Existence of homoclinic solutions for a class of second-order Hamiltonian systems with subquadratic growth

Boundary Value Problems20122012:132

DOI: 10.1186/1687-2770-2012-132

Received: 6 July 2012

Accepted: 25 October 2012

Published: 13 November 2012

Abstract

By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.

Keywords

homoclinic solutions critical point theory Hamiltonian systems nontrivial solution

1 Introduction

Consider the following second-order Hamiltonian system:
q ¨ ( t ) L ( t ) q ( t ) + W q ( t , q ( t ) ) = 0 , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ1_HTML.gif
(HS)

where q = ( q 1 , q 2 , , q n ) R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq1_HTML.gif, L C ( R , R n × n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq2_HTML.gif is a symmetric matrix-valued function, and W ( t , q ) C 1 ( R × R n , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq3_HTML.gif, W q ( t , q ) C 1 ( R × R n , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq4_HTML.gif is the gradient of W about q. As usual we say that a solution q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq5_HTML.gif of (HS) is homoclinic (to 0) if q C 2 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq6_HTML.gif such that q ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq7_HTML.gif and q ˙ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq8_HTML.gif as | t | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq9_HTML.gif. If q ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq10_HTML.gif, q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq5_HTML.gif is called a nontrivial homoclinic solution.

By now, the existence and multiplicity of homoclinic solutions for second-order Hamiltonian systems have been extensively investigated in many papers (see, e.g., [117] and the references therein) via variational methods. More precisely, many authors studied the existence and multiplicity of homoclinic solutions for (HS); see [517]. Some of them treated the case where L ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq11_HTML.gif and W ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq12_HTML.gif are either independent of t or periodic in t (see, for instance, [57]), and a more general case is considered in the recent paper [7]. If L ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq11_HTML.gif is neither constant nor periodic in t, the problem of the existence of homoclinic solutions for (HS) is quite different from the one just described due to the lack of compactness of the Sobolev embedding. After the work of Rabinowitz and Tanaka [8], many results [917] were obtained for the case where L ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq11_HTML.gif is neither constant nor periodic in t.

Recently, Zhang and Yuan [15] obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard minimizing argument. In this paper, ( , ) : R n × R n R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq13_HTML.gif denotes the standard inner product in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq14_HTML.gif, and subsequently, | | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq15_HTML.gif is the induced norm. If q = ( q 1 , q 2 , , q n ) R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq16_HTML.gif, then | q | = q 1 2 + q 2 2 + + q n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq17_HTML.gif.

Theorem 1.1 (See [[15], Theorem 1.1])

Assume that L and W satisfy the following conditions:

(H1) L ( t ) C ( R , R n × n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq18_HTML.gif is a symmetric matrix for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq19_HTML.gif, and there is a continuous function α : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq20_HTML.gif such that α ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq21_HTML.gif for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq19_HTML.gif and ( L ( t ) q , q ) α ( t ) | q | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq22_HTML.gif and α ( t ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq23_HTML.gif as | t | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq24_HTML.gif.

(H2) W ( t , q ) = a ( t ) | q | γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq25_HTML.gif where a ( t ) : R R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq26_HTML.gifis a positive continuous function such that a ( t ) L 2 ( R , R ) L 2 2 γ ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq27_HTML.gif and 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq28_HTML.gif is a constant.

Then (HS) possesses at least one nontrivial homoclinic solution.

In [1517], the authors considered the case where W ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq29_HTML.gif is subquadratic as | q | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq30_HTML.gif. However, there are many functions with subquadratic growth but they do not satisfy the condition (H2) in [15] and the corresponding conditions in [16, 17]. For example,
W ( t , q ) = a ( t ) | q | γ + b ( t ) e cos 3 | q | , ( t , q ) ( R , R n ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ2_HTML.gif
(1)

where 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq31_HTML.gif, a ( t ) , b ( t ) : R R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq32_HTML.gif are positive continuous functions such that a ( t ) , b ( t ) L 2 ( R , R ) L 2 2 γ ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq33_HTML.gif.

In this paper, our aim is to revisit (HS) and study the subquadratic case which is not included in [1517]. Now, we state our main result.

Theorem 1.2 Let the above condition (H1) hold. Moreover, assume that the following conditions hold:

(H3) W ( t , q ) a ( t ) | q | γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq34_HTML.gif, ( t , q ) ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq35_HTML.gif, where a ( t ) : R R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq26_HTML.gif is a positive continuous function such that a ( t ) L 2 ( R , R ) L 2 2 γ ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq27_HTML.gif and 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq36_HTML.gif is a constant.

(H4) | W q ( t , q ) | f 1 ( t ) | q | γ 1 + f 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq37_HTML.gif, ( t , q ) ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq38_HTML.gif where f 1 ( t ) , f 2 ( t ) : R R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq39_HTML.gif are positive continuous functions such that f 1 ( t ) , f 2 ( t ) L 2 ( R , R ) L 2 2 γ ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq40_HTML.gif.

Then (HS) possesses at least one nontrivial homoclinic solution.

Remark 1.1 Obviously, the condition (H2) is a special case of (H3)-(H4). If (H2) holds, so do (H3)-(H4); however, the reverse is not true. W ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq29_HTML.gif defined in (1) can satisfy the conditions (H3) and (H4), but W ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq29_HTML.gif cannot satisfy the condition (H2). So, we generalize and significantly improve Theorem 1.1 in [15].

Remark 1.2 We still consider the function W ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq29_HTML.gif defined in (1),
W ( t , q ) a ( t ) | q | γ + b ( t ) e 1 , ( t , q ) ( R , R n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equa_HTML.gif
Due to inf t R a ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq41_HTML.gif, there are no constants b , r 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq42_HTML.gif such that
W ( t , q ) b | q | γ , t R  and  | q | r 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equb_HTML.gif

so W ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq29_HTML.gif does not satisfy the conditions (W2) and (W3) in [16]. Moreover, for any given 1 < γ < 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq43_HTML.gif, W ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq29_HTML.gif does not satisfy the condition (W2) in [17]. Therefore, we also extend Theorem 1.2 in [16] and Theorem 1.1 in [17].

Example 1.1 Consider the following second-order Hamiltonian system with n = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq44_HTML.gif:
q ¨ L ( t ) q + W q ( t , q ) = 0 , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ3_HTML.gif
(2)
where
L ( t ) = ( 2 + t 2 0 0 0 2 + t 2 0 0 0 2 + t 2 ) , W ( t , q ) = ( 1 1 + | t | 3 ) | q | 5 4 + ( 1 1 + | t | 2 ) e sin 3 | q | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equc_HTML.gif

Let α ( t ) = t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq45_HTML.gif, γ = 5 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq46_HTML.gif and a 1 ( t ) = 1 1 + | t | 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq47_HTML.gif, a 2 ( t ) = 1 1 + | t | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq48_HTML.gif, W q ( t , q ) = 5 4 a 1 ( t ) | q | 3 4 q + 3 a 2 ( t ) e sin 3 | q | | q | 1 q sin 2 | q | cos | q | 5 4 a 1 ( t ) | q | 1 4 + 3 a 2 ( t ) e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq49_HTML.gif, W q ( t , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq50_HTML.gif. Clearly, (H1), (H3), and (H4) hold. Therefore, by applying Theorem 1.2, the Hamiltonian system (2) possesses at least one nontrivial homoclinic solution.

Remark 1.3 It is easy to see that (H2) in Theorem 1.1 is not satisfied, so we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1. On the other hand, W does not satisfy the conditions (W2) and (W5) of [17], then we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1 in [17].

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.

2 Preliminary results

In order to establish our result via the critical point theory, we firstly describe some properties of the space on which the variational associated with (HS) is defined. Like in [15], let
E = { q H 1 ( R , R n ) : R [ | q ˙ | 2 + ( L ( t ) q ( t ) , q ( t ) ) ] d t < } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equd_HTML.gif
Then the space E is a Hilbert space with the inner product
x , y = R [ ( x ˙ ( t ) , y ˙ ( t ) ) + ( L ( t ) x ( t ) , y ( t ) ) ] d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Eque_HTML.gif
and the corresponding norm x 2 = x , x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq51_HTML.gif. Note that
E H 1 ( R , R n ) L p ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equf_HTML.gif
for all p [ 2 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq52_HTML.gif with the embedding being continuous. Here L p ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq53_HTML.gif ( 2 p < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq54_HTML.gif) and H 1 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq55_HTML.gif denote the Banach spaces of functions on ℝ with values in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq14_HTML.gif under the norms
q p : = ( R | q | p d t ) 1 / p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equg_HTML.gif
and
q H 1 : = ( q 2 2 + q ˙ 2 2 ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equh_HTML.gif
respectively. In particular, for p = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq56_HTML.gif, there exists a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq57_HTML.gif such that
q C q , q E , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ4_HTML.gif
(3)

here q : = ess sup { | q ( t ) | : t R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq58_HTML.gif.

Lemma 2.1 There exists a constant β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq59_HTML.gif such that if q E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq60_HTML.gif, then
q β q 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ5_HTML.gif
(4)
Proof From (H1), we can imply that there exists a constant β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq59_HTML.gif such that
( L ( t ) q , q ) β | q | 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equi_HTML.gif
for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq61_HTML.gif and q R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq62_HTML.gif. By the above inequality, one has
q 2 R ( L ( t ) q ( t ) , q ( t ) ) d t β R | q ( t ) | 2 d t = β q 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equj_HTML.gif

So, the lemma is proved. □

Lemma 2.2 ([[9], Lemma 1])

Suppose that L satisfies (H1). Then the embedding of E in L 2 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq63_HTML.gif is compact.

Lemma 2.3 Suppose that (H1) and (H4) are satisfied. If q k q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq64_HTML.gif (weakly) in E, then W q ( t , q k ) W q ( t , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq65_HTML.gif in L 2 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq63_HTML.gif.

Proof Assume that q k q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq64_HTML.gif in E. Then there exists a constant d 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq66_HTML.gif such that, by the Banach-Steinhaus theorem and (3),
sup k N q k d 1 , q d 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equk_HTML.gif
Since 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq31_HTML.gif, by (H4) there exists a constant d 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq67_HTML.gif such that
| W q ( t , q k ) | d 2 f 1 ( t ) + f 2 ( t ) , | W q ( t , q ) | d 2 f 1 ( t ) + f 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equl_HTML.gif
for all k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq68_HTML.gifand t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq61_HTML.gif. Hence,
| W q ( t , q k ) W q ( t , q ) | 2 d 2 f 1 ( t ) + 2 f 2 ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equm_HTML.gif

On the other hand, by Lemma 2.2, q k q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq69_HTML.gif in L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq70_HTML.gif, passing to a subsequence if necessary, which implies q k ( t ) q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq71_HTML.gif for almost every t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq61_HTML.gif . Then using Lebesgue’s convergence theorem, the lemma is proved. □

Now, we introduce more notation and some necessary definitions. Let E be a real Banach space, I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq72_HTML.gif, which means that I is a continuously Fréchet-differentiable functional defined on E. Recall that I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq72_HTML.gif is said to satisfy the (PS) condition if any sequence { u j } j N E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq73_HTML.gif, for which { I ( u j ) } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq74_HTML.gif is bounded and I ( u j ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq75_HTML.gif as j + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq76_HTML.gif, possesses a convergent subsequence in E.

Lemma 2.4 ([[18], Theorem 2.7])

Let E be a real Banach space, and let us have I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq72_HTML.gif satisfying the (PS) condition. If I is bounded from below, then
c inf E I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equn_HTML.gif

is a critical value of I.

3 Proof of Theorem 1.2

Now, we are going to establish the corresponding variational framework to obtain homoclinic solutions of (HS). Define the functional I : E R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq77_HTML.gif
I ( q ) = R [ 1 2 | q ˙ ( t ) | 2 + 1 2 ( L ( t ) q ( t ) , q ( t ) ) W ( t , q ( t ) ) ] d t = 1 2 q 2 R W ( t , q ( t ) ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ6_HTML.gif
(5)
Lemma 3.1 Under the assumptions of Theorem  1.2, we have
I ( q ) v = R [ ( q ˙ ( t ) , v ˙ ( t ) ) + ( L ( t ) q ( t ) , v ( t ) ) ( W q ( t , q ( t ) ) , v ( t ) ) ] d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ7_HTML.gif
(6)
which yields that
I ( q ) q = q 2 R ( W q ( t , q ( t ) ) , q ( t ) ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ8_HTML.gif
(7)

Moreover, I is a continuously Fréchet-differentiable functional defined on E, i.e., I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq72_HTML.gif and any critical point of I on E is a classical solution of (HS) with q ( ± ) = 0 = q ˙ ( ± ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq78_HTML.gif.

Proof We firstly show that I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq72_HTML.gif. Let q E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq60_HTML.gif, by (3), (H4), and the Hölder inequality, we have
0 R W ( t , q ( t ) ) d t R ( f 1 ( t ) | q ( t ) | γ + f 2 ( t ) | q ( t ) | ) d t ( R | f 1 ( t ) | 2 2 γ d t ) 2 γ 2 ( R | q ( t ) | γ 2 γ d t ) γ 2 + ( R | f 2 ( t ) | 2 d t ) 1 2 ( R | q ( t ) | 2 d t ) 1 2 = f 1 2 2 γ q 2 γ + f 2 2 q 2 1 ( β ) γ f 1 2 2 γ q γ + 1 β f 2 2 q < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ9_HTML.gif
(8)
Combining (5) and (8), we show that I : E R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq77_HTML.gif. Next, we prove that I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq72_HTML.gif. Rewrite I as follows:
I = I 1 I 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equo_HTML.gif
where
I 1 : = R [ 1 2 | q ˙ ( t ) | 2 + 1 2 ( L ( t ) q ( t ) , q ( t ) ) ] d t , I 2 : = R W ( t , q ( t ) ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equp_HTML.gif
It is easy to check that I 1 C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq79_HTML.gif and
I 1 ( q ) v = R [ ( q ˙ ( t ) , v ˙ ( t ) ) + ( L ( t ) q ( t ) , v ( t ) ) ] d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ10_HTML.gif
(9)
Thus, it is sufficient to show that this is the case for I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq80_HTML.gif. In the process we will see that
I 2 ( q ) v = R ( W q ( t , q ( t ) ) , v ( t ) ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ11_HTML.gif
(10)
which is defined for all q , v E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq81_HTML.gif. For any given q E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq60_HTML.gif, let us define J ( q ) : E R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq82_HTML.gif as follows:
J ( q ) v = R ( W q ( t , q ( t ) ) , v ( t ) ) d t , v E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equq_HTML.gif
It is obvious that J ( q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq83_HTML.gif is linear. Now, we show that J ( q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq83_HTML.gif is bounded. Indeed, for any given q E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq60_HTML.gif, by (3) and (H4), there exists a constant d 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq84_HTML.gif such that
| W q ( t , q ( t ) ) | f 1 ( t ) | q | γ 1 + f 2 ( t ) d 3 f 1 ( t ) + f 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equr_HTML.gif
for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq61_HTML.gif, which yields that by (4) and the Hölder inequality,
| J ( q ) v | = | R ( W q ( t , q ( t ) ) , v ( t ) ) d t | R [ d 3 f 1 ( t ) | v ( t ) | + f 2 ( t ) | v ( t ) | ] d t d 3 f 1 2 v 2 + f 2 2 v 2 1 β ( d 3 f 1 2 + f 2 2 ) v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ12_HTML.gif
(11)
Moreover, for any q , v E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq85_HTML.gif, by the mean value theorem, we have
R W ( t , q ( t ) + v ( t ) ) d t R W ( t , q ( t ) ) d t = R ( W q ( t , q ( t ) + h ( t ) v ( t ) ) , v ( t ) ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equs_HTML.gif
where h ( t ) ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq86_HTML.gif. Therefore, by Lemma 2.3 and the Hölder inequality, one has
R ( W q ( t , q ( t ) + h ( t ) v ( t ) ) , v ( t ) ) d t R ( W q ( t , q ( t ) ) , v ( t ) ) d t = R ( W q ( t , q ( t ) + h ( t ) v ( t ) ) W q ( t , q ( t ) ) , v ( t ) ) d t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ13_HTML.gif
(12)
as v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq87_HTML.gif in E. Combining (11) and (12), we see that (10) holds. It remains to prove that I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq88_HTML.gif is continuous. Suppose that q q 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq89_HTML.gif in E and note that
I 2 ( q ) v I 2 ( q 0 ) v = R ( W q ( t , q ( t ) ) W q ( t , q 0 ( t ) ) , v ( t ) ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equt_HTML.gif
By Lemma 2.3 and the Hölder inequality, we obtain that
I 2 ( q ) v I 2 ( q 0 ) v 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equu_HTML.gif

as q q 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq89_HTML.gif, which implies the continuity of I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq88_HTML.gif and I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq90_HTML.gif.

Lastly, we check that critical points of I are classical solutions of (HS) satisfying q ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq7_HTML.gif and q ˙ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq91_HTML.gif as | t | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq24_HTML.gif. We know that E H 1 ( R , R n ) C 0 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq92_HTML.gif, the space of continuous functions q on ℝ such that q ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq7_HTML.gif as | t | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq24_HTML.gif. Moreover, if q is one critical point of I, by (6) we have
q ¨ ( t ) = L ( t ) q W q ( t , q ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equv_HTML.gif
which yields that q C 2 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq6_HTML.gif, i.e., q is a classical solution of (HS). Since q is one critical point of I, we have
I ( q ) q = R [ ( q ˙ ( t ) , q ˙ ( t ) ) + ( L ( t ) q ( t ) , q ( t ) ) ( W q ( t , q ( t ) ) , q ( t ) ) ] d t = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equw_HTML.gif
It follows from q ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq7_HTML.gif as | t | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq24_HTML.gif and the above equality that
R ( q ˙ ( t ) , q ˙ ( t ) ) d t 0 , as  | t | + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equx_HTML.gif

Hence, q satisfies q ˙ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq91_HTML.gif as | t | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq93_HTML.gif. This proof is complete. □

Lemma 3.2 Under the assumptions of Theorem  1.2, I satisfies the (PS) condition.

Proof In fact, assume that { q j } j N E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq94_HTML.gif is a sequence such that { I ( q j ) } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq95_HTML.gif is bounded and I ( q j ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq96_HTML.gif as j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq97_HTML.gif. Then there exists a constant C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq98_HTML.gif such that
| I ( q j ) | C 1 , I ( q j ) E C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ14_HTML.gif
(13)

for every j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq99_HTML.gif.

We firstly prove that { q j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq100_HTML.gif is bounded in E. By (5) and (8), we have
1 2 q j 2 = I ( q j ) + R W ( t , q j ( t ) ) d t C 1 + 1 ( β ) γ f 1 2 2 γ q j γ + 1 β f 2 2 q j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ15_HTML.gif
(14)
Combining (13) and (14), we obtain that
1 2 q j 2 1 ( β ) γ f 1 2 2 γ q j γ 1 β f 2 2 q j C 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ16_HTML.gif
(15)

Since 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq31_HTML.gif, the above inequality shows that { q j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq100_HTML.gif is bounded in E. By Lemma 2.2, the sequence { q j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq100_HTML.gif has a subsequence, again denoted by { q j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq100_HTML.gif, and there exists q E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq60_HTML.gif such that

q j q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq101_HTML.gif, weakly in E,

q j q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq102_HTML.gif, strongly in L 2 ( R , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq63_HTML.gif.

Hence,
( I ( q j ) I ( q ) , q j q ) 0 , R ( W q ( t , q j ( t ) ) W q ( t , q ( t ) ) , q j ( t ) q ( t ) ) d t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equy_HTML.gif
as j + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq103_HTML.gif. Moreover, an easy computation shows that
( I ( q j ) I ( q ) , q j q ) = q j q 2 R ( W q ( t , q j ( t ) ) W q ( t , q ( t ) ) , q j ( t ) q ( t ) ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equz_HTML.gif

So, q j q 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq104_HTML.gif as j + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq76_HTML.gif, i.e., I satisfies the Palais-Smale condition. □

Now, we can give the proof of Theorem 1.2.

Proof of Theorem 1.2 By (5) and (8), for every r R { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq105_HTML.gif and q E { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq106_HTML.gif, we have
I ( r q ) = r 2 2 q 2 R W ( t , r q ( t ) ) d t r 2 2 q 2 R [ f 1 ( t ) | r q ( t ) | γ + f 2 ( t ) | r q ( t ) | ] d t r 2 2 q 2 | r | γ 1 ( β ) γ f 1 2 2 γ q γ | r | 1 β f 2 2 q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equ17_HTML.gif
(16)
Since 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq107_HTML.gif, (16) implies that I ( r q ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq108_HTML.gif as | r | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq109_HTML.gif. Consequently, I is a functional bounded from below. By Lemmas 3.2 and 2.4, I possesses a critical value c = inf q E I ( q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq110_HTML.gif, i.e., there is a q E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq60_HTML.gif such that
I ( q ) = c , I ( q ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equaa_HTML.gif
On the other hand, take c 0 R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq111_HTML.gif with | c 0 | 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq112_HTML.gif, and let φ E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq113_HTML.gif be given by
φ ( t ) = { c 0 sin ( π t 2 t 1 ( t t 1 ) ) if  t [ t 1 , t 2 ] , 0 if  t R [ t 1 , t 2 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equab_HTML.gif
where < t 1 < t 2 < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq114_HTML.gif. Then we obtain that
I ( r φ ) = r 2 2 φ 2 R W ( t , r φ ( t ) ) d t r 2 2 φ 2 | r | γ R a ( t ) | φ ( t ) | γ d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_Equac_HTML.gif

which yields that I ( r φ ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq115_HTML.gif as | r | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq116_HTML.gif small enough since 1 < γ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-132/MediaObjects/13661_2012_Article_233_IEq117_HTML.gif, i.e., the critical point obtained above is nontrivial. □

Author’s contributions

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

Declarations

Acknowledgements

This work is supported by the Research Foundation of Education Bureau of Hunan Province, China (No.11C0594). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.

Authors’ Affiliations

(1)
Department of Mathematics, Hunan University of Science and Engineering

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