Abstract
By using the least action principle and the minimax methods, the existence of periodic solutions for a class of second order Hamiltonian systems is considered. The results obtained in this paper extend some previous results.
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1 Introduction and main results
Consider the second order Hamiltonian system
where \(T>0\) and \(F:[0,T]\times\mathbb{R}^{N}\rightarrow\mathbb{R}\) satisfies the following assumption:
(A) \(F(t,x)\) is measurable in t for every \(x\in\mathbb{R}^{N}\), continuously differentiable in x for a.e. \(t\in[0,T]\), and there exist \(a\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(b\in L^{1}([0,T];\mathbb {R}^{+})\) such that
for all \(x\in\mathbb{R}^{N}\) and a.e. \(t\in[0,T]\).
The corresponding functional \(\varphi:H_{T}^{1}\rightarrow\mathbb {R}\),
is continuously differentiable and weakly lower semi-continuous on \(H_{T}^{1}\), where \(H_{T}^{1}\) is the usual Sobolev space with the norm
for \(u\in H^{1}_{T}\), and
for all \(u, v\in H^{1}_{T}\), It is well known that the solutions of problem (1.1) correspond to the critical points of φ.
The existence of periodic solutions for problem (1.1) is obtained in [1–22] with many solvability conditions by using the least action principle and the minimax methods, such as the coercive type potential condition (see [2]), the convex type potential condition (see [5]), the periodic type potential conditions (see [16]), the even type potential condition (see [4]), the subquadratic potential condition in Rabinowitz’s sense (see [9]), the bounded nonlinearity condition (see [6]), the subadditive condition (see [11]), the sublinear nonlinearity condition (see [3, 13]), and the linear nonlinearity condition (see [7, 15, 19, 20]).
In particular, when the nonlinearity \(\triangledown F(t,x)\) is bounded, that is, there exists \(g(t)\in L^{1}([0,T],\mathbb{R}^{+})\) such that \(\vert \triangledown F(t,x)\vert \leq g(t)\) for all \(x\in\mathbb{R}^{N}\) and a.e. \(t\in[0,T]\), and that
Mawhin and Willem [6] proved that problem (1.1) has at least one periodic solution.
In [3, 13], Han and Tang generalized these results to the sublinear case:
and
where \(f(t),g(t)\in L^{1}([0,T],\mathbb{R}^{+})\) and \(\alpha\in[0,1)\).
Subsequently, when \(\alpha=1\) Zhao and Wu [19, 20] and Meng and Tang [7, 15] also proved the existence of periodic solutions for problem (1.1), i.e. \(\nabla F(t,x)\) was linear:
where \(f(t),g(t)\in L^{1}([0,T],\mathbb{R}^{+})\).
Recently, Wang and Zhang [21] used a control function \(h(\vert x\vert )\) instead of \(\vert x\vert ^{\alpha}\) in (1.2) and (1.3) and got some new results, where h satisfied the following conditions:
(B) \(h\in C([0,\infty),[0,\infty))\) and there exist constants \(C_{0}>0\), \(K_{1}>0\), \(K_{2}>0\), \(\alpha\in[0,1)\) such that
-
(i)
\(h(s)\leq h(t) \) \(\forall s\leq t\), \(s, t\in[0,\infty)\),
-
(ii)
\(h(s+t)\leq C_{0}(h(s)+h(t))\) \(\forall s, t\in[0,\infty)\),
-
(iii)
\(0\leq h(s)\leq K_{1}s^{\alpha}+K_{2}\) \(\forall s\in[0,\infty )\),
-
(iv)
\(h(s)\rightarrow\infty \) as \(s\rightarrow\infty\).
Motivated by the results mentioned above, we will consider the periodic solutions for problem (1.1). The following are our main results.
Theorem 1.1
Suppose that \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy assumption (A) and the following conditions:
-
(1)
there exist \(f,g\in L^{1}([0,T];\mathbb{R}^{+})\) such that
$$\bigl\vert \nabla F_{1}(t,x)\bigr\vert \leq f(t)h\bigl( \vert x\vert \bigr)+g(t), $$for all \(x\in\mathbb{R}^{N}\) and a.e. \(t\in[0,T]\), here h satisfies (B);
-
(2)
there exist constants \(r>0\) and \(\gamma\in[0,2)\) such that
$$\bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y \bigr) \geq-r\vert x-y\vert ^{\gamma}, $$for all \(x,y \in\mathbb{R}^{N}\);
-
(3)
$$\liminf_{\vert x\vert \rightarrow\infty}h^{-2}\bigl(\vert x\vert \bigr)\int_{0}^{T}F(t,x)\,dt> \frac {T^{2}C_{0}^{2}}{8\pi^{2}}\int _{0}^{T}f^{2}(t)\,dt. $$
Then problem (1.1) has at least one periodic solution which minimizes φ on \(H_{T}^{1}\).
Theorem 1.2
Suppose that \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy assumption (A), (1), (2), and the following conditions:
-
(4)
there exist \(\delta\in[0,2)\) and \(\mu>0\) such that
$$\bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y \bigr) \leq \mu \vert x-y\vert ^{\delta}, $$for all \(x,y \in R^{N}\);
-
(5)
$$\limsup_{\vert x\vert \rightarrow\infty}h^{-2}\bigl(\vert x\vert \bigr)\int_{0}^{T}F(t,x)\,dt< -\frac {3T^{2}C_{0}^{2}}{8\pi^{2}} \int_{0}^{T}f^{2}(t)\,dt. $$
Then problem (1.1) has at least one periodic solution which minimizes φ on \(H_{T}^{1}\).
Theorem 1.3
Suppose that \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy assumption (A), (1), and the following conditions:
-
(6)
there exists a constant \(0< r<4\pi^{2}/T^{2}\), such that
$$\bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y \bigr) \geq-r\vert x-y\vert ^{2}, $$for all \(x,y \in R^{N}\);
-
(7)
$$\liminf_{\vert x\vert \rightarrow\infty}h^{-2}\bigl(\vert x\vert \bigr)\int_{0}^{T}F(t,x)\,dt> \frac {T^{2}}{2(4\pi^{2}-rT^{2})}\int _{0}^{T}f^{2}(t)\,dt. $$
Then problem (1.1) has at least one periodic solution which minimizes φ on \(H_{T}^{1}\).
Theorem 1.4
Suppose that \(F=F_{1}+F_{2}\), where \(F_{1}\) and \(F_{2}\) satisfy assumption (A), (1), and the following conditions:
-
(8)
there exist \(k\in L^{1}([0,T];\mathbb{R}^{+})\) and (\(\lambda,\mu \))-subconvex potential \(G: \mathbb{R}^{N}\rightarrow\mathbb{R}\) with \(\lambda>1/2\) and \(0<\mu<2\lambda^{2}\), such that
$$\bigl(\nabla F_{2}(t,x),y \bigr)\geq-k(t)G(x-y), $$for all \(x,y \in\mathbb{R}^{N}\);
-
(9)
$$\begin{aligned}& \limsup_{\vert x\vert \rightarrow\infty} h^{-2}\bigl(\vert x\vert \bigr)\int_{0}^{T}F_{1}(t,x)\,dt < - \frac{3T^{2}C_{0}^{2}}{8\pi^{2}}\int_{0}^{T}f^{2}(t) \,dt, \\& \limsup_{\vert x\vert \rightarrow\infty} \vert x\vert ^{-\beta}\int _{0}^{T}F_{2}(t,x) \,dt \leq-8\mu\max _{\vert s\vert \leq1}G(s)\int_{0}^{T}k(t) \,dt, \end{aligned}$$
where \(\beta=\log_{2\lambda}(2\mu)\).
Then problem (1.1) has at least one periodic solution which minimizes φ on \(H_{T}^{1}\).
Remark 1.5
Theorems 1.1-1.4 extend some existing results: (i) [22], Theorems 1.1-1.4, are special cases of Theorems 1.1-1.4 with control function \(h(t)=t^{\alpha}, \alpha\in[0,1)\), \(t\in[0,+\infty)\); (ii) if \(F_{2}=0\), [15], Theorems 1 and 2, are special cases of Theorem 1.1 and Theorem 1.2, respectively; (iii) If \(F_{2}=0\), Theorem 1.1 and Theorem 1.2 extend [21], Theorems 1.1 and 1.2, since we weaken the so-called Ahmad-Lazer-Paul type conditions with the control function \(h(t)\).
2 Proof of theorems
For \(u\in H_{T}^{1}\), let \(\bar{u}=\frac{1}{T}\int_{0}^{T}\vert \dot {u}(t)\vert \,dt\) and \(\tilde{u}(t)=u(t)-\bar{u}\). Then one has
For the sake of convenience, we denote \(M_{1}=(\int_{0}^{T}f^{2}(t)\,dt)^{1/2}\), \(M_{2}=\int_{0}^{T}f(t)\,dt\), \(M_{3}=\int_{0}^{T}g(t)\,dt\).
Proof of Theorem 1.1
Due to (3), we can choose an \(a_{1}>T^{2}/(4\pi^{2})\) such that
For (B) and the Sobolev inequality, for any \(u\in H_{T}^{1}\) we have
Similarly, from (2) and the Sobolev inequality, for any \(u\in H_{T}^{1} \) we get
for all \(u\in H_{T}^{1}\). So, by (2.1) we get \(\varphi(u)\rightarrow \infty\) as \(\Vert u\Vert \rightarrow\infty\).
Hence, applying the least action principle (see [6], Theorem 1.1 and Corollary 1.1), the proof is complete. □
Proof of Theorem 1.2
Step 1. First, we assert that φ satisfies the (PS) condition. Suppose that \(\{u_{n}\}\) is a (PS) sequence, that is, \(\varphi '(u_{n})\rightarrow0\) as \(n\rightarrow\infty\) and \(\{\varphi(u_{n})\} \) is bounded. For (5), we can choose an \(a_{2}>T^{2}/(4\pi^{2})\) such that
Similar to the proof of Theorem 1.1, we have
and
for all n. Hence we have
for large n. So, by Wirtinger’s inequality we get
where
Note that \(a_{2}>T^{2}/(4\pi^{2})\) implies \(-\infty < C_{1} <0\). Hence, it follows from (2.8) that
and then
where \(0 < C_{2} < +\infty\). Similar to the proof of Theorem 1.1, we have
By (4), we obtain
From the boundedness of \({\varphi(u_{n})}\) and (2.9)-(2.11), we have
for large n. So, by (2.4) we see that \(\vert \bar{u}\vert \) is bounded. Hence \(\{u_{n}\}\) is bounded by (2.9). Arguing as in the proof of Proposition 4.1 of [6], we conclude that the (PS) condition is satisfied.
Step 2. Let \(\tilde{H}_{T}^{1}=\{u\in H_{T}^{1}:\bar{u}=0\}\). We assert that for \(u\in\tilde{H}_{T}^{1}\),
In fact, from (1) and Sobolev’s inequality, we get
for all \(u\in\tilde{H}_{T}^{1}\). It follows from (2) that
So, we get
By Wirtinger’s inequality, \(\Vert u\Vert \rightarrow \infty\) if and only if \(\Vert \dot{u}\Vert _{L^{2}} \rightarrow \infty\) in \(\tilde{H}_{T}^{1}\). Hence (2.12) holds.
Step 3. By (5), we can easily see that\(\int_{0}^{T}F(t,x)\,dt \rightarrow -\infty \) as \(\vert x\vert \rightarrow \infty\) for all \(x\in\mathbb{R}^{N}\). Thus, for all \(u\in (\tilde {H}_{T}^{1})^{\perp} =\mathbb{R}^{N}\),
Now, by saddle point theorem (see, [10], Theorem 4.6), the proof is completed. □
Proof of Theorem 1.3
By (7), we can choose an \(a_{3}>\frac{T^{2}}{4\pi^{2}-rT^{2}}\) such that
By (6) and the Sobolev inequality, we have
By a similar method to that of the proof of Theorem 1.1, we get
for all \(u\in{H}_{T}^{1}\), which implies that \(\varphi(u)\rightarrow\infty\) as \(\Vert u\Vert \rightarrow\infty\) by (2.13), due to the facts that \(r<\frac{4\pi^{2}}{T^{2}}\) and \(\Vert u\Vert \rightarrow\infty\) if and only if \((\vert \bar{u}\vert ^{2}+\Vert \dot{u}\Vert _{L^{2}}^{2})^{1/2}\rightarrow\infty\). So, applying the least action principle, Theorem 1.3 holds. □
Proof of Theorem 1.4
First, we assert that φ satisfies the (PS) condition. Suppose that \(\{u_{n}\}\) satisfies \(\varphi'(u_{n})\rightarrow0\) as \(n\rightarrow\infty\) and \(\{\varphi (u_{n})\}\) is bounded. By (9), we can choose an \(a_{4}>\frac{T^{2}}{4\pi^{2}}\) such that
By the (\(\lambda,\mu\))-subconvexity of \(G(x)\), we have
for all \(x\in\mathbb{R}^{N}\), and a.e. \(t\in[0,T]\), where \(G_{0}=\max_{\vert s\vert \leq1}G(s)\), \(\beta=\log_{2\lambda}(2\mu)<2\) Then
where \(M_{4}=G_{0}\int_{0}^{T}k(t)\,dt\). From (2.5) and (2.16), for large n, we have
So, from (2.7) and (2.17) we have
where
Note that \(-\infty< C_{4}<0\) due to \(a_{4}>\frac{T^{2}}{4\pi^{2}}\), by (2.18), one has
and then
where \(C_{5} >0\). From (8) and (2.15), we have
for all \(u\in H_{T}^{1}\). By the boundedness of \(\{\varphi(u_{n})\}\) and the inequalities (2.19)-(2.21), we get
for large n. The above inequality and (2.14) imply that \(\{\vert \overline {u}\vert \}\) is bounded. Hence \(\{u_{n}\}\) is bounded by (2.19). By using the standard method, the (PS) condition holds.
Since the rest of the proof is similar to that of Theorem 1.2, we omit the details here. □
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The author thanks the referees and the editors for their helpful comments and suggestions. The research was supported by NSFC (11561043).
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The main idea of this paper was proposed by D-BW, D-BW prepared the manuscript initially, and KY performed a part of the steps of the proofs in this research. All authors read and approved the final manuscript.
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Wang, DB., Yang, K. Existence of periodic solutions for a class of second order Hamiltonian systems. Bound Value Probl 2015, 199 (2015). https://doi.org/10.1186/s13661-015-0460-z
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DOI: https://doi.org/10.1186/s13661-015-0460-z