Abstract
We study the p-Hamiltonian systems , . Three periodic solutions are obtained by using a three critical points theorem.
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Introduction
Consider the p-Hamiltonian 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 p-Hamiltonian 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 p-Laplacian by using the following Theorem A. In this paper, we generalize the results in [[7]] on problem (1.1).
Theorem A
Let X be a separable and reflexive real Banach space, and letbe 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 someand for each, the functionalsatisfies the Palais-Smale condition and
Finally, assume that there existssuch that
Then, for every, there exist an open intervaland a positive real number σ, such that for every, the equation
admits at least three solutions whose norms are smaller than σ.
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 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,
where.
Theorem 2.1
Suppose that F and G satisfy 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 than σ.
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 have
Now, 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 that
Owing 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
If F and G satisfy 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 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 , 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 get
By 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. □
References
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Acknowledgements
Supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China.
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Meng, Q. Three periodic solutions for a class of ordinary p-Hamiltonian systems. Bound Value Probl 2014, 150 (2014). https://doi.org/10.1186/s13661-014-0150-2
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DOI: https://doi.org/10.1186/s13661-014-0150-2