Differential Equations, Vol. 40, No. 4, 2004, pp. 502–514. Translated from Differentsial’nye Uravneniya, Vol. 40, No.

4, 2004, pp. 465–476.

Original Russian Text Copyright
c 2004 by Marino, Toskano.

ORDINARY
DIFFERENTIAL EQUATIONS

Periodic Solutions of a Class of Nonlinear Systems

of Ordinary Differential Equations
G. Marino and R. Toskano
Università di Napoli “Federico II”, Naples, Italy
Seconda Università di Napoli, Caserta, Italy
Received May 20, 2003

1. INTRODUCTION
Let V be the space of functions v = (v1 , . . . , vn ), whose components vi ∈ W 1,p (]0, T [) are
T -periodic functions, equipped with the norm
 1/p
1/p
n T 
  n
v ≡ vV =  |vi (t)| + |vi (t)| dt =

p p p
vi 1,p for 1 < p.
i=1 0 i=1

 
Let fi ∈ (W 1,p (]0, T [)) ((W 1,p (]0, T [)) be the dual space of W 1,p (]0, T [)), and let ai (t, ξ) and
bij (t, ξ) (j = 1, . . . , m) be functions are defined on R × Rn [where ξ = (ξ1 , . . . , ξn )], T -periodic with
respect to t, and satisfying the Carathéodory condition and the requirement ai (t, u ) ,


bij (t, u) ∈ Lp (]0, T [) for all u = (u1 , . . . , un ) ∈ V ; here u = du/dt and p = p/(p − 1). We consider
the problem

d  m
− (ai (t, u1 (t), . . . , un (t))) + λ |ui (t)|
p−2
ui (t) = bij (t, u1 (t), . . . , un (t)) + fi ,
dt j=1 (1.1)
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},

where λ is a real nonnegative parameter. Let us introduce the following notion.

Definition 1.1. We say that u = (u1 , . . . , un ) ∈ V is a weak solution of system (1.1) if

n 

T n 

T

ai (t, u1 (t), . . . , un (t))vi (t)dt

p−2
+λ |ui (t)| ui (t)vi (t)dt
i=1 0 i=1 0

 m 
n  T

n
= bij (t, u1 (t), . . . , un (t)) vi (t)dt + fi , vi 
i=1 j=1 0 i=1

for all v = (v1 , . . . , vn ) ∈ V .

In the last thirty years, numerous authors have been studying the existence of periodic solutions
of second-order ordinary differential equations (e.g., see [1] and the more recent papers [2–6])
with the use of various variational techniques. In particular, in the last ten years, studies were
concentrated on the existence of multiple periodic solutions (e.g., see [7–9]). The present paper is
also in this direction. It is based on using the Lagrange multipliers in combination with fibering
method if necessary. This idea was suggested by S.I. Pokhozhaev (for basic notions illustrated by
examples, see [10]) and was later used by numerous researchers for the investigation of nonlinear
boundary value problems. In [11], Pokhozhaev used his method for proving the existence of at

0012-2661/04/4004-0502

c 2004 MAIK “Nauka/Interperiodica”
 PERIODIC SOLUTIONS OF A CLASS OF NONLINEAR SYSTEMS 503

least one periodic solution of nonlinear systems of ordinary differential equations. We have another
situation, which will be explained below.
We introduce the following assumptions:
(i1 ) there exist functions A : V → R and Bj : V → R of class C 1 in V \{0} such that

A(·) = 0, Bj (·) = 0 and Bj (·) ≥ 0,

n 

T

dA(u), v = ai (t, u1 , . . . , un ) vi dt,

i=1 0

n 

T

dBj (u), v = bij (t, u1 , . . . , un ) vi dt ∀u ∈ V \{0}, ∀v ∈ V ;

i=1 0

(i2 ) there exists a constant c1 > 0 such that

n 

T

|ui (t)| dt ≤ A(u)

p
c1 ∀u ∈ V.
i=1 0

Moreover, we impose the following additional conditions on A(·) and Bj (·) :

(i3 ) A (ru1 , . . . , run ) = |r|p A (u1 , . . . , un ) for all u ∈ V and r ∈ R;
(i4 ) Bj (ru1 , . . . , run ) = |r|γj Bj (u1 , . . . , un ) for all u ∈ V and r ∈ R, where 1 < γj for
j = 1, . . . , m;
(i5 ) Bj (·) is weakly continuous on V ;
(i6 ) A(·) is weakly lower semicontinuous on V .
Consequently, the Euler functional corresponding to (1.1) has the form

T
λ  
n m n
p
E(u) = A(u) + |ui (t)| dt − Bj (u) − fi , ui  .
p i=1 j=1 i=1
0

Moreover, from (i3 ) and (i4 ), we have

dA(u), u = pA(u), dBj (u), u = γj Bj (u) ∀u ∈ V \{0},

dA(ru), v = |r| rdA(u), v, dBj (ru), v = |r|γj −2 r dBj (u), v
p−2

∀u ∈ V \{0}, ∀v ∈ V, ∀r ∈ R\{0}.

In Sections 2 and 3, problem (1.1) is considered in the case in which λ > 0. In Section 2, we prove
the existence of at least one nontrivial solution of the homogeneous problem [f = (f1 , . . . , fn ) = 0]
all of whose components are nonnegative if γ1 = · · · = γm = γ = p as well as if p < γj , where all γj
do not necessarily coincide. Moreover, the existence of countably many solutions of this problem is
provided under appropriate assumptions about A(·) and Bj (·). Since the nonhomogeneous problem
(f = 0) is considered, it follows that one can prove the existence of at least three different solutions
if p < γj with γj equal or not equal to each other.
The homogeneous problem (with 2 < γj < 3 < p) and the nonhomogeneous problem (with p > γ
or 2 < γj < 3 < p) are considered in Section 3. For the first, we prove the existence of at least
one solution just as above; for the other, we justify the existence of at least two different solutions.
Some of these results require additional assumptions. For the nonhomogeneous problem, important
is the requirement of the sufficiently small sum of norms of the functionals fi .
The case in which λ = 0 is considered in Section 4. Since A(·) is coercive, we have to consider a
variational problem with constraints expressed by n+1 equations. Thus we prove (under additional
assumptions including those for norms of fi ) only the solvability of problem (1.1) for γ1 = · · · =

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

504 MARINO, TOSKANO

γm = γ : the existence of at least one nontrivial solution of the homogeneous problem with p = γ;
the existence of at least two or three different solutions of the nonhomogeneous problem with p < γ
(see Theorem 4.2). In Section 5, we consider problem (1.1) in some special cases.

2. THE CASE IN WHICH λ > 0

To consider this case, we set
 

m
V1 = v∈V : Bj (v) > 0 ,
j=1
 
 n  T

λ p
Σ= v ∈ V : H(v) = A(v) + |vi (t)| dt = 1 .
 p i=1 0


Note that V1 is an open set and V1 ∩ Σ = ∅.

Consider the homogeneous problem

d  m
− (ai (t, u1 (t), . . . , un (t))) + λ |ui (t)|
p−2
ui (t) = bij (t, u1 (t), . . . , un (t)) ,
dt j=1 (2.1)
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},

and distinguish two cases:

(h1 ) γ1 = γ2 = · · · = γm = γ, γ = p;
(h2 ) 1 < p < γj , where γ1 , . . . , γm are not all equal to the same value.
Let us first prove the following theorem.

Theorem 2.1. In both cases (h1 ) and (h2 ), under assumptions (i1 )–(i6 ), problem (2.1) has at
least two nontrivial solutions with opposite signs.
m
Proof. In case (h1 ), we set F (u) = j=1 Bj (u) and consider the variational problem P1 with
constants: find a maximizing element u0 = (u01 , . . . , u0n ) ∈ V , M = sup {F (u) : u ∈ V1 ∩ Σ}.
   
Obviously, M > 0. Let uk be a maximizing sequence of problem P1 ; thus F uk → M and
 
H uk = 1. (2.2)
 
By (2.2) and condition (i2 ), uk is a bounded sequence in V . Then there exists a u0 ∈ V such that
uk → u0 weakly in V (along a sequence), and H (u0 ) ≤ 1 by virtue of assumption (i6 ). It follows
from assumption (i5 ) that F (·) is a weakly continuous functional on V ; therefore, F uk → F (u0 ).
This implies that F (u0 ) = M . To show that H (u0 ) = 1, we firstly note that H (u0 ) > 0, since
otherwise, we would have u0 = 0 and condition (i4 ) would imply that F (u0 ) = 0. On the other
−1/p γ
hand, if H (u0 ) < 1, then, by setting t0 = [H (u0 )] , we obtain F (t0 u0 ) = (t0 ) F (u0 ) > M .
This contradicts the definition of M . Therefore, there exists a Lagrange multiplier τ ∈ R such that
  0     
dF u , v = τ dH u0 , v ∀v ∈ V.

Hence it follows that τ = γM/p > 0; consequently, τ 1/(p−γ) u0 is a nontrivial solution of prob-
lem (2.1).
In case (h2 ), it is impossible to use the above-represented proof, since some of the γj are not
equal to each other. Therefore, we use the fibering method. It also can be used in case (h1 ) with
p < γ as well. In accordance with that method, we find critical points of the Euler functional E(·)
of the form u = rv, where r ∈ R\{0} and v ∈ V , with the corresponding constraints for v.

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

 PERIODIC SOLUTIONS OF A CLASS OF NONLINEAR SYSTEMS 505

For any v ∈ V1 ∩ Σ, we consider the above-represented functional


m
Ē(r, v) = E(rv) = |r|p − |r|γj Bj (v)
j=1

and the bifurcation equation


m
p|r|p−2 r − γj |r|γj −2 rBj (v) = 0, (2.3)
j=1

which has two nonzero roots r(v) > 0 and −r(v) for each v ∈ V1 ∩ Σ. By the implicit function
theorem, these roots are C 1 in V1 .
Setting Ẽ(v) = Ē(r(v), v),1 we consider the variational problem P2 with constraints: find a
minimizing element v 0 ∈ V , m = inf{Ẽ(v) : v ∈ V1 ∩ Σ}.
 
Note that if v k is a minimizing sequence of problem P2 andv 0is 
its weak limit in V (along a
subsequence), then, by taking account of Eq. (2.3), we find that r v k
is bounded, v 0 ∈ V1 , and
0
hence H (v ) > 0. Since

d   0  m
  γj γj −1  0     −1/p 
Ē r τ v , τ v 0 = − γj r τ v 0 τ Bj v < 0 ∀τ ∈ 0, H v 0 ,
dτ j=1

we have H (v 0 ) = 1. Consequently, v 0 is a solution

 of problem P2 . Therefore, there exists a
Lagrange multiplier τ ∈ R such that dẼ (v ) , v = τ dH (v 0 ) , v for all v ∈ V . This, together
0

with (2.3), implies that r (v 0 ) v 0 is a nontrivial solution of problem (2.1). The proof of the theorem
is complete.
Remark 2.1. If

A (u1 , . . . , un ) = A (|u1 | , . . . , |un |) , Bj (u1 , . . . , un ) = Bj (|u1 | , . . . , |un |)

for all u = (u1 , . . . , un ) ∈ V , then all components of the solution of problem (2.1) found by methods
of Theorem 2.1 are nonnegative.
m
Remark 2.2. Let H(·) be a norm in V equivalent to the norm  · , and let j=1 Bj (v) > 0
for all v ∈ V \{0}. Since Ẽ(·) is even, positive, weakly continuous, and belongs to the class C 1
in V \{0}, it follows from the Lyusternik–Shnirelman theory that this functional has a countable
   
set v k of conditional critical points on Σ. Consequently, ±r v k v k are solutions of problem (2.1)
in case (h1 ) with p < γ as well as in case (h2 ).
Let us now proceed to the investigation of the nonhomogeneous problem

d  m
(ai (t, u1 (t), . . . , un (t))) + λ |ui (t)|
p−2
− ui (t) = bij (t, u1 (t), . . . , un (t)) + fi ,
dt j=1 (2.4)
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},

under the assumption

n 

T
γ
∃c > 0 : Bj (u) ≤ c |ui (t)| j dt ∀u ∈ V, ∀j ∈ {1, . . . , m}, (2.5)
i=1 0

1
This is a functional of class C 1 in V1 .

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

506 MARINO, TOSKANO
n
and set V2 = {v ∈ V : i=1 fi , vi  = 0} and S = {v ∈ V : H(v) ≤ 1}. It is natural to assume
that V2 is nonempty; otherwise fi = 0. We use the fibering method in case (h1 )2 with p < γ as well
as in case (h2 ).
In case (h1 ), for each v ∈ S, we consider the reduced functional


m 
n
Ē(r, v) = |r| − |r|
p γ
Bj (v) − r fi , vi 
j=1 i=1

and the bifurcation equation


m 
n
p|r|p−2
r − γ|r| γ−2
r Bj (v) = fi , vi  . (2.6)
j=1 i=1

By using conditions (2.5) and (i2 ), we find that there exist positive constants k1 and k2 such that


m
Bj (u) ≤ k1 uγ ∀u ∈ V, u ≤ k2 ∀u ∈ S. (2.7)
j=1

Thus we introduce the function

ψ(r) = p|r|p−2 r − γk|r|γ−2 r, r ∈ R,

where k = k1 k2γ , and set M = loc maxR ψ.

n 
If i=1 fi ∗ < M/k2 , fi ∗ is the norm of fi in (W 1,p (]0, T [)) , then we obtain the following:
if v ∈ S ∩ V1 , then Eq. (2.6) has three distinct roots, r1 (v), r2 (v) > 0, and r3 (v) = −r2 (v); if
v ∈ S\V1 , then Eq. (2.6) has a unique root r1 (v). Furthermore, we note that −t1 ≤ r1 (v) ≤ t1
for all v ∈S and r2 (v) ≥ t2 for all v ∈ S ∩ V1 , where t1 < t2 are positive roots of the equation
n
ψ(r) = k2 i=1 fi ∗ . By the implicit function theorem, r1 (·) ∈ C 1 (S ∩ V2 ) and r2 (·) ∈ C 1 (S ∩ V1 );
moreover, r1 (·) [respectively, r2 (·)] is weakly continuous in S (respectively, in S ∩ V1 ). By setting
Ẽi (v) = Ē (ri (v), v), i = 1, 2, 3, we find that Ẽ1 (·) [respectively, Ẽi (·), i = 2, 3] is weakly continuous
on S and belongs to the class C 1 in S ∩ V2 (respectively, in S ∩ V1 ). We consider the variational
problems with the following constraints.
 
Problem P̄1 . Find a minimizing element v̄ 1 = (v̄11 , . . . , v̄n1 ) ∈ V , m̄1 = inf Ẽ1 (v) : v ∈ Σ .
 
Problem P̄i . Find a minimizing element v̄ i = (v̄1i , . . . , v̄ni ) ∈ V , m̄i = inf Ẽi (v) : v ∈ Σ ∩ V1 ,
i = 2, 3.
One can readily prove the following assertions.
 
(α1 ) If v 1k is a minimizing sequence of Problem P̄1 and v̄ 1 is its weak limit in V (along a
subsequence), then r1 (v̄ 1 ) = 0 and v̄ 1 ∈ Σ ∩ V2 .
 
(α2 ) If v ik is a minimizing   sequence
 of Problem P̄i (i = 2, 3) and v̄ i is its weak limit in V
(along a subsequence), then ri v ik is bounded and v̄ i ∈ Σ ∩ V1 .
Just as in Problem P2 , it follows from (α1 ) and (α2 ) that v̄ i is a solution of Problem P̄i
for i = 1, 2, 3.
In conclusion, by appropriately using Lagrange multipliers, just as above, we verify that
p
ri (v̄ i ) × v̄ i (with i = 1, 2, 3) is a solution of problem (2.4). Since H (ri (v̄ i ) × v̄ i ) = |ri (v̄ i )| ,
we have r1 (v̄ ) × v̄ = ri (v̄ ) × v̄ for i = 2, 3. Suppose the contrary: r2 (v̄ ) × v̄ and r3 (v̄ ) × v̄ are
1 1 i i 2 2 3 3

2
In case (h1 ), the existence of a solution of problem (2.4) cannot be proved with the use of only Lagrange multipliers
(just as in the case of the homogeneous problem). To use the fibering method, we should assume that p < γ.
The case in which p > γ is to be considered in the following section with the use of the same method but under
other constraints for v.

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

 PERIODIC SOLUTIONS OF A CLASS OF NONLINEAR SYSTEMS 507

equal. This is the caseif r3 (v̄ 3 ) = −r2 (v̄ 2 ) and v̄ 3 = −v̄ 2 , i.e., r2 (−v̄ 2 ) = r2 (v̄ 2 ), and this relation
n
is valid if and only if i=1 fi , v̄i2  = 0. Consequently, −v̄ 2 ∈ V1 ∩ Σ and

    p   2 γ 
m
   
Ẽ2 −v̄ 2 = r2 −v̄ 2 − r2 −v̄ Bj −v̄ 2 = Ẽ2 v̄ 2 ≤ Ẽ2 (v) ∀v ∈ V1 ∩ Σ.
j=1

Therefore, r2 (−v̄ 2 ) (−v̄ 2 ) = −r2 (v̄ 2 ) v̄ 2 is a solution of problem (2.4) other than r2 (v̄ 2 ) v̄ 2 , but this
is impossible, since V2 = ∅.
In case (h2 ), we note that there exist positive constants k̄1 , . . . , k̄m such that

m
Bj (u) ≤ k̄1 uγ1 + · · · + k̄m uγm ∀u ∈ V.
j=1

We introduce the function

ψ̄(r) = p|r|p−2 r − γ k̄|r|γ−2 r, r ∈ R,
where !
k̄ = max k̄j (k2γ1 + · · · + k2γm ) ,
j
n
and k2 is the constant occurring in (2.7). If i=1 fi ∗ < M̄ /k2 , where M̄ = loc maxR ψ̄, then one
argue as in case (h1 ) and obtain the same results. We have thereby proved the following assertion.

Theorem 2.2. In case (h1 ), with p < γ as well as in case (h2 ), problem (2.4) has at least
threedistinct solutions provided that assumptions (i1 )–(i6 ) are valid, condition (2.5) is satisfied,
n
and i=1 fi ∗ is sufficiently small.

3. THE CASE IN WHICH λ > 0. CONTINUATION

We consider the homogeneous problem (2.1) in case (h3 ) with 2 < γj < 3 < p. We set
" m #
Σ1 = v ∈ V : Bj (v) = 1 , Bσ = {v ∈ V : H(v) ≤ σ},
j=1

where σ is a positive constant such that Σ1 ∩ Bσ = ∅. Following the fibering method, for each
v ∈ V1 ∩ Bσ , we consider the reduced functional

m
Ē(r, v) = |r|p H(v) − |r|γj Bj (v)
j=1

and the bifurcation equation


m
p|r|p−2 rH(v) − γj |r|γj −2 rBj (v) = 0. (3.1)
j=1

Let r(v) > 0 and −r(v) be nonzero roots of Eq. (3.1). They belong to the class C 1 in V1 ∩ Bσ .
Setting Ẽ(v) = Ē(r(v), v), we consider
  problem Q1 with constraints: find a mini-
the variational
mizing element v ∈ V , m = inf Ẽ(v) : v ∈ Bσ ∩ Σ1 .
0

It follows from Eq. (3.1) that

m $
 %
γj
Ẽ(v) = − 1 (r(v))γj Bj (v) ∀v ∈ Σ1 ∩ Bσ . (3.2)
j=1
p

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

508 MARINO, TOSKANO
 
Let v k be a minimizing sequence of problem Q1 , and let v 0 be its weak limit in  V (along
 k  a
subsequence). One can show that, by assumption (i5 ), v ∈ Bσ ∩ Σ1 , and, by (3.1), r v
0
is
bounded. It follows from assumption (i6 ) and condition (3.1) that r(·) is weakly upper semicontin-
uous on Bσ ∩ Σ1 ; therefore, by (3.2), Ẽ(·) is weakly lower semicontinuous on Bσ ∩ Σ1 . It follows
that Ẽ (v 0 ) ≤ m. Since v 0 ∈ Σ1 ∩ Bσ , we have Ẽ (v 0 ) = m. Therefore, there exists a Lagrange
multiplier τ ∈ R such that
  0  
m
   
dẼ v , v = τ dBj v 0 , v ∀v ∈ V.
j=1

This relation implies that r (v 0 ) v 0 is a nontrivial solution of problem (2.1). We have thereby
justified the following assertion.

Theorem 3.1. In case (h3 ) under assumptions (i1 )–(i6 ), problem (2.1) has at least two distinct
nontrivial solutions with opposite signs.

Remark 3.1. Obviously, Remark 2.1 is also valid for Theorem 3.1.
Let us proceed to the investigation of the nonhomogeneous problem (2.4) in case (h1 ) but with
p > γ. For each v ∈ Bσ \{0}, we consider the functional


n
Ē(r, v) = |r|p H(v) − |r|γ − r fi , vi 
i=1

and the bifurcation equation


n
p|r|p−2 rH(v) − γ|r|γ−2 r = fi , vi  . (3.3)
i=1

We introduce the function

χ(r) = p|r|p−2 rσ − γ|r|γ−2 r, r ∈ R,

and set M = loc maxR χ. It follows from assumption (i2 ) that

∃σ̄ : v ≤ σ̄ ∀v ∈ Bσ . (3.4)

Consequently, if

n
M
fi ∗ < , (3.5)
i=1
σ̄

then Eq. (3.3) has three distinct roots, r1 (v), r2 (v) > 0, and r3 (v) = −r2 (v). Note that

−t1 ≤ r1 (v) ≤ t1 , r2 (v) ≥ t2 ∀v ∈ Bσ \{0},

n
where t1 < t2 are positive roots of the equation χ(r) = σ̄ i=1 fi ∗ . We set Ẽi (v) = Ē (ri (v), v),
i = 1, 2, 3. By the implicit function theorem, these are functionals of class C 1 in Bσ \{0}. Setting
   
n n
V2+ = v ∈ V : fi , vi  ≥ 0 and V2− = v ∈ V : fi , vi  ≤ 0 ,
i=1 i=1

we consider the following problems.

 
Problem Q̄2 . Find a minimizing element v̄ 2 ∈ V , m̄2 = inf Ẽ2 (v) : v ∈ Bσ ∩ Σ1 ∩ V2+ .

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

 PERIODIC SOLUTIONS OF A CLASS OF NONLINEAR SYSTEMS 509
 
Problem Q̄3 . Find a minimizing element v̄ 3 ∈ V , m̄3 = inf Ẽ3 (v) : v ∈ Bσ ∩ Σ1 ∩ V2− .
One can readily prove the following assertions:
(β1 ) r2 (·) and r3 (·) are bounded in Bσ ∩ Σ1 ;
(β2 ) r2 (·) is weakly upper semicontinuous on Bσ ∩ Σ1 ∩ V2+ ;
(β3 ) r3 (·) is weakly lower semicontinuous on Bσ ∩ Σ1 ∩ V2− ;
 
(β4 ) if v ik (for i = 2, 3) is a minimizing sequence of Problem Q̄i and v̄ i is its weak limit in V
(along a subsequence), then v̄ 2 ∈ Bσ ∩ Σ1 ∩ V2+ and v̄ 3 ∈ Bσ ∩ Σ1 ∩ V2− .
In accordance with assumptions (β1 ), (β2 ), (β3 ), and (β4 ), by (3.3), the functional Ẽ2 (·) [respec-
tively, Ẽ3 (·)] is weakly lower semicontinuous on Bσ ∩ Σ1 ∩ V2+ (respectively, on Bσ ∩ Σ1 ∩ V2− )3 .
Hence it follows that Ẽ (v̄ i ) = m̄i , i = 2, 3. Therefore, r2 (v̄ 2 ) v̄ 2 and r3 (v̄ 3 ) v̄ 3 (i = 2, 3) are two
solutions of problem (2.4). By using the same procedure as in the preceding section, one can show
that these solutions are distinct. This implies the following assertion.

Theorem 3.2. In case (h1 ) with p > γ, if assumptions (i1 )–(i6 ) and condition (3.5) are valid,
then problem (2.4) has at least two distinct solutions.

Now consider the same problem (2.4) in case (h3 ). Suppose that there exists a constant c3 > 0
such that
n T
 γ
c3 |ui | j dt ≤ Bj (u) ∀u ∈ V, ∀j ∈ {1, . . . , m}. (3.6)
i=1 0

Let us first prove the following assertion.

Lemma 3.1. There exists a constant σ1 > 0 such that Bj (u) > σ1 for all u ∈ Bσ ∩ Σ1 and
all j ∈ {1, . . . , m}.

Proof. If we suppose the contrary,  we find that, for all k ∈ N, there exist j(k) ∈ {1,
 then  k.. . , m}
and v ∈ Bσ ∩ Σ1 such that Bj(k) v ≤ 1/k. Hence it follows that limk→+∞ Bj(k) v = 0.
k k

From (3.6), we obtain

 T  T
  n
& k &γj(k) n
& k &γ̄
Bj(k) v k ≥ c3 &vi & dt ≥ c4 &vi & dt,
i=1 0 i=1 0

n
where γ̄ = min1≤j≤m γj . Consequently, v k → 0 strongly in (Lγ̄ (]0, T [)) . Moreover, by assump-
tion (i2 ), there exists a v 0 ∈ V such that v k → v 0 weakly in V (along a subsequence), which,
n
together with the Rellich–Kondrashov theorem, implies that v k → v 0 strongly in (Lγ̄ (]0, T [)) .
Now we have v 0 = 0 and hence v k → 0 weakly in V . Therefore, by using assumption (i5 ), we ob-
tain a contradiction, which completes the proof of the lemma.
We set B (σ, σ1 ) = {v ∈ V : σ1 < Bj (v), H(v) ≤ σ}. For each v ∈ B (σ, σ1 ), we consider the
functional

m 
n
E(r, v) = |r|p H(v) − |r|γj Bj (v) − r fi , vi 
j=1 i=1

and the bifurcation equation


m 
n
γj −2
p|r| p−2
rH(v) − γj |r| rBj (v) = fi , vi  . (3.7)
j=1 i=1

3
The analysis of the bifurcation equation does not permit one to obtain properties of the weak semicontinuity of Ẽ1 (·).
Therefore, we do not consider the variational problem with constraints for Ẽ1 (·).

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

510 MARINO, TOSKANO

We introduce the function


m
χ̄(r) = p|r| p−2
rσ − σ1 γj |r|γ−2 r, r ∈ R,
j=1

and set M̄ = loc maxR χ̄. If


n
M̄
fi ∗ < , (3.8)
i=1
σ̄

where σ̄ is the constant occurring in (3.4), then Eq. (3.7) has three roots r1 (v), r2 (v) > 0, and
r3 (v) = −r2 (v). We set Ẽi (v) = Ē (ri (v), v), i = 1, 2, 3. Just as above, we consider only variational
problems with constraints.
 
Problem Q∗2 . Find a minimizing element v̄ 2 ∈ V , m̄∗2 = inf Ẽ2 (v) : v ∈ B (σ, σ1 ) ∩ Σ1 ∩ V2+ .
 
Problem Q∗3 . Find a minimizing element v̄ 3 ∈ V , m̄∗3 = inf Ẽ3 (v) : v ∈ B (σ, σ1 ) ∩ Σ1 ∩ V2− .
Arguing as in the preceding case, one can show that Problems Q∗2 and Q∗3 are solvable. Let
r2 (v̄ 2 ) v̄ 2 and r3 (v̄ 3 ) v̄ 3 be two solutions of problem (2.4). We claim that they are distinct. Indeed,
if they would coincide, then we would obtain v̄ 3 = −
v̄ 2 , where
= |r2 (v̄ 2 ) /r3 (v̄ 3 )| > 0. Since
v̄ 3 ∈ Σ1 , it would follow from assumption (i4 ) that


m
  
m
 
1=
γj Bj v̄ 2 ≥
γ̄ Bj v̄ 2
j=1 j=1

with
γ̄ = min1≤j≤m
γj ; therefore,
≤ 1. On the other hand, v̄ 2 = −(1/
)v̄ 3 . Thus, we would
obtain
≥ 1 and hence
= 1. Therefore, v̄ 3 = −v̄ 2 and r3 (v̄ 3 ) = −r2 (v̄ 2 ), and these equalities
would lead to a contradiction just as in the proof of Theorem 2.2. We have thereby justified the
following assertion.

Theorem 3.3. In case (h3 ), if assumptions (i1 )–(i6 ) are valid and conditions (3.6) and (3.8)
are satisfied, then problem (2.4) has at least two different solutions.

4. THE CASE IN WHICH λ = 0

By ei = (0, . . . , 1, 0, . . . , 0) we denote the ith vector of the normal coordinate system and set
 
m
∗
Σ = v ∈ V : A(v) = 1, Φi (v) = dBj (v), ei  = 0 ∀i ∈ {1, . . . , n} .
j=1

In addition to assumptions (i1 )–(i6 ), we state the following requirements:

(i7 ) Σ∗ ∩ V1 = ∅;
(i8 ) Φi (·) belongs to the C 1 -class in V \{0};
(i9 ) Φi (·) is weakly continuous on V \{0};
(i10 ) for each (c1 , . . . , cn ) ∈ Rn \{0}, there exists an i ∈ {1, . . . , n} such that Φi (c1 , . . . , cn ) = 0;
(i11 ) det dΦi (u), ek  = 0 for every u ∈ Σ∗ ∩ V1 .
It follows from assumption (i4 ) that Φi (ru) = |r|γ−2 rΦi (u) for all u ∈ V \{0} and all r ∈ R\{0},
and dΦi (u), u = (γ − 1)Φi (u) for all u ∈ V \{0}.  kFirst,
 we note that Σ∗ is a bounded ' k 'set in V .
Indeed, supposing the contrary, we assume that v ⊆ Σ ∗
is a sequence such that 'v ' → +∞.
 k 
By setting w = v1 / vk  , . . . , vn / vk  , we obtain
k k

' k'
'w ' = 1. (4.1)

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

 PERIODIC SOLUTIONS OF A CLASS OF NONLINEAR SYSTEMS 511

Therefore, there exists a w ∈ V such that

wk → w weakly in V (along a subsequence), (4.2)

which, together with the Rellich–Kondrashov theorem, implies that

n
wk → w strongly (Lp (]0, T [)) . (4.3)
 
Since limk→+∞ A wk = 0, we have
  n
wk → 0 strongly in (Lp (]0, T [)) (4.4)

by assumption (i2 ). Therefore, by (4.2), wi (t) = ci for all t ∈ [0, T ] and all i ∈ {1, . . . , n};
consequently, by (4.3) and (4.4), wk → w strongly in V . Consequently, by (4.1), w = 0, and, by
assumption (i9 ), Φi (w) = 0; but this contradicts assumption (i10 ).
Let us the investigation of the homogeneous problem

d  m
− (ai (t, u1 (t), . . . , un (t))) = bij (t, u1 (t), . . . , un (t)) ,
dt j=1 (4.5)
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n}.

Let us restrict our considerations to case (h1 ).

Theorem 4.1. In case (h1 ) under assumptions (i1 )–(i11 ), problem (4.5) has at least two non-
trivial solutions with opposite signs.
m
Proof. We set F (u) = j=1 Bj (u) and consider the variational problem with constraints:
Problem P ∗ . Find a maximizing element u∗ ∈ V , M ∗ = sup {F (u) : u ∈ V1 ∩ Σ∗ }.
   
Obviously, M ∗ > 0. Let uk be a maximizing sequence of Problem P ∗ ; thus F uk → M ∗ .
Since Σ∗ is a bounded set in V , it follows that there exists a u∗ ∈ V such that uk → u∗ weakly in V
(along a subsequence); moreover, A (u∗ ) ≤ 1. This, together with assumption (i5 ), implies that

F (u∗ ) = M ∗ . (4.6)

Let us now show that u∗ ∈ V1 ∩ Σ∗ . It follows from (4.6) that u∗ ∈ V1 . By using assumptions (i4 ),
(i9 ), and (i10 ), one can show that Φi (u∗ ) = 0 for all i ∈ {1, . . . , n}, A (u∗ ) = 1. Therefore,
the function u∗ is a solution of problem P ∗ .
By assumption (i11 ), v ∈ V → (dA (u∗ ) , v , dΦ1 (u∗ ) , v , . . . , dΦn (u∗ ) , v) ∈ Rn+1 is a sur-
jective mapping. Therefore, there exist Lagrange multipliers τ0 , τ1 , . . . , τn such that

dF (u∗ ) , v = τ0 dA (u∗ ) , v + τ1 dΦ1 (u∗ ) , v + · · · + τn dΦn (u∗ ) , v

for all v ∈ V . This relation written out with v = u∗ permits one to obtain τ0 = γM/p > 0; this
relation with v = ei (for i = 1, . . . , n), together with assumption (i11 ), implies that τi = 0 for all
1/(p−γ) ∗
i ∈ {1, . . . , n}. Finally, one can readily see that τ0 u is a nontrivial solution of problem (4.5).
The proof of the theorem is complete.
Now consider the nonhomogeneous problem

d  m
− (ai (t, u1 (t), . . . , un (t))) = bij (t, u1 (t), . . . , un (t)) + fi ,
dt j=1 (4.7)
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},
DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004
512 MARINO, TOSKANO

in case (h1 ) with p < γ. One can readily see that

 1/γ
n 

T

∃σ > 0 : ∀v ∈ Σ∗  |vi | dt

γ
<σ
i=1 0

provided that Σ∗ is bounded in V . Further, we set

  1/γ 

 n T 

Sσ = v ∈ V : A(v) ≤ 1,  γ
|vi | dt  ≤σ

 i=1


0

and suppose that (i12 ) Σ∗ ∩ V2 = ∅, (i13 ) fi , 1 = 0, and condition (2.5) is satisfied.
By relation (2.5) and assumption (i2 ), there exist two positive constants k3 and k4 such that

m 
n
Bj (u) ≤ k3 ∀u ∈ Sσ , ui 1,p ≤ k4 ∀u ∈ Sσ .
j=1 i=1

We introduce the function χ̃(r) = p|r|p−2 r − γk3 |r|γ−2 r, r ∈ R, and set M̃ = loc maxR χ̃. If


n
M̃
fi ∗ < , (4.8)
i=1
k4

then we obtain the same results as for λ > 0 (see Section 2) with Sσ instead of S for roots of the
bifurcation equation and for the following variational problems with constraints.
 
Problem P1∗ . Find a minimizing element v ∗1 ∈ V , m∗1 = inf Ẽ1 (v) : v ∈ Σ∗ .
 
Problem Pi∗ . Find a minimizing element v ∗i ∈ V , m∗i = inf Ẽi (v) : v ∈ Σ∗ ∩ V1 for i = 2, 3.
Here Ẽi (v) = Ē (ri (v), v), i = 1, 2, 3. By using the Lagrange multiplier in an appropriate way
and by taking account of assumptions (i11 ) and (i13 ), one can see that ri (vi∗ ) · vi∗ (with i = 2, 3)
are solutions of problem (4.7). If assumption (i11 ) is additionally introduced  for
 each v ∈ Σ∗ , then
r1 (v1∗ ) · v1∗ is also a solution of problem (4.7). Moreover, ri (vi∗ ) × vi∗ = rj vj∗ × vj∗ for i = j, just
as for the case in which λ > 0. We have thereby justified the following assertion.

Theorem 4.2. Consider case (h1 ) with p < γ. If assumptions (i1 )–(i13 ) are valid and condi-
tions (2.5) and (4.8) are satisfied, then problem (4.7) has at least two distinct solutions. Moreover,
if assumption (i11 ) holds for all u ∈ Σ∗ , then there exist at least three distinct solutions of prob-
lem (4.7).

Remark 4.1. The nonhomogeneous problem in case (h1 ) with p > γ as well as the homogeneous
and nonhomogeneous problems in cases (h2 ) and (h3 ) have not been solved yet.

5. EXAMPLES
Example 5.1. Consider the problem

d ! m
a(t) |u (t)|
p−2 
− u (t) + λ|u(t)|p−2 u(t) = bj (t)|u(t)|γj −2 u(t) + f,
dt j=1 (5.1)
u(0) = u(T ), u (0) = u (T )

with λ ≥ 0, 1 < p, 1 < γj , a, bj ∈ L∞ (]0, T [), inf a > 0, bj ≥ 0, bj = 0, and f ∈ (W 1,p (]0, T [)) .

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

 PERIODIC SOLUTIONS OF A CLASS OF NONLINEAR SYSTEMS 513

For λ > 0, we have the following.

1. If f = 0, then in cases (h1 ), (h2 ), and (h3 ), problem (5.1) has at least two nontrivial solutions
with opposite signs, and all components of one of them are nonnegative (see Theorems 2.1 and
3.1 and Remarks 2.1 and 3.1); in addition, if bj > 0, then there exist countably many solutions of
problem (5.1) (see Remark 2.2).
2. If f = 0 and f ∗ is sufficiently small, then problem (5.1) in case (h1 ) with p < γ as well
as in case (h2 ) has at least three distinct solutions (see Theorem 2.2). But in case (h1 ) with
p > γ (Theorem 3.2) and in case (h3 ), it has at least two distinct solutions under the additional
assumption minj inf bj > 0 (Theorem 3.3).
For λ = 0, we obtain the following.
3. If f = 0, then in case (h1 ) with γ ≥ 2, problem (5.1) has at least two nontrivial solutions with
opposite signs (Theorem 4.1).
If f = 0, f, 1 = 0, and f ∗ is sufficiently small, then in case (h1 ) with p < γ and γ ≥ 2,
problem (5.1) has at least two distinctsolutions, and there exist at least three distinct solutions
m
under the additional assumption that j=1 bj > 0 (Theorem 4.2).

Example 5.2. Consider the system of differential equations

d !
ai (t) |ui (t)|
p−2  p−2
− ui (t) + λ |ui (t)| ui (t)
dt
m 
µ µ γ /µ −2 µ
= |b1j (t) (u1 (t)) j + · · · + bnj (t) (un (t)) j | j j b1j (t) (u1 (t)) j + · · ·
j=1
(5.2)
! 
µ µ −1
+ bnj (t) (un (t)) j bij (t) (ui (t)) j + fi ,
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},
where λ > 0, 1 < p, µj ∈ N, µj < γj , with the conditions
ai ∈ L∞ (]0, T [), min inf ai > 0, bij ∈ L∞ (]0, T [),
i
(5.3)
∀j ∈ {1, . . . , m} ∃i ∈ {1, . . . , n} : bij = 0.
If f = (f1 , . . . , fn ) = 0, then in cases (h1 ), (h2 ), and (h3 ), problem (5.2) has at least two nontrivial
n
solutions with opposite signs (Theorems 2.1 and 3.1); if f = 0 and i=1 fi ∗ is sufficiently small,
then problem (5.2) has at least three distinct solutions in case (h1 ) with p < γ as well as in case (h2 )
(Theorem 2.2). In case (h1 ) with p > γ, there exist at least two distinct solutions of problem (5.2)
(Theorem 3.2).

Example 5.3. Consider the system

d !
ai (t) |ui (t)|
p−2  p−2
− ui (t) + λ |ui (t)| ui (t)
dt
m 
µ µ γ /µ −2
= |b1j (t) |u1 (t)| j + · · · + bnj (t) |un (t)| j | j j
j=1
(5.4)

µj µ µ −2
× (b1j (t) |u1 (t)| + · · · + bnj (t) |un (t)| j ) bij (t) |ui (t)| j ui (t) + fi ,
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},
where λ > 0, 1 < p, µj > 0, γj > max {1, µj }, with conditions (5.3). Therefore, if
f = (f1 , . . . , fn ) = 0,
then in cases (h1 ), (h2 ), and (h3 ), problem (5.4) has at least two nontrivial solutions with opposite
signs and all components of one of them are nonnegative (Theorems 2.1 and 3.1; Remarks 2.1
and 3.1); moreover,
n if rg bij  = n, then problem (5.4) has countably many solutions (Remark 2.2);
if f = 0 and i=1 fi ∗ is sufficiently small, then problem (5.4) has at least three distinct solutions

DIFFERENTIAL EQUATIONS Vol. 40 No. 4 2004

514 MARINO, TOSKANO

in case (h1 ) with p < γ as well as in case (h2 ) (Theorem 2.2). But it has at least two distinct
solutions in case (h1 ) with p > γ (Theorem 3.2) and in case (h3 ) provided that mini,j inf bij > 0
(Theorem 3.3).
Example 5.4. Consider the problem
d !  m
 p−2 
− ai (t) |ui (t)| ui (t) = (b1j (t)u1 (t) + · · · + bnj (t)un (t)) bij (t) + fi ,
dt j=1 (5.5)
ui (0) = ui (T ), ui (0) = ui (T ), i ∈ {1, . . . , n},
where 1 < p, m ≥ n, with conditions (5.3). Suppose that
' '
' m T '
' '
' bij (t)bkj (t)dt'
det ' ' = 0 (5.6)
' j=1 '
0

under assumption (i11 ) provided that

rg bij  = n. (5.7)
We first show that the assumption (i7 ) is valid. Let ū ∈ V satisfy the requirements A (ū) = 0 and
−p
v̄ = r̄ū, where r̄ = [A (ū)] . Since
m 

T

Φi (u) = (b1j (t)u1 (t) + · · · + bnj (t)un (t)) bij (t)dt,

j=1 0

it follows from (5.6) that the linear system

Φ1 ((v̄1 + c1 ) , . . . , (v̄n + cn )) = 0,
Φ2 ((v̄1 + c1 ) , . . . , (v̄n + cn )) = 0,
.................................
Φn ((v̄1 + c1 ) , . . . , (v̄n + cn )) = 0
has the unique solution c̄ = (c̄1 , c̄2 , . . . , c̄n ). Further, one can readily see that, by (5.7),
w̄ = v̄ + c̄ ∈ V1 ; therefore, w̄ ∈ V1 ∩ Σ∗ . Assumption (i10 ) is valid by virtue of (5.6). More-
over, if f = (f1 , . . . , fn ) = 0, then, by (5.6), assumption (i13 ) implies assumption (i12 ). Therefore,
if f = 0 and p = 2, then problem (5.5) has at least two nontrivial solutions with opposite signs
n
(Theorem 4.1); if f = 0, fi , 1 = 0, i=1 fi ∗ is sufficiently small, and p < 2, then there exist at
least three distinct solutions of problem (5.5) (Theorem 4.2).

ACKNOWLEDGMENTS
The work was financially supported by Seconda Università di Napoli, Italy.

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