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Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space

Xiubo Shi1, Qiongfen Zhang1* and Qi-Ming Zhang2

Author Affiliations

1 College of Science, Guilin University of Technology, Guilin, Guangxi, 541004, P.R. China

2 College of Science, Hunan University of Technology, Zhuzhou, Hunan, 412000, P.R. China

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Boundary Value Problems 2013, 2013:137  doi:10.1186/1687-2770-2013-137

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/137


Received:5 December 2012
Accepted:7 May 2013
Published:24 May 2013

© 2013 Shi et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, the existence of at least one or infinitely many homoclinic solutions is obtained for the following p-Laplacian system:

d d t ( | u ˙ ( t ) | p 2 u ˙ ( t ) ) a ( t ) | u ( t ) | q p u ( t ) + W ( t , u ( t ) ) = 0 ,

where 1 < p < ( q + 2 ) / 2 , q > 2 , t R , u R N , a C ( R , R ) and W C 1 ( R × R N , R ) are not periodic in t.

MSC: 34C37, 35A15, 37J45, 47J30.

Keywords:
homoclinic solutions; variational methods; weighted L q p + 2 space; p-Laplacian systems

1 Introduction

Consider homoclinic solutions of the following p-Laplacian system:

d d t ( | u ˙ ( t ) | p 2 u ˙ ( t ) ) a ( t ) | u ( t ) | q p u ( t ) + W ( t , u ( t ) ) = 0 , t R , (1.1)

where 1 < p < ( q + 2 ) / 2 , q > 2 , t R , u R N , a : R R , W : R × R N R . As usual, we say that a solution u of (1.1) is a nontrivial homoclinic (to 0) if u C 2 ( R , R N ) such that u 0 , u ( t ) 0 as t ± .

When p = 2 , (1.1) reduces to the following second-order Hamiltonian system:

u ¨ ( t ) a ( t ) | u ( t ) | q 2 u ( t ) + W ( t , u ( t ) ) = 0 , t R . (1.2)

If we take p = 2 and q = 2 , then (1.2) reduces to the following second-order Hamiltonian system:

u ¨ ( t ) a ( t ) u ( t ) + W ( t , u ( t ) ) = 0 , t R . (1.3)

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré [1]. Up to the year of 1990, a few of isolated results can be found, and the only method for dealing with such a problem was the small perturbation technique of Melnikov.

Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For example, see [2-20] and references therein. However, few results [21,22] have been obtained in the literature for system (1.2). In [22], by introducing a suitable Sobolev space, Salvatore established the following existence results for system (1.2) when q > 2 .

Theorem A[22]

Assume thataandWsatisfy the following conditions:

(A) Let q > 2 , a ( t ) is a continuous, positive function onsuch that for all t R

a ( t ) α 0 | t | β 0 , α 0 > 0 , β 0 > ( q 2 ) / 2 .

(W1) W C 1 ( R × R N , R ) and there exists a constant μ > q such that

0 < μ W ( t , x ) ( W ( t , x ) , x ) , ( t , x ) R × R N { 0 } .

(W2) | W ( t , x ) | = o ( | x | q 1 ) as | x | 0 uniformly with respect to t R .

(W3) There exists W ¯ C ( R N , R ) such that

| W ( t , x ) | + | W ( t , x ) | | W ¯ ( x ) | , ( t , x ) R × R N .

Then problem (1.2) has one nontrivial homoclinic solution.

When W ( t , x ) is an even function in x, Salvatore [22] obtained the following existence theorem of an unbounded sequence of homoclinic orbits for problem (1.2) by the symmetric mountain pass theorem.

Theorem B[22]

Assume thataandWsatisfy (A), (W1)-(W3) and the following condition:

(W4) W ( t , x ) = W ( t , x ) , ( t , x ) R × R N .

Then problem (1.2) has an unbounded sequence of homoclinic solutions.

In [21], Chen and Tang improved Theorem A and Theorem B by relaxing conditions (W1) and (W2) and removing condition (W3). Motivated mainly by the ideas of [18,21-23], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and symmetric mountain pass theorem. Precisely, we obtain the following main results.

Theorem 1.1Suppose thataandWsatisfy the following conditions:

(A)′ Let 1 < p < ( q + 2 ) / 2 and q > 2 , a ( t ) is a continuous, positive function onsuch that for all t R

a ( t ) α | t | β , α > 0 , β > ( q 2 p + 2 ) / p .

(W5) W ( t , x ) = W 1 ( t , x ) W 2 ( t , x ) , W 1 , W 2 C 1 ( R × R N , R ) , and there exists a constant R > 0 such that

1 a ( t ) | W ( t , x ) | = o ( | x | q p + 1 ) as   x 0

uniformly in t ( , R ] [ R , + ) .

(W6) There is a constant μ > q p + 2 such that

0 < μ W 1 ( t , x ) ( W 1 ( t , x ) , x ) , ( t , x ) R × R N { 0 } .

(W7) W 2 ( t , 0 ) = 0 and there exists a constant ϱ ( q p + 2 , μ ) such that

W 2 ( t , x ) 0 , ( W 2 ( t , x ) , x ) ϱ W 2 ( t , x ) , ( t , x ) R × R N .

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.2Suppose thataandWsatisfy (A)′, (W6) and the following conditions:

(W5)′ W ( t , x ) = W 1 ( t , x ) W 2 ( t , x ) , W 1 , W 2 C 1 ( R × R N , R ) , and

1 a ( t ) | W ( t , x ) | = o ( | x | q p + 1 ) as   x 0

uniformly in t R .

(W7)′ W 2 ( t , 0 ) = 0 and there exists a constant ϱ ( q p + 2 , μ ) such that

( W 2 ( t , x ) , x ) ϱ W 2 ( t , x ) , ( t , x ) R × R N .

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.3Suppose thataandWsatisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Theorem 1.4Suppose thataandWsatisfy (A)′, (W4), (W5)′, (W6), (W7)′. Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Remark 1.1 When p = 2 , condition (A)′ reduces to condition (A). Obviously, Theorem 1.1-Theorem 1.4 generalize and improve Theorem A, Theorem B and the corresponding results in [21]. It is easy to see that our results hold true even if p = 2 . To the best of our knowledge, similar results for problem (1.1) cannot be seen in the literature; from this point, our results are new.

Remark 1.2 If p = 2 and q = 2 , then problem (1.1) reduces to problem (1.3). As pointed out in [23], Theorem A can be proved by replacing (A) with the more general assumption: a ( t ) + as | t | + .

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2 Preliminaries

We set, for any real number 1 h < + ,

L h = L h ( R , R N ) , L = L ( R , R N )

with the usual norms

u h = ( R | u ( t ) | h d t ) 1 / h , u = max t R | u ( t ) | .

Let

W 1 , p = W 1 , p ( R , R N ) = { u : R R N u  is absolutely continuous , u , u ˙ L p ( R , R N ) }

be the Sobolev space with the norm given by

u W 1 , p = ( R [ | u ˙ ( t ) | p + | u ( t ) | p ] d t ) 1 / p .

If σ is a positive, continuous function on ℝ and 1 < s < + , let

L σ s = L σ s ( R , R N ; σ ) = { u L loc 1 ( R , R N ) | R σ ( t ) | u ( t ) | s d t < + } .

L σ s equipped with the norm

u s , σ = ( R σ ( t ) | u ( t ) | s d t ) 1 / s

is a reflexive Banach space. When s = + , we set

L σ = L σ ( R , R N ; σ ) = { u | max t R σ ( t ) | u ( t ) | < + }

with the norm given by

u , σ = max t R σ ( t ) | u ( t ) | .

Set E = W 1 , p L a q p + 2 , where a is the function given in condition (A)′. Then E with its standard norm is a reflexive Banach space. The functional φ corresponding to (1.1) on E is given by

φ ( u ) = R [ 1 p | u ˙ ( t ) | p + a ( t ) q p + 2 | u ( t ) | q p + 2 W ( t , u ( t ) ) ] d t , u E . (2.1)

Clearly, it follows from (W5) or (W5)′ that φ : E R . By Theorem 2.1 of [24], we can deduce that the map

u a ( t ) | u ( t ) | q p u ( t )

is continuous from L a q p + 2 in the dual space L a 1 / ( q p + 1 ) p 1 , where p 1 = q p + 2 q p + 1 . As the embeddings E W 1 , p L γ for all γ p are continuous, if (A)′ and (W5) or (W5)′ hold, then φ C 1 ( E , R ) and one can easily check that

φ ( u ) , v = R [ | u ˙ ( t ) | p 2 ( u ˙ ( t ) , v ˙ ( t ) ) + a ( t ) | u ( t ) | q p ( u ( t ) , v ( t ) ) ( W ( t , u ( t ) ) , v ( t ) ) ] d t , u E . (2.2)

Furthermore, the critical points of φ in E are classical solutions of (1.1) with u ( ± ) = 0 .

To prove our results, we need the following generalization of the Lebesgue dominated convergence theorem.

Lemma 2.1[25]

Let { f n ( t ) } and { g n ( t ) } be two sequences of measurable functions on a measurable setA, and let

| f n ( t ) | g n ( t ) for a.e.   t A .

If

lim n f n ( t ) = f ( t ) , lim n g n ( t ) = g ( t ) for a.e.   t A

and

lim n A g n ( t ) d t = A g ( t ) d t < + ,

then

lim n A f n ( t ) d t = A f ( t ) d t .

The following lemma is an improvement result of [23] in which the author considered the case p = 2 .

Lemma 2.2Ifasatisfies assumption (A)′, then

the embedding   L a q p + 2 L p   is continuous . (2.3)

Moreover, there exists a Sobolev spaceZsuch that

the embeddings   L a q p + 2 Z L p   are continuous , (2.4)

the embedding   W 1 , p Z L p   is compact . (2.5)

Proof Let θ = ( q p + 2 ) / ( q 2 p + 2 ) , θ = ( q p + 2 ) / p , we have

u p p = R a 1 / θ a 1 / θ | u | p d t ( R a θ / θ d t ) 1 / θ ( R a | u | p θ d t ) 1 / θ = a 1 ( R a | u | q p + 2 d t ) p / q p + 2 = a 1 u q p + 2 , a p ,

where from (A)′, a 1 = ( R a p / ( q 2 p + 2 ) d t ) ( q 2 p + 2 ) / ( q p + 2 ) < + . Then (2.3) holds.

By (A)′, there exists a continuous positive function ρ such that ρ ( t ) + as | t | + and

a 2 = ( R ρ θ a θ / θ d t ) 1 / θ < + .

Since

u p , ρ p = R ρ | u | p d t = R ρ a 1 / θ a 1 / θ | u | p d t ( R ρ θ a θ / θ d t ) 1 / θ ( R a | u | q p + 2 d t ) 1 / θ = a 2 u q p + 2 , a p ,

(2.4) holds by taking Z = L ρ p .

Finally, as W 1 , p Z is the weighted Sobolev space Γ 1 , p ( R , ρ , 1 ) , it follows from [24] that (2.5) holds. □

The following two lemmas are the mountain pass theorem and symmetric mountain pass theorem, which are useful in the proofs of our theorems.

Lemma 2.3[26]

LetEbe a real Banach space and I C 1 ( E , R ) satisfying (PS)-condition. Suppose I ( 0 ) = 0 and:

(i) There exist constants ρ , α > 0 such that I B ρ ( 0 ) α .

(ii) There exists an e E B ¯ ρ ( 0 ) such that I ( e ) 0 .

ThenIpossesses a critical value c α which can be characterized as

c = inf h Φ max s [ 0 , 1 ] I ( h ( s ) ) ,

where Φ = { h C ( [ 0 , 1 ] , E ) h ( 0 ) = 0 , h ( 1 ) = e } and B ρ ( 0 ) is an open ball inEof radiusρcentered at 0.

Lemma 2.4[26]

LetEbe a real Banach space and I C 1 ( E , R ) withIeven. Assume that I ( 0 ) = 0 andIsatisfies (PS)-condition, (i) of Lemma 2.3 and the following condition:

(iii) For each finite dimensional subspace E E , there is r = r ( E ) > 0 such that I ( u ) 0 for u E B r ( 0 ) , B r ( 0 ) is an open ball inEof radiusrcentered at 0.

ThenIpossesses an unbounded sequence of critical values.

Lemma 2.5Assume that (W6) and (W7) or (W7)′ hold. Then, for every ( t , x ) R × R N ,

(i) s μ W 1 ( t , s x ) is nondecreasing on ( 0 , + ) ;

(ii) s ϱ W 2 ( t , s x ) is nonincreasing on ( 0 , + ) .

The proof of Lemma 2.5 is routine and we omit it.

3 Proofs of theorems

Proof of Theorem 1.1 Step 1. The functional φ satisfies (PS)-condition. Let { u n } E satisfying φ ( u n ) be bounded and φ ( u n ) 0 as n . Hence, there exists a constant C 1 > 0 such that

| φ ( u n ) | C 1 , φ ( u n ) E μ C 1 . (3.1)

It is well known [27] that there exists a constant C 2 > 0 such that

u C 2 u , u E . (3.2)

From (2.1), (2.2), (3.1), (W6) and (W7), we have

p C 1 + p C 1 u n p φ ( u n ) p μ φ ( u n ) , u n = μ p μ u ˙ n p p + p R [ W 2 ( t , u n ( t ) ) 1 μ ( W 2 ( t , u n ( t ) ) , u n ( t ) ) ] d t p R [ W 1 ( t , u n ( t ) ) 1 μ ( W 1 ( t , u n ( t ) ) , u n ( t ) ) ] d t + ( p q p + 2 p μ ) R a ( t ) | u n ( t ) | q p + 2 d t μ p μ u ˙ n p p + ( p q p + 2 p μ ) u n q p + 2 , a q p + 2 .

It follows from Lemma 2.2, p < ( q + 2 ) / 2 , μ > q p + 2 and the above inequalities that there exists a constant C 3 > 0 such that

u n C 3 , n N . (3.3)

Now we prove that u n u 0 in E. Passing to a subsequence if necessary, it can be assumed that u n u 0 in E. From Lemma 2.2, we have u n u 0 in L p . From (3.2) and (3.3), we have

u n C 2 u n C 2 C 3 , u n E . (3.4)

Inequality (3.4) implies that | u n ( t ) | C 2 C 3 for all t R . By (W5), we know that

| W ( t , x ) | a ( t ) | x | q p + 1 0 as  x 0 ,

which implies that for any given constant C > 0 , there exists a constant C > 0 related to C such that

| W ( t , x ) | a ( t ) | x | q p + 1 C for  | x | C .

Hence, there exists a constant C 4 > 0 such that

| W ( t , x ) | C 4 a ( t ) | x | q p + 1 for  | x | C 2 C 3 . (3.5)

Hence, from (3.5), we have

| W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) | p [ C 4 a ( t ) ( | u n ( t ) | q p + 1 + | u 0 ( t ) | q p + 1 ) ] p [ C 4 2 q p + 1 a ( t ) | u n ( t ) u 0 ( t ) | q p + 1 + C 4 ( 1 + 2 q p + 1 ) a ( t ) | u 0 ( t ) | q p + 1 ] p 2 p ( q p + 2 ) C 4 p a p ( t ) | u n ( t ) u 0 ( t ) | p ( q p + 1 ) + 2 p C 4 p ( 1 + 2 q p + 1 ) p a p ( t ) | u 0 ( t ) | p ( q p + 1 ) : = g n ( t ) , (3.6)

where p = p p 1 . Moreover, since a ( t ) is a positive continuous function on ℝ, p < q p + 2 and u n ( t ) u 0 ( t ) for almost every t R , we have

lim n g n ( t ) = 2 p C 4 p ( 1 + 2 q p + 1 ) p a p ( t ) | u 0 ( t ) | p ( q p + 1 ) : = g ( t ) for a.e.  t R

and

lim n R g n ( t ) d t = lim n R [ 2 p ( q p + 2 ) C 4 p a p ( t ) | u n ( t ) u 0 ( t ) | p ( q p + 1 ) + 2 p C 4 p ( 1 + 2 q p + 1 ) p a p ( t ) | u 0 ( t ) | p ( q p + 1 ) ] d t = 2 p ( q p + 2 ) C 4 p lim n R a p ( t ) | u n ( t ) u 0 ( t ) | p ( q p + 1 ) d t + 2 p C 4 p ( 1 + 2 q p + 1 ) p R a p ( t ) | u 0 ( t ) | p ( q p + 1 ) d t = 2 p C 4 p ( 1 + 2 q p + 1 ) p R a p ( t ) | u 0 ( t ) | p ( q p + 1 ) d t = R g ( t ) d t < + .

It follows from Lemma 2.1, (3.6) and the above inequalities that

lim n R | W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) | p d t = 0 .

This shows that

W ( t , u n ) W ( t , u 0 ) in  L p ( R , R N ) . (3.7)

From (2.2), we have

φ ( u n ) φ ( u 0 ) , u n u 0 ) = R ( | u ˙ n ( t ) | p 2 u ˙ n ( t ) | u ˙ 0 ( t ) | p 2 u ˙ 0 ( t ) , u ˙ n ( t ) u ˙ 0 ( t ) ) d t + R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t R ( W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) , u n ( t ) u 0 ( t ) ) d t u ˙ n p p + u ˙ 0 p p u ˙ 0 p u ˙ n p p 1 u ˙ n p u ˙ 0 p p 1 + R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t R ( W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) , u n ( t ) u 0 ( t ) ) d t = ( u ˙ n p p 1 u ˙ 0 p p 1 ) ( u ˙ n p u ˙ 0 p ) + R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t R ( W ( t , u n ( t ) ) W ( t , u 0 ( t ) ) , u n ( t ) u 0 ( t ) ) d t . (3.8)

It is easy to see that for any α > 1 there exists a constant C 5 > 0 such that

( | x | α 1 x | y | α 1 y ) ( x y ) C 5 | x y | α + 1 , x , y R . (3.9)

Inequality (3.9) implies that

( u ˙ n p p 1 u ˙ 0 p p 1 ) ( u ˙ n p u ˙ 0 p ) C 5 | u ˙ n p u ˙ 0 p | p (3.10)

and

R a ( t ) ( | u n ( t ) | q p u n ( t ) | u 0 ( t ) | q p u 0 ( t ) ) ( u n ( t ) u 0 ( t ) ) d t C 5 R a ( t ) | u n ( t ) u 0 ( t ) | q p + 2 d t , (3.11)

where C 5 and C 5 are positive constants. Since φ ( u n ) 0 as n + , u n u in E and the embeddings E W 1 , p L γ for all γ p are continuous, it follows from Lemma 2.2, (3.7), (3.8), (3.10) and (3.11) that

u ˙ n p u ˙ 0 p as  n (3.12)

and

R a ( t ) | u n ( t ) | q p + 2 d t R a ( t ) | u 0 ( t ) | q p + 2 d t as  n . (3.13)

Hence, by (3.12) and (3.13), u n u 0 in E. This shows that φ satisfies (PS)-condition.

Step 2. From (W5), there exists δ ( 0 , 1 ) such that

| W ( t , x ) | 1 p a ( t ) | x | q p + 1 for  | t | R , | x | δ . (3.14)

By (3.14), we have

| W ( t , x ) | 1 p ( q p + 2 ) a ( t ) | x | q p + 2 for  | t | R , | x | δ . (3.15)

Let

C 6 = sup { W 1 ( t , x ) a ( t ) | t [ R , R ] , x R , | x | = 1 } . (3.16)

Set σ = min { 1 / ( p ( q p + 2 ) C 6 + 1 ) 1 / ( μ q + p 2 ) , δ } and u = σ / C 2 : = ρ , it follows from (3.2) that

u C 2 u σ ,

which shows that | u ( t ) | σ δ < 1 . From Lemma 2.5(i) and (3.16), we have

R R W 1 ( t , u ( t ) ) d t { t [ R , R ] : u ( t ) 0 } W 1 ( t , u ( t ) | u ( t ) | ) | u ( t ) | μ d t C 6 R R a ( t ) | u ( t ) | μ d t C 6 σ μ q + p 2 R R a ( t ) | u ( t ) | q p + 2 d t 1 p ( q p + 2 ) R R a ( t ) | u ( t ) | q p + 2 d t . (3.17)

It follows from (W7), (3.15), (3.17) that

φ ( u ) = 1 p R | u ˙ ( t ) | p d t + R a ( t ) q p + 2 | u ( t ) | q p + 2 d t R W ( t , u ( t ) ) d t = 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 R [ R , R ] W ( t , u ( t ) ) d t R R W ( t , u ( t ) ) d t 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 R R W 1 ( t , u ( t ) ) d t R [ R , R ] 1 p ( q p + 2 ) a ( t ) | u ( t ) | q p + 2 d t 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 1 p ( q p + 2 ) R R a ( t ) | u ( t ) | q p + 2 d t R [ R , R ] 1 p ( q p + 2 ) a ( t ) | u ( t ) | q p + 2 d t = 1 p u ˙ p p + p 1 p ( q p + 2 ) u q p + 2 , a q p + 2 .

Therefore, we can choose a constant α > 0 depending on ρ such that φ ( u ) α for any u E with u = ρ .

Step 3. From Lemma 2.5(ii) and (3.2), we have for any u E

2 2 W 2 ( t , u ( t ) ) d t = { t [ 2 , 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) ) d t + { t [ 2 , 2 ] : | u ( t ) | 1 } W 2 ( t , u ( t ) ) d t { t [ 2 , 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) | u ( t ) | ) | u ( t ) | ϱ d t + 2 2 max | x | 1 W 2 ( t , x ) d t u ϱ 2 2 max | x | = 1 W 2 ( t , x ) d t + 2 2 max | x | 1 W 2 ( t , x ) d t C 2 ϱ u ϱ 2 2 max | x | = 1 W 2 ( t , x ) d t + 2 2 max | x | 1 W 2 ( t , x ) d t = C 7 u ϱ + C 8 , (3.18)

where C 7 = C 2 ϱ 2 2 max | x | = 1 W 2 ( t , x ) d t , C 8 = 2 2 max | x | 1 W 2 ( t , x ) d t . Take ω E such that

| ω ( t ) | = { 1 for  | t | 1 , 0 for  | t | 2 (3.19)

and | ω ( t ) | 1 for | t | ( 1 , 2 ] . For s > 1 , from Lemma 2.5(i) and (3.19), we get

1 1 W 1 ( t , s ω ( t ) ) d t s μ 1 1 W 1 ( t , ω ( t ) ) d t = C 9 s μ , (3.20)

where C 9 = 1 1 W 1 ( t , ω ( t ) ) d t > 0 . From (W7), (2.1), (3.18), (3.19), (3.20), we get for s > 1

φ ( s ω ) = s p p ω ˙ p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + R [ W 2 ( t , s ω ( t ) ) W 1 ( t , s ω ( t ) ) ] d t s p p ω ˙ p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + 2 2 W 2 ( t , s ω ( t ) ) d t 1 1 W 1 ( t , s ω ( t ) ) d t s p p ω ˙ p p + s q p + 2 q p + 2 ω q p + 2 , a q p + 2 + C 7 s ϱ ω ϱ + C 8 C 9 s μ . (3.21)

Since μ > ϱ > q p + 2 and C 9 > 0 , it follows from (3.21) that there exists s 1 > 1 such that s 1 ω > ρ and φ ( s 1 ω ) < 0 . Let e = s 1 ω ( t ) , then e E , e = s 1 ω > ρ and φ ( e ) = φ ( s 1 ω ) < 0 . By Lemma 2.3, φ has a critical value d > α given by

d = inf g Φ max s [ 0 , 1 ] φ ( g ( s ) ) , (3.22)

where

Φ = { g C ( [ 0 , 1 ] , E ) : g ( 0 ) = 0 , g ( 1 ) = e } .

Hence, there exists u E such that

φ ( u ) = d , φ ( u ) = 0 .

The function u is a desired solution of problem (1.1). Since d > 0 , u is a nontrivial homoclinic solution. The proof is complete. □

Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition W 2 ( t , x ) 0 in (W7) is only used in the proofs of (3.3) and Step 2. Therefore, we only need to prove that (3.3) and Step 2 still hold if we use (W5)′ and (W7)′ instead of (W5) and (W7). We first prove that (3.3) holds. From (W6), (W7)′, (2.1), (2.2) and (3.1), we have

p ( q p + 2 ) C 1 + p ( q p + 2 ) C 1 μ ϱ u n p ( q p + 2 ) φ ( u n ) p ( q p + 2 ) ϱ φ ( u n ) , u n = ( ϱ p ) ( q p + 2 ) ϱ u ˙ n p p + p ( q p + 2 ) R [ W 2 ( t , u n ( t ) ) 1 ϱ ( W 2 ( t , u n ( t ) ) , u n ( t ) ) ] d t p ( q p + 2 ) R [ W 1 ( t , u n ( t ) ) 1 ϱ ( W 1 ( t , u n ( t ) ) , u n ( t ) ) ] d t + p ( 1 q p + 2 ϱ ) R a ( t ) | u n ( t ) | q p + 2 d t ( ϱ p ) ( q p + 2 ) ϱ u ˙ n p p + p ( 1 q p + 2 ϱ ) u n q p + 2 , a q p + 2 ,

which implies that there exists a constant C 3 > 0 such that (3.3) holds. Next, we prove Step 2 still holds. From (W5)′, there exists δ ( 0 , 1 ) such that

| W ( t , x ) | 1 p a ( t ) | x | q p + 1 for  t R , | x | δ . (3.23)

By (3.23), we have

| W ( t , x ) | 1 p ( q p + 2 ) a ( t ) | x | q p + 2 for  t R , | x | δ . (3.24)

Let 0 < σ δ and u = σ / C 2 : = ρ , it follows from (3.2) that

u C 2 u σ ,

which shows that | u ( t ) | σ δ < 1 . It follows from (2.1) and (3.24) that

φ ( u ) = 1 p R | u ˙ ( t ) | p d t + R a ( t ) q p + 2 | u ( t ) | q p + 2 d t R W ( t , u ( t ) ) d t 1 p u ˙ p p + 1 q p + 2 u q p + 2 , a q p + 2 R 1 p ( q p + 2 ) a ( t ) | u ( t ) | q p + 2 d t = 1 p u ˙ p p + p 1 p ( q p + 2 ) u q p + 2 , a q p + 2 .

Therefore, we can choose a constant α > 0 depending on ρ such that φ ( u ) α for any u E with u = ρ . The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that φ C 1 ( E , R ) and satisfies (PS)-condition and assumptions (i) of Lemma 2.3. Now, we prove that (iii) of Lemma 2.4. Let E be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists c > 0 such that

u c u . (3.25)

Assume that dim E = m and { u 1 , u 2 , , u m } is a base of E such that

u i = c , i = 1 , 2 , , m . (3.26)

For any u E , there exists λ i R , i = 1 , 2 , , m such that

u ( t ) = i = 1 m λ i u i ( t ) for  t R . (3.27)

Let

u = i = 1 m | λ i | u i . (3.28)

It is easy to see that is a norm of E . Hence, there exists a constant c > 0 such that c u u . Since u i E , by Lemma 2.2, we can choose R 1 > R such that

| u i ( t ) | < c δ 1 + c , | t | > R 1 , i = 1 , 2 , , m , (3.29)

where δ is given in (3.24). Let

Θ = { i = 1 m λ i u i ( t ) : λ i R , i = 1 , 2 , , m ; i = 1 m | λ i | = 1 } = { u E : u = c } . (3.30)

Hence, for u Θ , let t 0 = t 0 ( u ) R such that

| u ( t 0 ) | = u . (3.31)

Then by (3.25)-(3.28), (3.30) and (3.31), we have

c c = c c i = 1 m | λ i | = c i = 1 m | λ i | u i = c u u c u = c | u ( t 0 ) | c i = 1 m | λ i | | u i ( t 0 ) | , u Θ . (3.32)

This shows that | u ( t 0 ) | c and there exists i 0 { 1 , 2 , , m } such that | u i 0 ( t 0 ) | c , which together with (3.29), implies that | t 0 | R 1 . Let R 2 = R 1 + 1 and

γ = min { W 1 ( t , x ) : R 2 t R 2 , c 2 1 / p | x | c C 2 } . (3.33)

Since W 1 ( t , x ) > 0 for all t R and x R N { 0 } , and W 1 C 1 ( R × R N , R ) , it follows that γ > 0 . For any u E , from Lemma 2.5(i) and (3.2), we have

R 2 R 2 W 2 ( t , u ( t ) ) d t = { t [ R 2 , R 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) ) d t + { t [ R 2 , R 2 ] : | u ( t ) | 1 } W 2 ( t , u ( t ) ) d t { t [ R 2 , R 2 ] : | u ( t ) | > 1 } W 2 ( t , u ( t ) | u ( t ) | ) | u ( t ) | ϱ d t + R 2 R 2 max | x | 1 W 2 ( t , x ) d t u ϱ R 2 R 2 max | x | = 1 W 2 ( t , x ) d t + R 2 R 2 max | x | 1 W 2 ( t , x ) d t C 2 ϱ u ϱ R 2 R 2 max | x | = 1 W 2 ( t , x ) d t + R 2 R 2 max | x | 1 W 2 ( t , x ) d t = C 10 u ϱ + C 11 , (3.34)

where C 10 = C 2 ϱ R 2 R 2 max | x | = 1 W 2 ( t , x ) d t , C 11 = R 2 R 2 max | x | 1 W 2 ( t , x ) d t . Since u ˙ i L p ( R ) , i = 1 , 2 , , m , it follows that there exists ε ( 0 , 1 ) such that

t + ε t ε | u ˙ i ( s ) | d s ( 2 ε ) 1 / p ( t + ε t ε | u ˙ i ( s ) | p d s ) 1 / p ( 2 ε ) 1 / p u ˙ i p c 2 p for  t R , i = 1 , 2 , , m , (3.35)

where 1 / p + 1 / p = 1 . Then, for u Θ with | u ( t 0 ) | = u and t [ t 0 ε , t 0 + ε ] , it follows from (3.27), (3.30), (3.31), (3.32) and (3.35) that

| u ( t ) | p = | u ( t 0 ) | p + p t 0 t | u ( s ) | p 2 ( u ˙ ( s ) , u ( s ) ) d s | u ( t 0 ) | p p t 0 ε t 0 + ε | u ( s ) | p 1 | u ˙ ( s ) | d s | u ( t 0 ) | p p | u ( t 0 ) | p 1 t 0 ε t 0 + ε | u ˙ ( s ) | d s c 2 | u ( t 0 ) | p 1 c p 2 . (3.36)

On the other hand, since u c for u Θ , then

| u ( t ) | u C 2 c , t R , u Θ . (3.37)

Therefore, from (3.33), (3.36) and (3.37), we have

R 2 R 2 W 1 ( t , u ( t ) ) d t t 0 ε t 0 + ε W 1 ( t , u ( t ) ) d t 2 ε γ for  u Θ . (3.38)

From (3.29) and (3.30), we have

| u ( t ) | i = 1 m | λ i | | u i ( t ) | δ for  | t | R 1 , u Θ . (3.39)

By (2.1), (3.15), (3.34), (3.38), (3.39) and Lemma 2.5, we have for u Θ and r > 1

φ ( r u ) = r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + R [ W 2 ( t , r u ( t ) ) W 1 ( t , r u ( t ) ) ] d t r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ R W 2 ( t , u ( t ) ) d t r μ R W 1 ( t , u ( t ) ) d t = r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ R ( R 2 , R 2 ) W 2 ( t , u ( t ) ) d t r μ R ( R 2 , R 2 ) W 1 ( t , u ( t ) ) d t + r ϱ R 2 R 2 W 2 ( t , u ( t ) ) d t r μ R 2 R 2 W 1 ( t , u ( t ) ) d t r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 r ϱ R ( R 2 , R 2 ) W ( t , u ( t ) ) d t r μ R 2 R 2 W 1 ( t , u ( t ) ) d t + r ϱ R 2 R 2 W 2 ( t , u ( t ) ) d t r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ p ( q p + 2 ) R ( R 2 , R 2 ) a ( t ) | u ( t ) | q p + 2 d t + r ϱ ( C 10 u ϱ + C 11 ) 2 ε γ r μ r p p u ˙ p p + r q p + 2 q p + 2 u q p + 2 , a q p + 2 + r ϱ p ( q p + 2 ) u q p + 2 , a q p + 2 + r ϱ ( C 10 u ϱ + C 11 ) 2 ε γ r μ r p p c p + r q p + 2 q p + 2 c q p + 2 + r ϱ p ( q p + 2 ) c q p + 2 + C 10 ( r c ) ϱ + C 11 r ϱ 2 ε γ r μ . (3.40)

Since μ > ϱ > q p + 2 > p , we deduce that there exists r 0 = r 0 ( c , c , C 10 , C 11 , R 1 , R 2 , ε , γ ) = r 0 ( E ) > 1 such that

φ ( r u ) < 0 for  u Θ  and  r r 0 .

It follows that

φ ( u ) < 0 for  u E  and  u c r 0 ,

which shows that (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence { d n } n = 1 of critical values with d n = φ ( u n ) , where u n is such that φ ( u n ) = 0 for n = 1 , 2 ,  . If { u n } is bounded, then there exists C 12 > 0 such that

u n C 12 for  n N . (3.41)

By (3.2) and (3.41), we get

| u n ( t ) | C 2 C 12 for  n N . (3.42)

From (W5), we can choose C 13 > 0 and R 3 > R such that

| W ( t , x ) | C 13 a ( t ) | x | q p + 1 for  | t | R 3 , | x | C 2 C 12 ,

which implies that

| W ( t , x ) | C 13 q p + 2 a ( t ) | x | q p + 2 for  | t | R 3 , | x | C 2 C 12 . (3.43)

Hence, by (2.1) and (3.43), we have

1 p u ˙ n p p + 1 q p + 2 u n q p + 2 , a q p + 2 = d n + R W ( t , u n ( t ) ) d t = d n + R [ R 3 , R 3 ] W ( t , u n ( t ) ) d t + R 3 R 3 W ( t , u n ( t ) ) d t d n C 13 q p + 2 R [ R 3 , R 3 ] a ( t ) | u n ( t ) | q p + 2 d t R 3 R 3 | W ( t , u n ( t ) ) | d t d n C 13 q p + 2 u n q p + 2 , a q p + 2 R 3 R 3 max | x | C 2 C 12 | W ( t , x ) | d t ,

which, together with (3.41), implies that

d n 1 p u ˙ n p p + C 13 + 1 q p + 2 u n q p + 2 , a q p + 2 + R 3 R 3 max | x | C 2 C 12 | W ( t , x ) | d t < + .

This contradicts the fact that { d n } n = 1 is unbounded, and so { u n } is unbounded. The proof is complete. □

Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □

4 Examples

Example 4.1 Consider the following system:

d d t ( | u ˙ ( t ) | u ˙ ( t ) ) a ( t ) | u ( t ) | 3 u ( t ) + W ( t , u ( t ) ) = 0 , a.e.  t R , (4.1)

where p = 3 , q = 6 , t R , u R N , a C ( R , ( 0 , ) ) and a satisfies (A)′. Let

W ( t , x ) = a ( t ) ( i = 1 m a i | x | μ i j = 1 n b j | x | ϱ j ) ,

where μ 1 > μ 2 > > μ m > ϱ 1 > ϱ 2 > > ϱ j > 5 , a i , b j > 0 , i = 1 , , m , j = 1 , , n . Let

W 1 ( t , x ) = a ( t ) i = 1 m a i | x | μ i , W 2 ( t , x ) = a ( t ) j = 1 n b j | x | ϱ j .

Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with μ = μ m and ϱ = ϱ 1 . Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.

Example 4.2 Consider the following system:

d d t ( | u ˙ ( t ) | 1 / 2 u ˙ ( t ) ) a ( t ) | u ( t ) | u ( t ) + W ( t , u ( t ) ) = 0 , a.e.  t R , (4.2)

where p = 3 / 2 , q = 5 / 2 , t R , u R N , a C ( R , ( 0 , ) ) and a satisfies (A)′. Let

W ( t , x ) = a ( t ) [ a 1 | x | μ 1 + a 2 | x | μ 2 b 1 ( cos t ) | x | ϱ 1 b 2 | x | ϱ 2 ] ,

where μ 1 > μ 2 > ϱ 1 > ϱ 2 > 3 , a 1 , a 2 > 0 , b 1 , b 2 > 0 . Let

W 1 ( t , x ) = a ( t ) ( a 1 | x | μ 1 + a 2 | x | μ 2 ) , W 2 ( t , x ) = a ( t ) [ b 1 ( cos t ) | x | ϱ 1 + b 2 | x | ϱ 2 ] .

Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with μ = μ 2 and ϱ = ϱ 1 . Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

XS and QZ are supported by the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093), Guangxi Natural Science Foundation (Nos. 2013GXNSFBA019004 and 2012GXNSFBA053013) and the Scientific Research Foundation of Guilin University of Technology. QMZ is supported by the NNSF of China (No. 11201138) and the Scientific Research Fund of Hunan Provincial Education Department (No. 12B034).

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