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Ground state homoclinic orbits of superquadratic damped vibration systems

Advances in Difference Equations20142014:230

https://doi.org/10.1186/1687-1847-2014-230

  • Received: 10 June 2014
  • Accepted: 7 August 2014
  • Published:

Abstract

In the case where nonlinearities are superquadratic at infinity, we study the existence of ground state homoclinic orbits for damped vibration systems without periodic conditions by using variational methods. Here the (local) Ambrosetti-Rabinowitz superquadratic condition is replaced by a general superquadratic condition.

MSC:49J40, 70H05.

Keywords

  • damped vibration systems
  • ground state homoclinic orbits
  • superlinear
  • variant weak linking theorem

1 Introduction and main result

We shall study the existence of ground state homoclinic orbits for the following damped vibration system:
u ¨ ( t ) + M u ˙ ( t ) L ( t ) u ( t ) + H u ( t , u ( t ) ) = 0 , t R ,
(1.1)

where M is an antisymmetric N × N constant matrix, L ( t ) C ( R , R N × N ) is a symmetric matrix, H ( t , u ) C 1 ( R × R N , R ) and H u ( t , u ) denotes its gradient with respect to the u variable. We say that a solution u ( t ) of (1.1) is homoclinic (to 0) if u ( t ) C 2 ( R , R N ) such that u ( t ) 0 and u ˙ ( t ) 0 as | t | . If u ( t ) 0 , then u ( t ) is called a nontrivial homoclinic solution.

If M = 0 (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 .
(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 [117] and the references therein.

The periodicity assumption is 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 [10] introduced a type of coercive condition on the matrix L ( t ) :
l ( t ) : = inf | u | = 1 ( L ( t ) u , u ) + as  | t |
(1.3)
and obtained the existence of 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 } ,

where μ > 2 is a constant, ( , ) denotes the standard inner product in R N and the associated norm is denoted by | | .

We should mention that the case where M 0 , i.e., the damped vibration system (1.1), only a few authors have studied homoclinic orbits of (1.1); see [1822]. Zhu [22] considered the periodic case of (1.1) (i.e., L ( t ) and H ( t , u ) are T-periodic in t with T > 0 ) and obtained the existence of nontrivial homoclinic solutions of (1.1). The authors [1821] considered the non-periodic case of (1.1): Zhang and Yuan [21] 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 [20] obtained the existence and multiplicity of homoclinic orbits for (1.1) when H satisfies the local ( A R ) superquadratic growth condition:
0 < μ H ( t , u ) ( H u ( t , u ) , u ) , t R , | u | r ,
(1.4)

where μ > 2 and r > 0 are two constants. Notice that the authors [20, 21] all used the condition (1.3). Recently, Chen [18, 19] obtained infinitely many homoclinic orbits for (1.1) when H satisfies the subquadratic [18] and asymptotically quadratic [19] condition at infinity by the following weaker conditions than (1.3):

(L1) There is a constant β > 1 such that
meas { t R : | t | β L ( t ) < b I N } < + , b > 0 .
(L2) There is a constant γ 0 such that
l ( t ) : = inf | u | = 1 ( L ( t ) u , u ) γ , t R ,

which were firstly used in [14]. It is not hard to check that the matrix-valued function L ( t ) : = ( t 4 sin 2 t + 1 ) I N satisfying (L1) and (L2), but not satisfying (1.3).

We define an operator J : H 1 ( R , R N ) H 1 ( R , R N ) by
( J u , v ) : = R ( M u ( t ) , v ˙ ( t ) ) d t , u , v H 1 ( R , R N ) .

Since M is an antisymmetric N × N constant matrix, J is self-adjoint on H 1 ( R , R N ) . Let χ denote the self-adjoint extension of the operator d 2 d t 2 + L ( t ) + J . We are interested in the indefinite case:

(J1) sup ( σ ( χ ) ( , 0 ) ) < 0 < inf ( σ ( χ ) ( 0 , ) ) .

Let H ˜ ( t , u ) : = 1 2 ( H u ( t , u ) , u ) H ( t , u ) . We assume the following.

(H1) H ( t , u ) C 1 ( R × R N , R ) and | H u ( t , u ) | = o ( | u | ) as | u | 0 uniformly in t.

(H2) H ( t , u ) | u | 2 + as | u | + uniformly in t, and H ( t , u ) 0 , ( t , u ) R × R N .

(H3) H ˜ ( t , u ) > 0 if u 0 , and for any a , b > 0 we have
inf { H ˜ ( t , u ) | u | 2 : t R  and  u R N  with  a | u | < b } > 0 .
(H4) There are constants c 0 , r 0 > 0 and σ > 1 such that
| H u ( t , u ) | σ | u | σ c 0 H ˜ ( t , u ) if  | u | r 0 , t R .

Now, our main result reads as follows.

Theorem 1.1 If (J1), (L1)-(L2), and (H1)-(H4) hold, then (1.1) has a ground state homoclinic orbit.

Remark 1.1 Although the authors [20] have studied the superquadratic case of (1.1), it is not hard to check that our superquadratic condition (H2) is weaker than the condition (1.4) (see Example 1.1). Moreover, we obtain the existence of ground state homoclinic orbits of (1.1), i.e., nontrivial homoclinic orbits with least energy of the action functional of (1.1).

Example 1.1 Let
H ( t , u ) = g ( t ) ( | u | p + ( p 2 ) | u | p ε sin 2 ( | u | ε / ε ) ) ,

where p > 2 , 0 < ε < p 2 and g ( t ) > 0 is continuous. It is not hard to check that H satisfies (H1)-(H4) but does not satisfy (1.4).

The following abstract critical point theorem plays an important role in proving our main result. Let W be a Hilbert space with norm and have an orthogonal decomposition W = N N , N W is a closed and separable subspace. There exists a norm | V | ω that satisfies | v | ω v for all v N and induces a topology equivalent to the weak topology of N on bounded subset of N. For u = v + z W = N N with v N , z N , we define | u | ω 2 = | v | ω 2 + z 2 . Particularly, if ( u n = v n + z n ) is -bounded and u n | | ω u , then v n v weakly in N, z n z strongly in N , u n v + z weakly in W (cf. [23]).

Let W = W W + , z 0 W + with z 0 = 1 . Let N : = W R z 0 and W 1 + : = N = ( W R z 0 ) . For R > 0 , let
Q : = { u : = u + s z 0 : s R + , u W , u < R }
with P 0 = s 0 z 0 Q , s 0 > 0 . We define
D : = { u : = s z 0 + z + : s R , z + W 1 + , s z 0 + z + = s 0 } .
For Φ C 1 ( W , R ) , we define
Γ : = { h : [ 0 , 1 ] × Q ¯ W  is  | | ω -continuous ; h ( 0 , u ) = u  and  Φ ( h ( s , u ) ) Φ ( 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 ¯ ) } W fin } ,

where W fin denotes various finite-dimensional subspaces of W , Γ 0 , since i d Γ .

We shall use the following variant weak linking theorem to prove our result.

Theorem A ([23])

The family of C 1 -functional { Φ λ } has the form
Φ λ ( u ) : = J ( u ) λ K ( u ) , λ [ 1 , 2 ] .
Assume that
  1. (a)

    K ( u ) 0 , u W , Φ 1 = Φ ;

     
  2. (b)

    J ( u ) or K ( u ) as u ;

     
  3. (c)

    Φ λ is | | ω -upper semicontinuous, Φ λ is weakly sequentially continuous on W. Moreover, Φ λ maps bounded sets to bounded sets;

     
  4. (d)

    sup Q Φ λ < inf D Φ λ , λ [ 1 , 2 ] .

     
Then for almost all λ [ 1 , 2 ] , there exists a sequence { u n } such that
sup n u n < , Φ λ ( u n ) 0 , Φ λ ( u n ) c λ ,

where c λ : = inf h Γ sup u Q Φ λ ( h ( 1 , u ) ) [ inf D Φ λ , sup Q ¯ Φ ] .

The rest of the present paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proof of our main result. In Section 3, we give the detailed proof of our main result.

2 Preliminaries

In this section, we firstly give the variational frameworks of our problem and some related preliminary lemmas, and then give the detailed proof of the main result.

In the following, we use p to denote the norm of L p ( R , R N ) for any p [ 1 , ] . Let E : = H 1 ( R , R N ) be a Hilbert space with the inner product and norm given, respectively, by
u , v E = R [ ( u ˙ ( t ) , v ˙ ( t ) ) + ( u ( t ) , v ( t ) ) ] d t , u E = u , u E 1 / 2 , u , v E .
It is well known that E is continuously embedded in L p ( R , R N ) for p [ 2 , ) . We define an operator J : E E by
( J u , v ) : = R ( M u ( t ) , v ˙ ( t ) ) d t , u , v E .

Since M is an antisymmetric N × N constant matrix, J is self-adjoint on E. Moreover, we denote by χ the self-adjoint extension of the operator d 2 d t 2 + L ( t ) + J with the domain D ( χ ) L 2 ( R , R N ) .

Let W : = D ( | χ | 1 / 2 ) , the domain of | χ | 1 / 2 . We define, respectively, on W the inner product and the norm
u , v W : = ( | χ | 1 / 2 u , | χ | 1 / 2 v ) 2 + ( u , v ) 2 and u W = u , u W 1 / 2 ,

where ( , ) 2 denotes the inner product in L 2 ( R , R N ) .

By a similar proof of Lemma 3.1 in [14], we can prove the following lemma.

Lemma 2.1 If conditions (L1) and (L2) hold, then W is compactly embedded into L p ( R , R N ) for all 1 p + .

By Lemma 2.1, it is easy to prove that the spectrum σ ( χ ) has a sequence of eigenvalues (counted with their multiplicities)
λ 1 λ 2 λ k ,

and the corresponding system of eigenfunctions { e k : k N } ( χ e k = λ k e k ) forms an orthogonal basis in L 2 ( R , R N ) .

By (J1), we may let
k 1 : = { j : λ j < 0 } , W : = span { e 1 , , e k 1 } , W + : = cl W ( span { e k 1 , } ) .
Then one has the orthogonal decomposition
W = W W +
with respect to the inner product , W . Now, we introduce, respectively, on W the following new inner product and norm:
u , v : = ( | χ | 1 / 2 u , | χ | 1 / 2 v ) 2 , u = u , u 1 / 2 ,

where u , v W = W W + with u = u + u + and v = v + v + . Clearly, the norms and W are equivalent (see [3]), and the decomposition W = W W + is also orthogonal with respect to both inner products , and ( , ) 2 .

For problem (1.1), we consider the following functional:
Φ ( 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 W .
Then Φ can be rewritten as
Φ ( u ) = 1 2 u + 2 1 2 u 2 R H ( t , u ) d t , u = u + u + W .
(2.1)
Let I ( u ) : = R H ( t , u ) d t . In view of the assumptions of H, we know Φ , I C 1 ( W , R ) and the derivatives are given by
I ( u ) v = R ( H u ( t , u ) , v ) d t , Φ ( u ) v = u + , v + u , v I ( u ) v ,

for any u , v W = W W + with u = u + u + and v = v + v + . By the discussion of [24], the (weak) solutions of system (1.1) are the critical points of the C 1 functional Φ : W R . Moreover, it is easy to verify that if u 0 is a solution of (1.1), then u ( t ) 0 and u ˙ ( t ) 0 as | t | (see Lemma 3.1 in [25]).

In order to apply Theorem A, we consider
Φ λ ( u ) : = 1 2 u + 2 λ ( 1 2 u 2 + R H ( t , u ) d t ) .
(2.2)
It is easy to see that Φ λ satisfies conditions (a), (b) in Theorem A. To see (c), if u n | | ω u , then u n + u + and u n u in W, going to a subsequence if necessary, u n u a.e. on . By Fatou’s lemma and the weak lower semicontinuity of the norm, we have
lim ¯ n Φ λ ( u n ) Φ λ ( u ) ,

which means that Φ λ is | | ω -upper semicontinuous. Φ λ is weakly sequentially continuous on W is due to [26]. To continue the discussion, we still need to verify condition (d) in Theorem A.

Lemma 2.2 Under the assumptions of Theorem  1.1, we have the following facts:
  1. (i)
    There exists ρ > 0 independent of λ [ 1 , 2 ] such that κ : = inf Φ λ ( S ρ W + ) > 0 , where
    S ρ W + : = { z W + : z = ρ } .
     
  2. (ii)

    For fixed z 0 W + with z 0 = 1 and any λ [ 1 , 2 ] , there is R > ρ > 0 such that sup Φ λ ( Q ) 0 , where Q : = { u : = u + s z 0 : s R + , u W , u < R } .

     
Proof (i) Under assumptions (H1) and (H4), we know for any ε > 0 there exists C ε > 0 such that
| H u ( t , u ) | ε | u | + C ε | u | p 1
(2.3)
and
| H ( t , u ) | ε | u | 2 + C ε | u | p ,
(2.4)
where p 2 σ σ 1 > 2 with σ > 1 . Hence, for any u W + ,
Φ λ ( u ) 1 2 u 2 λ ε u 2 C ε u p ,
which implies the conclusion.
  1. (ii)
    Suppose by contradiction that there exist u n W R + z 0 such that Φ λ ( u n ) > 0 for all n and u n as n . Let v n : = u n u n = s n z 0 + v n , then
    0 < Φ λ ( u n ) u n 2 = 1 2 ( s n 2 λ v n 2 ) λ R H ( t , u n ) | u n | 2 | v n | 2 d t .
    (2.5)
     
It follows from H ( t , u ) 0 (see (H2)) that
v n 2 λ v n 2 < s n 2 = 1 v n 2 ,

therefore, v n 1 2 and 1 1 2 s n 1 .

Thus s n s 0 after passing to a subsequence, v n v and v n v a.e. on . Hence, v = s z 0 + v 0 and, since | v n | if v 0 , it follows from (H2) and Fatou’s lemma that
R H ( t , u n ) | u n | 2 | v n | 2 d t + ,
(2.6)

contrary to (2.5). The proof is finished. □

Applying Theorem A, we soon obtain the following facts.

Lemma 2.3 Under the assumptions of Theorem  1.1, for almost all λ [ 1 , 2 ] , there exists a sequence { u n } such that
sup n u n < , Φ λ ( u n ) 0 , Φ λ ( u n ) c λ [ κ , sup Q ¯ Φ ] .
Lemma 2.4 Under the assumptions of Theorem  1.1, for almost all λ [ 1 , 2 ] , there exists a u λ such that
Φ λ ( u λ ) = 0 , Φ λ ( u λ ) sup Q ¯ Φ .

Proof Let { u n } be the sequence obtained in Lemma 2.3, write u n = u n + u n + with u n ± W ± . Since { u n } is bounded, { u n + } is also bounded, then u n u λ and u n + u λ + in W, after passing to a subsequence.

We claim that u λ + 0 . If not, then Lemma 2.1 implies u n + 0 in L q ( R , R N ) for all q [ 1 , + ] . It follows from the definition of Φ, Hölder’s inequality, and (2.3) that
0 R | ( H u ( t , u n ) , u n + ) | d t ε R | u n | | u n + | d t + C ε R | u n | p 1 | u n + | d t ε u n 2 u n + 2 + C ε u n p p 1 u n + p 0 .
It follows from (2.2) and Lemma 2.3 that
Φ λ ( u n ) u n + 2 = Φ λ ( u n ) u n + + λ R ( H u ( t , u n ) , u n + ) d t 0 ,
which contradicts with the fact that Φ λ ( u n ) κ . Hence, u λ + 0 , and thus u λ 0 . Note that Φ λ is weakly sequentially continuous on W, thus
Φ λ ( u λ ) z = lim n Φ λ ( u n ) z = 0 , z W .
By (H3), Fatou’s lemma, and Lemma 2.3, we have
sup Q ¯ Φ c λ = lim n ( Φ λ ( u n ) 1 2 Φ λ ( u n ) u n ) = lim n λ R ( 1 2 ( H u ( t , u n ) , u n ) H ( t , u n ) ) d t λ R ( 1 2 ( H u ( t , u λ ) , u λ ) H ( t , u λ ) ) d t = Φ λ ( u λ ) .

Thus we get Φ λ ( u λ ) sup Q ¯ Φ . □

Lemma 2.5 Under the assumptions of Theorem  1.1, there exists λ n 1 and { u λ n } such that
Φ λ n ( u λ n ) = 0 , Φ λ n ( u λ n ) sup Q ¯ Φ .

Moreover, { u λ n } is bounded.

Proof The existence of { u λ n } such that
Φ λ n ( u λ n ) = 0 , Φ λ n ( u λ n ) sup Q ¯ Φ

is the direct consequence of Lemma 2.4. To prove the boundedness of { u λ n } , arguing by contradiction, suppose that u λ n . Let v λ n : = u λ n u λ n . Then v λ n = 1 , v λ n v in W and v λ n v a.e. in , after passing to a subsequence.

Recall that Φ λ n ( u λ n ) = 0 . Thus for any φ W , we have
u λ n + , φ λ n u λ n , φ = λ n R ( H u ( t , u λ n ) , φ ) d t .
Consequently { v λ n } satisfies
v λ n + , φ λ n v λ n , φ = λ n R ( H u ( t , u λ n ) , φ ) u λ n d t .
(2.7)
Let φ = v λ n ± in (2.7), respectively. Then we have
v λ n + , v λ n + = λ n R ( H u ( t , u λ n ) , v λ n + ) u λ n d t
and
λ n v λ n , v λ n = λ n R ( H u ( t , u λ n ) , v λ n ) u λ n d t .
Since 1 = v λ n 2 = v λ n + 2 + v λ n 2 , we have
1 = R ( H u ( t , u λ n ) , λ n v λ n + v λ n ) u λ n d t .
(2.8)
For r 0 , let
h ( r ) : = inf { H ˜ ( t , u ) : t R  and  u R N  with  | u | r } .
By (H1) and (H3), we have h ( r ) > 0 for all r > 0 . By (H3) and (H4), for | u | r 0 ,
c 0 H ˜ ( t , u ) | H u ( t , u ) | σ | u | σ = ( | H u ( t , u ) | | u | | u | 2 ) σ ( ( H u ( t , u ) , u ) | u | 2 ) σ ( 2 H ( t , u ) | u | 2 ) σ ,
it follows from (H2) and the definition of h ( r ) that
h ( r ) as  r .
For 0 < a < b , let
Ω n ( a , b ) : = { t R : a | u λ n ( t ) | < b }
and
C a b : = inf { H ˜ ( t , u ) | u | 2 : t R  and  u R N  with  a | u | < b } .
By (H3), we have C a b > 0 and
H ˜ ( t , u λ n ) C a b | u λ n | 2 for all  t Ω n ( a , b ) .
Since Φ λ n ( u λ n ) = 0 and Φ λ n ( u λ n ) sup Q ¯ Φ , there exists a constant C 0 > 0 such that for all n
C 0 Φ λ n ( u λ n ) 1 2 Φ λ n ( u λ n ) u λ n = R H ˜ ( t , u λ n ) d t ,
(2.9)
from which we have
C 0 Ω n ( 0 , a ) H ˜ ( t , u λ n ) d t + Ω n ( a , b ) H ˜ ( t , u λ n ) d t + Ω n ( b , ) H ˜ ( t , u λ n ) d t Ω n ( 0 , a ) H ˜ ( t , u λ n ) d t + C a b Ω n ( a , b ) | u λ n | 2 d t + h ( b ) | Ω n ( b , ) | .
(2.10)
Invoking (H4), set τ : = 2 σ / ( σ 1 ) and σ = τ / 2 . Since σ > 1 one sees τ ( 2 , + ) . Fix arbitrarily τ ˆ ( τ , + ) . By (2.10) and the fact h ( r ) as r , we have
| Ω n ( b , ) | C 0 h ( b ) 0
as b uniformly in n, it follows from v λ n = 1 , Hölder’s inequality, and Sobolev’s embedding theorem that
Ω n ( b , ) | v λ n | τ d t C | Ω n ( b , ) | 1 τ τ ˆ 0
(2.11)
as b uniformly in n. By (2.10) and u λ n , for any fixed 0 < a < b ,
Ω n ( a , b ) | v λ n | 2 d t = 1 u λ n 2 Ω n ( a , b ) | u λ n | 2 d t C 0 C a b u λ n 2 0 as  n .
(2.12)
Let 0 < ε < 1 3 . Sobolev’s embedding theorem implies v λ n 2 2 C v λ n 2 = C and | λ n | C 1 . It follows from the fact that there is a ε > 0 such that | H u ( t , u ) | < ε C 1 C | u | for all | u | a ε (see (H1)) that
Ω n ( 0 , a ε ) ( H u ( t , u λ n ) , λ n v λ n + v λ n ) u λ n d t Ω n ( 0 , a ε ) | H u ( t , u λ n ) | | u λ n | | v λ n | | λ n v λ n + v λ n | d t ε C 1 C Ω n ( 0 , a ε ) | v λ n | | λ n v λ n + v λ n | d t ε C 1 C ( R | v λ n | 2 d t ) 1 / 2 ( R | λ n v λ n + v λ n | 2 d t ) 1 / 2 ε C v λ n 2 2 ε
(2.13)
for all n. By (H4), (2.9), and (2.11), we can take b ε r 0 large so that
Ω n ( b ε , ) ( H u ( t , u λ n ) , λ n v λ n + v λ n ) u λ n d t Ω n ( b ε , ) | H u ( t , u λ n ) | | u λ n | | v λ n | | λ n v λ n + v λ n | d t ( Ω n ( b ε , ) | H u ( t , u λ n ) | σ | u λ n | σ d t ) 1 / σ ( Ω n ( b ε , ) ( | v λ n | | λ n v λ n + v λ n | ) σ d t ) 1 / σ ( Ω c 0 H ˜ ( t , u λ n ) d t ) 1 / σ ( R | λ n v λ n + v λ n | τ d t ) 1 / τ ( Ω n ( b ε , ) | v λ n | τ d t ) 1 / τ < ε
(2.14)
for all n. Note that there is γ = γ ( ε ) > 0 independent of n such that | H u ( t , u λ n ) | γ | u λ n | for t Ω n ( a ε , b ε ) . By (2.12) there is n 0 such that
Ω n ( a ε , b ε ) ( H u ( t , u λ n ) , λ n v λ n + v λ n ) u λ n d t Ω n ( a ε , b ε ) | H u ( t , u λ n ) | | u λ n | | v λ n | | λ n v λ n + v λ n | d t γ Ω n ( a ε , b ε ) | v λ n | | λ n v λ n + v λ n | d t γ ( R | v λ n | 2 d t ) 1 / 2 ( Ω n ( a ε , b ε ) | λ n v λ n + v λ n | 2 d t ) 1 / 2 γ λ n v λ n 2 ( Ω n ( a ε , b ε ) | v λ n | 2 d t ) 1 / 2 < ε
(2.15)
for all n n 0 . Therefore, the combination of (2.13)-(2.15) implies that for n n 0 , we have
R ( H u ( t , u λ n ) , λ n v λ n + v λ n ) u λ n d t < 3 ε < 1 ,

which contradicts with (2.8). Thus { u λ n } is bounded. □

Lemma 2.6 If { u λ n } is the sequence obtained in Lemma  2.5, then it is also a ( P S ) sequence for Φ satisfying
lim n Φ ( u λ n ) = 0 , lim n Φ ( u λ n ) sup Q ¯ Φ .
Proof Note that u λ n is bounded. From
lim n Φ ( u λ n ) = lim n [ Φ λ n ( u λ n ) + ( λ n 1 ) ( 1 2 u λ n 2 + R H ( t , u λ n ) d t ) ]
and noting that
lim n Φ ( u λ n ) φ = lim n [ Φ λ n ( u λ n ) φ + ( λ n 1 ) ( u λ n , φ + R ( H u ( t , u λ n ) , φ ) d t ) ]

uniformly in φ W , we obtain the conclusion. □

3 Proof of main result

We are now in a position to prove our main result.

Proof of Theorem 1.1 Note that Lemma 2.5 implies { u λ n } is bounded, thus u λ n u in W, and u λ n u in L q ( R , R N ) for all q [ 1 , + ] by Lemma 2.1, after passing to a subsequence.

By (2.3), Φ λ n ( u λ n ) u λ n + = 0 , Hölder’s inequality, and Sobolev’s embedding theorem,
u λ n + 2 = λ R ( H u ( t , u λ n ) , u λ n + ) d t ε R | u λ n | | u λ n + | d t + C ε R | u λ n | p 1 | u λ n + | d t ε u λ n u λ n + + C ε u λ n p p 1 u λ n + ε u λ n u λ n + + C ε u λ n p p 2 u λ n u λ n + ε u λ n 2 + C ε u λ n p p 2 u λ n 2 .
(3.1)
Similarly, we have
u λ n 2 ε u λ n 2 + C ε u λ n p p 2 u λ n 2 .
(3.2)
From (3.1) and (3.2), we get
u λ n 2 2 ε u λ n 2 + 2 C ε u λ n p p 2 u λ n 2 ,

which means u λ n p c for some constant c, it follows from u λ n u in L p ( R , R N ) that u 0 . The facts that Φ is weakly sequentially continuous on W and u λ n u in W imply Φ ( u ) = 0 .

Let K : = { u W : Φ ( u ) = 0 } be the critical set of Φ and
C : = inf { Φ ( z ) : z K { 0 } } .
For any critical point u of Φ, assumption (H3) implies that
Φ ( u ) = Φ ( u ) 1 2 Φ ( u ) u = R ( 1 2 ( H u ( t , u ) , u ) H ( t , u ) ) d t > 0 if  u 0 .
(3.3)
Therefore C 0 . We prove that C > 0 and there is u K such that Φ ( u ) = C . Let u j K { 0 } be such that Φ ( u j ) C . Then the proof in Lemma 2.5 shows that { u j } is bounded, and by the concentration compactness principle discussion above we know u j u K { 0 } . Thus
C = lim j Φ ( u j ) = lim j R ( 1 2 ( H u ( t , u j ) , u j ) H ( t , u j ) ) d t Ω ( 1 2 ( H u ( t , u ) , u ) H ( t , u ) ) d t = Φ ( u ) C ,

where the first inequality is due to (H3) and Fatou’s lemma. So Φ ( u ) = C and C > 0 because u 0 . □

Declarations

Acknowledgements

Research 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, Anyang, Henan Province, 455000, P.R., China
(2)
Petrochina Pipeline Company, Langfang, Hebei Province, 5000, P.R., China

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© Chen and Zhao; licensee Springer 2014

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