J (v) > 0, v ∈ Nm © Ml © Br \{0}

(16.76)

by Lemma 16.23. Thus J has a positive maximum on Nm . We can now apply Theorem

16.26 and Lemma 16.9 to obtain the desired conclusion. With respect to Theorem 16.4,

we note that by Lemma 16.18, it suf¬ces to show that J (w) given by (16.46) has two

nontrivial solutions. Now J is bounded from below by Lemma 16.21 and satis¬es (PS)

by (16.56). Moreover,

J (w) < 0, w ∈ Nl © Mm © Br \{0},

(16.77)

by Lemma 16.24, and

J (w) > 0, w ∈ Ml © Br \{0},

(16.78)

by Lemma 16.25. Thus, J has a negative minimum on Mm . We can now apply Theo-

rem 16.26 and Lemma 16.9 to obtain the desired conclusion.

16.7 Notes and remarks

In his studies of semilinear elliptic problems with jumping nonlinearities, C´ c [29]“

a

[34] proved the following.

210 16. Multiple Solutions

Theorem 16.27. Let be a bounded domain in Rn , n ≥ 2, with smooth boundary

‚ . Let 0 < »0 < »1 < · · · < »k < · · · be the sequence of distinct eigenvalues of the

eigenvalue problem

’ u = »u i n , u = 0 on ‚ .

(16.79)

Let p(t) be a continuous function such that p(0) = 0 and

p(t)/t ’’ a as t ’’ ’∞

and

p(t)/t ’’ b as t ’’ +∞.

Assume that for some k ≥ 1, we have a ∈ (»k’1 , »k ), b ∈ (»k , »k+1 ), and the only

solution of

’ u = bu + ’ au ’ i n , u = 0 on ‚

(16.80)

is u ≡ 0, where u ± = max[±u, 0]. Assume further that

p(s) ’ p(t)

¤ ν < »k+1 , s, t ∈ R, s = t.

(16.81)

s’t

Assume also that p (0) exists and satis¬es p (0) ∈ (» j ’1 , » j ) for some j ¤ k. Then

’ u = p(u) i n , u = 0 on ‚

(16.82)

has at least two nontrivial solutions.

This theorem generalizes the work of Gallou¨ t and Kavian [72], [73] which

e

required »k to be a simple eigenvalue and the left-hand side of (16.81) to be sand-

wiched in between »k’1 and »k+1 and bounded away from both of them. C´ c proves a

a

counterpart of the theorem in which the inequalities are reversed.

In the present chapter we generalized this theorem and its reverse-inequality coun-

terpart by not requiring p(t)/t to converge to limits at either ±∞ or ±0. Rather, we

worked with the primitive

t

f (x, s)ds

F(x, t) :=

0

and bounded 2F(x, t)/t 2 near ±∞ and ±0 [we replaced p(t) with a function f (x, t)

depending on x as well]. Our main assumptions were

t[ f (x, t1 ) ’ f (x, t0 )] ¤ a(t ’ )2 + b(t + )2 , t j ∈ R, t = t1 ’ t0 ,

(16.83)

a0 (t ’ )2 + b0 (t + )2 ¤ 2F(x, t) ¤ a1 (t ’ )2 + b1 (t + )2 , |t| < δ,

(16.84)

for some δ > 0,

a2 (t ’ )2 + b2 (t + )2 ’ W1 (x) ¤ 2F(x, t), |t| > K ,

(16.85)

16.7. Notes and remarks 211

for some K > 0 and W1 ∈ L 1 ( ), where the constants a, a0, a1 , a2 , b, b0 , b1 , b2 are

suitably chosen (they include the cases considered by C´ c). The advantage of such

a

inequalities is that they do not restrict the expression 2F(x, t)/t 2 or f (x, t)/t to any

particular interval.

The results of this chapter come from [101] with changes in the proofs. Theo-

rem 16.26 is from [28] with variations made in the proof. Lemma 16.9 is due to Castro

[38].

Chapter 17

Second-Order Periodic Systems

17.1 Introduction

In this chapter we study a general system of second-order differential equations, and we

look for periodic solutions. We show that for several sets of hypotheses such systems

can be solved by the methods used in the book.

We consider the following problem. One wishes to solve

’x(t) = ∇x V (t, x(t)),

¨

(17.1)

where

x(t) = (x 1 (t), · · · , x n (t))

(17.2)

is a map from I = [0, T ] to Rn such that each component x j (t) is a periodic function

with period T, and the function V (t, x) = V (t, x 1 , . . . , x n ) is continuous from Rn+1

to R with

∇x V (t, x) = (‚ V /‚ x 1, . . . , ‚ V /‚ x n ) ∈ C(Rn+1 , Rn ).

(17.3)

For each x ∈ Rn , the function V (t, x) is periodic in t with period T.

We shall study this problem under the following assumptions:

1.

V (t, x) ≥ 0, t ∈ I, x ∈ Rn .

2. There are constants m > 0, ± ¤ 6m 2 /T 2 such that

V (t, x) ¤ ±, |x| ¤ m, t ∈ I, x ∈ Rn .

3. There is a constant μ > 2 such that

Hμ(t, x)

¤ W (t) ∈ L 1 (I ), |x| ≥ C, t ∈ I, x ∈ Rn ,

(17.4)

|x| 2

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_17,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

214 17. Second-Order Periodic Systems

and

Hμ (t, x)

¤ 0,

(17.5) lim sup

|x|2

|x|’∞

where