Open Access

Multiplicity of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

Boundary Value Problems20132013:200

DOI: 10.1186/1687-2770-2013-200

Received: 17 June 2013

Accepted: 20 August 2013

Published: 5 September 2013

Abstract

In this paper, we consider the existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Infinitely many solutions are obtained by using a version of the symmetric mountain-pass theorem, and this sequence of solutions converge to zero. Some recent results are extended.

MSC:34B37, 35B38.

Keywords

impulsive effects variational methods Dirichlet boundary value problem critical points

1 Introduction

In this paper, we study the following nonlinear impulsive differential equations with Dirichlet boundary conditions
{ u ( t ) + a ( t ) u ( t ) = μ | u | p 2 u ( t ) + f ( t , u ( t ) ) , a.e.  t [ 0 , T ] , Δ u ( t j ) = u ( t j + ) u ( t j ) = I j ( u ( t j ) ) , j = 1 , 2 , , N , u ( 0 ) = u ( T ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ1_HTML.gif
(1.1)

where p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq1_HTML.gif, T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq2_HTML.gif, μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq3_HTML.gif, f : [ 0 , T ] × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq4_HTML.gif is continuous, a L [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq5_HTML.gif, N is a positive integer, 0 = t 0 < t 1 < t 2 < < t N < t N + 1 = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq6_HTML.gif, Δ u ( t j ) = u ( t j + ) u ( t j ) = lim t t j + u ( t ) lim t t j u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq7_HTML.gif, I j : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq8_HTML.gif are continuous. With the help of the symmetric mountain-pass lemma due to Kajikiya [1], we prove that there are infinitely many small weak solutions for equations (1.1) with the general nonlinearities f ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq9_HTML.gif.

In recent years, a great deal of works have been done in the study of the existence of solutions for impulsive boundary value problems, by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [2]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [313]. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory [14], the method of upper and lower solutions with a monotone iterative technique [15], and some fixed point theorems in cones [16, 17].

On the other hand, in the last few years, many authors have used a variational method to study the existence and multiplicity of solutions for boundary value problems without impulsive effects [1821]. For related basic information, we refer the reader to [22, 23].

For a second order differential equation u = f ( t , u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq10_HTML.gif, one usually considers impulses in the position u and the velocity u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq11_HTML.gif. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in the position [2427].

A new approach via critical point and variational methods is proved to be very effective in studying the boundary problem for differential equations. For some general and recent works on the theory of critical point theory and variational methods, we refer the reader to [2837].

More precisely, in [28] the authors studied the following equations with Dirichlet boundary conditions:
{ u ¨ ( t ) + λ u ( t ) = f ( t , u ( t ) ) , a.e.  t [ 0 , T ] , Δ u ˙ ( t j ) = I j ( u ( t j ) ) , j = 1 , 2 , , p , u ( 0 ) = u ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ2_HTML.gif
(1.2)

They obtained the existence of solutions for problems by using the variational method. Zhang and Yuan [30] extended the results in [28]. They obtained the existence of solutions for problem (1.2) with a perturbation term. Also, they obtained infinitely many solutions for problem (1.2) under the assumption that the nonlinearity f is a superlinear case. Soon after that, Zhou and Li [29] extended problem (1.2). In all the above-mentioned works, the information on the sequence of solutions was not given.

Motivated by the fact above, the aim of this paper is to show the existence of infinitely many solutions for problem (1.1), and that there exists a sequence of infinitely many arbitrarily small solutions, converging to zero, by using a new version of the symmetric mountain-pass lemma due to Kajikiya [1]. Our main results extend the existing study.

Throughout this paper, we assume that I j : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq12_HTML.gif is continuous, and f ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq13_HTML.gif satisfies the following conditions:

(I1) I j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq14_HTML.gif ( j = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq15_HTML.gif) are odd and satisfy
0 u ( t j ) I j ( s ) d s 1 2 I j ( u ( t j ) ) u ( t j ) 0 , 0 u ( t j ) I j ( s ) d s 0 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equa_HTML.gif
(I2) There exist δ j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq16_HTML.gif, j = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq15_HTML.gif such that
0 u ( t j ) I j ( s ) d s δ j | u | 2 , for  u R { 0 } ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equb_HTML.gif

(H1) f ( t , u ) C ( [ 0 , T ] × R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq17_HTML.gif, f ( t , u ) = f ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq18_HTML.gif for all u R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq19_HTML.gif;

(H2) lim | u | f ( t , u ) | u | p 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq20_HTML.gif uniformly for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq21_HTML.gif;

(H3) lim | u | 0 + f ( t , u ) u = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq22_HTML.gif uniformly for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq23_HTML.gif.

The main result of this paper is as follows.

Theorem 1.1 Suppose that (I1)-(I2) and (H1)-(H3) hold. Then problem (1.1) has a sequence of nontrivial solutions { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif and u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq25_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq26_HTML.gif.

Remark 1.1 Without the symmetry condition (i.e., f ( x , u ) = f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq27_HTML.gif and I ( s ) = I ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq28_HTML.gif), we can obtain at least one nontrivial solution by the same method in this paper.

Remark 1.2 We should point out that Theorem 1.1 is different from the previous results of [2837] in three main directions:
  1. (1)

    We do not make the nonlinearity f satisfy the well-known Ambrosetti-Rabinowitz condition [23];

     
  2. (2)

    We try to use Lusternik-Schnirelman’s theory for Z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq29_HTML.gif-invariant functional. But since the functional is not bounded from below, we could not use the theory directly. So, we follow [38] to consider a truncated functional.

     
  3. (3)

    We can obtain a sequence of nontrivial solutions { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif and u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq25_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq26_HTML.gif.

     

Remark 1.3 There exist many functions I j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq14_HTML.gif and f ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq9_HTML.gif satisfying conditions (I1)-(I2) and (H1)-(H3), respectively. For example, when p = 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq30_HTML.gif, I j ( s ) = s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq31_HTML.gif and f ( t , u ) = e t u 1 / 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq32_HTML.gif.

2 Preliminary lemmas

In this section, we first introduce some notations and some necessary definitions.

Definition 2.1 Let E be a Banach space and J : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq33_HTML.gif. J is said to be sequentially weakly lower semi-continuous if lim n inf J ( u n ) J ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq34_HTML.gif as u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq35_HTML.gif in E.

Definition 2.2 Let E be a real Banach space. For any sequence { u n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq36_HTML.gif, if { J ( u n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq37_HTML.gif is bounded and J ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq38_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq39_HTML.gif possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition (denoted by (PS) condition for short).

In the Sobolev space H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq40_HTML.gif, consider the inner product
u , v H 0 1 ( 0 , T ) = 0 T u ( t ) v ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equc_HTML.gif
which induces the norm
u H 0 1 ( 0 , T ) = ( 0 T ( u ( t ) ) 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equd_HTML.gif
It is a consequence of Poincaré’s inequality that
( 0 T ( u ( t ) ) 2 d t ) 1 2 1 λ 1 ( 0 T ( u ( t ) ) 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Eque_HTML.gif
Here, λ 1 = π 2 / T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq41_HTML.gif is the first eigenvalue of the Dirichlet problem
{ u ( t ) = λ u ( t ) , t [ 0 , T ] , u ( 0 ) = u ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ3_HTML.gif
(2.1)
In this paper, we will assume that inf t [ 0 , T ] a ( t ) = m > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq42_HTML.gif. We can also define the inner product
u , v = 0 T u ( t ) v ( t ) d t + 0 T a ( t ) u ( t ) v ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equf_HTML.gif
which induces the equivalent norm
u = ( 0 T ( u ( t ) ) 2 d t + 0 T a ( t ) ( u ( t ) ) 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equg_HTML.gif

Lemma 2.1 [29]

If ess inf t [ 0 , T ] a ( t ) = m > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq43_HTML.gif, then the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq44_HTML.gif and the norm H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq45_HTML.gif are equivalent.

Lemma 2.2 [29]

There exists c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq46_HTML.gif such that if u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq47_HTML.gif, then
u c u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ4_HTML.gif
(2.2)

where u = max t [ 0 , T ] | u ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq48_HTML.gif.

For u H 2 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq49_HTML.gif, we have that u and u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq11_HTML.gif are both absolutely continuous, and u L 2 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq50_HTML.gif, hence, Δ u ( t j ) = u ( t j + ) u ( t j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq51_HTML.gif for any t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq21_HTML.gif. If u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq47_HTML.gif, then u is absolutely continuous and u L 2 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq52_HTML.gif. In this case, the one-side derivatives u ( t j + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq53_HTML.gif and u ( t j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq54_HTML.gif may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that u C [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq55_HTML.gif satisfies the Dirichlet condition u ( 0 ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq56_HTML.gif. Assume that, for every j = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq15_HTML.gif, u j = u | ( t j , t j + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq57_HTML.gif and u j H 2 ( t j , t j + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq58_HTML.gif. Let 0 = t 0 < t 1 < t 2 < < t N < t N + 1 = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq59_HTML.gif.

Taking v H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq60_HTML.gif and multiplying the two sides of the equality
u ( t ) + a ( t ) u ( t ) = μ | u | p 2 u ( t ) + f ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equh_HTML.gif
by v and integrating between 0 and T, we have
0 T [ u ( t ) + a ( t ) u ( t ) μ | u | p 2 u ( t ) f ( t , u ( t ) ) ] v ( t ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ5_HTML.gif
(2.3)
Moreover, since u ( 0 ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq56_HTML.gif, one has
0 T u ( t ) v ( t ) d t = j = 0 N t j t j + 1 u ( t ) v ( t ) d t = j = 0 N u ( t ) v ( t ) | t j + t j + 1 + 0 T u ( t ) v ( t ) d t = ( j = 1 N Δ u ( t j ) v ( t j ) u ( 0 ) v ( 0 ) + u ( T ) v ( T ) ) + 0 T u ( t ) v ( t ) d t = j = 1 N Δ u ( t j ) v ( t j ) + 0 T u ( t ) v ( t ) d t = j = 1 N I j ( u ( t j ) ) v ( t j ) + 0 T u ( t ) v ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equi_HTML.gif
Combining (2.3), we get
0 T u ( t ) v ( t ) d t + 0 T a ( t ) u ( t ) v ( t ) d t μ 0 T | u | p 2 u ( t ) v ( t ) d t 0 T f ( t , u ( t ) ) v ( t ) d t + j = 1 N I j ( u ( t j ) ) v ( t j ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equj_HTML.gif
Lemma 2.3 A weak solution of (1.1) is a function u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq61_HTML.gif such that
0 T u ( t ) v ( t ) d t + 0 T a ( t ) u ( t ) v ( t ) d t μ 0 T | u | p 2 u ( t ) v ( t ) d t 0 T f ( t , u ( t ) ) v ( t ) d t + j = 1 N I j ( u ( t j ) ) v ( t j ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ6_HTML.gif
(2.4)

for any v H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq60_HTML.gif.

Consider J : H 0 1 ( 0 , T ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq62_HTML.gif defined by
J ( u ) = 1 2 u 2 μ p 0 T | u | p d t 0 T F ( t , u ( t ) ) d t + j = 1 N 0 u ( t j ) I j ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ7_HTML.gif
(2.5)
where F ( t , u ) = 0 u f ( t , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq63_HTML.gif. Using the continuity of f and I j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq14_HTML.gif, j = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq64_HTML.gif, we obtain the continuity and differentiability of J and J C 1 ( H 0 1 ( 0 , T ) , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq65_HTML.gif. For any v H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq60_HTML.gif, one has
J ( u ) v = 0 T u ( t ) v ( t ) d t + 0 T a ( t ) u ( t ) v ( t ) d t μ 0 T | u | p 2 u ( t ) v ( t ) d t 0 T f ( t , u ( t ) ) v ( t ) d t + j = 1 N I j ( u ( t j ) ) v ( t j ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ8_HTML.gif
(2.6)

Thus, the solutions of problem (1.1) are the corresponding critical points of J.

Lemma 2.4 If u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq47_HTML.gif is a weak solution of problem (1.1), then u is a classical solution of problem (1.1).

Proof Obviously, we have u ( 0 ) = u ( T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq56_HTML.gif since u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq47_HTML.gif. By the definition of weak solution, for any v H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq60_HTML.gif, one has
0 T u ( t ) v ( t ) d t + 0 T a ( t ) u ( t ) v ( t ) d t μ 0 T | u | p 2 u ( t ) v ( t ) d t 0 T f ( t , u ( t ) ) v ( t ) d t + j = 1 N I j ( u ( t j ) ) v ( t j ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ9_HTML.gif
(2.7)
For j { 0 , 1 , 2 , , N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq66_HTML.gif, choose v H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq60_HTML.gif with v ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq67_HTML.gif for every t [ 0 , t j ] [ t j + 1 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq68_HTML.gif. Then
t j t j + 1 u ( t ) v ( t ) d t + t j t j + 1 a ( t ) u ( t ) v ( t ) d t = μ t j t j + 1 | u | p 2 u ( t ) v ( t ) d t + 0 T f ( t , u ( t ) ) v ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equk_HTML.gif
By the definition of weak derivative, the equality above implies that
u ( t ) + a ( t ) u ( t ) = μ | u | p 2 u ( t ) + f ( t , u ( t ) ) , a.e.  t ( t j , t j + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ10_HTML.gif
(2.8)
Hence u j H 2 ( t j , t j + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq58_HTML.gif and u satisfies the equation in (1.1) a.e. on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq69_HTML.gif. By integrating (2.7), we have
j = 1 N Δ u ( t j ) v ( t j ) + u ( T ) v ( T ) u ( 0 ) v ( 0 ) + j = 1 N I j ( u ( t j ) ) v ( t j ) + 0 T [ u ( t ) + a ( t ) u ( t ) μ | u | p 2 u ( t ) f ( t , u ( t ) ) ] v ( t ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equl_HTML.gif
Combining this fact with (2.8), we get
j = 1 N Δ u ( t j ) v ( t j ) = j = 1 N I j ( u ( t j ) ) v ( t j ) for any  v H 0 1 ( 0 , T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equm_HTML.gif

Hence, Δ u ( t j ) = I j ( u ( t j ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq70_HTML.gif for every j = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq64_HTML.gif, and the impulsive condition in (1.1) is satisfied. This completes the proof. □

Lemma 2.5 If ess inf t [ 0 , T ] a ( t ) = m > λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq71_HTML.gif, then the functional J is sequentially weakly lower semi-continuous.

Proof Let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif be a weakly convergent sequence to u in H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq72_HTML.gif, then
u lim n inf u n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equn_HTML.gif
We have that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif converges uniformly to u on C [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq73_HTML.gif. Then
lim n inf J ( u n ) = lim n [ 1 2 u n 2 μ p 0 T | u n | p d t 0 T F ( t , u n ( t ) ) d t + j = 1 N 0 u n ( t j ) I j ( s ) d s ] 1 2 u 2 μ p 0 T | u | p d t 0 T F ( t , u ( t ) ) d t + j = 1 N 0 u ( t j ) I j ( s ) d s = J ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equo_HTML.gif

This completes the proof. □

Under assumptions (H1) and (H2), we have
f ( t , u ) u = o ( | u | p ) , F ( t , u ) = o ( | u | p ) as  | u | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equp_HTML.gif
which means that for all ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq74_HTML.gif, there exist a ( ε ) , b ( ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq75_HTML.gif such that
| f ( t , u ) u | a ( ε ) + ε | u | p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ11_HTML.gif
(2.9)
| F ( x , u ) | b ( ε ) + ε | u | p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ12_HTML.gif
(2.10)
Hence, for every positive constant k, we have
F ( x , u ) k f ( x , u ) u c ( ε ) + ε | u | p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ13_HTML.gif
(2.11)

where c ( ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq76_HTML.gif.

Lemma 2.6 Suppose that (I1)-(I2) and (H1)-(H3) hold, then J ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq77_HTML.gif satisfies the (PS) condition.

Proof Let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif be a sequence in H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq40_HTML.gif such that { J ( u n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq37_HTML.gif is bounded and J ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq38_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq26_HTML.gif. First, we prove that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif is bounded. By (2.5), (2.6) and (2.11), one has
J ( u n ) 1 2 J ( u n ) u n = ( 1 2 1 p ) μ 0 T | u n | p d t + 0 T [ 1 2 f ( t , u n ( t ) ) u n ( t ) F ( t , u n ( t ) ) ] d t + j = 1 N 0 u n ( t j ) I j ( s ) d s 1 2 j = 1 N I j ( u n ( t j ) ) u n ( t j ) ( ( p 2 ) μ 2 p ε ) 0 T | u n | p d t c ( ε ) T + j = 1 N 0 u n ( t j ) I j ( s ) d s 1 2 j = 1 N I j ( u n ( t j ) ) u n ( t j ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equq_HTML.gif
By condition (I1), we can deduce that
j = 1 N 0 u n ( t j ) I j ( s ) d s 1 2 j = 1 N I j ( u n ( t j ) ) u n ( t j ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equr_HTML.gif
Setting ε = ( p 2 ) μ 4 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq78_HTML.gif, we get
0 T | u n | p d t M + o ( 1 ) u n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ14_HTML.gif
(2.12)
where o ( 1 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq79_HTML.gif and M is a positive constant. On the other hand, by (I1), (2.5) and (2.10), we have
> J ( u n ) = 1 2 u n 2 μ p 0 T | u n | p d t 0 T F ( t , u n ( t ) ) d t + j = 1 N 0 u n ( t j ) I j ( s ) d s 1 2 u n 2 ( μ p + ε ) 0 T | u n | p d t b ( ε ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ15_HTML.gif
(2.13)
Thus, (2.12) and (2.13) imply that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif is bounded in H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq40_HTML.gif. Going if necessary to a subsequence, we can assume that there exists u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq47_HTML.gif such that
u n u weakly in  H 0 1 ( 0 , T ) , u n u strongly in  C ( [ 0 , T ] , R ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equs_HTML.gif
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq80_HTML.gif. Hence,
( J ( u n ) J ( u ) ) ( u n u ) 0 , 0 T [ f ( t , u n ( t ) ) f ( t , u ( t ) ) ] ( u n ( t ) u ( t ) ) d t 0 , 0 T ( | u n | p 2 u n ( t ) | u | p 2 u ( t ) ) ( u n ( t ) u ( t ) ) d t 0 , j = 1 N [ I j ( u n ( t j ) ) I j ( u ( t j ) ) ] ( u n ( t j ) u ( t j ) ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equt_HTML.gif
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq80_HTML.gif. Moreover, one has
( J ( u n ) J ( u ) ) ( u n u ) = u n u 2 0 T ( | u n | p 2 u n ( t ) | u | p 2 u ( t ) ) ( u n ( t ) u ( t ) ) d t 0 T [ f ( t , u n ( t ) ) f ( t , u ( t ) ) ] ( u n ( t ) u ( t ) ) d t j = 1 N [ I j ( u n ( t j ) ) I j ( u ( t j ) ) ] ( u n ( t j ) u ( t j ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equu_HTML.gif

Therefore, u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq81_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq82_HTML.gif. That is { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif converges strongly to u in H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq40_HTML.gif. That is J satisfies the (PS) condition. □

3 Existence of a sequence of arbitrarily small solutions

In this section, we prove the existence of infinitely many solutions of (1.1), which tend to zero. Let X be a Banach space and denote
Σ : = { A X { 0 } : A  is closed in  X  and symmetric with respect to the orgin } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equv_HTML.gif
For A Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq83_HTML.gif, we define genus γ ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq84_HTML.gif as
γ ( A ) : = inf { m N : φ C ( A , R m { 0 } , φ ( x ) = φ ( x ) ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equw_HTML.gif

If there is no mapping φ as above for any m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq85_HTML.gif, then γ ( A ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq86_HTML.gif. We list some properties of the genus (see [1]).

Proposition 3.1 Let A and B be closed symmetric subsets of X, which do not contain the origin. Then the following hold.
  1. (1)

    If there exists an odd continuous mapping from A to B, then γ ( A ) γ ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq87_HTML.gif;

     
  2. (2)

    If there is an odd homeomorphism from A to B, then γ ( A ) = γ ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq88_HTML.gif;

     
  3. (3)

    If γ ( B ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq89_HTML.gif, then γ ( A B ) ¯ γ ( A ) γ ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq90_HTML.gif;

     
  4. (4)

    Then n-dimensional sphere S n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq91_HTML.gif has a genus of n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq92_HTML.gif by the Borsuk-Ulam theorem;

     
  5. (5)

    If A is compact, then γ ( A ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq93_HTML.gif and there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq94_HTML.gif such that U δ ( A ) Σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq95_HTML.gif and γ ( U δ ( A ) ) = γ ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq96_HTML.gif, where U δ ( A ) = { x X : x A δ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq97_HTML.gif.

     

Let Σ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq98_HTML.gif denote the family of closed symmetric subsets A of X such that 0 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq99_HTML.gif and γ ( A ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq100_HTML.gif. The following version of the symmetric mountain-pass lemma is due to Kajikiya [1].

Lemma 3.1 Let E be an infinite-dimensional space and I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq101_HTML.gif, and suppose the following conditions hold.

(C1) I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq102_HTML.gif is even, bounded from below, I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq103_HTML.gif and I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq102_HTML.gif satisfies the Palais-Smale condition;

(C2) For each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq104_HTML.gif, there exists an A k Σ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq105_HTML.gif such that sup u A k I ( u ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq106_HTML.gif.

Then either (R1) or (R2) below holds.

(R1) There exists a sequence { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq107_HTML.gif such that I ( u k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq108_HTML.gif, I ( u k ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq109_HTML.gif and { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq107_HTML.gif converges to zero;

(R2) There exist two sequences { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq107_HTML.gif and { v k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq110_HTML.gif such that I ( u k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq108_HTML.gif, I ( u k ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq109_HTML.gif, u k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq111_HTML.gif, lim k u k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq112_HTML.gif, I ( v k ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq113_HTML.gif, I ( v k ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq114_HTML.gif, lim k v k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq115_HTML.gif, and { v k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq116_HTML.gif converges to a nonzero limit.

Remark 3.1 From Lemma 3.1, we have a sequence { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq107_HTML.gif of critical points such that I ( u k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq117_HTML.gif, u k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq111_HTML.gif and lim k u k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq118_HTML.gif.

In order to get infinitely many solutions, we need some lemmas. Under the assumptions of Theorem 1.1, let ε = μ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq119_HTML.gif, we have
J ( u ) = 1 2 u 2 μ p 0 T | u | p d t 0 T F ( t , u ( t ) ) d t + j = 1 N 0 u ( t j ) I j ( s ) d s 1 2 u 2 ( μ p + ε ) 0 T | u | p d t b ( ε ) T 1 2 u 2 ( μ p + ε ) c p T u p b ( ε ) T = 1 2 u 2 2 μ p c p T u p b ( μ p ) T = A u 2 B u p C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equx_HTML.gif
where
A = 1 2 , B = 2 μ p c p T , C = b ( μ p ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equy_HTML.gif
Let P ( t ) = A t 2 B t p C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq120_HTML.gif. As P ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq121_HTML.gif attains a local but not a global minimum (P is not bounded below), we have to perform some sort of truncation. To this end, let R 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq122_HTML.gif, R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq123_HTML.gif be such that m < R 0 < M < R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq124_HTML.gif, where m is the local minimum of P ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq121_HTML.gif, and M is the local maximum and P ( R 1 ) > P ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq125_HTML.gif. For these values R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq123_HTML.gif and R 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq122_HTML.gif, we can choose a smooth function χ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq126_HTML.gif defined as follows
χ ( t ) = { 1 , 0 t R 0 , 0 , t R 1 , C , χ ( t ) [ 0 , 1 ] , R 0 t R 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equz_HTML.gif
Then it is easy to see χ ( t ) [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq127_HTML.gif and χ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq126_HTML.gif is C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq128_HTML.gif. Let φ ( u ) = χ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq129_HTML.gif and consider the perturbation of J ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq77_HTML.gif:
G ( u ) = 1 2 u 2 φ ( u ) μ p 0 T | u | p d t φ ( u ) 0 T F ( t , u ( t ) ) d t + j = 1 N 0 u ( t j ) I j ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equ16_HTML.gif
(3.1)
Then
G ( u ) A u 2 B φ ( u ) u p C = P ¯ ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equaa_HTML.gif
where P ¯ ( t ) = A t 2 B χ ( t ) t p C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq130_HTML.gif and
P ¯ ( t ) = { P ( t ) , 0 t ρ 0 , m , t ρ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equab_HTML.gif

From the arguments above, we have the following.

Lemma 3.2 Let G ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq131_HTML.gif is defined as in (3.1). Then
  1. (i)

    G C 1 ( H 0 1 ( 0 , T ) , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq132_HTML.gif and G is even and bounded from below;

     
  2. (ii)

    If G ( u ) < m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq133_HTML.gif, then P ¯ ( u ) < m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq134_HTML.gif, consequently, u < ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq135_HTML.gif and J ( u ) = G ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq136_HTML.gif;

     
  3. (iii)

    Suppose that (I1)-(I2) and (H1)-(H3) hold, then G ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq131_HTML.gif satisfies the (PS) condition.

     

Proof It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 2.6. □

Lemma 3.3 Assume that (I2) and (H3) hold. Then for any k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq104_HTML.gif, there exists δ = δ ( k ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq137_HTML.gif such that γ ( { u H 0 1 ( 0 , T ) : G ( u ) δ ( k ) } { 0 } ) k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq138_HTML.gif.

Proof Firstly, by (H3) of Theorem 1.1, for any fixed u H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq139_HTML.gif, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq140_HTML.gif, we have
F ( x , ρ u ) M ( ρ ) ( ρ u ) 2 with  M ( ρ )  as  ρ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equac_HTML.gif
Secondly, from Lemma 5 of [33], we have that for any finite dimensional subspace E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq141_HTML.gif of H 0 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq40_HTML.gif and any u E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq142_HTML.gif, there exists a constant d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq143_HTML.gif such that
| u | s = ( 0 T | u ( t ) | s d t ) 1 s d u , s 1 , u E k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equad_HTML.gif
Therefore, for any u E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq142_HTML.gif with u = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq144_HTML.gif and ρ small enough, we have
G ( ρ u ) = J ( ρ u ) 1 2 ρ 2 μ p ρ p 0 T | u | p d t M ( ρ ) ρ 2 0 T | u | 2 d t + ρ 2 j = 1 N δ j c 2 ( 1 2 + j = 1 N δ j c 2 M ( ρ ) d 2 ) ρ 2 = δ ( k ) < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equae_HTML.gif
since lim | ρ | 0 M ( ρ ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq145_HTML.gif. That is,
{ u E k : u = ρ } { u H 0 1 ( 0 , T ) : G ( u ) δ ( k ) } { 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equaf_HTML.gif

This completes the proof. □

Now, we give the proof of Theorem 1.1 as following.

Proof of Theorem 1.1 Recall that
Σ k = { A H 0 1 ( 0 , T ) { 0 } : A  is closed and  A = A , γ ( A ) k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equag_HTML.gif
and define
c k = inf A Σ k sup u A G ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_Equah_HTML.gif

By Lemma 3.2(i) and Lemma 3.3, we know that < c k < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq146_HTML.gif. Therefore, assumptions (C1) and (C2) of Lemma 3.1 are satisfied. This means that G has a sequence of solutions { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-200/MediaObjects/13661_2013_Article_453_IEq24_HTML.gif converging to zero. Hence, Theorem 1.1 follows by Lemma 3.2(ii). □

Declarations

Acknowledgements

The authors are supported by the Research Foundation during the 12th Five-Year Plan Period of Department of Education of Jilin Province, China (Grant [2013] No. 252), the China Postdoctoral Science Foundation (Grant No. 2012M520665), the Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH), the open project program of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant No. 93K172013K03), the Natural Science Foundation of Changchun Normal University.

Authors’ Affiliations

(1)
College of Mathematics, Changchun Normal University
(2)
Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University

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