Abstract
We study the pHamiltonian systems , . Three periodic solutions are obtained by using a three critical points theorem.
Keywords:
pHamiltonian systems; three periodic solutions; three critical points theoremIntroduction
Consider the pHamiltonian systems
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 pHamiltonian systems
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 pLaplacian by using the following Theorem A. In this paper, we generalize the results in [[7]] on problem (1.1).
Theorem A
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 PalaisSmale condition and
Finally, assume that there existssuch thatThen, for every, there exist an open intervaland a positive real number σ, such that for every, the equationadmits at least three solutions whose norms are smaller thanσ.Proofs of theorems
First, we give some notations and definitions. Let
and is endowed with the norm Let be defined by the energy functional where , .Then and one can check that
for all . It is well known that the Tperiodic 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,
where.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.
Proof
By (H1) and (H2), given , we may find a constant such that
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 For , from the inequality (2.5) and (2.6), we deduce that Since , ε small enough, we haveNow, we prove that satisfies the (PS) condition.
Suppose is a (PS) sequence of , that is, there exists such that
Assume that . By (2.7), which contradicts . Thus is bounded. We may assume that there exists satisfying Observe that We already know that By (2.4) and (H4) we have Using this, (2.8), and (2.9) we obtain This together with the weak convergence of in implies that Hence, satisfies the (PS) condition. Next, we want to prove thatOwing to the assumption (H3), we can find , for , such that
We choose a function , put , and we take small. Then we obtain Thus (2.10) holds.From (H2), , such that
Thus Choose , one has Hence, there exists such that So we have 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
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.
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 , ψ is even and ϕ is odd. For every , the functional satisfies the (PS) condition and
To this end, we choose a function with . By condition (H3), a simple calculation shows that, as , Then (2.11) implies that for large enough. So, we choose large enough, , let , such that . Thus, we getBy the proof of Theorem 2.1 we know that there exists , such that
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.
Acknowledgements
Supported by the Natural Science Foundation of Shanxi Province (No. 20120110041) of China.
References

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

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

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

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

Li, C, Ou, ZQ, Tang, C: Three periodic solutions for pHamiltonian systems. Nonlinear Anal.. 74, 1596–1606 (2011). Publisher Full Text

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)

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