SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Open Badges Research

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

Qiong Meng

Author Affiliations

School of Mathematical Science, Shanxi University, Taiyuan 030006, Shanxi, P.R. China

Boundary Value Problems 2014, 2014:150  doi:10.1186/s13661-014-0150-2

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/150

Received:17 December 2013
Accepted:3 June 2014
Published:11 July 2014

© 2014 Meng; licensee Springer

Open Access 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 credited.


We study the p-Hamiltonian systems , . Three periodic solutions are obtained by using a three critical points theorem.

p-Hamiltonian systems; three periodic solutions; three critical points theorem


Consider the p-Hamiltonian systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M1">View MathML</a>

where , , , is a function such that is continuous in for all and is a -function in for almost every , and is measurable in and . is symmetric, , and there exists a positive constant such that for all and , that is, is positive definite for all .

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M2">View MathML</a>

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]]

LetXbe a separable and reflexive real Banach space, and letbe two continuously Gâteaux differentiable functionals. Assume thatψis sequentially weakly lower semicontinuous and even thatϕis sequentially weakly continuous and odd, and that, for someand for each, the functionalsatisfies the Palais-Smale condition and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M3">View MathML</a>

Finally, assume that there existssuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M4">View MathML</a>

Then, for every, there exist an open intervaland a positive real number σ, such that for every, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M5">View MathML</a>

admits at least three solutions whose norms are smaller thanσ.

Proofs of theorems

First, we give some notations and definitions. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M6">View MathML</a>

and is endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M7">View MathML</a>

Let be defined by the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M8">View MathML</a>

where , .

Then and one can check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M9">View MathML</a>

for all . It is well known that the T-periodic solutions of problem (1.1) correspond to the critical points of .

As is positive definite for all , we have Lemma 2.1.

Lemma 2.1

For each,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M10">View MathML</a>


Theorem 2.1

Suppose thatFandGsatisfy the following conditions:

(H1) , for a.e. ;

(H2) , for a.e. ;

(H3) , for a.e. ;

(H4) , , a.e. , for someand;

(H5) is even andis odd for a.e. .

Then, for every, there exist an open intervaland a positive real number σ, such that for every, problem (1.1) admits at least three solutions whose norms are smaller thanσ.


By (H1) and (H2), given , we may find a constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M11">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M12">View MathML</a>

and so the functional is continuously Gâteaux differentiable functional and sequentially weakly continuous in the space . Also, by (H4), we know is sequentially weakly continuous. According to (H4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M13">View MathML</a>

For , from the inequality (2.5) and (2.6), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M14">View MathML</a>

Since , ε small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M15">View MathML</a>


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

Suppose is a (PS) sequence of , that is, there exists such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M16">View MathML</a>

Assume that . By (2.7), which contradicts . Thus is bounded. We may assume that there exists satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M17">View MathML</a>

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M18">View MathML</a>

We already know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M19">View MathML</a>

By (2.4) and (H4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M20">View MathML</a>

Using this, (2.8), and (2.9) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M21">View MathML</a>

This together with the weak convergence of in implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M22">View MathML</a>

Hence, satisfies the (PS) condition. Next, we want to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M23">View MathML</a>


Owing to the assumption (H3), we can find , for , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M24">View MathML</a>

We choose a function , put , and we take small. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M25">View MathML</a>

Thus (2.10) holds.

From (H2), , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M26">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M27">View MathML</a>

Choose , one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M28">View MathML</a>

Hence, there exists such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M29">View MathML</a>

So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M30">View MathML</a>

The condition (H5) implies ψ is even and ϕ is odd. All the assumptions of Theorem A are verified. Thus, for every there exist an open interval and a positive real number σ, such that for every , problem (1.1) admits at least three weak solutions in whose norms are smaller than σ. □

Theorem 2.2

IfFandGsatisfy assumptions (H1)-(H2), (H4)-(H5), and the following condition (H3′):

(H3′): there is a constant, , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M31">View MathML</a>


Then, for every, there exist an open intervaland a positive real number σ, such that for every, problem (1.1) admits at least three solutions whose norms are smaller thanσ.


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 , ψ is even and ϕ is odd. For every , the functional satisfies the (PS) condition and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M32">View MathML</a>

To this end, we choose a function with . By condition (H3), a simple calculation shows that, as ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M33">View MathML</a>

Then (2.11) implies that for large enough. So, we choose large enough, , let , such that . Thus, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M34">View MathML</a>


By the proof of Theorem 2.1 we know that there exists , such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/150/mathml/M35">View MathML</a>

According to Theorem A, for every there exist an open interval and a positive real number σ, such that for every , problem (1.1) admits at least three weak solutions in whose norms are smaller than σ. □

Competing interests

The author declares that they have no competing interests.


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


  1. Bonanno, G: Some remarks on a three critical points theorem. Nonlinear Anal.. 54, 651–665 (2003). Publisher Full Text OpenURL

  2. Afrouzi, GA, Heidarkhani, S: Three solutions for a Dirichlet boundary value problem involving the p-Laplacian. Nonlinear Anal.. 66, 2281–2288 (2007). Publisher Full Text OpenURL

  3. Ricceri, B: On a three critical points theorem. Arch. Math.. 75, 220–226 (2000). Publisher Full Text OpenURL

  4. Ricceri, B: A three critical points theorem revisited. Nonlinear Anal.. 70, 3084–3089 (2009). Publisher Full Text OpenURL

  5. Li, C, Ou, Z-Q, Tang, C: Three periodic solutions for p-Hamiltonian systems. Nonlinear Anal.. 74, 1596–1606 (2011). Publisher Full Text OpenURL

  6. Averna, D, Bonanno, G: A three critical point theorems and its applications to the ordinary Dirichlet problem. Topol. Methods Nonlinear Anal.. 22, 93–103 (2003)

  7. Shang, X, Zhang, J: Three solutions for a perturbed Dirichlet boundary value problem involving the p-Laplacian. Nonlinear Anal.. 72, 1417–1422 (2010). Publisher Full Text OpenURL