New periodic solutions for a class of singular Hamiltonian systems

Abstract

We use the variational minimizing method to study the existence of new nontrivial periodic solutions with a prescribed energy for second order Hamiltonian systems with singular potential [InlineEquation not available: see fulltext.], which may have an unbounded potential well.

MSC: 34C15, 34C25, 58F.

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R E S E A R C H Open Access

New periodic solutions for a class of singular Hamiltonian systems

Xiong-rui Wang1* and Shengyi Shen2

*Correspondence: [email protected]

1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, ChinaFull list of author information is available at the end of the article

Abstract

We use the variational minimizing method to study the existence of new nontrivial periodic solutions with a prescribed energy for second order Hamiltonian systems with singular potential V C1(Rn\{0}, R), which may have an unbounded potential

well.

MSC: 34C15; 34C25; 58F

Keywords: singular Hamiltonian systems; periodic solutions; variational methods

1 Introduction and main results

For singular Hamiltonian systems with a xed energy h R,

q + V (q) = , (.) |q| + V(q) = h. (.)

Ambrosetti-Coti Zelati [, ] used Ljusternik-Schnirelmann theory on an C manifold to get the following theorem.

Theorem . (Ambrosetti-Coti Zelati []) Suppose V C(Rn\{}, R) satises(A)

V(u) , u ,

(A)

V (u) u + [parenleftbig]V

(u)u, u

[parenrightbig]

= ,

(A)

V (u) u > , u = ,

(A) > , s.t.

V (u) u V(u),

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(A) > , r > , s.t.

V (u) u V(u), < |u| < r,

(A)

lim sup

|u|+

V(u) +

V(q)

Theorem . Suppose h > , > and V C(Rn\{}, R) satises (B), (A), and (B ) lim|q|+ V (q) = ;

(B ) V(q) + V (q)q h, |q| .

Then (.)-(.) have at least one periodic solution with the given energy h and whose action is at most r.

Using variational minimizing methods, we get the following theorem.

Theorem . Suppose V C(Rn\{}, R) satises

(V) > , > , r > , s.t.

V(q) |q|, < |q| < r;

(V)

V(q) < , q = ;

(V)

V(q) = V(q), q = .

Then for any h > , (.)-(.) have at least one non-constant periodic solution with the given energy h.

V (u)u[bracketrightbigg] .

Then (.)-(.) have at least one non-constant periodic solution.

After Ambrosetti-Coti Zelati, a lot of mathematicians studied singular Hamiltonian systems. Here we only mention a related recent paper of Carminati-Sere-Tanaka [], in which they used complex variational and geometrical and topological methods to generalize Pisanis results []. They got the following theorems.

Theorem . Suppose h > , L > and V C(Rn\{}, R) satises (A), (A), and

(B) V(q) ;

(B) V(q) + V (q)q h, |q| eL; (B) V(q) + V (q)q h, |q| eL.

Then (.)-(.) have at least one periodic solution with the given energy h and whose action is at most r with

r = max

[braceleftbig][bracketleftbig][parenleftbig]h

[parenrightbig][bracketrightbig]

; |q| = [bracerightbig].

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2 A few lemmas

Let

H = W,

R/Z, Rn[parenrightbig]=

u : R Rn, u L, u L, u(t + ) = u(t)[bracerightbig].

Then the standard H norm is equivalent to

u = u H

=

[parenleftbigg][integraldisplay]

|u| dt[parenrightbigg]/ + [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay]

u(t) dt[vextendsingle][vextendsingle][vextendsingle][vextendsingle].

Let

=

u H|u(t) = , t[bracerightbig].

By symmetry condition (V), similar to Ambrosetti-Coti Zelati [], let

= u H = W,[parenleftbig]R/Z, Rn

, u(t + /) = u(t), u(t) = [bracerightbig].

We dene the equivalent norm in E = {u H = W,(R/Z, Rn), u(t + ) = u(t)}:

u = u E = [parenleftbigg][integraldisplay]

|u| dt[parenrightbigg]/.

Lemma . ([, ]) Let f (u) = [integraltext]

|u| dt [integraltext]

(h V(u)) dt and be such that f () =

and f () > . Set

T = [integraltext]

(h V()) dt

[integraltext]

. (.)

Then q(t) =(t/T) is a non-constant T-periodic solution for (.)-(.). Furthermore, if V(x) < h, x = , then f (u) on and f (u) = , u if and only if u is a nonzero constant.

If such that f () = and f () > , then we nd that q(t) =(t/T) is a non-constant T-periodic solution for (.)-(.).

Lemma . (Gordon []) Let V satisfy the so-called Gordon Strong Force condition: There exist a neighborhood N of and a function U C( ,

R) such that:

|| dt

(i) lims U(x) = ;(ii) V(x) |U (x)| for every x N {}. Let

=

u H = W,[parenleftbig]R/Z, Rn

, t, u(t) = [bracerightbig].

Then we have

[integraldisplay]

V(u) dt , un

u .

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Let

=

u H = W,[parenleftbig]R/Z, Rn

, u

t +

[parenrightbigg] = u(t), t, u(t) = [bracerightbigg].

Then we have

[integraldisplay]

V(u) dt , un

u .

Lemma . Let X be a Banach space, and let E X be a weakly closed subset. Suppose that (u) is dened on an open subset X and (u) = for any u . Let (u) = +

for u . Assume (u) + and is weakly lower semi-continuous on

E, and that it

is coercive on E:

(u) +, u +

and

(un) +, un u .

Then attains its inmum in E.

Proof We set

c = inf

E

(u).

Then

< c < +,

in fact, by the assumptions, it is obvious that c < +. Now if c = , then there exists {un} E such that (un) . Then we know that {un} is bounded, since is coercive. By the Eberlein-Schmulyan theorem, {un} has a weakly convergent subsequence.

Finally, by the denition for c and the assumption for the weakly lower semi-continuity for (u), we know (u) = . This is a contradiction.

Now we know that there exists minimizing sequence {un} such that (un) c. Furthermore by the coercivity of we know that {un} is bounded; then {un} has a weakly convergent subsequence. We claim the weak limit u , since otherwise (u) = + by the assumption. On the other hand, by the denition of the inmum c and the assumption for the weak lower semi-continuity for (u) on

E, we know (u) = c < +. This is a contradiction. So the weak limit u E and (u) = c.

3 The proof of Theorem 1.4

Lemma . Assume (V) hold, then for any weakly convergent sequence un u , we have

f (un) +.

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Proof Notice that (V) imply Gordons strong force condition. By the weak limit u and V satisfying Gordons strong force condition, we have

[integraldisplay]

V(un) dt +, un

u .

By un u in the Hilbert space H, we know that un is bounded.

() If u , then by Sobolevs embedding theorem, we have the uniform convergence property:

|un| , n +.

By the symmetry of u(t + /) = u(t), we have [integraltext]

u(t) dt = , then we have Sobolevs in-

equality:

[integraldisplay]

[vextendsingle][vextendsingle]

u(t)[vextendsingle][vextendsingle]

dt [vextendsingle][vextendsingle]u(t)[vextendsingle][vextendsingle]

.

Then we have

f (un) |un| +, n +.

So in this case we have

lim inf f (un) = + f (u).

() If u = , then we have the following. By the weakly lower semi-continuity for the norm, we have

lim inf un u > .

So, by Gordons lemma, we have

lim inf f (un) = lim inf

[integraldisplay]

|un| dt[parenrightbigg] [integraldisplay]

h V(un)

[parenrightbig]

dt = +

h V(u)

[parenrightbig]dt = f (u).

Lemma . f (u) is weakly lower semi-continuous on

.

[integraldisplay]

|u| dt [integraldisplay]

: un u, by Sobolevs embedding theorem, we have uniform convergence:

[vextendsingle]u

n(t) u(t)

Proof For any {un}

[vextendsingle]

.

(i) If u , then by V C(Rn\{}, R), we have

[vextendsingle]

V

un(t)

[parenrightbig]

V

u(t)

[parenrightbig][vextendsingle][vextendsingle]

.

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By the weakly lower semi-continuity for norm, we have

lim inf un u .

Hence

lim inf f (un) = lim inf

[integraldisplay]

|un| dt[parenrightbigg] [integraldisplay]

h V(un)

[parenrightbig]

dt

[integraldisplay]

|u| dt [integraldisplay]

h V(u)

[parenrightbig]dt = f (u).

(ii) If u , then by satisfying Gordons strong force condition, we have

[integraldisplay]

V(un) dt +, un

u .

() If u , then

|un| , n +.

Then we have

f (un) |un| +, n +.

So in this case we have

lim inf f (un) = + f (u).

() If u = . By the weakly lower semi-continuity for norm, we have

lim inf un u > .

So by Gordons lemma, we have

lim inf f (un) = lim inf

[integraldisplay]

|un| dt[parenrightbigg] [integraldisplay]

h V(un)

[parenrightbig]

dt = +

[integraldisplay]

|u| dt [integraldisplay]

h V(u)

[parenrightbig]dt = f (u).

is a weakly closed subset of H.

Proof By Sobolevs embedding theorems, the proof is obvious.

Lemma . The functional f (u) is coercive on .

Proof By the denition of f (u) and the assumption (V), we have

f (u) =

[integraldisplay]

|u| dt [integraldisplay]

Lemma .

h V(u)

[parenrightbig]dt h

[integraldisplay]

|u| dt, u

.

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Lemma . The functional f (u) attains the inmum on ; furthermore, the minimizer is non-constant.

Proof By Lemma . and Lemmas .-., we know that the functional f (u) attains the inmum in ; furthermore, we claim that

inf

f (u) > ,since otherwise, u(t) = const attains the inmum , then by the symmetry of , we have u(t) , which contradicts the denition of . Now we know that the minimizer is non-constant.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

XW proved the main theorem, SS participated in the proof and helped to draft the manuscript. Both authors read and approved the nal manuscript.

Author details

1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China. 2Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China.

Acknowledgements

This study was supported by the Scientic Research Fund of Sichuan Provincial Education Department (11ZA172) and the Scientic Research Fund of Science Technology Department of Sichuan Province (2011JYZ010).

Received: 14 November 2013 Accepted: 10 February 2014 Published: 19 February 2014

References

1. Ambrosetti, A, Coti Zelati, V: Closed orbits of xed energy for singular Hamiltonian systems. Arch. Ration. Mech. Anal. 112, 339-362 (1990)

2. Ambrosetti, A, Coti Zelati, V: Periodic Solutions for Singular Lagrangian Systems. Springer, Berlin (1993)3. Carminati, C, Sere, E, Tanaka, K: The xed energy problem for a class of nonconvex singular Hamiltonian systems.J. Dier. Equ. 230, 362-377 (2006)4. Tanaka, K: Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds. Ann. Inst. Henri Poincar, Anal. Non Linaire 17, 1-33 (2000)

5. Pisani, L: Periodic solutions with prescribed energy for singular conservative systems involving strong forces. Nonlinear Anal. TMA 21, 167-179 (1993)

6. Gordon, WB: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 204, 113-135 (1975)

doi:10.1186/1687-2770-2014-42Cite this article as: Wang and Shen: New periodic solutions for a class of singular Hamiltonian systems. Boundary Value Problems 2014 2014:42.