Existence of periodic solutionsfor some second order quasilinear

Hamiltonian systems

by

Mario Girardi

Dipartimento di Matematica

Universita degli Studi di Roma Tre

Largo S. Leonardo Murialdo — 00146 Roma, Italy

Michele Matzeu

Dipartimento di Matematica

Universita Roma “Tor Vergata”

Viale della Ricerca Scientifica — 00133 Roma, Italy

Supported by MURST, Project “Variational Methods and Nonlinear DifferentialEquations”

2005, December

Abstract

A class of second order nonautonomous quasilinear Hamiltonian systems (S) isconsidered. One states that, for any T < T0, depending on the growth coefficientsof the Hamiltonian function H, there exists a T–periodic and T/2–antiperiodicsolution of (S), in case that two symmetry conditions hold for H.

Keywords. Quasilinear Hamiltonian systems, periodic solutions, Mountain Passtechniques, iteration methods.

Subject classification 34C25, 49R99, 58E99, 47H15

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2

Introduction

In the present paper we are interested in the existence of periodic solutions to thefollowing second order quasilinear Hamiltonian system

(S) −u(t) = b(t)∇H(u(t), u(t)) , t ∈ R

where ∇H denotes the gradient of H arth respect to its first variable in RN , and bis a periodic function.

In [2] the authors, in collaboration with D. De Figueredo, introduced a methodin order to solve a quasilinear elliptic equaiton by a variational approach usingMountain Pass techniques and some estimates for Mountain Pass solutions of somesemilinear problems suitably approximating the quasilinear problem.

A similar kind of approach allows to solve another quasilinear problem wihtmore general assumptions (see [4]), a semilinear integro–differential equation withnon–symmetric kernel (see [5]), finally a fully nonlinear elliptic equation (see [3]).

The aim of the present paper is to use some basic ideas of [2], [3], [4], [5], inorder to find periodic solutions of (S).

As a matter of fact, in [2], [3], [4], [5] it is very important tha the Poincareinequality holds in the variational approach in order to state that the approximatingsolutions actually converge to a solution of the initial poblem. In the presentcase, we consider a variational approach in which, due to some suitable symmetryassumption on H, one case use the Wirtinger inequality, which still allows to provethe convergence of the approximating solutions.

The T–periodicity of the solution founded is assured for any T < T0, where T0

depends on the growth coefficients of H with respect to its two variables and onthe maximum of b(·), which is further supposed to be T/2–periodic. Indeed onefinds that this solution is even T/2–antiperiodic.

Obviously, as a particular case, one can consider also autonomous Hamiltoniansystems, if one chooses b(t) ≡ 1.

1. The result

Let H : RN × RN be a continuously differentiable function on the first variable(let ∇H denote the corresponding gradient) and continuous on the second variable.Let b a continuous periodic function on R.

The problem is to find a non–zero periodic solution to the following second orderquasilinear Hamiltonian system

(S) −u(t) = b(t)∇H(u(t), u(t)) t ∈ R

One can state the following

Theorem 1. Let H be as above and let the further assumptions be satisfied:(H0) ∇H is Lipschitz continuous w.r. to its two variables(H1) H(x, y) = H(−x, y) ∀x, y ∈ RN

(H2) H(x, y) = H(x,−y) ∀x, y ∈ RN

(H3) limx→0

H(x,y)|x|2 = 0 uniformly w.r. to y ∈ RN

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(H4) ∃ a1 > 0, p > 1, r ∈ (0, 1) such that

|∇H(x, y)| ≤ a1(1 + |x|p)(1 + |y|r) ∀x, y ∈ RN

(H5) ∃ϑ > 2 such that

0 < ϑH(x, y) ≤ x∇H(x, y) ∀x ∈ RN\{0} , ∀ y ∈ RN

(H6) ∃ a2, a3 > 0 : H(x, y) ≥ a2|y|ϑ − a3 ∀x, y ∈ RN

Then there exists a positive number T0, explicitely depending on p, r, ϑ, a1, a2,a3, N , B 1, such that, for all T ∈ (0, T0) and for all T/2–periodic functions b(·),there exists a non–zero classical T–periodic and T/2–antiperiodic solution of (S).

Remark 1. From (H4) and (H5) it follows ϑ ≥ p + 1.

Remark 2. Let us point out that one can always choose, for any T0 > 0, b(t) ≡ 1,then Theorem 1 assures the existence a T–periodic and T/2–antiperiodic non–zerosolution, for any T < T0, of the autonomous Hamiltonian system

−u(t) = ∇H(u(t), u(t)) t ∈ R

if (H0), . . . , (H6) hold.

Remark 3. Let us note that, as a consequence of the T–periodicity and T/2–antiperiodicity of the solution u(·), one gets

∫ T

0

u(t)dt = 0

As a standard example of a function H satisfying (H0), . . . , (H6), one can consider∇H(x, y) = h(x, y) with

h(x, y) ={

a1x|x|p−1(1 + |y|β) , |x| ≤ δ1 , |y| ≤ δ2

x|x|p−1|y|r , |x| ≥ δ1 , |y| ≥ δ2

where p, β > 1, r ∈ (0, 1), moreover ai δj δj (j = 1, 2) are fixed positive constantswith δj > δj .

1B = max{|b(t)| : t ∈ [0, T ]}.

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2. A variational approach to the problem

Let us consider the space

V = {v ∈ H1([0, T ]; RN ) : v(0) = v(T )}

and its L2 orthogonal decomposition

V = V0 ⊕ V1 ⊕ V2

where

V0 = RN =

{1T

∫ T

0

v(t)dt : v ∈ V

}

V1 =

v ∈ V : v(t) =∑

k=2h+1h∈Z

akei2kπ

T t , a−k = ak

V2 =

v ∈ V : v(t) =∑k=2h

h∈Z\{0}

akei2kπ

T t , a−k = ak

Let us fix w in V1 and consider the following functional

Iw(v) =12

∫ T

0

|v(t)|2dt−∫ T

0

b(t)H(v(t), w(t))dt ∀ v ∈ V

Proposition 1. Let b(·) be T/2–periodic, then any critical point uw of Iw on V1 isa weak solution of the following second order Hamiltonian system with T–periodicboundary conditions

(Sw){ −uw(t) = b(t)∇H(uw(t), uw(t))

uw(0) = uw(T ) , uw(0) = uw(T )

Moreover uw is T/2–antiperiodic.

Proof . Let uw be critical on V1, that is

〈I ′w(uw), v1〉 = 0 ∀ v1 ∈ V1

We have to prove that

〈I ′w(uw) , v0 + v1 + v2〉 = 0 ∀ v = v0 + v1 + v2 ∈ V = V0 ⊕ V1 ⊕ V2

(as the critical points of Iw on the whole space V are the weak solutions of (Sw),and, if uw ∈ V1, then uw is T/2–antiperiodic).

It is enough to prove that

〈I ′w(uw), v0〉 = 〈I ′w(uw), v2〉 = 0 ∀ v0 ∈ V0 , ∀ v1 ∈ V2

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On the other side,

I ′w(uw(t)) = −u′′w(t)− b(t)∇H(uw(t), w(t))

and one has, by (H1) (H2) and the T/2–periodicity of b(·),

b(t + T/2)∇H(uw(t + T/2), w(t + T/2)) = b(t + T/2)∇H(−uw(t),−w(t)) =

= −b(t)∇H(uw(t),−w(t)) = −b(t)∇H(uw(t)), w(t))

Therefore the functionb(t)∇H(uw(t), w(t))

is T/2–antiperiodic, so it is orthogonal to V2 as well as to V0, as it has zero mean.Since −u′′w obviously belongs to V2, the proof is complete. �

In the following we will put

‖v‖ =

(∫ T

0

|v(t)|2dt

)1/2

∀ v ∈ V1

which is an equivalent norm to the H1–norm of V , as the Wirtinger inequality holdsin the space V1.

Therefore the functional Iw, for any w ∈ V1, has the form

Iw(v) =12‖v‖2 −

∫ T

0

b(t)H(v(t), w(t))dt ∀ v ∈ V1

3. Proof of Theorem 1

First of all, let us fix R > 0 and put

CR = {v ∈ V1 ∩ C2([0, T ]) : ‖v‖C2(0,T ) ≤ R}

Proposition 2. For any w ∈ CR, there exists a Mountain Pass critical point ofuw ∈ CR for Iw on V1 (as defined in Step 3).

We prove Proposition 2 by steps.

Step 1. Let w ∈ CR. Then there exists ρR, αR > 0 depending on R, but not onw such that

Iw(v) ≥ αR∀ v ∈ V1 : ‖v‖ = ρR

Proof . From (H1) it follows that, for any ε > 0, there exists δ > 0 such that

H(v(t), w(t)) <12

ε|v(t)|2 ∀ t ∈ [0, T ] , ∀ t ∈ [0, T ] , |v(t)| ≥ δ

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hence ∫ T

0

H(v(t), w(t))dt < ε/2∫ T

0

|v(t)|2dt + K(1 + R)r

∫ T

0

|v(t)|p+1dt ≤

≤ K ′(ε/2 + K(1 + R)r‖v‖p−1)‖v‖2

with a constant K ′ depending on the Wirtinger and Sobolev inequalities. Choosing

‖v| =(

ε

2K(1 + R)r

)1/p−1

= ρR

one gets ∫ T

0

H(v, w) ≤ K ′ε‖v‖2

Recalling that B = maxt∈[0,T ]

|b(t)|, if one chooses ε < 12BK′ and αR =

(12 −K ′ε

)ρ2

R,

the thesis follows. �

Step 2. Let w ∈ CR and let us fix v in V1 with ‖v‖ = 1. Then there exists s > 0,independent of w and R, such that

(1) Iw(sv) ≤ 0 ∀ s ≥ s ,

then v = sv satisfies the relations

‖v‖ > ρR Iw(v) ≤ 0

Proof . It follows from (H6) that

Iw(sv) ≤ 12

s2 − a2|s|ϑ∫ T

0

|v|ϑdt + a3T

By the Sobolev embedding theorem as ϑ ≤ p + 1 (see Remark 1), one gets

Iw(sv) ≤ 12

s2 − a2|s|ϑ(Sϑ)ϑ + a3T

where Sϑ is the embedding constant of V1 in Lϑ([0, T ]). Since ϑ > 2, one gets somes such that (1) holds.

Step 3. Let w ∈ CR. Then there exists a Mountain Pass critical pont uw for Iw onV1, that is

(2) Iw(uw) = infγ∈T

maxt∈[0,1]

I(γ(t))

whereΓ = {γ ∈ C0([0, 1]; RN ) : γ(0) = 0 , γ(1) = v}

andIw(uw) ≥ αR > 0 (⇒ uw 6≡ 0)

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Proof . It is a consequence of the theorem by Ambrosetti and Robinowitz (see [1]),as Iw(0) = 0, Step 1 and Step 2 hold, and the Palais–Smale condition is triviallysatisfied, due to the continuous embedding of V1 into L∞(0, T ; RN ) and the factthat p > 2. �

Proposition 3. Let w ∈ CR and let uw be a Mountain Pass solution given byProposition 2. Then there exists a positive constant c1(R) depending on R, butnot on w, such that

‖uw‖ ≥ c1(R)

Proof . Actually the estimate holds for any critical point uw of Iw on V1 with uw 6≡ 0and one does not use the Mountain Pass nature of uw. Indeed, if one puts in therelation ∫ T

0

uwv =∫ T

0

b(t)∇H(uw, w)v ∀ v ∈ V1

v = uw, one gets

(3) ‖uw‖2 =∫ T

0

b(t)∇H(uw, w)uwdt

From (H3), (H4), (H5), it follows that, for any ε > 0, there exists a positiveconstant cε, R, depending on ε and R, but not on w, such that

|∇H(uw, w)| ≤ ε|uw|+ cε,R|uw|p

Using this inequality, (3) yields

‖uw‖2 ≤ B

(ε

∫ T

0

|uw|2 + cε,R

∫ T

0

|uw|p+1

),

hence, by the Wirtinger inequality and the continuous Sobolev embedding,

(1− ε(T/2π)2)‖uw‖2 ≤ cε,R‖uw‖p+1

which implies the thesis choosing ε <(

2πT

)2, as p + 1 > 2. �

Proposition 4. Let w ∈ CR and let uw be a Mountain Pass solution given byProposition 2. Then there exists a constant c2 > 0, independent of w and R, suchthat

‖uw‖ ≤ c2

Proof . From the inf–maximum characterization of uw, by choosing γ in Γ as thesegment line joining 0 and v, one gets

Iw(uw) ≤ sups≥0

Iw(sv) ,

hence, by (H6),

Iw(uw) ≤ B sups≥0

{s2

2

∫ T

0

|v|2 − a2|s|ϑ∫ T

0

|v|ϑ + a3T

}

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Since ϑ > 2, such an upper bound is a maximum and it does not depend on R andw, hence

Iw(uw) ≤ const ∀R > 0 , ∀w ∈ CR

At this point, using the criticality of uw for Iw, (H5) and (3), one gets

12‖uw‖2 ≤ const +

1ϑ

∫ T

0

b(t)∇H(uw, w)uw = const +1ϑ‖uw‖2

by which the estimate follows, as ϑ > 2. �

Actually, for any w ∈ CR, any Mountain Pass solution of (Sw) is not only weak,but just of class C2, so a classical solution, since it solves a problem of the type

(4){ −uw(t) = ϕ(t)(= b(t)∇H(uw(t), w(t))) t ∈ [0, T ]

uw(0) = uw(T ) , uw(0) = uw(T )

where ϕ belongs to C0([0, T ]).

Proposition 5. There exists a constant µ > 0 such that the Mountain Passsolutions uw of (4) verify, for any R > 0, and any w ∈ CR,

‖uw‖C2([0,T ]) ≤ µ(1 + Rr)

Proof . It is a consequence of (H4), of Proposition 4 and the fact that uw solves(4) in the classical sense. �

Proposition 6. There exists a constant R > 0 such that

w ∈ CR ⇒ uw ∈ CR

for any Mountain Pass solution uw of (4).

Proof . It easily follows from Proposition 5 and the fat that r < 1. �

At this point it is very natural to introduce an iterative scheme in the followingway. Let R be given by Proposition 6 and let u0 be arbitrarily fixed in CR. Let usdefine un as a Mountain Pass solution the following problem, for any n ∈ N,

(Sn){ −un(t) = b(t)∇H(un(t), un−1(t)) t ∈ [0, T ]

un(0) = un(T ) , un(0) = un(T )

Obviously, by Proposition 6, one has, for any n ∈ N,

un ∈ CR

Now we are in a position to give the

Proof of Theorem 1 . Let us consider un given as a Mountain Pass solution of (Sn)for any n ∈ N. First of all we prove that there exists some positive number T0, suchthat, if T < T0, then the whole sequence {un} strongly converges in V1.

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Indeed, using (sn) and (sn+1), one gets∫ T

0

un+1(t)(un+1(t)− un(t))dt =∫ T

0

b(t)∇H(un+1(t), un(t))(un+1(t)− un(t))dt∫ T

0

un(t)(un+1(t)− un(t))dt =∫ T

0

b(t)∇H(un(t), un−1(t))(un+1(t)− un(t))dt

which yields

‖un+1 − un‖2 ≤ B

∫ T

0

|∇H(un+1(t), un(t))−∇H(un(t), un−1(t))‖un+1(t)− un(t)|dt ≤

≤ B

∫ T

0

|∇H(un+1(t), un(t))−∇H(un(t), un(t))‖un+1(t)− un(t)|dt+(5)

+ B

∫ T

0

|∇H(un(t), un(t))−∇H(un(t), un−1(t))| |un+1(t)− un(t)|dt

Denoting by c′R, c′′

Rthe best Lipschitz constants of ∇H w.r. to its two variables in

the set BR ×BR where

BR = {x ∈ RN : |x| ≤ R} ,

one gets from (5) the relation

‖un+1−un‖2 ≤ B

c′R

∫ T

0

|un+1 − un|2dt + c′′R‖un − un−1‖

(∫ T

0

|un+1 − un|2dt

)1/2

Using the Wirtinger inequality (as un and un+1 are T–periodic with zero mean),one obtains

‖un+1−un‖2 ≤ B

(c′R

(T

2π

)2

‖un+1 − un‖2 + c′′R

(T

2π

)‖un − un−1‖ ‖un+1 − un‖

)from which

(6) ‖un+1 − un‖ ≤Bc′′

R

(T2π

)1−Bc′

R

(T2π

)2 ‖un − un−1‖ = γ‖un − un−1‖

ifT <

2π√Bc′R

Actually, by putting

(7) T0 = min

2π√Bc′R

,

√B2(c′′

R)2 − 4Bc′

R−Bc′′R

2Bc′R

and if T < T0, the constant γ in (6) is less than 1. Therefore if T < T0, given by(7), then (6) implies that {un} is a Cauchy sequence in V1, so it strongly convergesto some u in V1.

At this point, from the Ascoli–Arzela’s thereom and the fact that {un} is con-tained in CR, it follows that the whole sequence {un} converges in C2([0, T ]), then,as it easily verified, to a classical solution u of (S). The fact that u is not identicallyzero is an immediate consequence of Proposition 3, putting R = R. �

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References

[1] A. Ambrosetti, P.H. Rabinowitz, “Dual variational methods in critical pointtheory and applications”, J. Funct. Analysis 14, (1973), 349–381.

[2] D. De Figuereido, M. Girardi, M. Matzeu, “Semilinear elliptic equationswith dependence on the gradient via mountain–pass tecniques”, Differential andIntegral Equations 17, 1–2 (2004), 119–126.

[3] M. Girardi, M. Matzeu, “Positive and negative solutions of a quasi–linearequation by a Mountain–Pass method and truncature techniques”, NonlinearAnalysis 59 (2004), 199–210.

[4] M. Girardi, S. Mataloni, M. Matzeu, Mountain Pass techniques for someclasses of nonvariational problems, to appear on Proceedings ISAAC 2005.

[5] S. Mataloni, M. Matzeu, “Semilinear integrodifferential problems with non–symmetric kernels via mountain–pass techniques”, Advanced Nonlinear Sutides5 (2005), 23–31.