Skip to main content

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

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, tR, u R N , aC(R,R) and W C 1 (R× R N ,R) are not periodic in t.

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

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,tR,
(1.1)

where 1<p<(q+2)/2, q>2, tR, u R N , a:RR, 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 u0, 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,tR.
(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,tR.
(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 [220] 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 that a and W satisfy the following conditions:

  1. (A)

    Let q>2, a(t) is a continuous, positive function on such that for all tR

    a(t) α 0 | t | β 0 , α 0 >0, β 0 >(q2)/2.

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

0<μW(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 tR.

(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 that a and W satisfy (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, 2123], 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 1.1 Suppose that a and W satisfy the following conditions:

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

a(t)α | t | β ,α>0,β>(q2p+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 x0

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

(W6) There is a constant μ>qp+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(qp+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 1.2 Suppose that a and W satisfy (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 x0

uniformly in tR.

(W7)′ W 2 (t,0)=0 and there exists a constant ϱ(qp+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 1.3 Suppose that a and W satisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Theorem 1.4 Suppose that a and W satisfy (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 1h<+,

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 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 ) ) ] dt,uE.
(2.1)

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

ua(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 γp 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 set A, and let

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

If

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

and

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

then

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

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

Lemma 2.2 If a satisfies assumption (A)′, then

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

Moreover, there exists a Sobolev space Z such 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=(qp+2)/(q2p+2), θ =(qp+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]

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

  1. (i)

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

  2. (ii)

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

Then I possesses a critical value cα which can be characterized as

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

where Φ={hC([0,1],E)h(0)=0,h(1)=e} and B ρ (0) is an open ball in E of radius ρ centered at 0.

Lemma 2.4 [26]

Let E be a real Banach space and I C 1 (E,R) with I even. Assume that I(0)=0 and I satisfies (PS)-condition, (i) of Lemma  2.3 and the following condition:

  1. (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 in E of radius r centered at 0.

Then I possesses an unbounded sequence of critical values.

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

  1. (i)

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

  2. (ii)

    s ϱ W 2 (t,sx) 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,uE.
(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>qp+2 and the above inequalities that there exists a constant C 3 >0 such that

u n C 3 ,nN.
(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 tR. By (W5), we know that

| W ( t , x ) | a ( t ) | x | q p + 1 0as x0,

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 , p<qp+2 and u n (t) u 0 (t) for almost every tR, 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. tR

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 dt=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 ) (xy) C 5 | x y | α + 1 ,x,yR.
(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 γp 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 dt R a(t) | u 0 ( t ) | q p + 2 dtas 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 uσ,

which shows that |u(t)|σδ<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 uE with u=ρ.

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

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)dt, C 8 = 2 2 max | x | 1 W 2 (t,x)dt. Take ωE 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 ) ) dt s μ 1 1 W 1 ( t , ω ( t ) ) dt= C 9 s μ ,
(3.20)

where C 9 = 1 1 W 1 (t,ω(t))dt>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>qp+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 eE, 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 tR,|x|δ.
(3.23)

By (3.23), we have

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

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

u C 2 uσ,

which shows that |u(t)|σδ<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 uE 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

uc 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 tR.
(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 uΘ, 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 tR and x R N {0}, and W 1 C 1 (R× R N ,R), it follows that γ>0. For any uE, 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)dt, C 11 = R 2 R 2 max | x | 1 W 2 (t,x)dt. 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 uΘ 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 uc for uΘ, then

| u ( t ) | u C 2 c,tR,uΘ.
(3.37)

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

R 2 R 2 W 1 ( t , u ( t ) ) dt t 0 ε t 0 + ε W 1 ( t , u ( t ) ) dt2εγfor uΘ.
(3.38)

From (3.29) and (3.30), we have

| u ( t ) | i = 1 m | λ i | | u i ( t ) | δfor |t| R 1 ,uΘ.
(3.39)

By (2.1), (3.15), (3.34), (3.38), (3.39) and Lemma 2.5, we have for uΘ 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>qp+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(ru)<0for uΘ and r r 0 .

It follows that

φ(u)<0for u E  and uc 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 nN.
(3.41)

By (3.2) and (3.41), we get

| u n ( t ) | C 2 C 12 for nN.
(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 ) | dt<+.

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. tR,
(4.1)

where p=3, q=6, tR, u R N , aC(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. tR,
(4.2)

where p=3/2, q=5/2, tR, u R N , aC(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.

References

  1. Poincaré H: Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris; 1897–1899.

    Google Scholar 

  2. Alves CO, Carriao PC, Miyagaki OH: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. Lett. 2003, 16(5):639–642. 10.1016/S0893-9659(03)00059-4

    Article  MathSciNet  Google Scholar 

  3. Carriao PC, Miyagaki OH: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 1999, 230(1):157–172. 10.1006/jmaa.1998.6184

    Article  MathSciNet  Google Scholar 

  4. Coti ZV, Rabinowitz PH: Homoclinic orbits for second-order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 1991, 4(4):693–727.

    Article  Google Scholar 

  5. Chen CN, Tzeng SY: Existence and multiplicity results for homoclinic orbits of Hamiltonian systems. Electron. J. Differ. Equ. 1997., 1997: Article ID 7

    Google Scholar 

  6. Ding YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 1995, 25(11):1095–1113. 10.1016/0362-546X(94)00229-B

    Article  MathSciNet  Google Scholar 

  7. Izydorek M, Janczewska J: Homoclinic solutions for a class of the second-order Hamiltonian systems. J. Differ. Equ. 2005, 219(2):375–389. 10.1016/j.jde.2005.06.029

    Article  MathSciNet  Google Scholar 

  8. Korman P, Lazer AC: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electron. J. Differ. Equ. 1994., 1994: Article ID 1

    Google Scholar 

  9. Korman P, Lazer AC, Li Y: On homoclinic and heteroclinic orbits for Hamiltonian systems. Differ. Integral Equ. 1997, 10(2):357–368.

    MathSciNet  Google Scholar 

  10. Lu YF, Li CY, Zhong SZ, Zhang WJ: Homoclinic orbits for a class of Hamiltonian systems with potentials changing sign. Ann. Differ. Equ. 2005, 21(3):370–372.

    MathSciNet  Google Scholar 

  11. Lv X, Lu SP, Yan P: Existence of homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal. 2010, 72(1):390–398. 10.1016/j.na.2009.06.073

    Article  MathSciNet  Google Scholar 

  12. Omana W, Willem M: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 1992, 5(5):1115–1120.

    MathSciNet  Google Scholar 

  13. Rabinowitz PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. A 1990, 114(1–2):33–38. 10.1017/S0308210500024240

    Article  MathSciNet  Google Scholar 

  14. Rabinowitz PH, Tanaka K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 1991, 206(3):473–499.

    Article  MathSciNet  Google Scholar 

  15. Tang XH, Xiao L: Homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal. 2009, 71(3–4):1140–1152. 10.1016/j.na.2008.11.038

    Article  MathSciNet  Google Scholar 

  16. Tang XH, Xiao L: Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 2009, 351(2):586–594. 10.1016/j.jmaa.2008.10.038

    Article  MathSciNet  Google Scholar 

  17. Tang XH, Xiao L: Homoclinic solutions for ordinary p -Laplacian systems with a coercive potential. Nonlinear Anal. 2009, 71(3–4):1124–1132. 10.1016/j.na.2008.11.027

    Article  MathSciNet  Google Scholar 

  18. Tang XH, Lin XY: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb. A 2011, 141: 1103–1119. 10.1017/S0308210509001346

    Article  MathSciNet  Google Scholar 

  19. Zhang QF, Tang XH: Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems. Math. Nachr. 2012, 285(5–6):778–789. 10.1002/mana.201000096

    Article  MathSciNet  Google Scholar 

  20. Zhang XY, Tang XH: A note on the minimal periodic solutions of nonconvex superlinear Hamiltonian system. Appl. Math. Comput. 2013, 219: 7586–7590. 10.1016/j.amc.2013.01.044

    Article  MathSciNet  Google Scholar 

  21. Chen P, Tang XH: New existence of homoclinic orbits for a second-order Hamiltonian system. Comput. Math. Appl. 2011, 62(1):131–141. 10.1016/j.camwa.2011.04.060

    Article  MathSciNet  Google Scholar 

  22. Salvatore A: On the existence of homoclinic orbits for a second-order Hamiltonian system. Differ. Integral Equ. 1997, 10(2):381–392.

    MathSciNet  Google Scholar 

  23. Salvatore A: Homoclinic orbits for a class of strictly convex Hamiltonian systems. Dyn. Syst. Appl. 1997, 6: 153–164.

    MathSciNet  Google Scholar 

  24. Benci V, Fortunato D: Weighted Sobolev space and the nonlinear Dirichlet problem in unbounded domains. Ann. Mat. Pura Appl. 1979, 121: 319–336. 10.1007/BF02412010

    Article  MathSciNet  Google Scholar 

  25. Royden HL: Real Analysis. 2nd edition. Macmillan Co., New York; 1968.

    Google Scholar 

  26. Rabinowitz PH CBMS Regional Conf. Ser. in Math. 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.

    Google Scholar 

  27. Adams RA, Fournier JJF: Sobolev Spaces. 2nd edition. Academic Press, Amsterdam; 2003.

    Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Qiongfen Zhang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Shi, X., Zhang, Q. & Zhang, QM. Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space. Bound Value Probl 2013, 137 (2013). https://doi.org/10.1186/1687-2770-2013-137

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-137

Keywords