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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2014 | Research

New periodic solutions of singular Hamiltonian systems with fixed energies

verfasst von: Fengying Li, Qingqing Hua, Shiqing Zhang

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2014

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Abstract

By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second-order Hamiltonian systems with a singular potential

V \in C^{2} (R^{n} ∖ O, R)

and

V \in C^{1} (R^{2} ∖ O, R)

, which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as complements of the well-known theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.

MSC:35A15, 47J30.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The research and writing of this manuscript was a collaborative effort made by all the authors. All authors read and approved the final manuscript.

1 Introduction

Seifert [1] in 1948 and Rabinowitz [2, 3] in 1978 and 1979 studied classical second-order Hamiltonian systems without singularity, based on their work, Benci [4, 5] and Gluck and Ziller [6] and Hayashi [7] used a Jacobi metric and very complicated geodesic methods and algebraic topology to study the periodic solutions with a fixed energy of the following system:

\ddot{q} + V^{'} (q) = O,

(1.1)

\frac{1}{2} {| \dot{q} |}^{2} + V (q) = h .

(1.2)

They proved a very general theorem.

Theorem 1.1 Suppose

V \in C^{2} (R^{n}, R)

, if

{x \in R^{n} | V (x) \leq h}

is bounded and non-empty, then (1.1)-(1.2) has a periodic solution with energy h.

Furthermore, if

V^{'} (x) \neq O, \forall x \in {x \in R^{n} | V (x) = h},

then (1.1)-(1.2) has a nonconstant periodic solution with energy h.

For the existence of multiple periodic solutions for (1.1)-(1.2) with compact energy surfaces, we can refer to Groessen [8] and Long [9] and the references therein.

In the 1987 paper of Ambrosetti and Coti Zelati [10], Clark-Ekeland’s dual action principle, Ambrosetti-Rabinowitz’s mountain pass theorem etc. were used to study the existence of T-periodic solutions of the second-order equation

- \ddot{x} = \nabla U (x),

where

U = V \in C^{2} (Ω; R)

is such that

U (x) \to \infty, x \to Γ = \partial Ω;

here

Ω \subset R^{n}

is a bounded and convex domain, and they got the following result.

Theorem 1.2 Suppose that

(i)

U (O) = 0 = min U

;

(ii)

U (x) \leq θ (x, \nabla U (x))

for some

θ \in (0, \frac{1}{2})

and for all x near Γ (superquadraticity near Γ);

(iii)

(U^{″} (x) y, y) \geq k {| y |}^{2}

for some

k > 0

and for all

(x, y) \in Ω \times R^{N}

Let

ω_{N}

be the greatest eigenvalue of

U^{″} (0)

and

T_{0} = {(2 / ω_{N})}^{1 / 2}

. Then

- \ddot{x} = \nabla U (x)

has for each

T \in (0, T_{0})

a periodic solution with minimal period T.

For

C^{r}

systems, a natural interesting problem is if

{x \in R^{n} | V (x) \leq h}

is unbounded: can we get a nonconstant periodic solution for the system (1.1)-(1.2)?

In 1987, Offin [11] firstly generalized Theorem 1.1 to some non-compact cases under

V \in C^{3} (R^{n}, R)

and complicated geometrical assumptions on potential wells, but it seems to be difficult to verify this for concrete potentials under the geometrical conditions.

In 1988, Rabinowitz [12] studied multiple periodic solutions for classical Hamiltonian systems with potential

V \in C^{1} (R \times R^{n}, R)

, where

V (q_{1}, \dots, q_{n}; t)

T_{i}

-periodic in positions

q_{i} \in R

and is T-periodic in t.

In 1990, using Clark-Ekeland’s dual variational principle and Ambrosetti-Rabinowitz’s mountain pass lemma, Coti Zelati et al. [13] studied Hamiltonian systems with convex potential wells, they got the following result.

Theorem 1.3 Let Ω be a convex open subset of

R^{n}

containing the origin O. Let

V \in C^{2} (Ω, R)

be such that

(V1)

V (q) \geq V (O) = 0

\forall q \in Ω

(V2)

\forall q \neq O

V^{″} (q) > 0

(V3)

\exists ω > 0

, s.t.

V (q) \leq \frac{ω}{2} {∥ q ∥}^{2}

\forall ∥ q ∥ < ϵ

(V4)

V^{″} {(q)}^{- 1} \to 0

∥ q ∥ \to 0

, or

(V4)′

V^{″} {(q)}^{- 1} \to 0

q \to \partial Ω

Then, for every

T < \frac{2 π}{\sqrt{ω}}

, (1.1) has a solution with minimal period T.

In Theorems 1.2 and 1.3, the authors assumed the convex conditions for potentials and potential wells so that they can apply Clark-Ekeland’s dual variational principle; we notice that Theorems 1.1-1.3 essentially made the following assumption:

V (x) \to \infty, x \to Γ = \partial Ω .

So all the potential wells are bounded.

For singular Hamiltonian systems with a fixed energy

h \in R

, Ambrosetti and Coti Zelati in [14, 15] used Ljusternik-Schnirelmann theory on a

C^{1}

manifold to get the following theorem.

Theorem 1.4 (Ambrosetti and Coti Zelati [14])

Suppose

V \in C^{2} (R^{n} ∖ {O}, R)

satisfies

V (q) \to - \infty

q \to 0

and

(A1) $3 V^{'} (u) \cdot u + (V^{″} (u) u, u) \neq 0$ , $\forall u \neq 0$ ;
(A2) $V^{'} (u) \cdot u > 0$ , $\forall u \neq 0$ ;
(A3) $\exists α > 2$ , s.t. $V^{'} (u) \cdot u \leq - α V (u)$ , $\forall u \neq 0$ ;
(A4) $\exists β > 2$ , $r > 0$ , s.t. $V^{'} (u) \cdot u \geq - β V (u)$ , $0 < | u | < r$ ;
(A5) $V (u) + \frac{1}{2} V^{'} (u) u \leq 0$ , $\forall u \neq 0$ .

Then (1.1)-(1.2) has at least one nonconstant periodic solution.

Besides Ambrosetti-Coti Zelati, many other mathematicians [16‐34] studied singular Hamiltonian systems, here we only mention a related recent paper of Carminati, Sere and Tanaka [16]. They used complex variational and topological methods to generalize Pisani’s results [17], and they got the following theorem.

Theorem 1.5 Suppose

h > 0

L_{0} > 0

and

V \in C^{\infty} (R^{n} ∖ {O}, R)

satisfies

V (q) \to - \infty

q \to 0

and

(B1) $V (q) \leq 0$ , $\forall q \neq 0$ ;
(B2) $V (q) + \frac{1}{2} V^{'} (q) q \leq h$ , $\forall | q | \geq e^{L_{0}}$ ;
(B3) $V (q) + \frac{1}{2} V^{'} (q) q \geq h$ , $\forall | q | \leq e^{- L_{0}}$ ;
(A4) $\exists β > 2$ , $r > 0$ , s.t. $V^{'} (q) \cdot q \geq - β V (q)$ , $0 < | q | < r$ .

Then (1.1)-(1.2) has at least one periodic solution with the given energy h and whose action is at most

2 π r_{0}

with

r_{0} = max {{[2 (h - V (q))]}^{\frac{1}{2}}; | q | = 1} .

Theorem 1.6 Suppose

h > 0

ρ_{0} > 0

, and

V \in C^{\infty} (R^{n} ∖ {O}, R)

satisfies

V (q) \to - \infty

q \to 0

and (B1), (A4) and

(B2)′

{lim}_{| q | \to + \infty} V^{'} (q) = O

;

(B3)′

V (q) + \frac{1}{2} V^{'} (q) q \geq h

\forall | q | \leq ρ_{0}

Then (1.1)-(1.2) has at least one periodic solution with the given energy h whose action is at most

2 π r_{0}

By using the variational minimizing method with a special constraint, we obtain the following result.

Theorem 1.7 Suppose

V \in C^{2} (R^{n} ∖ {O}, R)

and

V (q) \to - \infty

q \to 0

and satisfies (A1)-(A3) and

(A4)′

\exists β > 2

, s.t.

V^{'} (q) \cdot q \geq - β V (q)

0 < | q | < + \infty

;

(A5)′

V (- q) = V (q)

\forall q \neq O

Then for any

h > 0

, (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Using the direct variational minimizing method, we get the following theorem.

Theorem 1.8 Suppose

V \in C^{1} (R^{2} ∖ {O}, R)

and

V (q) \to - \infty

q \to 0

and satisfies

(B1)′

V (q) < h

\forall q \neq O

;

(P1)′

V^{'} (u) \to O

∥ u ∥ \to + \infty

;

(A3)′

\exists α > 2

μ_{2} > 0

, s.t.

V^{'} (u) \cdot u \leq - α V (u) + μ_{2}

\forall u \neq 0

;

(A4)

\exists β > 2

r > 0

, s.t.

V^{'} (u) \cdot u \geq - β V (u)

0 < | u | < r

Then for any

h > \frac{μ_{2}}{α}

, (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Corollary 1.9 Suppose

α = β > 2

and

V (x) = - {| x |}^{- α} .

Then for any

h > 0

, (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Remark In Theorem 1.8, the assumption on regularity for potential V is weaker than Theorems 1.1-1.6. Comparing Theorem 1.5 with Theorem 1.8, our (B1)′ is also weaker than (B1), and (A3)′ is also different from (B2)-(B3) and (B3)′.

2 A few lemmas

Let

H^{1} = W^{1, 2} (R / Z, R^{n}) = {u : R \to R^{n}, u \in L^{2}, \dot{u} \in L^{2}, u (t + 1) = u (t)} .

Then the standard

H^{1}

norm is equivalent to

∥ u ∥ = {∥ u ∥}_{H^{1}} = {(\int_{0}^{1} {| \dot{u} |}^{2} d t)}^{1 / 2} + | u (0) | .

Let

Λ = {u \in H^{1} | u (t) \neq O, \forall t} .

Lemma 2.1 ([14])

Let

F = {u \in H^{1} | \int_{0}^{1} (V (u) + \frac{1}{2} V^{'} (u) u) d t = h} .

If (A1) holds, then F is a

C^{1}

manifold with codimension 1 in

H^{1}

. Let

f (u) = \frac{1}{4} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} V^{'} (u) u d t

and let

\tilde{u} \in F

be such that

f^{'} (\tilde{u}) = O

and

f (\tilde{u}) > 0

. Set

\frac{1}{T^{2}} = \frac{\int_{0}^{1} V^{'} (\tilde{u}) \tilde{u} d t}{\int_{0}^{1} {| \dot{\tilde{u}} |}^{2} d t} .

If (A2) holds, then

\tilde{q} (t) = \tilde{u} (t / T)

is a nonconstant T-periodic solution for (1.1)-(1.2). Moreover, if (A2) holds, then

f (u) \geq 0

on F and

f (u) = 0

u \in F

if and only if u is constant.

Lemma 2.2 ([8, 14])

Let

f (u) = \frac{1}{2} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} (h - V (u)) d t

and

\tilde{u} \in Λ

be such that

f^{'} (\tilde{u}) = O

and

f (\tilde{u}) > 0

. Set

\frac{1}{T^{2}} = \frac{\int_{0}^{1} (h - V (\tilde{u})) d t}{\frac{1}{2} \int_{0}^{1} {| \dot{\tilde{u}} |}^{2} d t} .

Then

\tilde{q} (t) = \tilde{u} (t / T)

is a nonconstant T-periodic solution for (1.1)-(1.2). Furthermore, if

V (x) < h

\forall x \neq O

, then

f (u) \geq 0

on Λ and

f (u) = 0

u \in Λ

if and only if u is a nonzero constant.

Lemma 2.3 (Sobolev-Rellich-Kondrachov [35, 36])

W^{1, 2} (R / Z, R^{n}) \subset C (R / Z, R^{n})

and the imbedding is compact.

Lemma 2.4 ([35, 36])

Let

q \in W^{1, 2} (R / T Z, R^{n})

(1)

q (0) = q (T) = O

, then we have the Friedrics-Poincaré inequality:

\int_{0}^{T} {| \dot{q} (t) |}^{2} d t \geq {(\frac{π}{T})}^{2} \int_{0}^{T} {| q (t) |}^{2} d t .

(2)

\int_{0}^{T} q (t) d t = 0

, then we have Wirtinger’s inequality:

\int_{0}^{T} {| \dot{q} (t) |}^{2} d t \geq {(\frac{2 π}{T})}^{2} \int_{0}^{T} {| q (t) |}^{2} d t

and Sobolev’s inequality:

\int_{0}^{T} {| \dot{q} (t) |}^{2} d t \geq \frac{12}{T} {| q (t) |}_{\infty}^{2} .

Lemma 2.5 (Eberlein-Shmulyan [37])

A Banach space X is reflexive if and only if any bounded sequence in X has a weakly convergent subsequence.

Definition 2.6 (Tonelli [35])

Let X be a Banach space,

f : X \to R

(i)

If for any

{x_{n}} \subset X

strongly converges to

x_{0} : x_{n} \to x_{0}

, we have

lim inf f (x_{n}) \geq f (x_{0}),

then we call

f (x)

lower semi-continuous at

x_{0}

(ii)

If for any

{x_{n}} \subset X

weakly converges to

x_{0} : x_{n} ⇀ x_{0}

, we have

lim inf f (x_{n}) \geq f (x_{0}),

then we call

f (x)

weakly lower semi-continuous at

x_{0}

Using the famous Ekeland variational principle, Ekeland proved the following.

Lemma 2.7 (Ekeland [38])

Let X be a Banach space,

F \subset X

be a closed (weakly closed) subset, let

δ (x_{1}, x_{2})

be the geodesic distance between two points

x_{1}

and

x_{2}

in X,

δ (x, F)

be the geodesic distance between x and the set F. Suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous (or weakly lower semi-continuous) and assume

Φ |_{F}

restricted on F is bounded from below. Then there is a sequence

{x_{n}} \subset F

such that

\begin{aligned} δ (x_{n}, F) \to 0, \\ Φ (x_{n}) \to inf_{F} Φ, \\ (1 + ∥ x_{n} ∥) ∥ Φ |_{F}^{'} (x_{n}) ∥ \to 0 . \end{aligned}

Definition 2.8 ([38, 39])

Let X be a Banach space,

F \subset X

be a closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence

{x_{n}} \subset F

is such that

\begin{aligned} δ (x_{n}, F) \to 0, \\ Φ (x_{n}) \to c, \\ (1 + ∥ x_{n} ∥) ∥ Φ |_{F}^{'} (x_{n}) ∥ \to 0, \end{aligned}

then

{x_{n}}

has a strongly convergent subsequence.

Then we say that f satisfies the

{(C P S)}_{c, F}

condition at the level c for the closed subset

F \subset X

We notice that if

F = X

, then the above condition is the classical Cerami-Palais-Smale condition [40].

We can give a weaker condition than the

{(C P S)}_{c, F}

condition.

Definition 2.9 Let X be a Banach space,

F \subset X

be a weakly closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence

{x_{n}} \subset F

such that

\begin{aligned} δ (x_{n}, F) \to 0, \\ Φ (x_{n}) \to c, \\ ∥ Φ |_{F}^{'} (x_{n}) ∥ \to 0, \end{aligned}

then

{x_{n}}

has a weakly convergent subsequence.

Then we say that f satisfies the

{(W C P S)}_{c, F}

condition.

Lemma 2.10 (Gordon [18])

Let V satisfy the so-called Gordon strong force condition:

There exists a neighborhood

N

of O and a function

U \in C^{1} (Ω, R)

such that:

(i)

{lim}_{s \to 0} U (x) = - \infty

;

(ii)

- V (x) \geq {| U^{'} (x) |}^{2}

for every

x \in N - {O}

Let

\partial Λ = {u \in H^{1} = W^{1, 2} (R / Z, R^{n}), \exists t_{0}, u (t_{0}) = O} .

Then we have

\int_{0}^{1} V (u) d t \to - \infty, \forall u_{n} ⇀ u \in \partial Λ .

Let

\partial Λ = {u \in H^{1} = W^{1, 2} (R / Z, R^{n}), \exists t_{0}, u (t_{0}) = 0} .

Then we have

\int_{0}^{1} V (u) d t \to - \infty, \forall u_{n} ⇀ u \in \partial Λ .

By Lemmas 2.7 and 2.10, it is easy to prove the following.

Lemma 2.11 Let X be a Banach space, let

F \subset X

be a weakly closed subset. Suppose that Φ defined on F is Gateaux-differentiable and weakly lower semi-continuous and bounded from below on F. If Φ satisfies the

{(C P S)}_{inf Φ, F}

condition or the

{(W C P S)}_{inf Φ, F}

condition, and suppose that

Φ (u_{n}) \to + \infty, u_{n} ⇀ u \in \partial Λ,

then Φ attains its infimum on F.

The next lemma is a variant on the classical Tonelli’s theorem, whose proof is easy, so we omit its proof.

Lemma 2.12 Let X be a Banach space,

F \subset X

be a weakly closed subset. Suppose that

ϕ (u)

is defined on an open subset

Λ \subset X

and is Gateaux-differentiable on Λ and weakly lower semi-continuous and bounded from below on

Λ \cap F

, if ϕ is coercive, that is,

ϕ (x) \to + \infty

∥ x ∥ \to + \infty

, and suppose that

ϕ (u_{n}) \to + \infty, u_{n} ⇀ u \in \partial Λ,

then ϕ attains its infimum on

Λ \cap F

3 The proof of Theorem 1.7

By the symmetrical condition (A5)′, it is easy to prove that the critical point of the functional f on

Λ_{0}

is also the critical point of the functional f on Λ.

Let

\partial Λ_{0} = {u \in H^{1} = W^{1, 2} (R / Z, R^{n}), u (t + 1 / 2) = - u (t), \exists t_{0}, u (t_{0}) = 0} .

Lemma 3.1 Assume (A4)′ holds, then for any weakly convergent sequence

u_{n} ⇀ u \in \partial Λ_{0}

, we have

f (u_{n}) \to + \infty .

Proof Similar to the proof of Zhang [19]. □

Lemma 3.2

F \cap Λ

is a weakly closed subset in

H^{1}

Proof Let

{u_{n}} \subset F \cap Λ

be a weakly convergent sequence, we use the embedding theorem to find which uniformly converges to

u \in H^{1}

Now we claim

u \in Λ

, and then it is obvious that

u \in F

. In fact, if

u \in \partial Λ

, by

V (q) \to - \infty

q \to 0

and the condition (A4)′ we have

- V (u) \geq C_{1} {| u |}^{- β}, 0 < | u | < r^{'} < r .

V (u)

satisfies Gordon’s strong force condition, and by his lemma, we have

\int_{0}^{1} - V (u_{n}) d t \to + \infty, \forall u_{n} ⇀ u \in \partial Λ .

The condition (A4)′ implies

V (u_{n}) + \frac{1}{2} 〈 V^{'} (u_{n}), u_{n} 〉 \geq (1 - \frac{β}{2}) V (u_{n}) .

Hence

h = \int_{0}^{1} [V (u_{n}) + \frac{1}{2} 〈 V^{'} (u_{n}), u_{n} 〉] d t \to + \infty .

This is a contradiction. □

Lemma 3.3

f (u)

is weakly lower semi-continuous on

F \cap Λ_{0}

Proof For any

{u_{n}} \subset F : u_{n} ⇀ u

, then by Sobolev’s embedding theorem and functional analysis, we have uniform convergence:

{| u_{n} (t) - u (t) |}_{\infty} \to 0 .

(i)

u \in Λ_{0}

, then by

V \in C^{1} (R^{n} ∖ {0}, R)

, we have

{| V (u_{n} (t)) - V (u (t) |}_{\infty} \to 0 .

It’s well known that the norm is weakly lower semi-continuous, we have

lim inf ∥ u_{n} ∥ \geq ∥ u ∥ .

Hence

\begin{aligned} lim inf f (u_{n}) & = lim inf (\frac{1}{2} \int_{0}^{1} {| {\dot{u}}_{n} |}^{2} d t) \int_{0}^{1} (h - V (u_{n})) d t, \\ \geq \frac{1}{2} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} (h - V (u)) d t = f (u) . \end{aligned}

(ii)

u \in \partial Λ_{0}

, then by our assumption on V which satisfies Gordon’s strong force condition, we have

\int_{0}^{1} - V (u_{n}) d t \to + \infty, \forall u_{n} ⇀ u \in \partial Λ_{0} .

(1)

(1) If

u \equiv 0

, then

{| u_{n} |}_{\infty} \to 0, n \to + \infty .

Then similar to the proof in [19], we have

f (u_{n}) \geq 6 {| u_{n} |}_{\infty}^{2 - β} \to + \infty, n \to + \infty .

So in this case we have

lim inf f (u_{n}) = + \infty \geq f (u) .

lim inf ∥ u_{n} ∥ \geq ∥ u ∥ > 0 .

(2)

(2) If

u \neq 0

, then by the weakly lower semi-continuity for norm, we have

So by Gordon’s lemma, we have

\begin{array}{rcl} lim inf f (u_{n}) & = & lim inf (\frac{1}{2} \int_{0}^{1} {| {\dot{u}}_{n} |}^{2} d t) \int_{0}^{1} (h - V (u_{n})) d t = + \infty \\ \geq & \frac{1}{2} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} (h - V (u)) d t = f (u) . \end{array}

□

Lemma 3.4 The functional

f (u)

has a positive lower bound on F.

Proof By the definitions of

f (u)

and F and the assumption (A2), we have

f (u) = \frac{1}{4} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} (V^{'} (u) u) d t \geq 0, \forall u \in F .

□

By the definitions of the functional

f (u)

and its domain

Λ_{0}

, and the conditions on the energy

h > 0

and the potential

V (u) < 0

, it is easy to prove the following lemma.

Lemma 3.5 The functional

f (u)

is coercive.

Furthermore, we claim that

c = inf_{F \cap Λ_{0}} f (u) > 0,

since otherwise,

u_{0} (t) = const

attains the infimum 0, then by the symmetry of

Λ_{0}

, we have

u_{0} (t) \equiv o

, which contradicts the definition of

Λ_{0}

. Now by Lemmas 3.1-3.4 and Lemmas 2.11 and 2.12, we know

f (u)

attains the infimum on F, furthermore we know that the minimizer is nonconstant.

4 The proof of Theorem 1.8

In order to prove the Cerami-Palais-Smale type condition and get a nonconstant periodic solution in non-symmetrical case, we need to add a topological condition, we know that there are winding numbers (degrees) in the planar case, so we define

Λ_{1} = {u \in Λ, deg (u) \neq 0} .

Lemma 4.1 If

u_{n} ⇀ u \in \partial Λ_{1}

, then

f (u_{n}) \to + \infty

Proof By V satisfying Gordon’s strong force condition, we have

\int_{0}^{1} - V (u_{n}) d t \to + \infty, \forall u_{n} ⇀ u \in \partial Λ_{1} .

(1)

u \equiv 0

, then by Sobolev’s embedding theorem, we have

{| u_{n} |}_{\infty} \to 0, n \to + \infty .

Then by

deg (u_{n}) \neq 0

, we have

c > 0

such that

c {| u_{n} |}_{\infty} \leq {∥ {\dot{u}}_{n} ∥}_{L^{2}}

and

{∥ {\dot{u}}_{n} ∥}_{L^{2}}

is an equivalent norm of

W^{1, 2}

and

f (u_{n}) \geq c {| u_{n} |}_{\infty}^{2 - β} \to + \infty, n \to + \infty .

So in this case, we have

lim inf f (u_{n}) = + \infty \geq f (u) .

(2)

u \neq 0

, then by the weakly lower semi-continuity for the norm, we have

lim inf ∥ u_{n} ∥ \geq ∥ u ∥ > 0 .

So by Gordon’s lemma, we have

\begin{array}{rcl} lim inf f (u_{n}) & = & lim inf (\frac{1}{2} \int_{0}^{1} {| {\dot{u}}_{n} |}^{2} d t) \int_{0}^{1} (h - V (u_{n})) d t = + \infty \\ = & \frac{1}{2} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} (h - V (u)) d t = f (u) . \end{array}

□

Lemma 4.2 Under the assumptions of Theorem 1.8,

f (u) = \frac{1}{2} \int_{0}^{1} {| \dot{u} |}^{2} d t \int_{0}^{1} (h - V (u)) d t

satisfies the

{(C P S)}^{+}

condition on

Λ_{1}

, that is, if

{u_{n}} \subset Λ_{1}

satisfies

f (u_{n}) \to c > 0, (1 + ∥ u_{n} ∥) f^{'} (u_{n}) \to O,

(4.1)

then

{u_{n}}

has a strongly convergent subsequence in

Λ_{1}

Proof Since

f^{'} (u_{n})

makes sense, we know

{u_{n}} \subset Λ_{1} .

We claim

\int_{0}^{1} {| {\dot{u}}_{n} |}^{2} d t

is bounded. In fact, by

f (u_{n}) \to c

, we have

- \frac{1}{2} {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2} \cdot \int_{0}^{1} V (u_{n}) d t \to c - \frac{h}{2} {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2} .

(4.2)

By (A3)′ we have

\begin{array}{rcl} 〈 f^{'} (u_{n}), u_{n} 〉 & = & {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2} \cdot \int_{0}^{1} (h - V (u_{n}) - \frac{1}{2} 〈 V^{'} (u_{n}), u_{n} 〉) d t \\ \geq & {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2} \int_{0}^{1} [h - \frac{μ_{2}}{2} - (1 - \frac{α}{2}) V (u_{n})] d t . \end{array}

(4.3)

By (4.2) and (4.3) we have

\begin{array}{rcl} 〈 f^{'} (u_{n}), u_{n} 〉 & \geq & (h - \frac{μ_{2}}{2}) {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2} + (1 - \frac{α}{2}) (2 c - h {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2}) \\ = & (\frac{α}{2} h - \frac{μ_{2}}{2}) {∥ {\dot{u}}_{n} ∥}_{L^{2}}^{2} + C_{1}, \end{array}

(4.4)

where

C_{1} = 2 (1 - \frac{α}{2}) c

α > 2

h > \frac{μ_{2}}{α}

. So

{∥ {\dot{u}}_{n} ∥}_{2} \leq C_{2}

Then we claim

| u_{n} (0) |

is bounded.

We notice that

\begin{aligned} f^{'} (u_{n}) \cdot (u_{n} - u_{n} (0)) \\ = \int_{0}^{1} {| {\dot{u}}_{n} |}^{2} d t \int_{0}^{1} (h - V (u_{n})) d t \\ - \frac{1}{2} \int_{0}^{1} {| {\dot{u}}_{n} |}^{2} d t \int_{0}^{1} 〈 V^{'} (u_{n}), u_{n} - u_{n} (0) 〉 d t \\ = 2 f (u_{n}) - \frac{1}{2} \int_{0}^{1} {| {\dot{u}}_{n} |}^{2} \int_{0}^{1} 〈 V^{'} (u_{n}), u_{n} - u_{n} (0) 〉 d t . \end{aligned}

(4.5)

| u_{n} (0) |

is unbounded, then there is a subsequence, still denoted by

u_{n}

s.t.

| u_{n} (0) | \to + \infty

. Since

∥ {\dot{u}}_{n} ∥ \leq M_{1},

we have

min_{0 \leq t \leq 1} | u_{n} (t) | \geq | u_{n} (0) | - {∥ {\dot{u}}_{n} ∥}_{2} \to + \infty, as n \to + \infty .

(4.6)

By Friedrics-Poincaré’s inequality and the condition (P1), we have

\int_{0}^{1} {| {\dot{u}}_{n} (t) |}^{2} d t \geq π^{2} \int_{0}^{1} {| u_{n} (t) - u_{n} (0) |}^{2} d t,

(4.7)

\int_{0}^{1} V^{'} (u_{n}) (u_{n} - u_{n} (0)) d t \to 0,

(4.8)

f^{'} (u_{n}) \cdot (u_{n} - u_{n} (0)) \to 0 .

(4.9)

f (u_{n}) \to 0

, which contradicts

f (u_{n}) \to c > 0

, hence

u_{n} (0)

is bounded, and

∥ u_{n} ∥ = {∥ {\dot{u}}_{n} ∥}_{L^{2}} + | u_{n} (0) |

is bounded. Furthermore, similar to the proof of Ambrosetti and Coti Zelati [15],

u_{n}

strongly converges to

u \in Λ

. □

It is easy to prove the following.

Lemma 4.3 Under the assumption (B1)′,

f (u) \geq 0

on Λ, that is, f has a lower bound.

Lemma 4.4 Under the assumptions of Theorem 1.8,

f (u)

is weakly lower semi-continuous on the closure

\bar{Λ}

of Λ.

Now we can prove our Theorem 1.8, in fact, by Lemma 4.1, we know that the infimum of f on

Λ_{1}

is equal to the infimum of f on the closure of

Λ_{1}

. Furthermore, we can prove the infimum of f on

Λ_{1}

is greater than zero, otherwise if it is zero, the corresponding minimizer must be constant, then the winding number is zero, which is a contradiction. Now by the above lemmas, especially Lemma 2.11, we know that f attains the positive infimum on

Λ_{1}

and the corresponding minimizer must be nonconstant.

Acknowledgements

The authors would like to thank the editor and the referees for their many valuable comments. This paper was partially supported by NSF of China and the Grant for the Advisors of PhD students.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The research and writing of this manuscript was a collaborative effort made by all the authors. All authors read and approved the final manuscript.

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Titel: New periodic solutions of singular Hamiltonian systems with fixed energies
verfasst von: Fengying Li
Qingqing Hua
Shiqing Zhang
Publikationsdatum: 01.12.2014
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2014
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2014-400

Springer Professional

New periodic solutions of singular Hamiltonian systems with fixed energies

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 A few lemmas

3 The proof of Theorem 1.7

4 The proof of Theorem 1.8

Acknowledgements

Competing interests

Authors’ contributions

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Abstract

Competing interests

Authors’ contributions

1 Introduction

2 A few lemmas

3 The proof of Theorem 1.7

4 The proof of Theorem 1.8

Acknowledgements

Competing interests

Authors’ contributions

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