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Applied Mathematics
Letters
Applied Mathematics Letters 16 (2003) 639-642
PERGAMON
www.elsevier.com/locate/aml
Existence of Homoclinic Orbits for Asymptotically Periodic Systems Involving Duffing-Like Equation C. 0. ALVES Universidade Federal da Paraiba, Departamento de MatemBtica 58100-970 Campina GrandePB, Brazil coalvesQdme.ufpb.br P. C. CARRI~O Universidade Federal de Minas Gerais, Departamento de Matemktica 31270-010 Belo Horizon&MG, Brazil carrion&nat.ufmg.br 0. H. MIYAGAKI* Universidade Federal de Vicosa, Departamento de Matemiitica 36571-OOO-ViCosa-MG,Brazil [email protected] (Received November 2001; accepted October 2002) Abstract-we
are concerned with perturbations of the Hamiltonian system of the type c - L(t)q + W&, q) = 0,
t E R,
WI
where q = (ql, , qN) E WN, W E C1(W x WN,R), and L(t) E C(W,WN2) is a positive definite symmetric matrix. Variational arguments are used to prove the existence of homoclinic solutions for system (HS). @ 2003 Elsevier Science Ltd. All rights reserved.
Keywords-
Homoclinic orbits, Duffing equations, Critical points.
1. INTRODUCTION Many papers have dealt with the existence of homoclinic a nontrivial function q E H’(W,l@) second-order
time-dependent
Hamiltonian
ii whereq=
(ql,...
orbits to the origin in H1(R, RN), i.e.,
such that q(t) and 4(t) + 0 as ItI + +co, which verifies the systems of the type
W)q + W,(t,
, qN) E WN, W E C1(W
x
q) = 0,
RN,W), and
t E w,
(HS)
L(t) E C(W, RN’) is a positive definite
symmetric matrix for all t. The case where L(t) and W(t, q) are either periodic in t or independent Research supported by CNPq-Brasil. *Author to whom all correspondenceshould be addressed. 0893-9659/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO893-9659(03)00059-4
Typeset by &+W
C. 0. ALVES et al
640
oft were studied in several articles, for example in [l-5], and references therein. The existence of a homoclinic solution can be obtained by going to the limit of periodic solutions of approximating problems on expanding interval; in this argument the variational method can be applied to solve the approximated problems as well as to obtain a good estimates for their solutions. This approach has been considered in [1,‘2,5]. System (HS) without periodicity assumption on both L and I& was considered in [S] assuming that the smallest eigenvalue of L(t) tends to +co as ]t] --f +zc (see also [7,8]), and in [9] assuming that L(t)q = a(t)lNq with a(t) ---f a0 > 0 as Jtj --) +CU, where 1, denotes the identity matrix. In articles [6-g], the authors used a variant of the Mountain Pass theorem without (PS) condition (see [lo]). For the particular case N = 1, system (HS) is related to an equation introduced by Duffing (see [ll]), namely
the equation
of the form ii - U + a(t)]+-lu.
t fs R.
= 0,
!JJj
The class of Duffing equation represents a model for the forced vibrations of a cantilever beam in the nonuniform field of two permanent magnets. Recently, in [12] (see also [13]) was proved the existence of homoclinic orbits of (D) for the asymptotically periodic case; that is, a : IR -+ IR is a positive function in Lo3(R), a(t) = am(t) + so(t) such that a, is a function T-periodic. at(t) -+ 0 as t + foe, liminfltl+M at(t) exp (@It]) > -co when 0 > 2 and p > 1. In the present paper, by employing the variational approach we consider a perturbation of a periodic system (HS) in RN involving more general time dependence, not necessarily with the nonlinearity of the power type, without any condition from above on growth at infinity in IV with respect to q. Here the main feature of problem (HS) is the lack of global compactness due to unboundedness of the domain implying in a failure of the Sobolev compact embedding of H1(IW, RN) into LP(Iw, RN). To overcome the above difficulties, we adapt some arguments explored by authors in [14] and a result due to Lions proved in [2]. In order to state our main result, we need to fix some notations and assumptions on L and W. We assume L(t)q.q := L(t)q2 = L”(t)q2 + L”(t)q2, where L”(t) and Lo(t) are symmetric matrices, satisfying L(t)8 2 G1412, LV)q2 > %1412, a,,> 0, t E R, CLJ L” is l-periodic
function
in t, and L”(t)qX 5 0,
With respect to W, we assume that l-periodic function in t, and
Lo(t) + 0,
=
ItI + co,
W = W”” + W”, with W”,
W(Gq)
2 W03(t>Q)l
iA,j
W” in C’(Iw x RN,IRj, W”’ IS
t E B,
WI)
where at least one of the inequalities in (Am) and (Hi) is strict on a subset of positive measure in R. Moreover, we suppose the following conditions. For all q # 0 fixed, the requirement below holds uniformly in t # 0
Wq% 7d4 7 and for each K c RN
compact
IS strictly
increasing
with respect to T > 0,
032)
set, given c > 0 there exists M > 0 such that
p”), (t+d~ 2
(*1) i*2)
641
Existence of Homoclinic Orbits
In our approach, the periodic case related to (HS) studied in [S], more precisely, the system ii -
L”(t)q + W,oo(& q) = 0,
tent,
qE@,
PWco
will be very important to prove the existence of solution for the perturbed problem. We are now ready to state our main result. THEOREM
1. Assume (L) and that W verifies (*).
(i) IfLo = W” = 0, then there exists a homoclinic orbit q of (HS), emanating from 0. (ii) Suppose that W” verifies (*) and that (Hi), (Hz), (H,), and (A,) hold. Then, there exists a homoclinic orbit q of (HS) emanating from 0. In spite of Theorem l(i) having already been proved in [S], we will present a new proof of this result employing the variational method, getting more information on the behaviours of the critical levels of the associated functional defined in whole R, which have important roles in our arguments. 2.
PROOFS
First, we will give some notations, definitions, and basic facts. From now on, J h denotes JRh(t) dt. Let E = H’(B,RN) be endowed with the norm given by 11q112 = J(lg12 + L(t)q2). For q E E, letting
I”(q) = i / ( M2+ t”(t)s2) - /WV,
4)
and
I(q) = f IId - / WC44,
then I”,
I E Cl(E,R) (see, e.g., [4]). Also we know that any critical point of I on E is a homoclinic orbit of (HS) (for a proof, check for instance IS]). Similarly, the critical points of 10° are solutions of periodic system (HS),. In the sequel, for a functional J on E, we say that {qn} C E is a (PS)d,J-sequence at the level d E R, if
J(q,) -
d and
J’(q,) -
0,
asn+co.
P%,J
l(i). Since Woo satisfies (*), from Hypothesis (L) we can show that the Mountain Pass geometry holds for functional P; thus (see, e.g., [lo]), there exists {qn} c E which is a (PS) .,I~-sequence and PROOF OF THEOREM
3 q E E, such that qn -
q, weakly in E, as n --+ 00 and I”‘(q)
= 0,
where c > 0 is the minimax level related to I”. If q # 0, then q is a solution of ((HS),) and it can be proved that I”(q) = c (see [14]). If not, since c > 0 we infer that jlqnll ft 0, as n + co. From the periodicity of L” and Wm, by applying a version of the compactness result of Lions due to Arioli and Szulkin [2, Lemma 81there exists yn E Z and R > 0 such that &(t) = qn(t + yn) is a (PS),,r--sequence and 3 @ E E, such that & Moreover, we can prove that P’(g)
S, weakly in E, and
= 0 and I”(Q)
J
ItI2 > 0.
= c. This proves (i).
PROOF OF THEOREM l(ii). By (*) for W and from (L) and (A,) follow that I satisfies Mountain Pass geometry, so that there exists a (PS),. ,I-sequence {qn} in E, where c* is the positive minimax level related to I. As in the proof of (i), there exists Q E E such that qn - q and I’(q) = 0. Therefore, Q is a weak solution of (HS). nrovided that B f 0. We will show that 4 # 0. Supnose on the contrarv
C. 0. ALVES et al.
642
that 4 = 0, and thus, qn - 0 weakly in E asn -+ 0.Using (A,) and (H,) we reach that {qn} is a (PS),.,fm-sequence. Now, by (A,) and (HI), and recalling that there exists q E E such that I”(q) = c (see (i)), we conclude that c* < c. Using again the same arguments explored in (i), there exists &, E Z such that the sequence &(t) = qn(t + &) verifies
- -GEE, qn Arguing
I”(@)
as in [15], the following inequality c < yl,yP
_
which is a contradiction.
=c*,
and
Ioo’ (Q) = 0.
follows from (Hz) and the Fatou lemma: (7-4) = IO0 (4) =
Then the proof of Theorem
c*,
1 is completed.
I
REFERENCES 1. A. Ambrosetti and M. Bertotti, Homoclinic for second order conservative systems, In P.D.E. and Related Subjects, (Edited by M. Miranda), pp. 21-37, Pitman, Essex, (1992). 2. G. Arioli and A. Szulkin, Homoclinic solutions for a class of systems of second order differential equations, Volume 5, Report of Dept. Math., Univ. Stockholm, Sweden, (1995). 3. V. Coti-Zelati, I. Ekeland and E. S&e, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288, 133-160, (1990). 4. V. Coti-Zelati and P.H. Babinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Sot. 4, 693-727, (1992). 5. P.H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Royal Sot. Edinburgh 114A, 33-38, (1990). 6. P.H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206, 473-499, (1991). 7. Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Analysis 25, 1095-1113, (1995). 8. W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, DQeerential and Integral Equations 5, 1115-1120, (1992). 9. P.C. Carri&o and O.H. Miyagaki, Existence of homoclinic solutions for a class of time-dependent Hamiltoman systems, J. Math. Anal. Appl. 230, 157-172, (1999). 10. M. Willem, Minimar Theorems, Birkhauser, Boston, MA, (1996). 11. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, (1983). 12. F. Alessio, P. Caldiori and P. Montecchiari, On the existence of homoclinic orbits for asymptotically periodic Duffing equation, Top. Meth. Nonl. Analysis 12, 275-292, (1998). 13. G. Spradlin, A Hamiltonian system with an even term, Top. Meth. Nonl. Analysis 10, 93-106, (1997). 14. C.O. Alves, P.C. Carriao and O.H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth, J. Math. Anal. Appl. 280, 133-146, (2001). 15. Y. Ding and W.Y. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Me&. Anal. 91, 283-308, (1986).
Letters
Applied Mathematics Letters 16 (2003) 639-642
PERGAMON
www.elsevier.com/locate/aml
Existence of Homoclinic Orbits for Asymptotically Periodic Systems Involving Duffing-Like Equation C. 0. ALVES Universidade Federal da Paraiba, Departamento de MatemBtica 58100-970 Campina GrandePB, Brazil coalvesQdme.ufpb.br P. C. CARRI~O Universidade Federal de Minas Gerais, Departamento de Matemktica 31270-010 Belo Horizon&MG, Brazil carrion&nat.ufmg.br 0. H. MIYAGAKI* Universidade Federal de Vicosa, Departamento de Matemiitica 36571-OOO-ViCosa-MG,Brazil [email protected] (Received November 2001; accepted October 2002) Abstract-we
are concerned with perturbations of the Hamiltonian system of the type c - L(t)q + W&, q) = 0,
t E R,
WI
where q = (ql, , qN) E WN, W E C1(W x WN,R), and L(t) E C(W,WN2) is a positive definite symmetric matrix. Variational arguments are used to prove the existence of homoclinic solutions for system (HS). @ 2003 Elsevier Science Ltd. All rights reserved.
Keywords-
Homoclinic orbits, Duffing equations, Critical points.
1. INTRODUCTION Many papers have dealt with the existence of homoclinic a nontrivial function q E H’(W,l@) second-order
time-dependent
Hamiltonian
ii whereq=
(ql,...
orbits to the origin in H1(R, RN), i.e.,
such that q(t) and 4(t) + 0 as ItI + +co, which verifies the systems of the type
W)q + W,(t,
, qN) E WN, W E C1(W
x
q) = 0,
RN,W), and
t E w,
(HS)
L(t) E C(W, RN’) is a positive definite
symmetric matrix for all t. The case where L(t) and W(t, q) are either periodic in t or independent Research supported by CNPq-Brasil. *Author to whom all correspondenceshould be addressed. 0893-9659/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO893-9659(03)00059-4
Typeset by &+W
C. 0. ALVES et al
640
oft were studied in several articles, for example in [l-5], and references therein. The existence of a homoclinic solution can be obtained by going to the limit of periodic solutions of approximating problems on expanding interval; in this argument the variational method can be applied to solve the approximated problems as well as to obtain a good estimates for their solutions. This approach has been considered in [1,‘2,5]. System (HS) without periodicity assumption on both L and I& was considered in [S] assuming that the smallest eigenvalue of L(t) tends to +co as ]t] --f +zc (see also [7,8]), and in [9] assuming that L(t)q = a(t)lNq with a(t) ---f a0 > 0 as Jtj --) +CU, where 1, denotes the identity matrix. In articles [6-g], the authors used a variant of the Mountain Pass theorem without (PS) condition (see [lo]). For the particular case N = 1, system (HS) is related to an equation introduced by Duffing (see [ll]), namely
the equation
of the form ii - U + a(t)]+-lu.
t fs R.
= 0,
!JJj
The class of Duffing equation represents a model for the forced vibrations of a cantilever beam in the nonuniform field of two permanent magnets. Recently, in [12] (see also [13]) was proved the existence of homoclinic orbits of (D) for the asymptotically periodic case; that is, a : IR -+ IR is a positive function in Lo3(R), a(t) = am(t) + so(t) such that a, is a function T-periodic. at(t) -+ 0 as t + foe, liminfltl+M at(t) exp (@It]) > -co when 0 > 2 and p > 1. In the present paper, by employing the variational approach we consider a perturbation of a periodic system (HS) in RN involving more general time dependence, not necessarily with the nonlinearity of the power type, without any condition from above on growth at infinity in IV with respect to q. Here the main feature of problem (HS) is the lack of global compactness due to unboundedness of the domain implying in a failure of the Sobolev compact embedding of H1(IW, RN) into LP(Iw, RN). To overcome the above difficulties, we adapt some arguments explored by authors in [14] and a result due to Lions proved in [2]. In order to state our main result, we need to fix some notations and assumptions on L and W. We assume L(t)q.q := L(t)q2 = L”(t)q2 + L”(t)q2, where L”(t) and Lo(t) are symmetric matrices, satisfying L(t)8 2 G1412, LV)q2 > %1412, a,,> 0, t E R, CLJ L” is l-periodic
function
in t, and L”(t)qX 5 0,
With respect to W, we assume that l-periodic function in t, and
Lo(t) + 0,
=
ItI + co,
W = W”” + W”, with W”,
W(Gq)
2 W03(t>Q)l
iA,j
W” in C’(Iw x RN,IRj, W”’ IS
t E B,
WI)
where at least one of the inequalities in (Am) and (Hi) is strict on a subset of positive measure in R. Moreover, we suppose the following conditions. For all q # 0 fixed, the requirement below holds uniformly in t # 0
Wq% 7d4 7 and for each K c RN
compact
IS strictly
increasing
with respect to T > 0,
032)
set, given c > 0 there exists M > 0 such that
p”), (t+d~ 2
(*1) i*2)
641
Existence of Homoclinic Orbits
In our approach, the periodic case related to (HS) studied in [S], more precisely, the system ii -
L”(t)q + W,oo(& q) = 0,
tent,
qE@,
PWco
will be very important to prove the existence of solution for the perturbed problem. We are now ready to state our main result. THEOREM
1. Assume (L) and that W verifies (*).
(i) IfLo = W” = 0, then there exists a homoclinic orbit q of (HS), emanating from 0. (ii) Suppose that W” verifies (*) and that (Hi), (Hz), (H,), and (A,) hold. Then, there exists a homoclinic orbit q of (HS) emanating from 0. In spite of Theorem l(i) having already been proved in [S], we will present a new proof of this result employing the variational method, getting more information on the behaviours of the critical levels of the associated functional defined in whole R, which have important roles in our arguments. 2.
PROOFS
First, we will give some notations, definitions, and basic facts. From now on, J h denotes JRh(t) dt. Let E = H’(B,RN) be endowed with the norm given by 11q112 = J(lg12 + L(t)q2). For q E E, letting
I”(q) = i / ( M2+ t”(t)s2) - /WV,
4)
and
I(q) = f IId - / WC44,
then I”,
I E Cl(E,R) (see, e.g., [4]). Also we know that any critical point of I on E is a homoclinic orbit of (HS) (for a proof, check for instance IS]). Similarly, the critical points of 10° are solutions of periodic system (HS),. In the sequel, for a functional J on E, we say that {qn} C E is a (PS)d,J-sequence at the level d E R, if
J(q,) -
d and
J’(q,) -
0,
asn+co.
P%,J
l(i). Since Woo satisfies (*), from Hypothesis (L) we can show that the Mountain Pass geometry holds for functional P; thus (see, e.g., [lo]), there exists {qn} c E which is a (PS) .,I~-sequence and PROOF OF THEOREM
3 q E E, such that qn -
q, weakly in E, as n --+ 00 and I”‘(q)
= 0,
where c > 0 is the minimax level related to I”. If q # 0, then q is a solution of ((HS),) and it can be proved that I”(q) = c (see [14]). If not, since c > 0 we infer that jlqnll ft 0, as n + co. From the periodicity of L” and Wm, by applying a version of the compactness result of Lions due to Arioli and Szulkin [2, Lemma 81there exists yn E Z and R > 0 such that &(t) = qn(t + yn) is a (PS),,r--sequence and 3 @ E E, such that & Moreover, we can prove that P’(g)
S, weakly in E, and
= 0 and I”(Q)
J
ItI2 > 0.
= c. This proves (i).
PROOF OF THEOREM l(ii). By (*) for W and from (L) and (A,) follow that I satisfies Mountain Pass geometry, so that there exists a (PS),. ,I-sequence {qn} in E, where c* is the positive minimax level related to I. As in the proof of (i), there exists Q E E such that qn - q and I’(q) = 0. Therefore, Q is a weak solution of (HS). nrovided that B f 0. We will show that 4 # 0. Supnose on the contrarv
C. 0. ALVES et al.
642
that 4 = 0, and thus, qn - 0 weakly in E asn -+ 0.Using (A,) and (H,) we reach that {qn} is a (PS),.,fm-sequence. Now, by (A,) and (HI), and recalling that there exists q E E such that I”(q) = c (see (i)), we conclude that c* < c. Using again the same arguments explored in (i), there exists &, E Z such that the sequence &(t) = qn(t + &) verifies
- -GEE, qn Arguing
I”(@)
as in [15], the following inequality c < yl,yP
_
which is a contradiction.
=c*,
and
Ioo’ (Q) = 0.
follows from (Hz) and the Fatou lemma: (7-4) = IO0 (4) =
Then the proof of Theorem
c*,
1 is completed.
I
REFERENCES 1. A. Ambrosetti and M. Bertotti, Homoclinic for second order conservative systems, In P.D.E. and Related Subjects, (Edited by M. Miranda), pp. 21-37, Pitman, Essex, (1992). 2. G. Arioli and A. Szulkin, Homoclinic solutions for a class of systems of second order differential equations, Volume 5, Report of Dept. Math., Univ. Stockholm, Sweden, (1995). 3. V. Coti-Zelati, I. Ekeland and E. S&e, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288, 133-160, (1990). 4. V. Coti-Zelati and P.H. Babinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Sot. 4, 693-727, (1992). 5. P.H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Royal Sot. Edinburgh 114A, 33-38, (1990). 6. P.H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206, 473-499, (1991). 7. Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Analysis 25, 1095-1113, (1995). 8. W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, DQeerential and Integral Equations 5, 1115-1120, (1992). 9. P.C. Carri&o and O.H. Miyagaki, Existence of homoclinic solutions for a class of time-dependent Hamiltoman systems, J. Math. Anal. Appl. 230, 157-172, (1999). 10. M. Willem, Minimar Theorems, Birkhauser, Boston, MA, (1996). 11. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, (1983). 12. F. Alessio, P. Caldiori and P. Montecchiari, On the existence of homoclinic orbits for asymptotically periodic Duffing equation, Top. Meth. Nonl. Analysis 12, 275-292, (1998). 13. G. Spradlin, A Hamiltonian system with an even term, Top. Meth. Nonl. Analysis 10, 93-106, (1997). 14. C.O. Alves, P.C. Carriao and O.H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth, J. Math. Anal. Appl. 280, 133-146, (2001). 15. Y. Ding and W.Y. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Me&. Anal. 91, 283-308, (1986).
