Open Access

Three periodic solutions for a class of ordinary p-Hamiltonian systems

Boundary Value Problems20142014:150

DOI: 10.1186/s13661-014-0150-2

Received: 17 December 2013

Accepted: 3 June 2014

Published: 11 July 2014

Abstract

We study the p-Hamiltonian systems ( | u | p 2 u ) + A ( t ) | u | p 2 u = F ( t , u ) + λ G ( t , u ) , u ( 0 ) u ( T ) = u ( 0 ) u ( T ) = 0 . Three periodic solutions are obtained by using a three critical points theorem.

p-Hamiltonian systems three periodic solutions three critical points theorem
34K13 34B15 58E30

Introduction

Consider the p-Hamiltonian systems
{ ( | u | p 2 u ) + A ( t ) | u | p 2 u = F ( t , u ) + λ G ( t , u ) , u ( 0 ) u ( T ) = u ( 0 ) u ( T ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ1_HTML.gif
(1)
where p > 1 , T > 0 , λ ( , + ) , F : [ 0 , T ] × R N R is a function such that F ( , x ) is continuous in [ 0 , T ] for all x R N and F ( , x ) is a C 1 -function in R N for almost every t [ 0 , T ] , and G : [ 0 , T ] × R N R is measurable in [ 0 , T ] and C 1 R N . A = ( a i j ( t ) ) N × N is symmetric, A C ( [ 0 , T ] , R N × N ) , and there exists a positive constant λ 1 such that ( A ( t ) | x | p 2 x , x ) λ 1 p | x | p for all x R N and t [ 0 , T ] , that is, A ( t ) is positive definite for all t [ 0 , T ] .

In recent years, the three critical points theorem of Ricceri [[1]] has widely been used to solve differential equations; see [[2]–[4]] and references therein.

In [[5]], Li et al. have studied the three periodic solutions for p-Hamiltonian systems
{ ( | u | p 2 u ) + A ( t ) | u | p 2 u = λ F ( t , u ) + μ G ( t , u ) , u ( 0 ) u ( T ) = u ( 0 ) u ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ2_HTML.gif
(2)
Their technical approach is based on two general three critical points theorems obtained by Averna and Bonanno [[6]] and Ricceri [[4]].

In [[7]], Shang and Zhang obtained three solutions for a perturbed Dirichlet boundary value problem involving the p-Laplacian by using the following Theorem A. In this paper, we generalize the results in [[7]] on problem (1.1).

Theorem A

[[1], [7]]

Let X be a separable and reflexive real Banach space, and let ϕ , ψ : X R be two continuously Gâteaux differentiable functionals. Assume that ψ is sequentially weakly lower semicontinuous and even that ϕ is sequentially weakly continuous and odd, and that, for some b > 0 and for each λ [ b , b ] , the functional ψ + λ ϕ satisfies the Palais-Smale condition and
lim x [ ψ ( x ) + λ ϕ ( x ) ] = + . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ3_HTML.gif
(3)
Finally, assume that there exists k > 0 such that
inf x X ψ ( x ) < inf | ϕ ( x ) | < k ψ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ4_HTML.gif
(4)
Then, for every b > 0 , there exist an open interval Λ [ b , b ] and a positive real number σ, such that for every λ Λ , the equation
ψ ( x ) + λ ϕ ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ5_HTML.gif
(5)
admits at least three solutions whose norms are smaller than σ.

Proofs of theorems

First, we give some notations and definitions. Let
W T 1 , p = { u : [ 0 , T ] R N u  is absolutely continuous , u ( 0 ) = u ( T ) , u L p ( 0 , T ; R N ) } https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ6_HTML.gif
(6)
and is endowed with the norm
u = ( 0 T | u ( t ) | p d t + 0 T ( A ( t ) | u ( t ) | p 2 u ( t ) , u ( t ) ) d t ) 1 p . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ7_HTML.gif
(7)
Let φ λ : W T 1 , p R be defined by the energy functional
φ λ ( u ) = ψ ( u ) + λ ϕ ( u ) , https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ8_HTML.gif
(8)
where ψ ( u ) = 1 p u p 0 T F ( t , u ( t ) ) d t , ϕ ( u ) = 0 T G ( t , u ( t ) ) d t .
Then φ λ C ( W T 1 , p , R ) and one can check that
φ λ ( u ) , v = 0 T [ ( | u ( t ) | p 2 u ( t ) , v ( t ) ) ( F ( t , u ( t ) ) , v ( t ) ) λ ( G ( t , u ( t ) ) , v ( t ) ) ] d t , https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ9_HTML.gif
(9)
for all u , v W T 1 , p . It is well known that the T-periodic solutions of problem (1.1) correspond to the critical points of φ λ .

As A ( t ) is positive definite for all t [ 0 , T ] , we have Lemma 2.1.

Lemma 2.1

For each u W T 1 , p ,
λ 1 u L p u , https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ10_HTML.gif
(10)
where u L p = 0 T | u ( t ) | p d t .

Theorem 2.1

Suppose that F and G satisfy the following conditions:

(H1) lim | x | | F ( t , x ) | | x | p 1 = 0 , for a.e. t [ 0 , T ] ;

(H2) lim | x | 0 | F ( t , x ) | | x | p 1 = 0 , for a.e. t [ 0 , T ] ;

(H3) lim | x | 0 F ( t , x ) | x | p = , for a.e. t [ 0 , T ] ;

(H4) | G ( t , x ) | c ( 1 + | x | q 1 ) , x R N , a.e. t [ 0 , T ] , for some c > 0 and 1 q < p ;

(H5) F ( t , ) is even and G ( t , ) is odd for a.e. t [ 0 , T ] .

Then, for every b > 0 , there exist an open interval Λ [ b , b ] and a positive real number σ, such that for every λ Λ , problem (1.1) admits at least three solutions whose norms are smaller than σ.

Proof

By (H1) and (H2), given ε > 0 , we may find a constant C ε > 0 such that
| F ( t , x ) | C ε + ε | x | p 1 , for every  x R N ,  a.e.  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ11_HTML.gif
(11)
| F ( t , x ) | C ε + ε p | x | p , for every  x R N ,  a.e.  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ12_HTML.gif
(12)
and so the functional ψ ( u ) is continuously Gâteaux differentiable functional and sequentially weakly continuous in the space W T 1 , p . Also, by (H4), we know ϕ ( u ) is sequentially weakly continuous. According to (H4), we get
| G ( t , x ) | c | x | + c p | x | q , for every  x R N ,  a.e.  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ13_HTML.gif
(13)
For λ R , from the inequality (2.5) and (2.6), we deduce that
ψ ( u ) + λ ϕ ( u ) 1 p u p 0 T ( C ε + ε p | u ( t ) | p ) d t λ 0 T ( c | u ( t ) | + c q | u ( t ) | q ) d t 1 p ( 1 ε λ 1 ) u p c λ q λ 1 T p q q u q c λ λ 1 T p 1 p u ε T . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ14_HTML.gif
(14)
Since p > q , ε small enough, we have
lim u [ ψ ( u ) + λ ϕ ( u ) ] = + . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ15_HTML.gif
(15)

Now, we prove that φ λ satisfies the (PS) condition.

Suppose { u n } is a (PS) sequence of φ λ , that is, there exists C > 0 such that
φ λ ( u n ) C , φ λ ( u n ) 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ16_HTML.gif
(16)
Assume that u n . By (2.7), which contradicts φ λ ( u n ) C . Thus { u n } is bounded. We may assume that there exists u 0 W T 1 , p satisfying
u n u 0 , weakly in  W T 1 , p , u n u 0 , strongly in  L p [ 0 , T ] , u n ( x ) u 0 ( x ) , a.e.  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ17_HTML.gif
(17)
Observe that
φ λ ( u n ) , u n u 0 = 0 T [ ( | u n ( t ) | p 2 u n ( t ) , u n ( t ) u 0 ( t ) ) + ( A ( t ) | u n ( t ) | p 2 u n ( t ) , u n ( t ) u 0 ( t ) ) ] d t 0 T ( ( F ( t , u n ( t ) ) , u n ( t ) u 0 ( t ) ) ) d t λ 0 T ( G ( t , u n ( t ) ) , u n ( t ) u 0 ( t ) ) d t . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ18_HTML.gif
(18)
We already know that
φ λ ( u n ) , u n u 0 0 , as  n . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ19_HTML.gif
(19)
By (2.4) and (H4) we have
0 T ( F ( t , u n ( t ) ) , u n ( t ) u 0 ( t ) ) d t 0 , as  n , 0 T ( G ( t , u n ( t ) ) , u n ( t ) u 0 ( t ) ) d t 0 , as  n . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ20_HTML.gif
(20)
Using this, (2.8), and (2.9) we obtain
0 T [ ( | u n ( t ) | p 2 u n ( t ) , u n ( t ) u 0 ( t ) ) + ( A ( t ) | u n ( t ) | p 2 u n ( t ) , u n ( t ) u 0 ( t ) ) ] d t 0 , as  n . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ21_HTML.gif
(21)
This together with the weak convergence of u n u 0 in W T 1 , p implies that
u n u 0 , strongly in  W T 1 , p . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ22_HTML.gif
(22)
Hence, φ λ satisfies the (PS) condition. Next, we want to prove that
inf u W T 1 , p ψ ( u ) < 0 . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ23_HTML.gif
(23)
Owing to the assumption (H3), we can find δ > 0 , for L > 0 , such that
| F ( t , x ) | > L | x | , for  0 < | x | δ ,  and a.e.  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ24_HTML.gif
(24)
We choose a function 0 v C 0 ( [ 0 , T ] ) , put L > v p / ( p 0 T | v | p d t ) , and we take ε > 0 small. Then we obtain
ψ ( ε v ) = 1 p ε v p 0 T F ( t , ε v ( t ) ) d t ε p p v p L ε p 0 T | v ( t ) | p d t < 0 . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ25_HTML.gif
(25)
Thus (2.10) holds.
From (H2), ε > 0 , ρ 0 ( ε ) > 0 such that
| F ( t , x ) | ε | x | p 1 , if  0 < ρ = | x | < ρ 0 ( ε ) . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ26_HTML.gif
(26)
Thus
0 T F ( t , u ( t ) ) d t ε p 0 T | u ( t ) | p d t ε p λ 1 u p . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ27_HTML.gif
(27)
Choose ε = λ 1 / 2 , one has
ψ ( u ) = 1 p u p ε p λ 1 u p = 1 2 p u p > 0 . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ28_HTML.gif
(28)
Hence, there exists k > 0 such that
inf | ϕ ( u ) | < k ψ ( u ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ29_HTML.gif
(29)
So we have
inf u W T 1 , p ψ ( u ) < inf | ϕ ( u ) | < k ψ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ30_HTML.gif
(30)
The condition (H5) implies ψ is even and ϕ is odd. All the assumptions of Theorem A are verified. Thus, for every b > 0 there exist an open interval Λ [ b , b ] and a positive real number σ, such that for every λ Λ , problem (1.1) admits at least three weak solutions in W T 1 , p whose norms are smaller than σ. □

Theorem 2.2

If F and G satisfy assumptions (H1)-(H2), (H4)-(H5), and the following condition (H3′):

(H3′): there is a constant B 1 = sup { 1 / 0 T | u ( t ) | p d t : u = 1 } , B 2 0 , such that
F ( t , x ) 2 B 1 | x | p p B 2 , for  x R N ,  a.e.  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ31_HTML.gif
(31)

Then, for every b > 0 , there exist an open interval Λ [ b , b ] and a positive real number σ, such that for every λ Λ , problem (1.1) admits at least three solutions whose norms are smaller than σ.

Proof

The proof is similar to the one of Theorem 2.1. So we give only a sketch of it. By the proof of Theorem 2.1, the functional ψ and ϕ are sequentially weakly lower semicontinuous and continuously Gâteaux differentiable in W T 1 , p , ψ is even and ϕ is odd. For every λ R , the functional ψ + λ ϕ satisfies the (PS) condition and
lim u ( ψ + λ ϕ ) = + . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ32_HTML.gif
(32)
To this end, we choose a function v W T 1 , p with v = 1 . By condition (H3), a simple calculation shows that, as s ,
ψ ( s v ) = 1 p s v p 0 T F ( t , s v ( t ) ) d t s p p v p 2 s p B 1 p 0 T | v ( t ) | p d t + B 2 T s p p + B 2 T . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ33_HTML.gif
(33)
Then (2.11) implies that ψ ( s v ) < 0 for s > 0 large enough. So, we choose large enough, s 0 > 0 , let u 1 = s 0 v , such that ψ ( u 1 ) < 0 . Thus, we get
inf u W T 1 , p ψ ( u ) < 0 . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ34_HTML.gif
(34)
By the proof of Theorem 2.1 we know that there exists k > 0 , such that
inf u W T 1 , p ψ ( u ) < inf | ϕ ( u ) | < k ψ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2Fs13661-014-0150-2/MediaObjects/13661_2014_Article_150_Equ35_HTML.gif
(35)
According to Theorem A, for every b > 0 there exist an open interval Λ [ b , b ] and a positive real number σ, such that for every λ Λ , problem (1.1) admits at least three weak solutions in W T 1 , p whose norms are smaller than σ. □

Declarations

Acknowledgements

Supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China.

Authors’ Affiliations

(1)
School of Mathematical Science, Shanxi University

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Copyright

© Meng; licensee Springer 2014

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