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Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent

Yuanze Wu* and Yisheng Huang

Author Affiliations

Department of Mathematics, Soochow University, Suzhou, Jiangsu, 215006, P.R. China

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Boundary Value Problems 2013, 2013:149  doi:10.1186/1687-2770-2013-149


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/149


Received:14 December 2012
Accepted:28 May 2013
Published:19 June 2013

© 2013 Wu and Huang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we study the following problem:

{ Δ p u = λ | u | p 2 u + | u | p 2 u in  Ω , u = 0 on  Ω ,

where Ω R N is a smooth bounded domain, 1 < p < N , Δ p u = div ( | u | p 2 u ) is the p-Laplacian, p = p N / ( N p ) is the critical Sobolev exponent and λ > 0 is a parameter. By establishing a new deformation lemma, we show that if N > p 2 + p , then for each λ > 0 , this problem has infinitely many sign-changing solutions, which extends the results obtained in (Cao et al. in J. Funct. Anal. 262: 2861-2902, 2012; Schechter and Zou in Arch. Ration. Mech. Anal. 197: 337-356, 2010).

1 Introduction

In this paper, we consider the following problem:

{ Δ p u = λ | u | p 2 u + | u | p 2 u in  Ω , u = 0 on  Ω , (1.1)

where Ω R N ( N 3 ) is a smooth bounded domain, 1 < p < N , Δ p u = div ( | u | p 2 u ) is the p-Laplacian, p = p N / ( N p ) is the critical Sobolev exponent and λ > 0 is a parameter.

The first existence result of Problem (1.1) with p = 2 was obtained by Brezis and Nirenberg in the celebrated paper [1]. In that paper, the authors proved that Problem (1.1) had a positive solution for N 4 and λ ( 0 , λ 1 ) or N = 3 and λ ( λ 1 / 4 , λ 1 ) , where λ 1 is the first eigenvalue of ( Δ , H 0 1 ( Ω ) ) . After that, many existence results have appeared for (1.1); one can see, for example, [2-7] and the references therein for case of p = 2 and [8-11] and the references therein for case of 1 < p < N . In particular, in the case of p = 2 , the authors in [2] proved that the number of solutions of Problem (1.1) is bounded below by the number of eigenvalues of ( Δ , H 0 1 ( Ω ) ) lying in the open interval ( λ , λ + S | Ω | 2 / N ) , where S is the best Sobolev constant and | Ω | is the Lebesgue measure of Ω. In [5], the existence of infinitely many sign-changing solutions of (1.1) with p = 2 has been obtained when N 4 , λ > 0 and Ω is a ball, while it has been shown in [6] that (1.1) with p = 2 has infinitely many sign-changing radial solutions when N 7 , λ > 0 and Ω also is a ball. We remark that the methods used in [5,6] are strongly dependent on the symmetry of the ball Ω. For a general bounded smooth domain Ω, by the method of invariant sets of the descending flow, the authors in [7] have shown that (1.1) with p = 2 has infinitely many sign-changing solutions when N 7 and λ > 0 , which extends the main result in [4].

The main purpose of this paper is to try to obtain the existence of infinitely many sign-changing solutions of Problem (1.1) for general p ( 1 , N ) . In a very recent work [9], the authors have proved that (1.1) has infinitely many solutions for λ > 0 and N > p 2 + p . However, by using the Picone identity (cf.[12,13]), we see that every nonzero solution of Problem (1.1) is sign-changing for λ λ 1 , where λ 1 is the first eigenvalue of ( Δ p , W 0 1 , p ( Ω ) ) (see Lemma 2.1 for more details). Hence, to achieve our purpose, we mainly consider the situation of λ ( 0 , λ 1 ) .

Our main result in this paper is the following.

Theorem 1.1Assume that N > p 2 + p and λ > 0 . Then Problem (1.1) has infinitely many sign-changing solutions.

Since p is the critical Sobolev exponent, in order to overcome the lack of compactness of the embedding of W 0 1 , p ( Ω ) in the Lebesgue space L p ( Ω ) , we follow the ideas of [4,7,9] to consider the following auxiliary problems:

where p n < p and p n is increasing to p . It has been shown by [[14], Theorem 1.2] that for every n, Problem ( P n ) has infinitely many sign-changing solutions { u n , k } k N . Hence, to prove Theorem 1.1, we will show that for every k N , { u n , k } converges to some sign-changing solution u k of (1.1) as n , and that { u k } are different. The convergence of { u n , k } can be done with the help of [[9], Theorem 1.2], which we show in Lemma 2.3. To distinguish { u k } , we shall establish a new deformation lemma on special sets in W 0 1 , p ( Ω ) ; see Lemma 2.5 for details.

Throughout this paper, we will always respectively denote u and u r by the usual norm in spaces W 0 1 , p ( Ω ) and L r ( Ω ) ( r 1 ). Let C be indiscriminately used to denote various positive constants.

2 Proof of Theorem 1.1

We first consider the case of λ λ 1 . Recall that λ 1 , the first eigenvalue of Δ p in W 0 1 , p ( Ω ) , given by λ 1 : = inf { Ω | u | p d x , Ω | u | p d x = 1 } , is simple and there exists a positive eigenfunction e 1 W 0 1 , p ( Ω ) corresponding to λ 1 such that Ω | e 1 | p 2 e 1 η d x = λ 1 Ω e 1 p 1 η d x for every η W 0 1 , p ( Ω ) (cf.[15]). Moreover, by [[16], Proposition 2.1], we know that e 1 L ( Ω ) C loc 1 , α ( Ω ) . On the other hand, we have the following proposition which is the so-called Picone identity.

Proposition 2.1 [[13], Lemma A.6]

Let u , v W loc 1 , p ( Ω ) C ( Ω ) be such that u 0 , v > 0 and u v W loc 1 , p ( Ω ) . Then

Ω ( u p v p 1 ) | v | p 2 v d x = Ω ( p ( u v ) p 1 | v | p 2 v u ( p 1 ) ( u v ) p | v | p ) d x .

Moreover,

Ω ( p ( u v ) p 1 | v | p 2 v u ( p 1 ) ( u v ) p | v | p ) d x Ω | u | p d x ,

and the equality holds if and only if u = c v for some constant c > 0 .

Lemma 2.1Assume that u W 0 1 , p ( Ω ) is a nonzero solution of (1.1) for λ λ 1 . Thenuis sign-changing.

Proof By a contradiction, we may assume u 0 . By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have u C 1 , α ( Ω ) for some α ( 0 , 1 ) . Thus, it follows from the strong maximum principle (cf.[18]) that u > 0 . Now, for every ε > 0 , by applying the above Picone identity (i.e., Proposition 2.1) to u + ε and e 1 , we see

Ω | e 1 | p d x Ω ( e 1 p ( u + ε ) p 1 ) | u | p 2 u d x .

Noting that u is a solution of (1.1), we have

Ω | e 1 | p d x Ω ( λ u p 1 ( u + ε ) p 1 + u p 1 ( u + ε ) p 1 ) e 1 p d x .

It follows from the Fatou lemma that

Ω | e 1 | p d x lim inf ε 0 Ω ( λ u p 1 ( u + ε ) p 1 + u p 1 ( u + ε ) p 1 ) e 1 p d x Ω ( λ + u p p ) e 1 p d x ,

which is impossible since Ω | e 1 | p d x = λ 1 Ω e 1 p , u > 0 , e 1 > 0 and λ λ 1 . Therefore, we have proved Lemma 2.1. □

Next, we consider the case of λ < λ 1 .

It is clear that the corresponding functional of ( P n ) I n : W 0 1 , p ( Ω ) R , given by

I n ( u ) = 1 p ( u p λ u p p ) 1 p n u p n p n ,

is C 1 Fréchet differentiable. Let X m = span { φ 1 , , φ m } , where { φ i } is a linearly independent sequence of W 0 1 , p ( Ω ) . It is easy to show that there exists R m > 0 such that I n ( u ) 1 for u X m B m , where B m : = { u X m : u R m } (cf. [[14], Lemma 3.9]). We denote

P ( P ) : = { u W 0 1 , p ( Ω ) : u 0 ( u 0 )  a.e. } , D μ ± : = { u W 0 1 , p ( Ω ) : dist ( u , ± P ) μ } , D μ : = D μ + D μ , G m : = { h C ( B m , W 0 1 , p ( Ω ) ) : h  is odd,  h ( x ) = x  for  x X m B m } .

Recall that the genus of a symmetric set A of W 0 1 , p ( Ω ) is defined by

gen ( A ) : = inf { k 0 : f C ( A , R k { 0 } ) , f  is odd } .

Here, we say that A is symmetric if x A implies x A .

By [[14], Theorem 1.2], we know that, for every n N , I n ( u ) has infinitely many critical points, denoted by { u n , k } k N , in X D μ for μ small enough. Moreover,

I n ( u n , k ) = d n , k : = inf Z Γ k sup u Z I n ( u ) , (2.1)

where Γ k : = { h ( B m B ) D μ : h G m  for  m n , B = B B m  open, gen ( B ) m n } .

Lemma 2.2For every k N , there exists d k > 0 such that u n , k d k for all n N .

Proof Consider the following auxiliary functional:

I ( u ) : = 1 p ( u p u p p ) 1 p u σ σ ,

where σ = ( p + p ) / 2 . Since p n p , we may assume p n > σ for all n N . Then 1 p u σ σ meas ( Ω ) p + 1 p n u p n p n for all n N . This means

I ( u ) = I n ( u ) + ( 1 p n u p n p n 1 p u σ σ ) I n ( u ) meas ( Ω ) p . (2.2)

Note that I ( u ) is the corresponding functional of the following equation:

{ Δ p u = λ | u | p 2 u + σ p | u | σ 2 u in  Ω , u = 0 on  Ω ,

and the nonlinearity satisfies the assumptions of [[14], Theorem 1.2]. Thus, this equation has a sequence of solutions { v k } W 0 1 , p ( Ω ) ( D μ + D μ ) such that

I ( v k ) = d ¯ k : = inf Z Γ k sup u Z I ( u )

for μ small enough. For every k N , the definitions of d ¯ k and d k , n , together with (2.2), imply d ¯ k + meas ( Ω ) p d k , n for all n N . On the other hand, since for every n, { u n , k } k N is a sequence of solutions for ( P n ) whose energies satisfy (2.1), it follows that d n , k ( 1 p 1 p 1 ) ( 1 λ λ 1 ) u n , k p . We complete the proof by choosing d k = ( ( d ¯ k p + meas ( Ω ) ) p 1 p λ 1 p ( p 1 p ) ( λ 1 λ ) ) 1 / p . □

By Lemma 2.2 and [[9], Theorem 1.2], we know that for each k N , there exists u k W 0 1 , p ( Ω ) such that u n , k u k as n in W 0 1 , p ( Ω ) . The next lemma will give more information about u k .

Lemma 2.3 u k is a sign-changing solution of Problem (1.1) for every k N .

Proof We first prove that u k is a solution of Problem (1.1) for every k N . Since u n , k u k as n in W 0 1 , p ( Ω ) ,

Ω | u n , k | p 2 u n , k φ d x Ω | u k | p 2 u k φ d x

and

Ω | u n , k | p 2 u n , k φ d x Ω | u k | p 2 u k φ d x

as n for every φ W 0 1 , p ( Ω ) . If we can prove

Ω | u n , k | p n 2 u n , k φ d x Ω | u k | 2 2 u k φ d x (2.3)

as n for every φ W 0 1 , p ( Ω ) , then u k is a solution of (1.1) for u n , k is a solution of ( P n ). Indeed, u n , k u k a.e. in Ω as n since u n , k u k in W 0 1 , p ( Ω ) . By the Egoroff theorem, for every δ > 0 , there exists Ω δ such that u n , k u k uniformly in Ω Ω δ and | Ω δ | < δ , where | Ω δ | is the Lebesgue measure of Ω δ . This, together with the Lebesgue dominated convergence theorem, implies

lim n Ω Ω δ | u n , k | p n 2 u n , k φ d x = Ω Ω δ | u k | p 2 u k φ d x for every  φ W 0 1 , p ( Ω ) . (2.4)

On the other hand, for every φ W 0 1 , p ( Ω ) , we have

Ω δ | | u n , k | p n 2 u n , k | u k | p 2 u k | | φ | d x Ω δ | | u n , k | p n 2 u n , k | u n , k | p 1 | u n , k | p 1 | | φ | d x + Ω δ | | u k | p 1 + | u k | p 1 | u k | p 2 u k | | φ | d x + Ω δ | | u k | p 1 + | u k | p 1 | u n , k | p 1 | u n , k | p 1 | | φ | d x 2 Ω δ | | u n , k | p 1 + | u n , k | p 1 | | φ | d x + Ω δ | | u k | p 1 + | u k | p 1 | u k | p 2 u k | | φ | d x + Ω δ | | u k | p 1 + | u k | p 1 | u n , k | p 1 | u n , k | p 1 | | φ | d x 2 Ω δ | | u k | p 1 + | u k | p 1 | | φ | d x + Ω δ | | u k | p 1 + | u k | p 1 | u k | p 2 u k | | φ | d x + 3 Ω δ | | u k | p 1 + | u k | p 1 | u n , k | p 1 | u n , k | p 1 | | φ | d x .

For every ε > 0 , by the above inequality and the absolute continuity of the integral, we can take δ small enough such that

2 Ω δ | | u k | p 1 + | u k | p 1 | | φ | d x + Ω δ | | u k | p 1 + | u k | p 1 | u k | p 2 u k | | φ | d x < ε / 3 .

For this δ, since u n , k u k in W 0 1 , p ( Ω ) ,

3 Ω δ | | u k | p 1 + | u k | p 1 | u n , k | p 1 | u n , k | p 1 | | φ | d x < ε / 3

for n large enough. By (2.4), for this δ, we have

Ω Ω δ | | u n , k | p n 2 u n , k φ | u k | p 2 u k φ | d x < ε / 3

for n large enough. So (2.3) holds. Moreover, by a similar proof, we can show d k : = lim n d n , k = I n ( u n , k ) = I ( u k ) .

Next, we will show u k is sign-changing for all k N . Since for each n N , u n , k is a sign-changing solution of ( P n ), multiplying ( P n ) by u n , k ± , we obtain u n , k ± p = λ u n , k ± p p + u n , k ± p n p n , where u ± = max { ± u , 0 } . Note that λ < λ 1 , by the Sobolev imbedding theorem, we have 0 < ( 1 λ λ 1 ) C u n , k ± p p n p . It follows that u n , k u k in L p ( Ω ) as n for u n , k u k in W 0 1 , p ( Ω ) as n . This gives 0 < ( 1 λ λ 1 ) C u k ± p p p , i.e., u k ± 0 for all k N . □

Let ε > 0 and c R , we denote

K : = { u W 0 1 , p ( Ω ) : I ( u ) = 0 } , K c : = { u W 0 1 , p ( Ω ) : I ( u ) = c , I ( u ) = 0 } , K μ : = K ( int ( D μ + ) int ( D μ ) ) , K c , μ : = K c ( int ( D μ + ) int ( D μ ) ) , N c , μ , ε : = { u W 0 1 , p ( Ω ) : dist ( u , K c ) < ε } .

Thanks to Lemma 2.3, u k K μ k for some μ k > 0 . We claim that { u k } K μ for some μ > 0 . Indeed, if not, then dist ( u k , P ) 0 as k without loss of generality. On the one hand, since u k is a solution of (1.1), I ( u k ) , u k S λ ( u k ) W 0 1 , p ( Ω ) , W 0 1 , p ( Ω ) = 0 , where S λ ( u k ) : ( Δ p ) 1 ( λ | u k | p 2 u k + | u k | p 2 u k ) . On the other hand, by [[17], Lemma 3.7], we have

I ( u k ) , u k S λ ( u k ) W 0 1 , p ( Ω ) , W 0 1 , p ( Ω ) C u k S λ ( u k ) 2 ( u k + S λ ( u k ) ) p 2

for 1 < p < 2 and

I ( u k ) , u k S λ ( u k ) W 0 1 , p ( Ω ) , W 0 1 , p ( Ω ) C u k S λ ( u k ) p

for p 2 . Note that by a similar proof of [[14], Lemma 3.3], we can see that S λ ( D μ ± ) int ( D μ ± ) for μ small enough. Thus, u k S λ ( u k ) > 0 for k large enough. This implies

I ( u k ) , u k S λ ( u k ) W 0 1 , p ( Ω ) , W 0 1 , p ( Ω ) C k > 0

for k large enough, which contradicts I ( u k ) , u k S λ ( u k ) W 0 1 , p ( Ω ) , W 0 1 , p ( Ω ) = 0 . For the sake of convenience, we denote K μ , K c , μ , N c , μ , ε by K , K c , N c , ε . Note that for every c R , K c is compact in W 0 1 , p ( Ω ) (cf. [[9], Theorem 1.2]). It follows from [[19], Proposition 7.5] that there exists ε > 0 such that

gen ( N c , 2 ε ) = gen ( K c ) < + . (2.5)

Let J n c : = { u W 0 1 , p ( Ω ) : I n ( u ) c } and Q n c : = D μ J n c . Let J c : = { u W 0 1 , p ( Ω ) : I ( u ) c } . For δ > 0 small enough, we define A n , ε c , δ : = ( Q n c + δ Q n c δ ) N c , ε , then we have the following.

Lemma 2.4Assume that there exists δ > 0 such that K J c + δ int ( J c δ ) = K c fornlarge. Then there exists γ > 0 such that I n ( u ) γ for u A n , ε c , δ and largen.

Proof Assume a contradiction. Then, for every n N , there exists { v n , k } A n , ε c , δ such that lim k I n ( v n , k ) = 0 . It is clear that I n satisfies the (PS) condition for every n N . Hence there exists v n W 0 1 , p ( Ω ) such that, up to a subsequence, v n , k v n in W 0 1 , p ( Ω ) as k with I n ( v n ) = 0 and I n ( v n ) [ c δ , c + δ ] . This implies

c + δ I n ( v n ) = ( 1 p 1 p n ) ( 1 λ λ 1 ) v n p ( 1 p 1 p 1 ) ( 1 λ λ 1 ) v n p .

Thus, by [[9], Theorem 1.2], up to a subsequence, we see that there exists v 0 W 0 1 , p ( Ω ) such that v n v 0 in W 0 1 , p ( Ω ) as n . Moreover, by using the arguments in the proof of Lemma 2.3, we have I ( v 0 ) = 0 and I ( v 0 ) [ c δ , c + δ ] . On the other hand, for large n, v n ( int ( D μ + ) int ( D μ ) ) N c , ε since v n , k A n , ε c , δ . It follows that v 0 ( int ( D μ + ) int ( D μ ) ) N c , ε . This contradicts the fact that K J n c + δ int ( J n c δ ) = K c . □

Lemma 2.5Assume that there exists γ > 0 such that I n ( u ) γ for every u A n , ε c , δ and largen. Then there exist δ > 0 and an odd continuous map η n such that η n : A n , 2 ε c , δ Q n c δ Q n c δ and η | Q n c δ = Id for largen.

Proof We first assume 1 < p < 2 . It is clear that there exists L > 0 such that

u + S n , λ ( u ) L for all  u N c , 2 ε , (2.6)

where

S n , λ ( u ) , φ : = Ω ( λ | u | p 2 u + | u | p n 2 u ) φ d x for  u W 0 1 , p ( Ω )  and  φ W 0 1 , p ( Ω ) .

Let T n , λ : W 0 1 , p ( Ω ) K W 0 1 , p ( Ω ) be the local Lipschitz continuous operator obtained in [[14], Lemma 2.1] and let ϕ u ( t ) be the solution of the following O.D.E.

{ d ϕ d t = ϕ + T n , λ ( ϕ ) , ϕ = u W 0 1 , p ( Ω ) K .

Denote τ ( u ) to be the maximal interval of existence of ϕ u ( t ) .

Claim 1: ϕ u ( t ) cannot enter N c , ε before it enters Q n c δ for small δ, large n and u A n , 2 ε c , δ .

Indeed, if the claim fails, then for every δ > 0 , ϕ u ( t ) will enter N c , ε before it enters Q n c δ . Since u A n , 2 ε c , δ W 0 1 , p ( Ω ) N c , 2 ε , there exist 0 t 1 < t 2 < τ ( u ) such that ϕ u ( t ) N c , 2 ε N c , ε for t ( t 1 , t 2 ] and

dist ( ϕ u ( t 1 ) , K c ) = 2 ε , dist ( ϕ u ( t 2 ) , K c ) = ε .

By [[14], Lemma 2.1], C u S n , λ ( u ) 2 ( u + S n , λ ( u ) ) p 2 I n ( u ) , u T n , λ ( u ) . On the other hand, by the choice of t 1 and t 2 , we know that ϕ u ( t ) A n , ε c , δ for t ( t 1 , t 2 ] . Thanks to [[17], Lemma 3.8], u S n , λ ( u ) ( γ C ) 1 / ( p 1 ) for large n. This, together with (2.6) and [[14], Lemma 2.1], implies

ε ϕ u ( t 2 ) ϕ u ( t 1 ) t 1 t 2 ϕ u ( t ) T n , λ ( ϕ u ( t ) ) d t C t 1 t 2 ϕ u ( t ) S n , λ ( ϕ u ( t ) ) d t C t 1 t 2 ϕ u ( t ) S n , λ ( ϕ u ( t ) ) 2 ( ϕ u ( t ) + S n , λ ( ϕ u ( t ) ) ) p 2 d t C t 1 t 2 I n ( ϕ u ( t ) ) , ϕ u ( t ) T n , λ ( ϕ u ( t ) ) d t = C ( I n ( t 1 ) I n ( t 2 ) ) 4 C δ .

A contradiction with δ < 4 C / ε .

Claim 2: There exists τ 1 ( t ) < τ ( u ) such that ϕ u ( τ 1 ( u ) ) Q n c δ for large n and u A n , 2 ε c , δ .

If the claim is not true, then ϕ u ( t ) Q n c + δ Q n c δ for all t ( 0 , τ ( u ) ) . We first consider the case of τ ( u ) < + . In fact, by Claim 1, ϕ u ( t ) N c , ε , i.e., ϕ u ( t ) A n , ε c , δ for all t ( 0 , τ ( u ) ) . Since I n ( u ) γ > 0 for u A n , ε c , δ and large n, we must have

ϕ u ( t ) as  t τ ( u ) . (2.7)

On the other hand, by [[14], Lemma 2.1] and [[17], Lemma 5.2], we have

ϕ u ( t ) ϕ u ( 0 ) 0 t ϕ u ( s ) T λ , n ( ϕ u ( s ) ) d s C 0 t ϕ u ( s ) S λ , n ( ϕ u ( s ) ) d s C 0 t ( 1 + ϕ u ( s ) S λ , n ( ϕ u ( s ) ) ) p d s C 0 t ( 1 + ϕ u ( s ) S λ , n ( ϕ u ( s ) ) ) 2 ( ϕ u ( s ) + S λ , n ( ϕ u ( s ) ) ) p 2 d s C 0 t ϕ u ( s ) S λ , n ( ϕ u ( s ) ) 2 ( ϕ u ( s ) + S λ , n ( ϕ u ( s ) ) ) p 2 d s C ( I n ( ϕ u ( 0 ) ) I n ( ϕ u ( t ) ) ) C .

This means ϕ u ( t ) u + C for all t ( 0 , τ ( u ) ) , which contradicts with (2.7). It follows that there must exist τ 1 ( u ) < τ ( u ) such that ϕ u ( τ 1 ( u ) ) Q n c δ for u A n , 2 ε c , δ , large n and τ ( u ) < + . Next, we consider the case of τ ( u ) = + . Since u S n , λ ( u ) ( γ C ) 1 / ( p 1 ) for all u A n , ε c , δ and large n, it follows from [[14], Lemma 2.1] and [[17], Lemma 5.2] that

d I n ( ϕ u ( t ) ) d t = I n ( ϕ u ( t ) ) , ϕ u ( t ) + T n , λ ( ϕ u ( t ) ) C ϕ u ( t ) S n , λ ( ϕ u ( t ) ) 2 ( ϕ u ( t ) + S n , λ ( ϕ u ( t ) ) ) p 2 C ϕ u ( t ) S n , λ ( ϕ u ( t ) ) 2 ( 1 + ϕ u ( t ) S n , λ ( ϕ u ( t ) ) ) p 2 C < 0 .

Thus, there also exists τ 1 ( u ) ( 0 , + ) such that ϕ u ( τ 1 ( u ) ) Q n c δ for u A n , 2 ε c , δ and τ ( u ) = + . Moreover, we must have ϕ u ( t ) Q n c δ for t ( τ 1 ( u ) , τ ( u ) ) since d I n ( ϕ u ( t ) ) d t 0 for all u W 0 1 , p ( Ω ) K .

Let

η n ( u ) = { ϕ u ( τ 1 ( u ) ) , u A n , 2 ε c , u , u Q n c δ .

Then, by the continuity of ϕ u ( t ) , η n ( u ) is continuous. Note that ϕ u ( t ) is odd and τ 1 ( u ) is even, we see that η n ( u ) is odd and it is the desired map. The situation of p 2 can be proved in a similar way. Therefore, we complete the proof of this lemma. □

Proof of Theorem 1.1 We first consider the case λ λ 1 . Thanks to Lemma 2.1 and [[9], Theorem 1.1], (1.1) has infinitely many sign-changing solutions. Next, we consider the case of λ ( 0 , λ 1 ) . Since for every n N , 0 d n , k d n , k + 1 for all k N , d k d k + 1 for all k N . It follows that two cases may occur:

Case 1: There are 1 < k 1 < k 2 < such that d k 1 < d k 2 <  .

In this case, Problem (1.1) has infinitely many sign-changing solutions.

Case 2: There exists k 0 > 0 such that d = d k for all k k 0 .

In this case, if ( K J d + δ J d δ ) K d for every δ > 0 small enough, then Problem (1.1) also has infinitely many sign-changing solutions. Otherwise, there exists δ 0 > 0 such that ( K J d + δ J d δ ) = K d for δ < δ 0 . Thanks to Lemmas 2.4 and 2.5, there exists η n such that η n ( A n , 2 ε d Q n d δ ) Q n d δ for small δ and large n. Fix l N and k k 0 , the definitions of d k and d k + l give that there exists a large n such that d n , k > d δ and d n , k + l < d + δ for small δ ( 0 , 1 ) . By the definition of d n , k + l , there exists Z Γ k + l such that sup Z I n ( u ) < d + δ , where Z = h ( B m B ) D μ , h G m and gen ( B ) m k l . It follows that h ( B m B ) N d , 2 ε A n , 2 ε c , δ Q n c δ . Thus, η n ( h ( B m B ) N d , 2 ε ) Q n d δ . By the choice of δ and B m , we have η n h G m . If gen ( B h 1 ( N d , 2 ε ) ) m k , then we have

d δ < d n , k sup η n h ( B m ( B h 1 ( N d , 2 ε ) ) ) I n ( u ) d δ .

A contradiction. By the properties of gen, we have

m k + 1 gen ( B h 1 ( N d , 2 ε ) ) gen ( B ) + gen ( N d , 2 ε ) m k l + gen ( N d , 2 ε ) .

This implies gen ( N d , 2 ε ) l + 1 . Since l N is arbitrary, we have gen ( N d , 2 ε ) = + , which contradicts with (2.5). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors typed, read and approved the final manuscript.

Acknowledgements

The work was supported by the Natural Science Foundation of China (11071180, 11171247) and College Postgraduate Research and Innovation Project of Jiangsu Province (CXZZ110082).

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