in (2.3).

Proof

We take r ¯ k = ( r ¯ 1 , … , r ¯ k ) ∈ Γ k and w i ∈ H i r ¯ k with w i ≠ 0 . We fix 0 < δ 1 < δ 2 < ⋯ < δ k + 1 < + ∞ and define some new radial functions by

v i ( x ) = δ i w i x δ i , i = 1 , … , k + 1 .

Then supp ( v i ) ⊂ { x : δ i r ¯ i − 1 ≤ ∣ x ∣ ≤ δ i r ¯ i } and thus v i ∈ H i r k with

r k = ( δ 1 r ¯ 1 , … , δ k r ¯ k ) .

A simple calculation gives that

(3.1) ⟨ E b , 4 ′ ( v 1 , … , v k + 1 ) , v i ⟩ = ‖ v i ‖ i 2 + b ∑ j = 1 k + 1 ∣ ∇ v i ∣ L 2 2 ∣ ∇ v j ∣ L 2 2 − ∫ B i r ¯ k ∣ v i ∣ 4 = a δ i N ∣ ∇ w i ∣ L 2 2 + δ i 2 + N ∫ B i r k V ( δ i ∣ x ∣ ) w i 2 + b δ i N ∑ j = 1 k + 1 ∣ ∇ w i ∣ L 2 2 ∣ ∇ w j ∣ L 2 2 δ j N − δ i 4 + N ∫ B i r k ∣ w i ∣ 4 ≕ g i ( δ 1 , … , δ k + 1 ) where N = 2 or 3 .

By the condition (V), it follows that g i ( s 1 , … , s k + 1 ) > 0 if ∣ ( s 1 , … , s k + 1 ) ∣ is nearby 0 and g i ( s 1 , … , s k + 1 ) < 0 if ∣ ( s 1 , … , s k + 1 ) ∣ is large enough. Then there exist small δ > 0 and large L > 0 such that

g i ( δ , … , δ ) > 0 and g i ( L , … , L ) < 0 , i = 1 , … , k + 1 .

This implies that for all s j ∈ [ δ , L ] ,

(3.2) g i ( s 1 , … , s i − 1 , δ , s i + 1 , … , s k + 1 ) ≥ g i ( δ , … , δ ) > 0 , g i ( s 1 , … , s i − 1 , L , s i + 1 , … , s k + 1 ) ≤ g i ( L , … , L ) < 0 .

Set

D ≔ { ( t 1 , … , t k + 1 ) ∈ ( R > 0 ) k + 1 : δ ≤ t i ≤ L , i = 1 , … , k + 1 } .

It follows from (3.2) and Lemma 2.1 that there exists ( δ ¯ 1 , … , δ ¯ k + 1 ) ∈ D such that

g i ( δ ¯ 1 , … , δ ¯ k + 1 ) = 0 , i = 1 , … , k + 1 .

This implies that ( v 1 , … , v k + 1 ) ∈ ℳ k , 4 r k due to (2.3). Thus, ℳ k , 4 r k ≠ ∅ .□

The following result gives some properties of ℳ k , 4 r k .

Lemma 3.2

If r k ∈ Γ k such that ℳ k , 4 r k ≠ ∅ , then for any ( u 1 , … , u k + 1 ) ∈ ℳ k , 4 r k and ( t 1 , … , t k + 1 ) ∈ ( R ≥ 0 ) k + 1 \ ( 1 , … , 1 ) , there holds

E b , 4 ( t 1 u 1 , … , t k + 1 u k + 1 ) < E b , 4 ( u 1 , … , u k + 1 ) .

Proof

For ( u 1 , … , u k + 1 ) ∈ ℳ k , 4 r k , we set ξ i ( t ) = t 2 2 − t 4 4 ‖ u i ‖ i 2 . Then ξ i ′ ( t ) = t ( 1 + t ) ( 1 − t ) ‖ u i ‖ i 2 for any t > 0 and thus ξ i ( t ) < ξ ( 1 ) for any t ∈ ( 0 , 1 ) ∪ ( 1 , + ∞ ) , namely,

t 2 2 − t 4 4 ‖ u i ‖ i 2 < 1 4 ‖ u i ‖ i 2 for any t ∈ ( 0 , 1 ) ∪ ( 1 , + ∞ ) .

Hence, for ( t 1 , … , t k + 1 ) ∈ ( R ≥ 0 ) k + 1 \ ( 1 , … , 1 ) , there holds

E b , 4 ( t 1 u 1 , … , t k + 1 u k + 1 ) = E b , 4 ( t 1 u 1 , … , t k + 1 u k + 1 ) − ∑ i = 1 k + 1 t i 4 4 ⟨ E b , 4 ′ ( u 1 , … , u k + 1 ) , u i ⟩ = ∑ i = 1 k + 1 t i 2 2 ‖ u i ‖ i 2 + b t i 2 4 ∑ j = 1 k + 1 t j 2 ∫ B i r k ∣ ∇ u i ∣ 2 d x ∫ B j r k ∣ ∇ u j ∣ 2 d x − t i 4 4 ∫ B i r k ∣ u i ∣ 4 d x − ∑ i = 1 k + 1 t i 4 4 ‖ u i ‖ i 2 + b t i 4 4 ∑ j = 1 k + 1 ∫ B i r k ∣ ∇ u i ∣ 2 d x ∫ B j r k ∣ ∇ u j ∣ 2 d x − t i 4 4 ∫ B i r k ∣ u i ∣ 4 d x = ∑ i = 1 k + 1 t i 2 2 − t i 4 4 ‖ u i ‖ i 2 + b ∑ j = 1 k + 1 t i 2 t j 2 − t i 4 4 + t i 2 t j 2 − t j 4 4 ∫ B i r k ∣ ∇ u i ∣ 2 d x ∫ B i r k ∣ ∇ u j ∣ 2 d x < ∑ i = 1 k + 1 1 4 ‖ u i ‖ i 2 − b 4 ∑ j = 1 k + 1 ( t i 2 − t j 2 ) 2 ∫ B i r k ∣ ∇ u i ∣ 2 d x ∫ B j r k ∣ ∇ u j ∣ 2 d x ≤ ∑ i = 1 k + 1 1 4 ‖ u i ‖ i 2 = E b , 4 ( u 1 , … , u k + 1 ) − 1 4 ∑ i = 1 k + 1 ⟨ E b , 4 ′ ( u 1 , … , u k + 1 ) , u i ⟩ = E b , 4 ( u 1 , … , u k + 1 ) .

The proof is completed.□

The following result shows the nonempty of N k , 4 , which is a consequence of Lemma 3.1.

Lemma 3.3

There holds

N k , 4 ≠ ∅ and 0 < c k , 4 < + ∞ ,

where N k , 4 and c k , 4 are defined in (1.8) and (1.9), respectively.

Proof

In view of Lemma 3.1, we can take ( v 1 , … , v k + 1 ) ∈ ℳ k , 4 r k with v i ≠ 0 . Let v = ∑ i = 1 k + 1 v i . Then

⟨ I b , 4 ′ ( v ) , v i ⟩ = ⟨ E b , 4 ′ ( v 1 , … , v k + 1 ) , v i ⟩ = 0 , i = 1 , … , k + 1 .

So it follows from (1.8) and (2.3) immediately that v ∈ N k , 4 . Thus, N k , 4 ≠ ∅ .

Note that N k , 4 ⊂ N . Then we conclude from (1.7) directly that

0 < m ≔ inf N I b , 4 ( u ) ≤ inf u ∈ N k , 4 I b , 4 ( u ) ≤ c k , 4 ≤ I b , 4 ( v ) < + ∞ .

The proof is completed.□

4 Existence of nodal solutions

In this section, we are devoted to the proof of Theorem 1.1 by a limit approach. Before that, we first introduce the energy functional of equation (1.2)

I b , q ( u ) ≔ 1 2 ‖ u ‖ V 2 + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 − 1 4 ∫ R N ∣ u ∣ q .

Similar as before, we define the Nehari-type set N k , q as in (1.8) by replacing 4 by q , that is,

N k , q ≔ { u ∈ H V : there exists r k ∈ Γ k s.t. u i ≠ 0 in B i r k , ⟨ I b , q ′ ( u ) , u i ⟩ = 0 , i = 1 , … , k + 1 } ,

and the least energy level

c k , q = inf u ∈ N k , q I b , q ( u ) .

Then by [19, Theorem 1.1], the following result holds true.

Proposition 4.1

[19, Theorem 1.1] For each k ∈ N + and q ∈ ( 4 , 6 ) , equation (1.2) admits a radial nodal solution U k , q ∈ H V with exactly k nodes 0 < r 1 , q < ⋯ < r k , q < + ∞ such that

I b , q ( U k , q ) = c k , q .

With the aid of Proposition 4.1, we are going to prove Theorem 1.1 via a limit approach. The main difficulty is to ensure the limit function of U k , q as q → 4 + has exactly k nodes, although each U k , q has exactly k nodes. We stress here that even if the strong convergence is proved, the limit function may still degenerate. This is a novel point of this article.

Proof of Theorem 1.1

For each k ∈ N + and q ∈ ( 4 , 6 ) , let U k , q be defined as in Proposition 4.1, which is a radial nodal solution of (1.2) with exactly k nodes 0 < r 1 , q < ⋯ < r k , q < + ∞ such that I b , q ( U k , q ) = c k , q . In the following, we denote by

r k , q = ( r 1 , q , … , r k , q ) .

We divide the whole proof into several steps.

Step 1: Prove

(4.1) limsup q → 4 + c k , q ≤ c k , 4 < + ∞ .

According to Lemma 3.3 and [19, Lemma 2.1], we know that for any W 4 ≔ ∑ i = 1 k + 1 w i 4 ∈ N k , 4 with annulus { B i } i = 1 k + 1 , there exists a unique k + 1 -tuple ( t 1 , q , … , t k + 1 , q ) ∈ ( R > 0 ) k + 1 such that

W q ≔ ∑ i = 1 k + 1 t i , q w i 4 ∈ N k , q ,

where ( t 1 , q , … , t k + 1 , q ) satisfies

(4.2) t i , q 2 ‖ w i 4 ‖ i 2 + b t i , q 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∑ j = 1 k + 1 t j , q 2 ∫ B j ∣ ∇ w j 4 ∣ 2 = t i , q q ∫ B i ∣ w i 4 ∣ q , i = 1 , … , k + 1 .

This implies that t i , q q − 2 ≥ ∣ w i 4 ∣ i 2 ∫ B i ∣ w i 4 ∣ q and thereby there is δ 0 > 0 such that for all q ∈ ( 4 , 6 ) and i ∈ { 1 , … , k + 1 } ,

(4.3) t i , q ≥ δ 0 > 0 .

We denote the minimum index by i q among all i , which such that t i , q is the greatest value, that is, i q = inf { i : t i , q = max { t j , q : j = 1 , … , k + 1 } } . Furthermore, we have

t i q , q = max { t j , q : j = 1 , … , k + 1 } .

First, we assert that

(4.4) { ( t 1 , q , … , t k + 1 , q ) } q is bounded for q nearby 4 + .

In fact, if NOT, we may suppose on the contrary that t i q , q → + ∞ as q → 4 + . Then by (4.2), it follows

0 = t i q , q 2 − q ‖ w i q 4 ‖ i q 2 + b t i q , q 4 − q ∑ j = 1 k + 1 t j , q 2 t i q , q 2 ∫ B i q ∣ ∇ w i q 4 ∣ 2 d x ∫ B j ∣ ∇ w j 4 ∣ 2 d x − ∫ B i q ∣ w i q 4 ∣ q d x ≤ t i q , q 2 − q ‖ w i q 4 ‖ i q 2 + b ∑ j = 1 k + 1 ∫ B i q ∣ ∇ w i q 4 ∣ 2 d x ∫ B j ∣ ∇ w j 4 ∣ 2 d x − ∫ B i q ∣ w i q 4 ∣ q d x → b ∑ j = 1 k + 1 ∫ B i q ∣ ∇ w i q 4 ∣ 2 d x ∫ B j ∣ ∇ w j 4 ∣ 2 d x − ∫ B i q ∣ w i q 4 ∣ 4 d x ( as q → 4 + ) = − ‖ w i q 4 ‖ i q 2 ( because W 4 ∈ N k , 4 ) < 0 ,

which leads to a contradiction. Thus, the assertion (4.4) is proved.

Second, by (4.3) and (4.4), there exist a sequence { q n } and ( t 1 , 4 , … , t k + 1 , 4 ) ∈ ( R > 0 ) k + 1 such that

( t 1 , q n , … , t k + 1 , q n ) → ( t 1 , 4 , … , t k + 1 , 4 ) as q n → 4 + .

This combined with (4.2) yields that

(4.5) t i , 4 2 ‖ w i 4 ‖ i 2 + b t i , 4 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∑ j = 1 k + 1 t j , 4 2 ∫ B j ∣ ∇ w j ∣ 2 = t i , 4 4 ∫ B i ∣ w i 4 ∣ 4 , i = 1 , … , k + 1 ,

which implies that

∑ i = 1 k + 1 t i , 4 w i 4 ∈ N k , 4 .

Next, we prove that

( t 1 , 4 , … , t k + 1 , 4 ) = ( 1 , … , 1 ) .

In fact, in view of (1.4), we define a function f : ( R > 0 ) k + 1 → R by

f ( s 1 , … , s k + 1 ) ≔ I b , 4 ∑ i = 1 k + 1 s i 1 / 4 w i = ∑ i = 1 k + 1 s i 1 2 2 ‖ w i 4 ‖ i 2 + b 4 ∑ j = 1 k + 1 s i 1 2 s j 1 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∫ B j ∣ ∇ w j 4 ∣ 2 − s i 4 ∫ B i ∣ w i 4 ∣ 4 .

Some direct computations give that the partial derivatives of f satisfy

(4.6) f s i ( s 1 , … , s k + 1 ) = 1 4 s i I b , 4 ′ ∑ i = 1 k + 1 s i 1 / 4 w i , s i 1 / 4 w i = 1 4 s i − 1 2 ‖ w i 4 ‖ i 2 + b 4 ∑ j = 1 k + 1 s i − 1 2 s j 1 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∫ B j ∣ ∇ w j 4 ∣ 2 − 1 4 ∫ B i ∣ w i 4 ∣ 4 f s i s i ( s 1 , … , s k + 1 ) = − 1 8 s i − 3 2 ‖ w i 4 ‖ i 2 − b 8 ∑ j ≠ i k + 1 s i − 3 2 s j 1 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∫ B j ∣ ∇ w j 4 ∣ 2 , f s i s j ( s 1 , … , s k + 1 ) = b 8 s i − 1 2 s j − 1 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∫ B j ∣ ∇ w j 4 ∣ 2 , ∀ i ≠ j .

We denote by

M i j = f s i s j ( s 1 , … , s k + 1 )

and

A i i = − 1 8 s i − 3 2 ‖ w i 4 ‖ i 2 , B i i = − ∑ i ≠ j k + 1 s j s i b 8 s i − 1 2 s j − 1 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∫ B j ∣ ∇ w j 4 ∣ 2 , A i j = 0 , B i j = b 8 s i − 1 2 s j − 1 2 ∫ B i ∣ ∇ w i 4 ∣ 2 ∫ B j ∣ ∇ w j 4 ∣ 2 if i ≠ j .

Then it follows from Lemma 2.2 that at each point ( s 1 , … , s k + 1 ) ∈ ( R > 0 ) k + 1 , the matrix ( B i j ) ( k + 1 , k + 1 ) is non-positive definite, and thus

( M i j ) ( k + 1 ) × ( k + 1 ) = ( A i j ) ( k + 1 ) × ( k + 1 ) + ( B i j ) ( k + 1 ) × ( k + 1 )

is negative definite. This implies that

f is a strictly concave function in ( R > 0 ) k + 1 .

From (4.5) and (4.6), we see that

0 = I b , 4 ′ ∑ i = 1 k + 1 t i , 4 w i , t i , 4 w i = 4 t i , 4 4 f s i ( t 1 , 4 4 , … , t k + 1 , 4 4 ) ,

which shows that ( t 1 , 4 4 , … , t k + 1 , 4 4 ) is a critical point of f . On the other hand, note that W 4 = ∑ i = 1 k + 1 w i 4 ∈ N k , 4 . Then ( 1 , … , 1 ) is a critical point of f . By Lemma 2.3, it follows that

( t 1 , 4 , … , t k + 1 , 4 ) = ( 1 , … , 1 ) .

So ( t 1 , q , … , t k + 1 , q ) → ( 1 , … , 1 ) and thus

I b , q ( W q ) = I b , q ∑ i = 1 k + 1 t i , q w i 4 → I b , 4 ∑ i = 1 k + 1 w i 4 as q → 4 + .

Hence,

limsup q → 4 + c k , q ≤ limsup q → 4 + I b , q ( W q ) = I b , 4 ( W 4 ) .

The arbitrariness of the choice of W 4 implies (4.1). Step 1 is completed.

Step 2: Prove

(4.7) U k , q → U k , 4 ≠ 0 in H V as q → 4 + .

In fact, by Proposition 4.1 and (4.1), it follows from the fact

c k , q = I b , q ( U k , q ) − 1 q ⟨ I b , q ′ ( U k , q ) , U k , q ⟩ = 1 2 − 1 q ‖ U k , q ‖ V 2 + b 4 − b q ∣ ∇ U k , q ∣ L 2 2 ≥ 1 2 − 1 q ‖ U k , q ‖ V 2

that ‖ U k , q ‖ V is bounded for q nearby 4 + . Then there is a sequence { U k , q n } such that U k , q n ⇀ U k , 4 weakly in H V as q n → 4 + . By the weakly lower semi-continuity of the norm, there holds

liminf n → ∞ ‖ U k , q n ‖ V 2 ≥ ‖ U k , 4 ‖ V 2 and liminf n → ∞ ∣ ∇ U k , q n ∣ L 2 2 ≥ ∣ ∇ U k , 4 ∣ L 2 2 .

Then by the compact embedding H V ↪ L s ( R N ) for any s ∈ ( 2 , 2 ∗ ) , it follows that

0 = lim n → ∞ ⟨ I b , q n ′ ( U k , q n ) , U k , q n − U k , 4 ⟩ = lim n → ∞ a ∫ R N ∇ U k , q n ∇ ( U k , q n − U k , 4 ) + b ∫ R N ∣ ∇ U k , q n ∣ 2 ∫ R N ∇ U k , q n ∇ ( U k , q n − U k , 4 )

+ ∫ R N V ( ∣ x ∣ ) U k , q n ( U k , q n − U k , 4 ) − ∫ R N ∣ U k , q n ∣ q n − 2 U k , q n ( U k , q n − U k , 4 ) = lim n → ∞ ( a ( ‖ U k , q n ‖ V 2 − ‖ U k , 4 ‖ V 2 ) + b ∣ ∇ U k , q n ∣ L 2 2 ( ∣ ∇ U k , q n ∣ L 2 2 − ∣ ∇ U k , 4 ∣ L 2 2 ) ) ≥ 0 .

This implies ∣ ∇ U k , q n ∣ L 2 → ∣ ∇ U k , 4 ∣ L 2 and

(4.8) ‖ U k , q n ‖ V → ‖ U k , 4 ‖ V as q n → 4 + .

In view of (1.2) and the Sobolev inequality, there exists C > 0 independent of n such that ‖ U k , q n ‖ V 2 ≤ ∫ R N ∣ U k , q n ∣ q n ≤ C ‖ U k , q n ‖ V q n . This yields that ‖ U k , q n ‖ V ≥ C − 2 > 0 due to q n nearby 4 + . Then by (4.8), it follows

U k , 4 ≠ 0 .

Therefore, (4.7) follows and Step 2 is completed.

This together with Proposition 4.1 yields that U k , 4 is a nontrivial weak solution of (1.3). By the standard elliptic regularity theory, U k , 4 ∈ C 2 ( R N ) and then U k , 4 can be viewed as a radial nodal function having at most k + 1 components { ( U k , 4 ) i } i = 1 k + 1 (possibly empty), because U k , q n has exactly k + 1 nodal domains. Without loss of generality, we assume that the nodes of U k , 4 are 0 ≤ r 1 , 4 ≤ ⋯ ≤ r k , 4 < + ∞ , and we denote by r k , 4 = ( r 1 , 4 , … , r k , 4 ) .

Step 3: Prove each component

(4.9) ( U k , 4 ) i ≠ 0 , i = 1 , … , k + 1 .

In fact, we prove it by contradiction. Suppose on the contrary that one of the following two cases may occur: either

Case 1: r k , q n → + ∞ as n → ∞ , or

Case 2: There is i 0 ∈ { 1 , … , k + 1 } such that

(4.10) either liminf q n → 4 + ‖ U k , q n ‖ i 0 2 ≠ 0 and liminf q n → 4 + ‖ U k , q n ‖ i 0 + 1 2 = 0 , or liminf q n → 4 + ‖ U k , q n ‖ i 0 2 = 0 and liminf q n → 4 + ‖ U k , q n ‖ i 0 + 1 2 ≠ 0 .

If case 1 happens, we recall the Strauss inequality: for all u ∈ H r 1 ( R N ) ,

∣ u ( x ) ∣ ≤ a 0 ‖ u ‖ H r 1 ( R N ) ∣ x ∣ N − 1 2 for almost every x ∈ R N ,

where a 0 > 0 depends only on N . This together with the Sobolev inequality and (1.2), yields that

‖ ( U k , q n ) k + 1 ‖ k + 1 2 ≤ ∫ B k + 1 r k , q n ∣ ( U k , q n ) k + 1 ∣ q n ≤ a 0 q n − 2 ∫ B k + 1 r k , q n ‖ ( U k , q n ) k + 1 ‖ k + 1 q n − 2 ∣ x ∣ ( N − 1 ) ( q n − 2 ) 2 ∣ ( U k , q n ) k + 1 ∣ 2 d x ≤ a 0 q n − 2 ‖ ( U k , q n ) k + 1 ‖ k + 1 q n ∣ r k , q n ∣ ( N − 1 ) ( q n − 2 ) 2 ,

which implies ‖ ( U k , q n ) k + 1 ‖ k + 1 2 → + ∞ as n → + ∞ . This contradicts with (4.7).

If case 2 happens, we only deal with the latter situation in (4.10), because the former situation can be settled by a similar argument. Then we assume

(4.11) ( U k , q n ) i 0 < 0 in B i 0 r k , q n and ( U k , q n ) i 0 + 1 > 0 in B i 0 + 1 r k , q n .

We denote by

Ω q n = B i 0 r k , q n ∪ B i 0 + 1 r k , q n ¯ and Ω 4 = lim n → ∞ Ω q n .

For each q n , we introduce a normalized function V q n : R N → R ,

V q n = ( U k , q n ) i 0 ‖ ( U k , q n ) i 0 ‖ i 0 .

Clearly, { V q n } n is bounded in H V and there exists a subsequence, still denoted by { V q n } , and V 4 ∈ H V such that V q n ⇀ V 4 in H V and V q n ( x ) → V 4 ( x ) a.e. in R N . Then by the compactly embedding theorem, there holds

∫ B i 0 r k , q n ∣ ( U k , q n ) i 0 ∣ q n − 2 V q n 2 → ∫ B i 0 r k , 4 ( U k , 4 ) i 0 2 V 4 2 as q n → 4 + .

This together with the fact

‖ ( U k , q n ) i 0 ‖ i 0 2 + b ∣ ∇ ( U k , q n ) i 0 ∣ L 2 4 = ∫ B i 0 r k , q n ∣ ( U k , q n ) i 0 ∣ q n

yields that

1 + b ∣ ∇ ( U k , q n ) i 0 ∣ L 2 4 ‖ ( U k , q n ) i 0 ‖ i 0 2 ≤ ∫ B i 0 r k , q n ∣ ( U k , q n ) i 0 ∣ q n − 2 V q n 2 → ∫ B i 0 r k , 4 ( U k , 4 ) i 0 2 V 4 2 as q n → 4 + .

So V 4 ≠ 0 and the set { x ∈ R N : V 4 ( x ) < 0 } ≠ ∅ . Since { x ∈ R N : V q n ( x ) < 0 } ⊂ { x ∈ R N : ( U k , q n ) i 0 ( x ) ≤ 0 } , it follows that

(4.12) ∅ ≠ { x ∈ R N : V 4 ( x ) < 0 } ⊂ { x ∈ R N : ( U k , 4 ) i 0 ( x ) ≤ 0 } .

On the other hand, note from (1.3) that

(4.13) − ( a + b ∣ ∇ U k , 4 ∣ L 2 2 ) Δ U k , 4 + V ( ∣ x ∣ ) U k , 4 = ∣ U k , 4 ∣ 2 U k , 4 in Ω 4 , U k , 4 = 0 on ∂ Ω 4 ,

and from (4.10) and (4.11) that 0 ≢ ( U k , 4 ) ( i 0 ⋃ i 0 + 1 ) ≥ 0 in Ω 4 . Then by the classical elliptic regularity theory and the strong maximum principle to (4.13), we get ( U k , 4 ) ( i 0 ⋃ i 0 + 1 ) > 0 in Ω 4 and thus

{ x ∈ R N : ( U k , 4 ) i 0 ( x ) ≤ 0 } = ∅ .

This leads to a contradiction with (4.12).

Hence, case 1 and case 2 would not happen. So (4.9) follows.

Step 4: Prove

I b , 4 ( U k , 4 ) = c k , 4 .

We see that each ( U k , 4 ) i satisfies

− ( a + b ∣ ∇ U k , 4 ∣ L 2 2 ) Δ ( U k , 4 ) i + V ( ∣ x ∣ ) ( U k , 4 ) i = ∣ ( U k , 4 ) i ∣ 2 ( U k , 4 ) i in B i r k , 4 , ( U k , 4 ) i = 0 on ∂ B i r k , 4 .

Then by the classical elliptic regularity argument and the strong maximum principle, we conclude either ( U k , 4 ) i < 0 or ( U k , 4 ) i > 0 in B i r k , 4 . Thus, U k , 4 has exactly k + 1 nodal domains. Moreover, by (4.1), it follows that

c k , 4 ≥ limsup q n → 4 I b , q n ( U k , q n ) = limsup q n → 4 1 2 − 1 q n ‖ U k , q n ‖ V 2 + b 4 − b q n ∣ ∇ U k , q n ∣ L 2 4 = 1 4 ‖ U k , 4 ‖ V 2 = I b , 4 ( U k , 4 ) − 1 4 ⟨ I b , 4 ′ ( U k , 4 ) , U k , 4 ⟩ = I b , 4 ( U k , 4 ) ≥ c k , 4 .

Thus, I b , 4 ( U k , 4 ) = c k , 4 .

Therefore, U k , 4 is a radial nodal solution of (1.3) such that I b , 4 ( U k , 4 ) = c k , 4 , which has exactly k nodes. The proof is completed.□

5 Energy comparison and the convergence properties of nodal solutions

In this section, we prove Theorems 1.2 and 1.3. We shall finish the energy comparison by subtle energy estimates via the Miranda theorem.

Proof of Theorem 1.2

According to Theorem 1.1, we know that equation (1.3) admits a radial nodal solution U k + 1 , 4 with exactly k + 1 nodes 0 < r ¯ 1 < ⋯ < r ¯ k + 1 < + ∞ for any fixed positive integer k ≥ 1 . For the sake of convenience, we denote by

r k + 1 ≔ ( r ¯ 1 , r ¯ 2 , … , r ¯ k + 1 )

and

w i r k + 1 ≔ χ B i r k + 1 U k + 1 , 4 ,

where χ B i r k + 1 is the characteristic function on B i r k + 1 . Clearly, ( w 1 r k + 1 , … , w k + 2 r k + 1 ) ∈ ℋ k + 1 r k + 1 satisfies

(5.1) − a + b ∑ j = 1 k + 1 ∫ B j r k + 1 ∣ ∇ w j r k + 1 ∣ 2 d x Δ w i r k + 1 + V ( ∣ x ∣ ) w i r k + 1 = ∣ w i r k + 1 ∣ 2 w i r k + 1 , x ∈ B i r k + 1 , w i r k + 1 = 0 , x ∉ B i r k + 1 .

Next, we set

r ˆ k ≔ ( r ¯ 2 , … , r ¯ k + 1 ) .

Observe that ( t 2 w 2 r k + 1 , … , t k + 2 w k + 2 r k + 1 ) ∈ ℳ k , 4 r ˆ k if and only if

0 = t i 2 ‖ w i r k + 1 ‖ i 2 + b ∑ j = 2 k + 2 t i 2 t j 2 ∫ B i r k + 1 ∣ ∇ w i r k + 1 ∣ 2 d x ∫ B j r k + 1 ∣ ∇ w j r k + 1 ∣ 2 d x − t i 4 ∫ B i r k + 1 ∣ w i r k + 1 ∣ 4 d x ≕ T i ( t 2 , … , t k + 2 ) , i = 2 , … , k + 2 .

Then there exists a small number δ ∈ ( 0 , 1 ) such that for all i ∈ { 2 , … , k + 2 } ,

T i ( δ , … , δ ) > 0 .

Note from (5.1) that

T i ( 1 , … , 1 ) < 0 .

By the definition of T i , it gives that

(5.2) T i ( t 2 , … , t i − 1 , δ , t i + 1 , … , t k + 2 ) > 0 , t j ∈ [ δ , 1 ] , j ≠ i ; T i ( t 2 , … , t i − 1 , 1 , t i + 1 , … , t k + 2 ) < 0 , t j ∈ [ δ , 1 ] , j ≠ i .

Let

D δ 1 ≔ { ( t 2 , … , t k + 2 ) ∈ ( R > 0 ) k + 1 : δ < t j < 1 , j = 2 , … , k + 2 } .

Then by Lemma 2.1, there exists s ˜ ≔ ( s 2 , … , s k + 2 ) ∈ D δ 1 such that

T i ( s ˜ ) = 0 , i = 2 , … , k + 2 ,

which implies

(5.3) ( s 2 w 2 r k + 1 , … , s k + 2 w k + 2 r k + 1 ) ∈ ℳ k , 4 r ˆ k .

This together with (1.8), gives that

∑ i = 2 k + 2 s i w i r k + 1 ∈ N k , 4 .

Then

I b , 4 ∑ i = 2 k + 2 s i w i r k + 1 ≥ I b , 4 ( U k , 4 ) .

Note from Lemma 3.2 and (5.3) that

E b , 4 ( w 1 r k + 1 , … , w k + 2 r k + 1 ) > E b , 4 ( 0 , s 2 w 2 r k + 1 , … , s k + 2 w k + 1 r k + 1 ) .

Then by (2.2), we conclude that

I b , 4 ( U k + 1 , 4 ) = I b , 4 ∑ i = 1 k + 2 w i r k + 1 > I b , 4 ∑ i = 2 k + 2 s i w i r k + 1 ≥ I b , 4 ( U k , 4 ) .

Thus, I b , 4 ( U k , 4 ) is strictly increasing in k .

Now, we turn to prove

I b , 4 ( U k + 1 , 4 ) > ( k + 2 ) I b , 4 ( U 0 ) .

In fact, ⟨ I b , 4 ′ ( U k + 1 , 4 ) , w i r k + 1 ⟩ = 0 shows that

‖ w i r k + 1 ‖ i 2 + b ∫ B i r k + 1 ∣ ∇ w i r k + 1 ∣ 2 d x 2 − ∫ B i r k + 1 ∣ w i r k + 1 ∣ 4 d x < 0 , i = 1 , … , k + 2 .

Note that there exists small δ ¯ > 0 such that for all i ,

δ ¯ 2 ‖ w i r k + 1 ‖ i 2 + b δ ¯ 4 ∫ B i r k + 1 ∣ ∇ w i r k + 1 ∣ 2 d x 2 − δ ¯ 4 ∫ B i r k + 1 ∣ w i r k + 1 ∣ 4 d x > 0 .

Then by the mean value theorem for continuous functions, for each i ∈ { 1 , … , k + 2 } , there exists δ ¯ i ∈ ( δ ¯ , 1 ) such that

δ ¯ i ‖ w i r k + 1 ‖ i 2 + δ ¯ i 4 ∫ B i r k + 1 ∣ ∇ w i r k + 1 ∣ 2 d x 2 − δ ¯ i 4 ∫ B i r k + 1 ∣ w i r k + 1 ∣ 4 d x = 0 ,

which implies δ ¯ i w i r k + 1 ∈ N , where N is defined in (1.5). Hence, I b , 4 ( δ ¯ i w i r k + 1 ) ≥ I b , 4 ( U 0 ) and thereby

( k + 2 ) I b , 4 ( U 0 ) ≤ ∑ i = 1 k + 2 I b , 4 ( δ ¯ i w i r k + 1 ) − 1 4 ⟨ I b , 4 ′ ( δ ¯ i w i r k + 1 ) , δ ¯ i w i r k + 1 ⟩ = ∑ i = 1 k + 2 1 4 δ ¯ i 2 ‖ w i r k + 1 ‖ i 2 < ∑ i = 1 k + 1 1 4 ‖ w i r k + 1 ‖ i 2

= I b , 4 ∑ i = 1 k + 2 w i r k + 1 − 1 4 I b , 4 ′ ∑ i = 1 k + 1 w i r k + 1 , w i r k + 1 = I b , 4 ( U k + 1 , 4 ) .

The proof is completed.□

Hereafter, in order to emphasize the dependence on b , we denote N k , 4 by N k , 4 b , and the nodal solution of (1.3) obtained in Theorem 1.1 by U k , 4 b = ∑ i = 1 k + 1 w i r k , b ∈ H V , which changes sign exactly k times with r k , b = ( r ¯ 1 , b , … , r ¯ k , b ) . In the following, we are going to show the asymptotic behaviors of U k , 4 b as b → 0 + by using the Miranda theorem.

Proof of Theorem 1.3

We divide the whole proof into three steps.

Step 1. We claim that for any sequence { b n } with b n → 0 + as n → ∞ , { U k , 4 b n } n is bounded in H V .

In fact, for fixed r k ∈ Γ k , we take nonzero radial functions φ i ∈ C c ∞ ( B i r k ) , i = 1 , … , k + 1 . We define S i : R k + 1 → R by

S i ( s 1 , … , s k + 1 ) = a s i N ∣ ∇ φ i ∣ L 2 2 + s i 2 + N V ( s i ∣ x ∣ ) ∣ φ i ∣ L 2 2 + b ∑ j = 1 k + 1 s i N s j N ∣ ∇ φ i ∣ L 2 2 ∣ ∇ φ j ∣ L 2 2 − s i 4 + N ∫ B i r k ∣ φ i ∣ 4 .

An elementary computation shows that there exist a small number δ > 0 and a large number L > 0 such that for all b ∈ [ 0 , 1 ] ,

S i ( δ , … , δ ) > 0 and S i ( L , … , L ) < 0 , i = 1 , … , k + 1 ,

This combined with the definition of S i , gives that

S i ( s 1 , … , s i − 1 , δ , s i + 1 , … , s k + 1 ) > 0 , for any s j ∈ ( δ , L ) , j ≠ i , S i ( s 1 , … , s i − 1 , L , s i + 1 , … , s k + 1 ) < 0 , for any s j ∈ ( δ , L ) , j ≠ i .

Then by Lemma 2.1, there are k + 1 tuples ( a 1 ( b ) , … , a k + 1 ( b ) ) ∈ ( 0 , 1 ] k + 1 depending on b such that a i ( b ) L ∈ [ δ , L ] and

(5.4) S i ( a 1 ( b ) L , … , a k + 1 ( b ) L ) = 0 , i = 1 , … , k + 1 .

Let φ ¯ i ( x ) ≔ a i ( b ) L φ i x a i ( b ) L . Then by (5.4), we obtain

∑ i = 1 k + 1 φ ¯ i ( x ) ∈ N k , 4 .

Thus, there is C 0 > 0 such that for n large enough,

(5.5) 1 4 ‖ U k , 4 b n ‖ V 2 = I b n , 4 ( U k , 4 b n ) − 1 4 ⟨ I b n , 4 ′ ( U k , 4 b n ) , U k , 4 b n ⟩ ≤ I b n , 4 ∑ i = 1 k + 1 φ ¯ i ( x ) = I b n , 4 ∑ i = 1 k + 1 φ ¯ i ( x ) − 1 4 I b n , 4 ′ ∑ i = 1 k + 1 φ ¯ i ( x ) , φ ¯ i ( x ) = 1 4 ∑ i = 1 k + 1 a a i 3 ( b n ) L 3 ∣ ∇ φ i ∣ L 2 2 + a i 3 ( b n ) L 5 ∫ B i r k V ( a i ( b n ) L ∣ x ∣ ) φ i 2 d x

≤ 1 4 ∑ i = 1 k + 1 a L 3 ∣ ∇ φ i ∣ L 2 2 + L 5 ∫ B i r k sup t ∈ [ 0 , 1 ] V ( t L ∣ x ∣ ) φ i 2 d x ≕ C 0 .

This implies that { U k , 4 b n } n is bounded in H V . So the claim follows immediately.

Thus, there exists a subsequence { b n j } of { b n } and U k , 4 0 ∈ H V such that U k , 4 b n j ⇀ U k , 4 0 and ( U k , 4 b n j ) i ⇀ ( U k , 4 0 ) i weakly in H V as n j → + ∞ . Then U k , 4 0 is a weak solution of (1.10).

Step 2. We prove that U k , 4 0 is a radial nodal solution of (1.10) with exactly k + 1 nodal domains.

In fact, by the compactly embedding H V ↪ L s ( R N ) for any 2 < s < 6 , it follows that

‖ U k , 4 b n j − U k , 4 0 ‖ V 2 = ⟨ I b n j ′ ( U k , 4 b n j ) − I 0 ′ ( U k , 4 0 ) , U k , 4 b n j − U k , 4 0 ⟩ − ∫ R N ( U k , 4 0 ) 3 ( U k , 4 b n j − U k , 4 0 ) d x − b n j ∫ R N ∣ ∇ U k , 4 b n j ∣ 2 d x ∫ R N ∇ U k , 4 b n j ( ∇ U k , 4 b n j − ∇ U k , 4 0 ) d x + ∫ R N ( U k , 4 b n j ) 3 ( U k , 4 b n j − U k , 4 0 ) d x → 0 as j → ∞ .

Then U k , 4 b n j → U k , 4 0 strongly in H V as n j → + ∞ . Moreover, similar arguments can deduce that ( U k , 4 b n j ) i → ( U k , 4 0 ) i strongly in H V as n j → + ∞ .

Next, we prove ( U k , 4 0 ) i ≠ 0 . In fact, note from ⟨ I b n j ′ ( U k , 4 b n j ) , ( U k , 4 b n j ) i ⟩ = 0 that there is a number η > 0 such that

liminf j → ∞ ‖ ( U k , 4 b n j ) i ‖ i ≥ η > 0 .

Then it combined with the compactly embedding H V ↪ L s ( R N ) , gives that

η 2 ≤ ‖ ( U k , 4 b n j ) i ‖ i 2 ≤ ∫ R N ∣ ( U k , 4 b n j ) i ∣ 4 → ∫ R N ∣ ( U k , 4 0 ) i ∣ 4 .

This shows that ( U k , 4 0 ) i ≠ 0 . Hence, U k , 4 0 is a radial nodal solution of (1.10) with exactly k + 1 nodal domains.

Step 3. We prove that U k , 4 0 is a least energy radial solution of (1.10) among all the radial solutions changing sign exactly k times.

In fact, according to [24, Theorem 2.1], we assume that there is r ¯ k ∈ Γ k and v i supported on annuli B i r ¯ k such that V k , 4 ≔ v 1 + ⋯ + v k + 1 is a least energy radial solution of (1.10) among all the nodal solutions changing sign exactly k times. Obviously, for each b n > 0 , ∑ i = 1 k + 1 a i , n v i ∈ N k , 4 b n if and only if

(5.6) f i n ( a 1 , n , … , a k + 1 , n ) ≔ a i , n 2 ‖ v i ‖ i 2 + b n a i , n 2 ∫ B i r ¯ k ∣ ∇ v i ∣ 2 d x ∑ j = 1 k + 1 ∫ B j r ¯ k a j , n 2 ∣ ∇ v j ∣ 2 d x − a i , n 4 ∫ B i r ¯ k ∣ v i ∣ 4 d x = 0 , i = 1 , … , k + 1 ,

where N k , 4 b n is defined in (1.8) for b = b n . Note that

(5.7) f i n ( 1 , … , 1 ) > ‖ v i ‖ i 2 − ∫ B i r ¯ k ∣ v i ∣ 4 d x = 0 , i = 1 , … , k + 1

and

L 2 ‖ v i ‖ i 2 − L 4 ∫ B i r ¯ k ∣ v i ∣ 4 d x = L 4 ‖ v i ‖ i 2 L 2 − ∫ B i r ¯ k ∣ v i ∣ 4 d x < 0 for any L > 1 .

We take L j ≔ 1 + 1 j with j ≥ 1 . Then L j → 1 as j → + ∞ and for each j ≥ 1 , there exists n j > 0 such that b n j → 0 as j → ∞ and

(5.8) f i n j ( L j , … , L j ) = L j 2 ‖ v i ‖ i 2 + b n j L j 4 ∑ j = 1 k + 1 ∫ B i r ¯ k ∣ ∇ v i ∣ 2 d x ∫ B j r ¯ k ∣ ∇ v j ∣ 2 d x − L j 4 ∫ B i r ¯ k ∣ v i ∣ 4 d x < 0 .

This together with (5.7) and Lemma 2.1, yields that for each n j , there exists ( a 1 , n j , … , a k + 1 , n j ) ∈ ( 1 , L j ) k + 1 such that

f i n j ( a 1 , n j , … , a k + 1 , n j ) = 0 , i = 1 , … , k + 1 .

Hence, ∑ i = 1 k + 1 a i , n j v i ∈ N k , 4 b n j and ( a 1 , n j , … , a k + 1 , n j ) → ( 1 , … , 1 ) as j → ∞ . Then

I 0 , 4 ( V k ) ≤ I 0 , 4 ( U k , 4 0 ) = lim b n j → 0 I b n j , 4 ( U k , 4 b n j ) ≤ lim n j → ∞ I b n j , 4 ∑ i = 1 k + 1 a i , n j v i = I 0 , 4 ∑ i = 1 k + 1 v i = I 0 , 4 ( V k , 4 ) .

Therefore, U k , 4 0 is a least energy radial solution of (1.10) which changes sign exactly k times. The proof is completed.□

Funding information: Tao Wang was supported by National Natural Science Foundation of China (Grant No. 12001188) and the Postgraduate Science Research Innovation Project of Hunan Province (Grant No. CX20211002). Hui Guo was supported by Natural Science Foundation of Hunan Province (Grant No. 2020JJ5151).
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-07-22

Accepted: 2022-01-02

Published Online: 2022-03-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Multiple nodal solutions of the Kirchhoff-type problem with a cubic term

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

Theorem 1.3

Remark 1.4

2 Preliminaries

Lemma 2.1

Lemma 2.2

Lemma 2.3

Proof

3 Properties of the Nehari-type set

Lemma 3.1

Proof

Lemma 3.2

Proof

Lemma 3.3

Proof

4 Existence of nodal solutions

Proposition 4.1

Proof of Theorem 1.1

5 Energy comparison and the convergence properties of nodal solutions

Proof of Theorem 1.2

Proof of Theorem 1.3

References

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