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Periodic solutions for second order Hamiltonian system with a p-Laplacian.

1. Introduction

Consider the ordinary p-Laplacian system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

T > 0, p > 1,q > 1,1/p + 1/q = 1, and F : [0, T] x [R.sup.N] [right arrow] R, (t,x) [right arrow] F(t,x) is measurable in t for every x [member of] [R.sup.N] and continuously differentiable and convex in x for almost every t [member of] [0, T].

When p = 2, there are many existence results of periodic solutions for system (1.1) (see [1-6] and references therein). However, when p > 1, there are few papers to study these problems. In [7] and [8], the authors considered system (1.1) by using the dual least action principle and a generalized Mountain pass Lemma, respectively, and they obtained some existence results of solutions for system (1.1). In [9], we also considered system (1.1) by using the generalized Saddle point Theorem and obtained that system (1.1) has multiple solutions. Especially, in [7], Tian and Ge obtained the following results:

Theorem A Suppose F satisfies the following conditions:

([A.sub.1]) there exists l [member of] [L.sup.2max{q,p-1}](0,T;[R.sup.N]) such that for ally [member of] [R.sup.N] and a.e. t [member of] [0,T],

F(t,y) [greater than or equal to] (l(t), [[absolute value of y].sup.p-2/2]y);

([A.sub.2]) there are constants [alpha] [member of] (0, [T.sup.-p/q]) [[alpha].sup.q-1] [member of](0, [T.sup.q/p]) p > 1, [gamma] [member of] [L.sup.max{q,p-1}] (0, T; [R.sup.N]) such that for y [member of] [R.sup.N], and a.e. t [member of] [0, T],

F(t,y) [less than or equal to] [[alpha].sup.2]/p [[absolute value of y].sup.p] + [gamma](t);

([A.sub.3]) [[integral].sup.T.sub.0] F(t,y)dt [right arrow] + [infinity], as [absolute value of y] [right arrow] [infinity], y [member of] [R.sup.N]. Then, system (1.1) has at least one solution.

In our paper, by using the improved inequality, we improve the condition ([A.sub.2]) and also obtain an estimate of periodic solution for system (1.1).

2. Preliminaries

In the following, we use [absolute value of x] to denote the Euclidean norm in [R.sup.N]. Let

[W.sup.1,p.sub.T] = {u : [0, T] [right arrow] [R.sup.N] | u(t) is absolutely continuous on [0, T], u(0)=u(T) and [??] [member of] [L.sup.p] (0, T;[R.sup.N])}.

Then, it follows from [2] that [W.sup.1,p.sub.T] is a Banach space with the norm defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from [2] that [W.sup.1,p.sub.T] is also reflexive and uniformly convex Banach space.

Let

X={v = ([v.sub.1], [v.sub.2]) : [v.sub.1] [member of] [W.sup.1,p.sub.T] (0,T;[R.sup.N]), [v.sub.2] [member of] [W.sup.1,p.sub.T] (0,T;[R.sup.N])}

with the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is clear that X is a reflexive Banach space.

Let

[[??].sup.1,p.sub.T] = {u [member of] [W.sup.1,p.sub.T] | [[integral].sup.T.sub.0] u(t)dt = 0}.

It is easy to know that [[??].sup.1,p.sub.T] is a subset of [W.sup.1,p.sub.T] and [W.sup.1,p.sub.T] = [R.sup.N] [direct sum] [[??].sup.1,p.sub.T]. Then [??] stands for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and [([W.sup.1,p.sub.T]).sup.*] stands for the conjugate space of [W.sup.1,p.sub.T]. Then

[X.sup.*] = {f = ([f.sub.1], [f.sub.2]) : [f.sub.1] [member of] [([W.sup.1,q.sub.T]).sup.*], [f.sub.2] [member of] [([W.sup.1,p.sub.T]).sup.*]}

is the conjugate space of X. Furthermore, we define

Y = {u = ([u.sub.1], [u.sub.2]) : [u.sub.1] [member of] [W.sup.1,p.sub.T](0,T; [R.sup.N]), [u.sub.2] [member of] [W.sup.1,p.sub.T] (0,T; [R.sup.N])}.

For h [member of] [L.sup.1]([0, T]; [R.sup.N]), the mean value is defined by [bar.h] = 1/T [[integral].sup.T.sub.0] h(t)dt. Besides this, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] stand for the norm in [C.sup.0]([0, T]), [L.sup.k]([0, T]) and [W.sup.1,k.sub.T], respectively.

[[GAMMA].sub.0]([R.sup.N]) denotes the set of all convex lower semi-continuous (l.s.c.) functions F : [R.sup.N] [right arrow] (- [infinity], + [infinity]] whose effective domain D(F) = {u [member of] [R.sup.N] : F(u) < + [infinity]} is nonempty. Let H : [0,T] x [R.sup.2N] [right arrow] R, (t, u) [right arrow] H(t,u) be a smooth Hamiltonian such that for each t [member of] [0, T], H(t, x) [member of] [[GAMMA].sub.0]([R.sup.2N]) is strictly convex and H(t,u)/[absolute value of u] [right arrow] + [infinity], if [absolute value of u] [right arrow] [infinity]. The Fenchel transform [H.sup.*](t, x) of H(t, x) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[H.sup.*](t,v) = (v,u)-H(t,u) v = [nabla]H(t,u), or u = [nabla][H.sup.*](t,v). (2.1)

If for u = ([u.sub.1], [u.sub.2]), [u.sub.1],[u.sub.2] [member of] [R.sup.N], H(t,u) can be split into parts H(t,u) = [H.sub.1]{t,[u.sub.1]) + [H.sub.2](t,[u.sub.2]), then by (2.1), [H.sup.*](t,v) = [H.sup.*.sub.1](t, [v.sub.1]) + [H.sup.*.sub.2] (t,[v.sub.2] ),v = ([v.sub.1], [v.sub.2]), [v.sub.1], [v.sub.2] [member of] [R.sup.N]. We denote by J the symplectic matrix. Then [J.sup.2] = -I and (Ju, v) = -(u, Jv) for all u,v [member of] [R.sup.2N]. It is clear that (Jv, v) = {[v.sub.2], [v.sub.1]) - ([v.sub.1], [v.sub.2]), where v = ([v.sub.i], [v.sub.2]), [v.sub.i] [member of] C(0, T; [R.sup.N]),i = 1,2. The above knowledge and statement come from [2,7] and the references therein.

Let x(t) = u\{t), Op(x(f)) = 0M2{t). Then system (1.1) is equivalent to the non-autonomous system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

where H : [0, T] x [R.sup.2N] [right arrow] R, [H.sub.i] : [0, T] x [R.sup.N] [right arrow] R, i = 1,2.

The dual action is defined on X by

[phi](v) = [[integral].sup.T.sub.0] [1/2(j[??](t),v(t)) + [H.sup.*.sub.1](t,[[??].sub.1](t)) + [H.sup.*.sub.2](t,[[??].sub.2](t))] dt,

where v = ([v.sub.1], [v.sub.2]), [H.sup.*](t,[??]) = [H.sup.*.sub.1](t,[[??].sub.1]) + [H.sup.*.sub.2](t,[[??].sub.2]).

Lemma 2.1. (also see [9], Lemma 2.2) Let u [member of] [[??].sup.1,p.sub.T]. Then

[parallel]u[[parallel].sub.[infinity]][less than or equal to][(T/q+1).sup.1/q][([[integral].sup.T.sub.0] [[absolute value of [??](s)].sup.p]ds).sup.1/p], (2.5)

and

[[integral].sup.T.sub.0][[absolute value of [u](s)].sup.p]ds[less than or equal to][T.sup.p][THETA](p,q)/[(q+1).sup.p/q][[integral].sup.T.sub.0][[absolute value of [??](s)].sup.p]ds, (2.6)

where

[THETA](p,q) = [[integral].sup.1.sub.0][[[s.sup.q+1] + [(1 - s).sup.q+1]].sup.p/q]ds.

Proof. Fix t [member of] [0, T]. For every [tau] [member of] [0, T], we have

u(t) = u([tau]) + [[integral].sup.t.sub.[tau]][??](s)ds. (2.7)

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Integrating (2.7) over [0, T] and using the Holder's inequality we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.8)

Since [t.sup.q+1] + [(T - t).sup.q+1] [less than or equal to] [T.sup.q+1] for t [member of] [0, T], it follows from (2.8) that (2.5) holds. On the other hand, from (2.8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that (2.6) holds. The proof is complete.

Remark 2.1. Obviously, our Lemma 2.1 improve Proposition 1.1 in [2] which shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Lemma 2.2. For every v = ([v.sub.1], [v.sub.2]) [member of] X,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

for every [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

where

C = T/[(q+1).sup.1/q] + T/[(p+1).sup.1/p].

Proof. Let v = [bar.v] + [??], where [bar.v] = 1/T [[integral].sup.T.sub.0] v(s)ds. Then by Lemma 2.1, Holder's inequality and Young's inequality, for v [member of] X, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly to the above process, the result (2.11) holds for u = ([u.sub.1], [u.sub.2]) [member of] Y.

Remark 2.2. Obviously, our Lemma 2.2 improve Lemma 3.3 in [7].

Lemma 2.3. [2, Proposition 1.4] Let G [member of] [C.sup.1]([R.sup.N], R) be a convex function. Then, for all x, y [member of] [R.sup.N], we have

G(x)[greater than or equal to]G(y) + ([nabla]G(y),x-y).

3. Main results and Proofs

Theorem 3.1 Suppose F satisfies {[A.sub.1]), ([A.sub.3]) and the following condition: ([A.sub.2])' there are constants [alpha] [member of] (0, [(C/2).sup.-p/q]), [[alpha].sup.q-1][member of](0, [(C/2).sup.-q/p]), [gamma][member of] [L.sup.max{q,p-1]} (0, T; [R.sup.N]) such that for all y [member of] [R.sup.N], and a.e. t [member of] [0, T],

F(t,y) [less than or equal to] [[alpha].sup.2]/p[[absolute value of y].sup.p] + [gamma](t),

where

C = T/[(q+1).sup.1/q] + T/[(p+1).sup.1/p].

Then, system (2.3) has at least one solution u [member of] Y such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

minimizes the dual action

[phi]: X [right arrow] (-[infinity] + [infinity]], v [right arrow] [[integral].sup.T.sub.0][1/2(J[??] (t),v(t)) + [H.sup.*](t,[??](t))]dt,

that is to say, system (1.1) has at least one solution x [member of] [W.sup.1,p.sub.T].

Proof. The proof is same as in [7]. We only need to replace Lemma 3.3 in [7] with Lemma 2.2 and replace (2.9) with (2.5) in the process of proof.

Next, we consider the estimate of solutions for system (1.1).

Theorem 3.2 Assume that there exist [alpha] [member of] (0,min {[C.sup.-1], [C.sup.-p/q]}), [beta][greater than or equal to] 0, [gamma][greater than or equal to] 0 and [delta] > 0 such that

[delta][absolute value of y] - [beta][less than or equal to] F(t,y)[less than or equal to] [[alpha].sup.2]/p[[absolute value of y].sup.p] + [gamma] (3.1)

for all t [member of] [0, T] and y [member of] [R.sup.N]. Then each solution x of system (1.1) satisfies

[[integral].sup.T.sub.0][absolute value of x(t)]dt [less than or equal to]([gamma] + [beta])T/[delta] + T[[alpha].sup.q][B.sup.1/p][D.sup.1/q]/[delta][(q+1).sup.1/q], (3.2)

[[integral].sup.T.sub.0][absolute value of [??](t)].sup.p]dt [less than or equal to] pT([gamma] + [beta])/1 - C[alpha], (3.3)

where

B = pT([gamma] + [beta])/[[alpha].sup.q] - C[[alpha].sup.q+1], D = qT([gamma] + [beta])/[[alpha].sup.1-q/p] = C[alpha].

Proof. By (3.1), for [u] = ([u.sub.1], [u.sub.2]) [member of] [R.sup.N] [R.sup.N], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Then, we have

(u, v) - H(t,u) [greater than or equal to] (u,v) - [alpha]/p[[absolute value of [u.sub.1]].sup.p] - [gamma]/ [alpha] - [[alpha].sup.q-1]/q [[absolute value of [u.sub.2]].sup.q].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence,

[H.sup.*](t,v)[greater than or equal to][[alpha].sup.-q/p][[absolute value of [v.sub.1]].sup.q]/q - [gamma]/[alpha] + 1/p[alpha][[absolute value of [v.sub.2]].sup.p]. (3.5)

By (2.1) and (3.4), we get

[H.sup.*](t,v) = (u,v) - H(t,u) [less than or equal to] (u,v) + [beta]/[alpha]. (3.6)

Then

[[alpha].sup.-q/p][[absolute value [v.sub.1]].sup.q]/q - [gamma]/[alpha] + 1/p[alpha][[absolute value of [v.sub.2]].sup.p][less than or equal to] (u,v) + [beta]/[alpha]. (3.7)

Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then by (2.1) and (3.7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is

[[alpha].sup.-q/p-q]/q[[absolute value of [nabla]F(t,[u.sub.1]].sup.q] - [gamma]/[alpha] + [[alpha].sup.q-1]/p [[absolute value of [u.sub.2]].sup.q] [less than or equal to] (u, [nabla]H(t,u)) + [beta]/[alpha].

For each solution u = ([u.sub.1], [u.sub.2]) of system (2.3), it is easy to know that [u.sub.1] is the solution of (1.1). By (2.2) and (2.3), we know [nabla]F(f,[u.sub.1](t)) = -[alpha][[??].sub.2](t) and [nabla]H(t,u(t)) = J[??](t). Hence

[[alpha].sup.-q/p]/q [[absolute value of [[??].sub.2](t)].sup.q] - [gamma]/[alpha] + [[alpha].sup.q-1]/p [[absolute value of [u.sub.2](t)].sup.q][less than or equal to](u(t), -J[??](t)) + [beta]/[alpha].

Integrating the above inequality over [0, T] and using Lemma 2.2 and (2.2), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [alpha] [member of] (0, min {[C.sup.-1], [C.sup.-p/q]}), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

It follows that (3.3) holds. Since F is continuously differentiable and convex in x, then by Lemma 2.3, (3.1), (2.2), (2.5), Holder's inequality and (3.8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, we get

[[integral].sup.T.sub.0][absolute value of [u.sub.1](t)]dt [less than or equal to]([gamma] + [beta])T/[delta] + T[[alpha].sup.9][B.sup.1/p][D.sup.1/q]/[delta][(q+1).sup.1/q].

It follows that (3.2) holds. The proof is complete.

References

[1] Y. M. Long, Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadritic potentials, Nonlinear Anal. 24(1995)1665-1671.

[2] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.

[3] J. Mawhin, M. Willem, Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. H. Poincare Anal. Non Lineaire, 3(1986)431-453.

[4] P. H. Rabinowitz, On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 33 (1980)609-633.

[5] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS Regional Conf. Ser. in Math., Vol. 65, American Mathematical Society, Providence, RI, 1986.

[6] C. L. Tang, X.P Wu, Notes on periodic solutions of subquadratic second order systems, J. Math. Anal. Appl. 285(1)(2003)8-16.

[7] Y. Tian, W. Ge, Periodic solutions of non-autonomous second-order systems with a p--Laplacian, Nonlinear Analysis, 66(2007)192-203.

[8] B. Xu, C. L. Tang, Some existence results on periodic solutions of ordinary p--Laplacian systems, J. Math. Anal. Appl., 333(2007)1228-1236.

[9] X. Zhang, X. Tang, Periodic solutions for an ordinary p--Laplacian system, Taiwanese Journal of Mathematics (in press).

* This work is supported by the Graduate degree thesis Innovation Foundation of Central South University (No: 3960-71131100014) and supported by the Outstanding Doctor degree thesis Implantation Foundation of Central South University (No: 2008yb032) and partially supported by the NNSF (No: 10771215) of China.

Received by the editors October 2009.

Communicated by J. Mawhin.

2000 Mathematics Subject Classification : 34C25,58E50.

School of Mathematical Sciences and Computing Technology,

Central South University, Changsha,

Hunan 410083, PR. China

E-mail: tangxh@mail.csu.edu.cn
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Author:Zhang, Xingyong; Tang, Xianhua
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:May 1, 2011
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