Open Access

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

Boundary Value Problems20132013:149

DOI: 10.1186/1687-2770-2013-149

Received: 14 December 2012

Accepted: 28 May 2013

Published: 19 June 2013

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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equa_HTML.gif

where Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq1_HTML.gif is a smooth bounded domain, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq2_HTML.gif, Δ p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq3_HTML.gif is the p-Laplacian, p = p N / ( N p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq4_HTML.gif is the critical Sobolev exponent and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif is a parameter. By establishing a new deformation lemma, we show that if N > p 2 + p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq6_HTML.gif, then for each λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ1_HTML.gif
(1.1)

where Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq1_HTML.gif ( N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq7_HTML.gif) is a smooth bounded domain, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq2_HTML.gif, Δ p u = div ( | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq3_HTML.gif is the p-Laplacian, p = p N / ( N p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq4_HTML.gif is the critical Sobolev exponent and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif is a parameter.

The first existence result of Problem (1.1) with p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq8_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq9_HTML.gif and λ ( 0 , λ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq10_HTML.gif or N = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq11_HTML.gif and λ ( λ 1 / 4 , λ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq12_HTML.gif, where λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq13_HTML.gif is the first eigenvalue of ( Δ , H 0 1 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq14_HTML.gif. After that, many existence results have appeared for (1.1); one can see, for example, [27] and the references therein for case of p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq8_HTML.gif and [811] and the references therein for case of 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq2_HTML.gif. In particular, in the case of p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq8_HTML.gif, 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 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq14_HTML.gif lying in the open interval ( λ , λ + S | Ω | 2 / N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq15_HTML.gif, where S is the best Sobolev constant and | Ω | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq16_HTML.gif is the Lebesgue measure of Ω. In [5], the existence of infinitely many sign-changing solutions of (1.1) with p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq8_HTML.gif has been obtained when N 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq9_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif and Ω is a ball, while it has been shown in [6] that (1.1) with p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq8_HTML.gif has infinitely many sign-changing radial solutions when N 7 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq17_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq8_HTML.gif has infinitely many sign-changing solutions when N 7 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq17_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq18_HTML.gif. In a very recent work [9], the authors have proved that (1.1) has infinitely many solutions for λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif and N > p 2 + p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq6_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq19_HTML.gif, where λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq20_HTML.gif is the first eigenvalue of ( Δ p , W 0 1 , p ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq21_HTML.gif (see Lemma 2.1 for more details). Hence, to achieve our purpose, we mainly consider the situation of λ ( 0 , λ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq22_HTML.gif.

Our main result in this paper is the following.

Theorem 1.1 Assume that N > p 2 + p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq6_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq5_HTML.gif. Then Problem (1.1) has infinitely many sign-changing solutions.

Since p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq23_HTML.gif is the critical Sobolev exponent, in order to overcome the lack of compactness of the embedding of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif in the Lebesgue space L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq25_HTML.gif, we follow the ideas of [4, 7, 9] to consider the following auxiliary problems:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equb_HTML.gif

where p n < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq26_HTML.gif and p n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq27_HTML.gif is increasing to p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq23_HTML.gif. It has been shown by [[14], Theorem 1.2] that for every n, Problem ( P n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq28_HTML.gif) has infinitely many sign-changing solutions { u n , k } k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq29_HTML.gif. Hence, to prove Theorem 1.1, we will show that for every k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq30_HTML.gif, { u n , k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq31_HTML.gif converges to some sign-changing solution u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif of (1.1) as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif, and that { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq34_HTML.gif are different. The convergence of { u n , k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq31_HTML.gif can be done with the help of [[9], Theorem 1.2], which we show in Lemma 2.3. To distinguish { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq34_HTML.gif, we shall establish a new deformation lemma on special sets in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif; see Lemma 2.5 for details.

Throughout this paper, we will always respectively denote u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq35_HTML.gif and u r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq36_HTML.gif by the usual norm in spaces W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif and L r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq37_HTML.gif ( r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq38_HTML.gif). 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq19_HTML.gif. Recall that λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq20_HTML.gif, the first eigenvalue of Δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq39_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq40_HTML.gif, given by λ 1 : = inf { Ω | u | p d x , Ω | u | p d x = 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq41_HTML.gif, is simple and there exists a positive eigenfunction e 1 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq42_HTML.gif corresponding to λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq20_HTML.gif such that Ω | e 1 | p 2 e 1 η d x = λ 1 Ω e 1 p 1 η d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq43_HTML.gif for every η W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq44_HTML.gif (cf. [15]). Moreover, by [[16], Proposition 2.1], we know that e 1 L ( Ω ) C loc 1 , α ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq45_HTML.gif. 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq46_HTML.gif be such that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq47_HTML.gif, v > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq48_HTML.gif and u v W loc 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq49_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equc_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equd_HTML.gif

and the equality holds if and only if u = c v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq50_HTML.gif for some constant c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq51_HTML.gif.

Lemma 2.1 Assume that u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq52_HTML.gif is a nonzero solution of (1.1) for λ λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq53_HTML.gif. Then u is sign-changing.

Proof By a contradiction, we may assume u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq47_HTML.gif. By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have u C 1 , α ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq54_HTML.gif for some α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq55_HTML.gif. Thus, it follows from the strong maximum principle (cf. [18]) that u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq56_HTML.gif. Now, for every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq57_HTML.gif, by applying the above Picone identity (i.e., Proposition 2.1) to u + ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq58_HTML.gif and e 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq59_HTML.gif, we see
Ω | e 1 | p d x Ω ( e 1 p ( u + ε ) p 1 ) | u | p 2 u d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Eque_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equf_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equg_HTML.gif

which is impossible since Ω | e 1 | p d x = λ 1 Ω e 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq60_HTML.gif, u > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq56_HTML.gif, e 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq61_HTML.gif and λ λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq19_HTML.gif. Therefore, we have proved Lemma 2.1. □

Next, we consider the case of λ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq62_HTML.gif.

It is clear that the corresponding functional of ( P n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq28_HTML.gif) I n : W 0 1 , p ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq63_HTML.gif, given by
I n ( u ) = 1 p ( u p λ u p p ) 1 p n u p n p n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equh_HTML.gif
is C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq64_HTML.gif Fréchet differentiable. Let X m = span { φ 1 , , φ m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq65_HTML.gif, where { φ i } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq66_HTML.gif is a linearly independent sequence of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif. It is easy to show that there exists R m > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq67_HTML.gif such that I n ( u ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq68_HTML.gif for u X m B m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq69_HTML.gif, where B m : = { u X m : u R m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq70_HTML.gif (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 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equi_HTML.gif
Recall that the genus of a symmetric set A of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif is defined by
gen ( A ) : = inf { k 0 : f C ( A , R k { 0 } ) , f  is odd } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equj_HTML.gif

Here, we say that A is symmetric if x A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq71_HTML.gif implies x A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq72_HTML.gif.

By [[14], Theorem 1.2], we know that, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif, I n ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq74_HTML.gif has infinitely many critical points, denoted by { u n , k } k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq29_HTML.gif, in X D μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq75_HTML.gif for μ small enough. Moreover,
I n ( u n , k ) = d n , k : = inf Z Γ k sup u Z I n ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ2_HTML.gif
(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 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq76_HTML.gif.

Lemma 2.2 For every k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif, there exists d k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq78_HTML.gif such that u n , k d k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq79_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif.

Proof Consider the following auxiliary functional:
I ( u ) : = 1 p ( u p u p p ) 1 p u σ σ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equk_HTML.gif
where σ = ( p + p ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq80_HTML.gif. Since p n p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq81_HTML.gif, we may assume p n > σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq82_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif. Then 1 p u σ σ meas ( Ω ) p + 1 p n u p n p n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq83_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ3_HTML.gif
(2.2)
Note that I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq84_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equl_HTML.gif
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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq85_HTML.gif such that
I ( v k ) = d ¯ k : = inf Z Γ k sup u Z I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equm_HTML.gif

for μ small enough. For every k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif, the definitions of d ¯ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq86_HTML.gif and d k , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq87_HTML.gif, together with (2.2), imply d ¯ k + meas ( Ω ) p d k , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq88_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif. On the other hand, since for every n, { u n , k } k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq29_HTML.gif is a sequence of solutions for ( P n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq28_HTML.gif) whose energies satisfy (2.1), it follows that d n , k ( 1 p 1 p 1 ) ( 1 λ λ 1 ) u n , k p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq89_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq90_HTML.gif. □

By Lemma 2.2 and [[9], Theorem 1.2], we know that for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif, there exists u k W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq91_HTML.gif such that u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif. The next lemma will give more information about u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif.

Lemma 2.3 u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif is a sign-changing solution of Problem (1.1) for every k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif.

Proof We first prove that u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif is a solution of Problem (1.1) for every k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif. Since u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif,
Ω | u n , k | p 2 u n , k φ d x Ω | u k | p 2 u k φ d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equn_HTML.gif
and
Ω | u n , k | p 2 u n , k φ d x Ω | u k | p 2 u k φ d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equo_HTML.gif
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif for every φ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq93_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ4_HTML.gif
(2.3)
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif for every φ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq93_HTML.gif, then u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif is a solution of (1.1) for u n , k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq94_HTML.gif is a solution of ( P n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq28_HTML.gif). Indeed, u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif a.e. in Ω as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif since u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif. By the Egoroff theorem, for every δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq95_HTML.gif, there exists Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq96_HTML.gif such that u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif uniformly in Ω Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq97_HTML.gif and | Ω δ | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq98_HTML.gif, where | Ω δ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq99_HTML.gif is the Lebesgue measure of Ω δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq96_HTML.gif. 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ5_HTML.gif
(2.4)
On the other hand, for every φ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq93_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equp_HTML.gif
For every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq57_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equq_HTML.gif
For this δ, since u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif,
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equr_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equs_HTML.gif

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq100_HTML.gif.

Next, we will show u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif is sign-changing for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif. Since for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif, u n , k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq94_HTML.gif is a sign-changing solution of ( P n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq28_HTML.gif), multiplying ( P n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq28_HTML.gif) by u n , k ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq101_HTML.gif, we obtain u n , k ± p = λ u n , k ± p p + u n , k ± p n p n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq102_HTML.gif, where u ± = max { ± u , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq103_HTML.gif. Note that λ < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq62_HTML.gif, by the Sobolev imbedding theorem, we have 0 < ( 1 λ λ 1 ) C u n , k ± p p n p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq104_HTML.gif. It follows that u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif in L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq25_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq105_HTML.gif for u n , k u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq92_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif. This gives 0 < ( 1 λ λ 1 ) C u k ± p p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq106_HTML.gif, i.e., u k ± 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq107_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif. □

Let ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq57_HTML.gif and c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq108_HTML.gif, 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 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equt_HTML.gif
Thanks to Lemma 2.3, u k K μ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq109_HTML.gif for some μ k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq110_HTML.gif. We claim that { u k } K μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq111_HTML.gif for some μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq112_HTML.gif. Indeed, if not, then dist ( u k , P ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq113_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq114_HTML.gif without loss of generality. On the one hand, since u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq32_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq115_HTML.gif, where S λ ( u k ) : ( Δ p ) 1 ( λ | u k | p 2 u k + | u k | p 2 u k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq116_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equu_HTML.gif
for 1 < p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq117_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equv_HTML.gif
for p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq118_HTML.gif. Note that by a similar proof of [[14], Lemma 3.3], we can see that S λ ( D μ ± ) int ( D μ ± ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq119_HTML.gif for μ small enough. Thus, u k S λ ( u k ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq120_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equw_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq121_HTML.gif. For the sake of convenience, we denote K μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq122_HTML.gif, K c , μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq123_HTML.gif, N c , μ , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq124_HTML.gif by K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq125_HTML.gif, K c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq126_HTML.gif, N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq127_HTML.gif. Note that for every c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq108_HTML.gif, K c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq128_HTML.gif is compact in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif (cf. [[9], Theorem 1.2]). It follows from [[19], Proposition 7.5] that there exists ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq57_HTML.gif such that
gen ( N c , 2 ε ) = gen ( K c ) < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ6_HTML.gif
(2.5)

Let J n c : = { u W 0 1 , p ( Ω ) : I n ( u ) c } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq129_HTML.gif and Q n c : = D μ J n c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq130_HTML.gif. Let J c : = { u W 0 1 , p ( Ω ) : I ( u ) c } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq131_HTML.gif. For δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq132_HTML.gif small enough, we define A n , ε c , δ : = ( Q n c + δ Q n c δ ) N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq133_HTML.gif, then we have the following.

Lemma 2.4 Assume that there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq132_HTML.gif such that K J c + δ int ( J c δ ) = K c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq134_HTML.gif for n large. Then there exists γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq135_HTML.gif such that I n ( u ) γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq136_HTML.gif for u A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq137_HTML.gif and large n.

Proof Assume a contradiction. Then, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif, there exists { v n , k } A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq138_HTML.gif such that lim k I n ( v n , k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq139_HTML.gif. It is clear that I n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq140_HTML.gif satisfies the (PS) condition for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif. Hence there exists v n W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq141_HTML.gif such that, up to a subsequence, v n , k v n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq142_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq114_HTML.gif with I n ( v n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq143_HTML.gif and I n ( v n ) [ c δ , c + δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq144_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equx_HTML.gif

Thus, by [[9], Theorem 1.2], up to a subsequence, we see that there exists v 0 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq145_HTML.gif such that v n v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq146_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq24_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq33_HTML.gif. Moreover, by using the arguments in the proof of Lemma 2.3, we have I ( v 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq147_HTML.gif and I ( v 0 ) [ c δ , c + δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq148_HTML.gif. On the other hand, for large n, v n ( int ( D μ + ) int ( D μ ) ) N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq149_HTML.gif since v n , k A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq150_HTML.gif. It follows that v 0 ( int ( D μ + ) int ( D μ ) ) N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq151_HTML.gif. This contradicts the fact that K J n c + δ int ( J n c δ ) = K c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq152_HTML.gif. □

Lemma 2.5 Assume that there exists γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq135_HTML.gif such that I n ( u ) γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq153_HTML.gif for every u A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq137_HTML.gif and large n. Then there exist δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq132_HTML.gif and an odd continuous map η n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq154_HTML.gif such that η n : A n , 2 ε c , δ Q n c δ Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq155_HTML.gif and η | Q n c δ = Id https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq156_HTML.gif for large n.

Proof We first assume 1 < p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq117_HTML.gif. It is clear that there exists L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq157_HTML.gif such that
u + S n , λ ( u ) L for all  u N c , 2 ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ7_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equy_HTML.gif
Let T n , λ : W 0 1 , p ( Ω ) K W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq158_HTML.gif be the local Lipschitz continuous operator obtained in [[14], Lemma 2.1] and let ϕ u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq159_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equz_HTML.gif

Denote τ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq160_HTML.gif to be the maximal interval of existence of ϕ u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq159_HTML.gif.

Claim 1: ϕ u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq159_HTML.gif cannot enter N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq161_HTML.gif before it enters Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq162_HTML.gif for small δ, large n and u A n , 2 ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq163_HTML.gif.

Indeed, if the claim fails, then for every δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq132_HTML.gif, ϕ u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq159_HTML.gif will enter N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq161_HTML.gif before it enters Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq162_HTML.gif. Since u A n , 2 ε c , δ W 0 1 , p ( Ω ) N c , 2 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq164_HTML.gif, there exist 0 t 1 < t 2 < τ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq165_HTML.gif such that ϕ u ( t ) N c , 2 ε N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq166_HTML.gif for t ( t 1 , t 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq167_HTML.gif and
dist ( ϕ u ( t 1 ) , K c ) = 2 ε , dist ( ϕ u ( t 2 ) , K c ) = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equaa_HTML.gif
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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq168_HTML.gif. On the other hand, by the choice of t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq169_HTML.gif and t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq170_HTML.gif, we know that ϕ u ( t ) A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq171_HTML.gif for t ( t 1 , t 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq172_HTML.gif. Thanks to [[17], Lemma 3.8], u S n , λ ( u ) ( γ C ) 1 / ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq173_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equab_HTML.gif

A contradiction with δ < 4 C / ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq174_HTML.gif.

Claim 2: There exists τ 1 ( t ) < τ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq175_HTML.gif such that ϕ u ( τ 1 ( u ) ) Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq176_HTML.gif for large n and u A n , 2 ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq177_HTML.gif.

If the claim is not true, then ϕ u ( t ) Q n c + δ Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq178_HTML.gif for all t ( 0 , τ ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq179_HTML.gif. We first consider the case of τ ( u ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq180_HTML.gif. In fact, by Claim 1, ϕ u ( t ) N c , ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq181_HTML.gif, i.e., ϕ u ( t ) A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq182_HTML.gif for all t ( 0 , τ ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq179_HTML.gif. Since I n ( u ) γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq183_HTML.gif for u A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq184_HTML.gif and large n, we must have
ϕ u ( t ) as  t τ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equ8_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equac_HTML.gif
This means ϕ u ( t ) u + C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq185_HTML.gif for all t ( 0 , τ ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq186_HTML.gif, which contradicts with (2.7). It follows that there must exist τ 1 ( u ) < τ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq187_HTML.gif such that ϕ u ( τ 1 ( u ) ) Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq176_HTML.gif for u A n , 2 ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq188_HTML.gif, large n and τ ( u ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq180_HTML.gif. Next, we consider the case of τ ( u ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq189_HTML.gif. Since u S n , λ ( u ) ( γ C ) 1 / ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq190_HTML.gif for all u A n , ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq184_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equad_HTML.gif

Thus, there also exists τ 1 ( u ) ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq191_HTML.gif such that ϕ u ( τ 1 ( u ) ) Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq192_HTML.gif for u A n , 2 ε c , δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq193_HTML.gif and τ ( u ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq189_HTML.gif. Moreover, we must have ϕ u ( t ) Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq194_HTML.gif for t ( τ 1 ( u ) , τ ( u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq195_HTML.gif since d I n ( ϕ u ( t ) ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq196_HTML.gif for all u W 0 1 , p ( Ω ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq197_HTML.gif.

Let
η n ( u ) = { ϕ u ( τ 1 ( u ) ) , u A n , 2 ε c , u , u Q n c δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equae_HTML.gif

Then, by the continuity of ϕ u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq159_HTML.gif, η n ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq198_HTML.gif is continuous. Note that ϕ u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq159_HTML.gif is odd and τ 1 ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq199_HTML.gif is even, we see that η n ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq198_HTML.gif is odd and it is the desired map. The situation of p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq118_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq19_HTML.gif. 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq200_HTML.gif. Since for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq73_HTML.gif, 0 d n , k d n , k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq201_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif, d k d k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq202_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq77_HTML.gif. It follows that two cases may occur:

Case 1: There are 1 < k 1 < k 2 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq203_HTML.gif such that d k 1 < d k 2 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq204_HTML.gif .

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

Case 2: There exists k 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq205_HTML.gif such that d = d k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq206_HTML.gif for all k k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq207_HTML.gif.

In this case, if ( K J d + δ J d δ ) K d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq208_HTML.gif for every δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq132_HTML.gif small enough, then Problem (1.1) also has infinitely many sign-changing solutions. Otherwise, there exists δ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq209_HTML.gif such that ( K J d + δ J d δ ) = K d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq210_HTML.gif for δ < δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq211_HTML.gif. Thanks to Lemmas 2.4 and 2.5, there exists η n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq154_HTML.gif such that η n ( A n , 2 ε d Q n d δ ) Q n d δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq212_HTML.gif for small δ and large n. Fix l N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq213_HTML.gif and k k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq207_HTML.gif, the definitions of d k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq214_HTML.gif and d k + l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq215_HTML.gif give that there exists a large n such that d n , k > d δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq216_HTML.gif and d n , k + l < d + δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq217_HTML.gif for small δ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq218_HTML.gif. By the definition of d n , k + l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq219_HTML.gif, there exists Z Γ k + l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq220_HTML.gif such that sup Z I n ( u ) < d + δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq221_HTML.gif, where Z = h ( B m B ) D μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq222_HTML.gif, h G m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq223_HTML.gif and gen ( B ) m k l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq224_HTML.gif. It follows that h ( B m B ) N d , 2 ε A n , 2 ε c , δ Q n c δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq225_HTML.gif. Thus, η n ( h ( B m B ) N d , 2 ε ) Q n d δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq226_HTML.gif. By the choice of δ and B m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq227_HTML.gif, we have η n h G m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq228_HTML.gif. If gen ( B h 1 ( N d , 2 ε ) ) m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq229_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equaf_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_Equag_HTML.gif

This implies gen ( N d , 2 ε ) l + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq230_HTML.gif. Since l N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq213_HTML.gif is arbitrary, we have gen ( N d , 2 ε ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-149/MediaObjects/13661_2012_Article_392_IEq231_HTML.gif, which contradicts with (2.5). □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Soochow University

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