Open Access

Ground state homoclinic orbits of damped vibration problems

Boundary Value Problems20142014:106

DOI: 10.1186/1687-2770-2014-106

Received: 6 January 2014

Accepted: 24 April 2014

Published: 9 May 2014

Abstract

In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case.

MSC: 49J40, 70H05.

Keywords

non-periodic damped vibration problems ground state homoclinic orbits superquadratic nonlinearity

1 Introduction and main results

We shall study the existence of ground state homoclinic orbits for the following non-periodic damped vibration system:
u ¨ ( t ) + M u ˙ ( t ) L ( t ) u ( t ) + H u ( t , u ( t ) ) = 0 , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ1_HTML.gif
(1.1)

where M is an antisymmetric N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq1_HTML.gif constant matrix, L ( t ) C ( R , R N × N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq2_HTML.gif is a symmetric matrix, H ( t , u ) C 1 ( R × R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq3_HTML.gif and H u ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq4_HTML.gif denotes its gradient with respect to the u variable. We say that a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq5_HTML.gif of (1.1) is homoclinic (to 0) if u ( t ) C 2 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq6_HTML.gif such that u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq7_HTML.gif and u ˙ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq8_HTML.gif as | t | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq9_HTML.gif. If u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq10_HTML.gif, then u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq5_HTML.gif is called a nontrivial homoclinic solution.

If M = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq11_HTML.gif (zero matrix), then (1.1) reduces to the following second-order Hamiltonian system:
u ¨ ( t ) L ( t ) u ( t ) + H u ( t , u ( t ) ) = 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ2_HTML.gif
(1.2)

This is a classical equation which can describe many mechanic systems such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (1.2) have been studied by many authors via variational methods; see [118] and the references therein.

The periodic assumptions are very important in the study of homoclinic orbits for (1.2) since periodicity is used to control the lack of compactness due to the fact that (1.2) is set on all . However, non-periodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka [11] introduced a type of coercive condition on the matrix L ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq12_HTML.gif,
l ( t ) : = inf | u | = 1 ( L ( t ) u , u ) + as  | t | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ3_HTML.gif
(1.3)
and obtained the existence of a homoclinic orbit for non-periodic (1.2) under the Ambrosetti-Rabinowitz (AR) superquadratic condition:
0 < μ H ( t , u ) ( H u ( t , u ) , u ) , t R , u R N { 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equa_HTML.gif

where μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq13_HTML.gif is a constant, ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq14_HTML.gif denotes the standard inner product in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq15_HTML.gif and the associated norm is denoted by | | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq16_HTML.gif.

We should mention that in the case where M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq17_HTML.gif, i.e., the damped vibration system (1.1), only a few authors have studied homoclinic orbits of (1.1); see [1923]. Zhu [23] considered the periodic case of (1.1) (i.e., L ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq12_HTML.gif and H ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq18_HTML.gif are T-periodic in t with T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq19_HTML.gif) and obtained the existence of nontrivial homoclinic solutions of (1.1). The authors [1922] considered the non-periodic case of (1.1): Zhang and Yuan [22] obtained the existence of at least one homoclinic orbit for (1.1) when H satisfies the subquadratic condition at infinity by using a standard minimizing argument; by a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [21] obtained the existence and multiplicity of homoclinic orbits for (1.1) when H satisfies the local (AR) superquadratic growth condition:
0 < μ H ( t , u ) ( H u ( t , u ) , u ) , t R , | u | r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ4_HTML.gif
(1.4)

where μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq13_HTML.gif and r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq20_HTML.gif are two constants. Notice that the authors [21, 22] all used condition (1.3). Recently, the author in [19, 20] obtained infinitely many homoclinic orbits for (1.1) when H satisfies the subquadratic[19] and asymptotically quadratic[20] condition at infinity by the following weaker conditions than (1.3):

(L1) There is a constant β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq21_HTML.gif such that
meas { t R : | t | β L ( t ) < b I N } < + , b > 0 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equb_HTML.gif
(L2) There is a constant γ 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq22_HTML.gif such that
l ( t ) : = inf | u | = 1 ( L ( t ) u , u ) γ 0 , t R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equc_HTML.gif

which were firstly used in [15]. It is not hard to check that the matrix-valued function L ( t ) : = ( t 4 sin 2 t + 1 ) I N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq23_HTML.gif satisfies (L1) and (L2), but does not satisfy (1.3).

We define an operator Γ : H 1 ( R , R N ) H 1 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq24_HTML.gif by
( Γ u , v ) : = R ( M u ( t ) , v ˙ ( t ) ) d t , u , v H 1 ( R , R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equd_HTML.gif

Since M is an antisymmetric N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq1_HTML.gif constant matrix, Γ is self-adjoint on H 1 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq25_HTML.gif. Let χ denote the self-adjoint extension of the operator d 2 d t 2 + L ( t ) + Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq26_HTML.gif. We are interested in the indefinite case:

(J1) a : = sup ( σ ( χ ) ( , 0 ) ) < 0 < b : = inf ( σ ( χ ) ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq27_HTML.gif.

To state our main result, we still need the following assumptions:

(H1) | H ( t , u ) | c ( 1 + | u | p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq28_HTML.gif for some c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq29_HTML.gif and p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq30_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq31_HTML.gif and u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq32_HTML.gif.

(H2) H ( t , u ) 1 2 a | u | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq33_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq31_HTML.gif and u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq32_HTML.gif.

(H3) For some δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq34_HTML.gif and γ ( 0 , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq35_HTML.gif,
| H ( t , u ) | γ | u | , | H ( t , u ) | 1 2 | H ( t , u ) | | u | , | u | < δ , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Eque_HTML.gif
(H4) H ( t , u ) | u | 2 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq36_HTML.gif as | u | + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq37_HTML.gif and there exists W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq38_HTML.gif such that
H ( t , u ) W 1 ( t ) , t R  and  u R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ5_HTML.gif
(1.5)
(H5) For all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq39_HTML.gif and u , z R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq40_HTML.gif, there holds
H ( t , u + z ) H ( t , u ) r ( H ( t , u ) , z ) + ( r 1 ) 2 2 ( H ( t , u ) , u ) W 1 ( t ) , r [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equf_HTML.gif

Our main results read as follows.

Theorem 1.1If (L1)-(L2), (J1) and (H1)-(H5) hold, then (1.1) has at least one nontrivial homoclinic orbit.

Theorem 1.2Let be the collection of solutions of (1.1), then there is a solution that minimizes the energy functional
I ( u ) = 1 2 R [ | u ˙ ( t ) | 2 + ( M u ( t ) , u ˙ ( t ) ) + ( L ( t ) u ( t ) , u ( t ) ) ] d t R H ( t , u ) d t , u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equg_HTML.gif
over , where the spaceEis defined in Section  2. In addition, if
| H ( t , u ) | = o ( | u | ) as  | u | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equh_HTML.gif

uniformly int, then there is a nontrivial homoclinic orbit that minimizes the energy functional over M { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq41_HTML.gif, i.e., a ground state homoclinic orbit.

Remark 1.1 Although the authors [21] have studied (1.1) with superquadratic nonlinearities, our superquadratic condition (H4) is weaker than (1.4) in [21]. Moreover, we study the ground state homoclinic orbit of (1.1). To the best of our knowledge, there has been no result published concerning the ground state homoclinic orbit of (1.1).

Example 1.1
  1. (1)

    H ( t , u ) = | u | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq42_HTML.gif,

     
  2. (2)

    H ( t , u ) = g ( t ) ( | u | p + ( p 2 ) | u | p ε sin 2 ( | u | ε ε ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq43_HTML.gif,

     

where p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq30_HTML.gif, g ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq44_HTML.gif is continuous and 0 < ε < p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq45_HTML.gif. It is easy to check that the above two functions satisfy assumptions (H1)-(H5) if we take 0 W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq46_HTML.gif, where W 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq47_HTML.gif is the function in (H4)-(H5).

The rest of the present paper is organized as follows. In Section 2, we establish the variational framework associated with (1.1), and we also give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

2 Preliminary lemmas

In the following, we use L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq48_HTML.gif to denote the norm of L p ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq49_HTML.gif for any p [ 1 , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq50_HTML.gif. Let W : = H 1 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq51_HTML.gif be a Hilbert space with the inner product and norm given respectively by
u , v W = R [ ( u ˙ ( t ) , v ˙ ( t ) ) + ( u ( t ) , v ( t ) ) ] d t , u W = u , u E 1 / 2 , u , v W . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equi_HTML.gif
It is well known that W is continuously embedded in L p ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq49_HTML.gif for p [ 2 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq52_HTML.gif. We define an operator Γ : W W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq53_HTML.gif by
( Γ u , v ) : = R ( M u ( t ) , v ˙ ( t ) ) d t , u , v W . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equj_HTML.gif

Since M is an antisymmetric N × N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq1_HTML.gif constant matrix, Γ is self-adjoint on W. Moreover, we denote by χ the self-adjoint extension of the operator d 2 d t 2 + L ( t ) + Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq26_HTML.gif with the domain D ( χ ) L 2 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq54_HTML.gif.

Let E : = D ( | χ | 1 / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq55_HTML.gif, the domain of | χ | 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq56_HTML.gif. We define respectively on E the inner product and the norm
u , v E : = ( | χ | 1 / 2 u , | χ | 1 / 2 v ) 2 + ( u , v ) 2 and u E = u , u E 1 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equk_HTML.gif

where ( , ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq57_HTML.gif denotes the inner product in L 2 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq58_HTML.gif.

By a similar proof of Lemma 3.1 in [15], we can prove that if conditions (L1) and (L2) hold, then
E  is compactly embedded into  L p ( R , R N ) , p [ 1 , + ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ6_HTML.gif
(2.1)
Therefore, it is easy to prove that the spectrum σ ( χ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq59_HTML.gif has a sequence of eigenvalues (counted with their multiplicities)
λ 1 λ 2 λ k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equl_HTML.gif

and the corresponding system of eigenfunctions { e k : k N } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq60_HTML.gif ( χ e k = λ k e k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq61_HTML.gif) forms an orthogonal basis in L 2 ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq58_HTML.gif.

By (J1), we may let
k 1 : = { j : λ j < 0 } , E : = span { e 1 , , e k 1 } , E + : = cl E ( span { e k 1 + 1 , } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equm_HTML.gif
Then one has the orthogonal decomposition
E = E E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equn_HTML.gif
with respect to the inner product , E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq62_HTML.gif. Now, we introduce respectively on E the following new inner product and norm:
u , v : = ( | χ | 1 / 2 u , | χ | 1 / 2 v ) 2 , u = u , u 1 / 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ7_HTML.gif
(2.2)
where u , v E = E E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq63_HTML.gif with u = u + u + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq64_HTML.gif and v = v + v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq65_HTML.gif. Clearly, the norms https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq66_HTML.gif and E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq67_HTML.gif are equivalent (see [4]), and the decomposition E = E E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq68_HTML.gif is also orthogonal with respect to both inner products , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq69_HTML.gif and ( , ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq57_HTML.gif. Hence, by (J1), E with equivalent norms, besides, we have
u 2 = ( χ u , u ) 2 a u L 2 2 , u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ8_HTML.gif
(2.3)
and
u + 2 = ( χ u + , u + ) 2 b u + L 2 2 , u + E + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ9_HTML.gif
(2.4)

where a and b are defined in (J1).

For problem (1.1), we consider the following functional:
I ( u ) = 1 2 R [ | u ˙ ( t ) | 2 + ( M u ( t ) , u ˙ ( t ) ) + ( L ( t ) u ( t ) , u ( t ) ) ] d t R H ( t , u ) d t , u E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equo_HTML.gif
Then I can be rewritten as
I ( u ) = 1 2 u + 2 1 2 u 2 R H ( t , u ) d t , u = u + u + E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equp_HTML.gif
Let Ψ ( u ) : = R H ( t , u ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq70_HTML.gif. In view of the assumptions of H, we know I , Ψ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq71_HTML.gif and the derivatives are given by
Ψ ( u ) v = R ( H u ( t , u ) , v ) d t , I ( u ) v = u + , v + u , v I ( u ) v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equq_HTML.gif

for any u , v E = E E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq72_HTML.gif with u = u + u + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq64_HTML.gif and v = v + v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq65_HTML.gif. By the discussion of [24], the (weak) solutions of system (1.1) are the critical points of the C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq73_HTML.gif functional I : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq74_HTML.gif. Moreover, it is easy to verify that if u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq75_HTML.gif is a solution of (1.1), then u ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq7_HTML.gif and u ˙ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq76_HTML.gif as | t | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq77_HTML.gif (see Lemma 3.1 in [25]).

The following abstract critical point theorem plays an important role in proving our main result. Let E be a Hilbert space with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq66_HTML.gif and have an orthogonal decomposition E = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq78_HTML.gif, N E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq79_HTML.gif is a closed and separable subspace. There exists a norm | v | ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq80_HTML.gif satisfying | v | ω v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq81_HTML.gif for all v N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq82_HTML.gif and inducing a topology equivalent to the weak topology of N on a bounded subset of N. For u = v + w E = N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq83_HTML.gif with v N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq82_HTML.gif, w N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq84_HTML.gif, we define | u | ω 2 = | v | ω 2 + w 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq85_HTML.gif. Particularly, if u n = v n + w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq86_HTML.gif is https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq66_HTML.gif-bounded and u n | | ω u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq87_HTML.gif, then v n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq88_HTML.gif weakly in N, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq89_HTML.gif strongly in N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq90_HTML.gif, u n v + w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq91_HTML.gif weakly in E (cf.[26]).

Let E : = E E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq92_HTML.gif, z 0 E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq93_HTML.gif with z 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq94_HTML.gif. Let N : = E R z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq95_HTML.gif and E 1 + : = N = ( E R z 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq96_HTML.gif. For R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq97_HTML.gif, let
Q : = { u : = u + s z 0 : s R + , u E , u < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equr_HTML.gif
with p 0 = s 0 z 0 Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq98_HTML.gif, s 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq99_HTML.gif. We define
D : = { u : = s z 0 + w + : s R , w + E 1 + , s z 0 + w + = s 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equs_HTML.gif
For I C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq100_HTML.gif, define
Γ : = { h : h : [ 0 , 1 ] × Q ¯ E  is  | | ω -continuous ; h ( 0 , u ) = u  and  I ( h ( s , u ) ) I ( u )  for all  u Q ¯ ; For any  ( s 0 , u 0 ) [ 0 , 1 ] × Q ¯ , there is a  | | ω -neighborhood U ( s 0 , u 0 )  s.t.  { u h ( t , u ) : ( t , u ) U ( s 0 , u 0 ) ( [ 0 , 1 ] × Q ¯ ) } E fin } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equt_HTML.gif

where E fin https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq101_HTML.gif denotes various finite-dimensional subspaces of E, Γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq102_HTML.gif since id Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq103_HTML.gif.

The variant weak linking theorem is as follows.

Lemma 2.1 ([26])

The family of C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq73_HTML.gif-functionals { I λ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq104_HTML.gifhas the form
I λ ( u ) : = λ K ( u ) J ( u ) , λ [ 1 , λ 0 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equu_HTML.gif
where λ 0 > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq105_HTML.gif. Assume that
  1. (a)

    K ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq106_HTML.gif, u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq107_HTML.gif, I 1 = I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq108_HTML.gif;

     
  2. (b)

    | J ( u ) | + K ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq109_HTML.gifas u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq110_HTML.gif;

     
  3. (c)

    I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq111_HTML.gifis | | ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq112_HTML.gif-upper semicontinuous, I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq113_HTML.gifis weakly sequentially continuous onE. Moreover, I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq111_HTML.gifmaps bounded sets to bounded sets;

     
  4. (d)

    sup Q I λ < inf D I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq114_HTML.gif, λ [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq115_HTML.gif.

     
Then, for almost all λ [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq116_HTML.gif, there exists a sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gifsuch that
sup n u n < , I λ ( u n ) 0 , I λ ( u n ) c λ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equv_HTML.gif

where c λ : = inf h Γ sup u Q ¯ I λ ( h ( t , u ) ) [ inf D I λ , sup Q ¯ I λ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq118_HTML.gif.

In order to apply Lemma 2.1, we shall prove a few lemmas. We pick λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq119_HTML.gif such that 1 < λ 0 < min [ 2 , b / γ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq120_HTML.gif. For 1 λ λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq121_HTML.gif, we consider
I λ ( u ) : = λ 2 u + 2 ( 1 2 u 2 + R H ( t , u ( t ) ) d t ) : = λ K ( u ) J ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equw_HTML.gif
It is easy to see that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq111_HTML.gif satisfies condition (a) in Lemma 2.1. To see (c), if u n | | ω u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq87_HTML.gif and I λ ( u n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq122_HTML.gif, then u n + u + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq123_HTML.gif and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq124_HTML.gif in E, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq125_HTML.gif a.e. on , going to a subsequence if necessary. It follows from the weak lower semicontinuity of the norm, Fatou’s lemma and the fact H ( t , u ) + W 1 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq126_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq39_HTML.gif and u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq127_HTML.gif by (1.5) in (H4) that
c lim sup n I λ ( u n ) = lim sup n [ λ 2 u n + 2 ( 1 2 u n 2 + R ( H ( t , u n ) + W 1 ( t ) ) d t ) + R W 1 ( t ) d t ] λ 2 u + 2 lim inf n [ 1 2 u n 2 + R ( H ( t , u n ) + W 1 ( t ) ) d t ] + R W 1 ( t ) d t λ 2 u + 2 ( 1 2 u 2 + R H ( t , u ) d t ) = I λ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equx_HTML.gif

Thus we get I λ ( u ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq128_HTML.gif. It implies that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq111_HTML.gif is | | ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq112_HTML.gif-upper semicontinuous. I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq113_HTML.gif is weakly sequentially continuous on E due to [27].

Lemma 2.2Under assumptions of Theorem  1.1, then
J ( u ) + K ( u ) as  u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equy_HTML.gif
Proof By the definition of I ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq129_HTML.gif and (H4), we have
J ( u ) + K ( u ) = 1 2 u + 2 + 1 2 u 2 + R H ( t , u ( t ) ) d t 1 2 u 2 R W 1 ( t ) d t + as  u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equz_HTML.gif

which is due to W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq38_HTML.gif. □

Therefore, Lemma 2.2 implies that condition (b) holds. To continue the discussion, we still need to verify condition (d), that is, the following two lemmas.

Lemma 2.3Under assumptions of Theorem  1.1, there are two positive constants ϵ , ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq130_HTML.gifsuch that
I λ ( u ) ϵ , u E + , u = ρ , λ [ 1 , λ 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equaa_HTML.gif
Proof By (H1), (H3), (2.4) and the Sobolev embedding theorem, for all u E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq131_HTML.gif,
I λ ( u ) 1 2 u 2 R H ( t , u ( t ) ) d t = 1 2 u 2 { t R : | u | < δ } H ( t , u ( t ) ) d t { t R : | u | δ } H ( t , u ( t ) ) d t 1 2 u 2 1 2 γ { t R : | u | < δ } | u | 2 d t c { t R : | u | δ } ( | u | p + | u | ) d t 1 2 u 2 γ b 1 2 u 2 C u p = 1 2 u 2 ( 1 γ b 2 C u p 2 ) , 0 γ < b , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equab_HTML.gif

where C is a positive constant. It implies the conclusion if we take u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq132_HTML.gif sufficiently small. □

Lemma 2.4Under assumptions of Theorem  1.1, then there is an R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq97_HTML.gifsuch that
I λ ( u ) 0 , u Q R , λ [ 1 , λ 0 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equac_HTML.gif

where Q R : = { u : = v + s z 0 : s 0 , v E , z 0 E +  with  z 0 = 1 , u R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq133_HTML.gif.

Proof Suppose by contradiction that there exist R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq134_HTML.gif, λ n [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq135_HTML.gif and u n = v n + s n z 0 Q R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq136_HTML.gif such that I λ n ( u n ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq137_HTML.gif. If s n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq138_HTML.gif, then by (H2) and (2.3), we have
I λ n ( v n ) = 1 2 v n 2 R H ( t , v n ) d t 1 2 v n 2 1 2 a v n L 2 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equad_HTML.gif
Therefore, s n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq139_HTML.gif and u n 2 = v n 2 + s n 2 = R n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq140_HTML.gif. Let u ˜ n = u n u n = s ˜ n z 0 + v ˜ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq141_HTML.gif, then
u ˜ n 2 = v ˜ n 2 + s ˜ n 2 = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equae_HTML.gif
It follows from I λ n ( u n ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq137_HTML.gif and the definition of I that
0 < I λ n ( u n ) u n 2 = 1 2 ( λ n s ˜ n 2 v ˜ n 2 ) R H ( t , u n ) | u n | 2 | u ˜ n | 2 d t = 1 2 [ ( λ n + 1 ) s ˜ n 2 1 ] R H ( t , u n ) | u n | 2 | u ˜ n | 2 d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ10_HTML.gif
(2.5)

There are renamed subsequences such that s ˜ n s ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq142_HTML.gif, λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq143_HTML.gif, and there is a renamed subsequence such that u ˜ n = u n u n = s ˜ n z 0 + v ˜ n u ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq144_HTML.gif in E and u ˜ n u ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq145_HTML.gif a.e. on .

We claim that
lim inf n R H ( t , u n ) | u n | 2 | u ˜ n | 2 d t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ11_HTML.gif
(2.6)
Case 1. If u ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq146_HTML.gif. Let Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq147_HTML.gif be the subset of where u ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq148_HTML.gif, then for all t Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq149_HTML.gif we have | u n | = | u ˜ n | u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq150_HTML.gif. It follows from (H4) and W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq38_HTML.gif that
R H ( t , u n ) | u n | 2 | u ˜ n | 2 d t Ω 0 H ( t , u n ) | u n | 2 | u ˜ n | 2 d t R Ω 0 W 1 ( t ) u n 2 d t + as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equaf_HTML.gif
Case 2. If u ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq151_HTML.gif, then by (H4) and W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq38_HTML.gif, we have
R H ( t , u n ) | u n | 2 | u ˜ n | 2 d t = R H ( t , u n ) u n 2 d t R W 1 ( t ) u n 2 d t 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equag_HTML.gif
Therefore, Cases 1 and 2 imply that (2.6) holds. Therefore, by (2.5), (2.6) and the facts s ˜ n s ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq142_HTML.gif, λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq143_HTML.gif, we have
( λ + 1 ) s ˜ 2 1 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equah_HTML.gif
that is, s ˜ 2 1 1 + λ 1 1 + λ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq152_HTML.gif. Thus, u ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq148_HTML.gif. It follows from (H4) that
R H ( t , u n ) | u n | 2 | u ˜ n | 2 d t + as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equai_HTML.gif

which contradicts (2.5). The proof is finished. □

Therefore, Lemmas 2.3 and 2.4 imply that condition (d) of Lemma 2.1 holds. Applying Lemma 2.1, we soon obtain the following fact.

Lemma 2.5Under assumptions of Theorem  1.1, for almost all λ [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq116_HTML.gif, there exists a sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gifsuch that
sup n u n < , I λ ( u n ) 0 , I λ ( u n ) c λ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equaj_HTML.gif

where the definition of c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq153_HTML.gifis given in Lemma  2.1.

Lemma 2.6Under assumptions of Theorem  1.1, for almost all λ [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq116_HTML.gif, there exists a u λ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq154_HTML.gifsuch that
I λ ( u λ ) = 0 , I λ ( u λ ) = c λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equak_HTML.gif
Proof Let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif be the sequence obtained in Lemma 2.5. Since { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif is bounded, we can assume u n u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq155_HTML.gif in E and u n u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq156_HTML.gif a.e. on . By (H1), (H3), (2.1) and Theorem A.4 in [27], we have
R 1 2 ( H ( t , u n ) , u n ) d t R 1 2 ( H ( t , u λ ) , u λ ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ12_HTML.gif
(2.7)
and
R H ( t , u n ) d t R H ( t , u λ ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ13_HTML.gif
(2.8)
By Lemma 2.5 and the fact that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq113_HTML.gif is weakly sequentially continuous, we have
I λ ( u λ ) φ = lim n I λ ( u n ) φ = 0 , φ E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equal_HTML.gif
That is, I λ ( u λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq157_HTML.gif. By Lemma 2.5, we have
I λ ( u n ) 1 2 I λ ( u n ) u n = R ( 1 2 ( H ( t , u n ) , u n ) H ( t , u n ) ) d t c λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equam_HTML.gif
It follows from (2.7), (2.8) and the fact I λ ( u λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq158_HTML.gif that
I λ ( u λ ) = I λ ( u λ ) 1 2 I λ ( u λ ) u λ = R ( 1 2 ( H ( t , u λ ) , u λ ) H ( t , u λ ) ) d t = c λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equan_HTML.gif

The proof is finished. □

Applying Lemma 2.6, we soon obtain the following fact.

Lemma 2.7Under assumptions of Theorem  1.1, for every λ [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq116_HTML.gif, there are sequences { u n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq159_HTML.gifand λ n [ 1 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq135_HTML.gifwith λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq143_HTML.gifsuch that
I λ n ( u n ) = 0 , I λ n ( u n ) = c λ n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equao_HTML.gif
Lemma 2.8Under assumptions of Theorem  1.1, then
R [ H ( t , u ) H ( t , r w ) + r 2 ( H ( t , u ) , w ) 1 + r 2 2 ( H ( t , u ) , u ) ] d t C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equap_HTML.gif

where u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq160_HTML.gif, w E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq161_HTML.gif, 0 r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq162_HTML.gifand the constant C : = R | W 1 ( t ) | d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq163_HTML.gifdoes not depend onu, wr.

Proof This follows from (H5) if we take u = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq164_HTML.gif and z = r w u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq165_HTML.gif. □

Lemma 2.9The sequences given in Lemma  2.7 are bounded.

Proof Write u n = u n + + u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq166_HTML.gif, where u n ± E ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq167_HTML.gif. Suppose that
u n as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equaq_HTML.gif

Let v n : = u n u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq168_HTML.gif, then v n + = u n + u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq169_HTML.gif, v n = u n u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq170_HTML.gif, v n 2 = v n + 2 + v n 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq171_HTML.gif and v n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq172_HTML.gif. Thus v n + v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq173_HTML.gif in E and v n + v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq174_HTML.gif a.e. on , after passing to a subsequence.

Case 1. If v + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq175_HTML.gif. Let Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq176_HTML.gif be the subset of where v + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq177_HTML.gif. Then v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq178_HTML.gif and | u n | = | v n | u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq179_HTML.gif on Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq176_HTML.gif. It follows from (H4) and W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq38_HTML.gif that
R H ( t , u n ) | u n | 2 | v n | 2 d t Ω 1 H ( t , u n ) | u n | 2 | v n | 2 d t R Ω 1 W 1 ( t ) u n 2 d t + as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equar_HTML.gif
which together with Lemmas 2.3 and 2.7 and v n ± v ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq180_HTML.gif in L q ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq181_HTML.gif for all 1 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq182_HTML.gif (by (2.1)) implies that
0 c λ n u n 2 = I λ n ( u n ) u n 2 = λ n 2 v n + 2 1 2 v n 2 R H ( t , u n ) | u n | 2 | v n | 2 d t as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equas_HTML.gif

It is a contradiction.

Case 2. If v + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq183_HTML.gif. We claim that there is a constant C independent of u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq184_HTML.gif and λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq185_HTML.gif such that
I λ n ( r u n + ) I λ n ( u n ) C , r [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ14_HTML.gif
(2.9)
Since
1 2 I λ n ( u n ) φ = 1 2 λ n u n + , φ + 1 2 u n , φ 1 2 R ( H ( t , u n ) , φ ) d t = 0 , φ E , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equat_HTML.gif
it follows from the definition of I that
I λ n ( r u n + ) I λ n ( u n ) = 1 2 λ n ( r 2 1 ) u n + 2 + 1 2 u n 2 + R ( H ( t , u n ) H ( t , r u n + ) ) d t + 1 2 λ n u n + , φ + 1 2 u n , φ 1 2 R ( H ( t , u n ) , φ ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ15_HTML.gif
(2.10)
Take φ : = ( r 2 + 1 ) u n ( r 2 1 ) u n + = ( r 2 + 1 ) u n 2 r 2 u n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq186_HTML.gif in (2.10), then it follows from Lemma 2.8 that
I λ n ( r u n + ) I λ n ( u n ) = r 2 2 u n 2 + R [ H ( t , u n ) H ( t , r u n + ) + r 2 ( H ( t , u n ) , u n + ) 1 + r 2 2 ( H ( t , u n ) , u n ) ] d t C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equau_HTML.gif

Thus (2.9) holds.

Let C 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq187_HTML.gif be a fixed constant and take
r n : = C 0 u n 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equav_HTML.gif
Therefore, (2.9) implies that
I λ n ( r n u n + ) I λ n ( u n ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equaw_HTML.gif
It follows from v n + = u n + u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq169_HTML.gif and Lemma 2.7 that
I λ n ( C 0 v n + ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ16_HTML.gif
(2.11)
Note that Lemmas 2.3 and 2.7 and (H4) imply that
0 c λ n u n 2 = I λ n ( u n ) u n 2 = λ n 2 v n + 2 1 2 v n 2 R H ( t , u n ) u n 2 d t λ 0 2 v n + 2 1 2 v n 2 + R W 1 ( t ) d t u n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equax_HTML.gif
It follows from the fact R W 1 ( t ) d t u n 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq188_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq189_HTML.gif due to W 1 ( t ) L 1 ( R , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq38_HTML.gif that
λ 0 2 v n + 2 1 2 v n 2 + ε 0 , ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ17_HTML.gif
(2.12)
for all sufficiently large n. We take ε = 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq190_HTML.gif, by (2.12) and v n 2 = v n + 2 + v n 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq171_HTML.gif, we have
v n + 2 1 2 ( 1 + λ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ18_HTML.gif
(2.13)
for all sufficiently large n. By (H1) and (H3), we have
R H ( t , C 0 v n + ) d t 1 2 γ C 0 2 { t R : | C 0 v n + | < δ } | v n + | 2 d t + 1 2 c { t R : | C 0 v n + | δ } ( C 0 | v n + | + C 0 p | v n + | p ) d t 1 2 γ C 0 2 { t R : | C 0 v n + | < δ } | v n + | 2 d t + C 1 C 0 p { t R : | C 0 v n + | δ } | v n + | p d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ19_HTML.gif
(2.14)
For all sufficiently large n, by (2.13) and (2.14), it follows from λ n λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq143_HTML.gif and v n + v + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq191_HTML.gif in L q ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq181_HTML.gif for all 1 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq182_HTML.gif (by (2.1)) that
I λ n ( C 0 v n + ) = 1 2 λ n C 0 2 v n + 2 R H ( t , C 0 v n + ) d t 1 2 λ n C 0 2 1 2 ( 1 + λ 0 ) 1 2 γ C 0 2 { t R : | C 0 v n + | < δ } | v n + | 2 d t C 1 C 0 p { t R : | C 0 v n + | δ } | v n + | p d t λ C 0 2 4 ( 1 + λ 0 ) as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equay_HTML.gif

This implies that I λ n ( C 0 v n + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq192_HTML.gif as C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq193_HTML.gif, contrary to (2.11).

Therefore, { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif are bounded. The proof is finished. □

3 Proofs of the main results

Proof of Theorem 1.1 From Lemma 2.7, there are sequences 1 < λ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq194_HTML.gif and { u n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq159_HTML.gif such that I λ n ( u n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq195_HTML.gif and I λ n ( u n ) = c λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq196_HTML.gif. By Lemma 2.9, we know that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif is bounded in E. Thus we can assume u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq197_HTML.gif in E, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq198_HTML.gif a.e. on . Therefore,
I λ n ( u n ) φ = λ n u n + , φ u n , φ R ( H ( t , u n ) , φ ) d t = 0 , φ E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equaz_HTML.gif
Hence, in the limit,
I ( u ) φ = u + , φ u , φ R ( H ( t , u ) , φ ) d t = 0 , φ E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equba_HTML.gif
Thus I ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq199_HTML.gif. Note that
I λ n ( u n ) 1 2 I λ n ( u n ) u n = R ( 1 2 ( H ( t , u n ) , u n ) H ( t , u n ) ) d t = c λ n c 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ20_HTML.gif
(3.1)
Similar to (2.7) and (2.8), we know
R ( 1 2 ( H ( t , u n ) , u n ) H ( t , u n ) ) d t R ( 1 2 ( H ( t , u ) , u ) H ( t , u ) ) d t as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbb_HTML.gif
It follows from I ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq199_HTML.gif, (3.1) and Lemma 2.3 that
I ( u ) = I ( u ) 1 2 I ( u ) u = R [ 1 2 ( H ( t , u ) , u ) H ( t , u ) ] d t = lim n R ( 1 2 ( H ( t , u n ) , u n ) H ( t , u n ) ) d t c 1 ϵ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbc_HTML.gif

Therefore, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq200_HTML.gif. □

Proof of Theorem 1.2 By Theorem 1.1, M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq201_HTML.gif, where is the collection of solutions of (1.1). Let
α : = inf u M I ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbd_HTML.gif
If u is a solution of (1.1), then by Lemma 2.8 (take r = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq202_HTML.gif),
I ( u ) = I ( u ) 1 2 I ( u ) u = R [ 1 2 ( H ( t , u ) , u ) H ( t , u ) ] d t C = R | W 1 ( t ) | d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Eqube_HTML.gif
Thus α > https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq203_HTML.gif. Let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif be a sequence in such that
I ( u n ) α . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ21_HTML.gif
(3.2)
By Lemma 2.9, the sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif is bounded in E. Therefore, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq197_HTML.gif in E, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq125_HTML.gif a.e. on and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq198_HTML.gif in L p ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq49_HTML.gif for all p [ 1 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq204_HTML.gif (by (2.1)), after passing to a subsequence. Therefore,
I ( u n ) φ = u n + , φ u n , φ R ( H ( t , u n ) , φ ) d t = 0 , φ E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbf_HTML.gif
Hence, in the limit,
I ( u ) φ = u + , φ u , φ R ( H ( t , u ) , φ ) d t = 0 , φ E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbg_HTML.gif
Thus I ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq199_HTML.gif. Similar to (2.7) and (2.8), we have
I ( u n ) 1 2 I ( u n ) u n = R ( 1 2 ( H ( t , u n ) , u n ) H ( t , u n ) ) d t R ( 1 2 ( H ( t , u ) , u ) H ( t , u ) ) d t as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbh_HTML.gif
It follows from I ( u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq199_HTML.gif and (3.2) that
I ( u ) = I ( u ) 1 2 I ( u ) u = R [ 1 2 ( H ( t , u ) , u ) H ( t , u ) ] d t = lim n R ( 1 2 ( H ( t , u n ) , u n ) H ( t , u n ) ) d t = lim n I ( u n ) = α . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbi_HTML.gif
Now suppose that
| H ( t , u ) | = o ( | u | ) as  | u | 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbj_HTML.gif
It follows from (H1) that for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq205_HTML.gif, there is a constant C ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq206_HTML.gif such that
| H ( t , u ) | ε | u | + C ε | u | p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ22_HTML.gif
(3.3)
Let
β : = inf u M I ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbk_HTML.gif
where M : = M { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq207_HTML.gif. Let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq117_HTML.gif be a sequence in M { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq41_HTML.gif such that
I ( u n ) β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ23_HTML.gif
(3.4)
Note that
0 = I ( u n ) u n + = u n + 2 R ( H ( t , u n ) , u n + ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbl_HTML.gif
which together with (3.3), Hölder’s inequality and the Sobolev embedding theorem implies
u n + 2 = R ( H ( t , u n ) , u n + ) d t ε 0 T | u n | | u n + | d t + C ε 0 T | u n | p 1 | u n + | d t ε u n u n + + C ε u n L p p 1 u n + ε u n u n + + C ε u n L p p 2 u n u n + ε u n 2 + C ε u n L p p 2 u n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ24_HTML.gif
(3.5)
Similarly, we have
u n 2 ε u n 2 + C ε u n L p p 2 u n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equ25_HTML.gif
(3.6)
From (3.5) and (3.6), we get
u n 2 2 ε u n 2 + 2 C ε u n L p p 2 u n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_Equbm_HTML.gif

which means u n L p C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq208_HTML.gif for some constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq209_HTML.gif. Since u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq198_HTML.gif in L p ( R , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq49_HTML.gif, we know u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq200_HTML.gif. As before, I ( u n ) I ( u ) = β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq210_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2014-106/MediaObjects/13661_2014_Article_577_IEq189_HTML.gif. □

Declarations

Acknowledgements

The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Anyang Normal University

References

  1. Ambrosetti A, Coti Zelati V: Multiple homoclinic orbits for a class of conservative systems.Rend. Semin. Mat. Univ. Padova 1993, 89: 177-194.MathSciNet
  2. Chen G, Ma S: Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum.J. Math. Anal. Appl. 2011, 379: 842-851. 10.1016/j.jmaa.2011.02.013MathSciNetView Article
  3. Chen G, Ma S: Ground state periodic solutions of second order Hamiltonian systems without spectrum 0.Isr. J. Math. 2013, 198: 111-127. 10.1007/s11856-013-0016-9MathSciNetView Article
  4. Ding Y: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems.Nonlinear Anal. 1995, 25: 1095-1113. 10.1016/0362-546X(94)00229-BMathSciNetView Article
  5. Izydorek M, Janczewska J: Homoclinic solutions for a class of second order Hamiltonian systems.J. Differ. Equ. 2005, 219: 375-389. 10.1016/j.jde.2005.06.029MathSciNetView Article
  6. Kim Y: Existence of periodic solutions for planar Hamiltonian systems at resonance.J. Korean Math. Soc. 2011, 48: 1143-1152. 10.4134/JKMS.2011.48.6.1143MathSciNetView Article
  7. Mawhin J, Willem M Applied Mathematical Sciences 74. In Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.View Article
  8. Omana W, Willem M: Homoclinic orbits for a class of Hamiltonian systems.Differ. Integral Equ. 1992, 5: 1115-1120.MathSciNet
  9. Paturel E: Multiple homoclinic orbits for a class of Hamiltonian systems.Calc. Var. Partial Differ. Equ. 2001, 12: 117-143. 10.1007/PL00009909MathSciNetView Article
  10. Rabinowitz PH: Homoclinic orbits for a class of Hamiltonian systems.Proc. R. Soc. Edinb., Sect. A 1990, 114: 33-38. 10.1017/S0308210500024240MathSciNetView Article
  11. Rabinowitz PH, Tanaka K: Some results on connecting orbits for a class of Hamiltonian systems.Math. Z. 1991, 206: 473-499. 10.1007/BF02571356MathSciNetView Article
  12. Séré E: Existence of infinitely many homoclinic orbits in Hamiltonian systems.Math. Z. 1992, 209: 133-160.View Article
  13. Sun J, Chen H, Nieto JJ: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems.J. Math. Anal. Appl. 2011, 373: 20-29. 10.1016/j.jmaa.2010.06.038MathSciNetView Article
  14. Tang X, Xiao L: Homoclinic solutions for non-autonomous second-order Hamiltonian systems with a coercive potential.J. Math. Anal. Appl. 2009, 351: 586-594. 10.1016/j.jmaa.2008.10.038MathSciNetView Article
  15. Wan L, Tang C: Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition.Discrete Contin. Dyn. Syst., Ser. B 2011, 15: 255-271.MathSciNet
  16. Xiao J, Nieto JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations.J. Franklin Inst. 2011, 348: 369-377. 10.1016/j.jfranklin.2010.12.003MathSciNetView Article
  17. Zhang P, Tang C: Infinitely many periodic solutions for nonautonomous sublinear second-order Hamiltonian systems.Abstr. Appl. Anal. 2010., 2010: Article ID 620438 10.1155/2010/620438
  18. Zhang Q, Liu C: Infinitely many homoclinic solutions for second order Hamiltonian systems.Nonlinear Anal. 2010, 72: 894-903. 10.1016/j.na.2009.07.021MathSciNetView Article
  19. Chen G: Non-periodic damped vibration systems with sublinear terms at infinity: infinitely many homoclinic orbits.Nonlinear Anal. 2013, 92: 168-176.MathSciNetView Article
  20. Chen G: Non-periodic damped vibration systems with asymptotically quadratic terms at infinity: infinitely many homoclinic orbits.Abstr. Appl. Anal. 2013., 2013: Article ID 937128
  21. Wu X, Zhang W: Existence and multiplicity of homoclinic solutions for a class of damped vibration problems.Nonlinear Anal. 2011, 74: 4392-4398. 10.1016/j.na.2011.03.059MathSciNetView Article
  22. Zhang Z, Yuan R: Homoclinic solutions for some second-order nonautonomous systems.Nonlinear Anal. 2009, 71: 5790-5798. 10.1016/j.na.2009.05.003MathSciNetView Article
  23. Zhu W: Existence of homoclinic solutions for a class of second order systems.Nonlinear Anal. 2012, 75: 2455-2463. 10.1016/j.na.2011.10.043MathSciNetView Article
  24. Costa DG, Magalhães CA: A variational approach to subquadratic perturbations of elliptic systems.J. Differ. Equ. 1994, 111: 103-122. 10.1006/jdeq.1994.1077View Article
  25. Wang J, Xu J, Zhang F: Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials.Commun. Pure Appl. Anal. 2011, 10: 269-286.MathSciNetView Article
  26. Schechter M, Zou W: Weak linking theorems and Schrödinger equations with critical Sobolev exponent.ESAIM Control Optim. Calc. Var. 2003, 9: 601-619.MathSciNetView Article
  27. Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View Article

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© Chen and Wang; licensee Springer. 2014

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