Research

# 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

For all author emails, please log on.

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).

### References

1. Brezis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Commun. Pure Appl. Math.. 36, 437–478 (1983). Publisher Full Text

2. Cerami, G, Fortunato, D, Struwe, M: Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 341–350 (1984)

3. Clapp, M, Weth, T: Multiple solutions for the Brezis-Nirenberg problem. Adv. Differ. Equ.. 10, 463–480 (2005)

4. Devillanova, G, Solimini, S: Concentration estimates and multiple solutions to elliptic problems at critical growth. Adv. Differ. Equ.. 7, 1257–1280 (2002)

5. Fortunato, D, Jannelli, E: Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains. Proc. R. Soc. Edinb. A. 105, 205–213 (1987). Publisher Full Text

6. Solimini, S: A note on compactness-type properties with respect to Lorenz norms of bounded subsets of a Sobolev spaces. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 12, 319–337 (1995)

7. Schechter, M, Zou, W: On the Brézis-Nirenberg problem. Arch. Ration. Mech. Anal.. 197, 337–356 (2010). Publisher Full Text

8. Alves, C, Ding, Y: Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity. J. Math. Anal. Appl.. 279, 508–521 (2003). Publisher Full Text

9. Cao, D, Peng, S, Yan, S: Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth. J. Funct. Anal.. 262, 2861–2902 (2012). Publisher Full Text

10. Cingolani, S, Vannella, G: Multiple positive solutions for a critical quasilinear equation via Morse theory. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 26, 397–413 (2009). Publisher Full Text

11. Degiovanni, M, Lancelotti, S: Linking solutions for p-Laplace equations with nonlinearity at critical growth. J. Funct. Anal.. 256, 3643–3659 (2009). Publisher Full Text

12. Allegretto, W, Huang, Y: A Picone’s identity for the p-Laplacian and applications. Nonlinear Anal.. 32, 819–830 (1998). Publisher Full Text

13. Iturriaga, L, Massa, E, Sanchez, J, Ubilla, P: Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros. J. Differ. Equ.. 248, 309–327 (2010). Publisher Full Text

14. Bartsch, T, Liu, Z, Weth, T: Nodal solutions of p-Laplacian equation. Proc. Lond. Math. Soc.. 91, 129–152 (2005). Publisher Full Text

15. Lindqvist, P: On the equation div ( | u | p 2 u ) + λ | u | p 2 u = 0 . Proc. Am. Math. Soc.. 109, 157–164 (1990)

16. Cuesta, M: Eigenvalue problem for the p-Laplacian with indefinite weights. Electron. J. Differ. Equ.. 2001, 1–9 (2001)

17. Bartsch, T, Liu, Z: On a superlinear elliptic p-Laplacian equation. J. Differ. Equ.. 198, 149–175 (2004). Publisher Full Text

18. Tolksdorf, P: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ.. 51, 126–150 (1984). Publisher Full Text

19. Rabinowitz, P: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)