Schechter Minimax Systems and Critical Point Theory (Springer, 2009)

210

16. Multiple Solutions

Theorem 16.27. Let be a bounded domain in Rn ,

n ≥ 2, with smooth boundary

∂ . Let 0 < λ0 < λ1 < · · · < λk < · · ·

be the sequence of distinct eigenvalues of the

eigenvalue problem

(16.79)

− u = λu i n ,

u = 0 on ∂ .

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

p(t)/t −→ a

as

t −→ −∞

and

p(t)/t −→ b

t −→ +∞.

as

Assume that for some k ≥ 1, we have a (λk−1, λk ),

b (λk , λk+1), and the only

solution of

(16.80)

− u = bu+ − au− i n ,

u = 0 on ∂

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

(16.81)

p(s) − p(t)

≤

ν < λ

,

s, t

R, s

=

t.

s

−

t

k+1

Assume also that p (0) exists and satisfies p (0) (λ j −1, λ j ) for some j ≤ k. Then

(16.82) − u = p(u) i n , u = 0 on ∂

has at least two nontrivial solutions.

This theorem generalizes the work of Gallou¨et and Kavian [72], [73] which required λk to be a simple eigenvalue and the left-hand side of (16.81) to be sandwiched in between λk−1 and λk+1 and bounded away from both of them. C´ac proves a counterpart of the theorem in which the inequalities are reversed.

In the present chapter we generalized this theorem and its reverse-inequality counterpart by not requiring p(t)/t to converge to limits at either ±∞ or ±0. Rather, we worked with the primitive

t

F(x, t) := f (x, s)ds

0

and bounded 2F(x, t)/t2 near ±∞ and ±0 [we replaced p(t) with a function f (x, t) depending on x as well]. Our main assumptions were

(16.83)	t[ f (x, t1) − f (x, t0)] ≤ a(t−)2 + b(t+)2,	t j R, t = t1 − t0,
(16.84)	a0(t−)2 + b0(t+)2 ≤ 2F(x, t) ≤ a1(t−)2	+ b1(t+)2, |t| < δ,
for some δ > 0,
(16.85)	a2(t−)2 + b2(t+)2 − W1(x) ≤ 2F(x, t),	|t| > K ,

16.7. Notes and remarks

211

for some K > 0 and W1 L1( ), where the constants a, a0, a1, a2, b, b0, b1, b2 are suitably chosen (they include the cases considered by C´ac). The advantage of such inequalities is that they do not restrict the expression 2F(x, t)/t2 or f (x, t)/t to any particular interval.

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

(17.1)	−x¨(t) = x V (t, x(t)),
where
(17.2)	x(t) = (x1(t), · · · , xn(t))

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, x1, . . . , xn ) is continuous from Rn+1 to R with

(17.3)

x V (t, x) = (∂ V /∂ x1, . . . , ∂ V /∂ xn) C(Rn+1, Rn ).

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, α ≤ 6m2/ T 2 such that
			V (t, x) ≤ α, |x| ≤ m, t I, x Rn.
3.	There is a constant μ > 2 such that
	(17.4)	Hμ(t, x)		≤ W (t) L1(I ),	|x| ≥ C, t I, x Rn ,
		|x|2

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_17, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

214							17.			Second-Order Periodic Systems
and
(17.5)	lim sup		Hμ(t, x)						≤ 0,
				|	x		2
	|x|→∞				|
where
(17.6)	Hμ(t, x) = μV (t, x) − x V (t, x) · x.
4. There is a subset e I of positive measure such that
(17.7)	lim inf	V (t, x)								t e.
		|x|2				> 0,
	|x|→∞

We have

Theorem 17.1. Under the above hypotheses, system (17.1) has a solution.

As a variant of Theorem 17.1, we have

Theorem 17.2. The conclusion in Theorem 17.1 is the same if we replace hypothesis 2 with

2A. There is a constant q > 2 such that

V (t, x) ≤ C(|x|q + 1), t I, x Rn , and there are constants m > 0, α < 2π 2/ T 2 such that

V (t, x) ≤ α|x|2, |x| ≤ m, t I, x Rn .

We also have

Theorem 17.3. The conclusion of Theorem 17.1 holds if we replace hypothesis 3 with

3A. There is a constant μ < 2 such that

(17.8)	Hμ(t, x)		≥ −W (t) L1(I ),			|x| ≥ C, t I, x Rn ,
	|x|2
and
(17.9)			lim inf	Hμ(t, x)		≥	0	.
				|x|2
			|x|→∞

And we have

Theorem 17.4. The conclusion of Theorem 17.1 holds if we replace hypothesis 1 with

1A.

0 ≤ V (t, x) ≤ C(|x|2 + 1), t I, x Rn .

and hypothesis 3 with

17.2.	Proofs of the theorems		215
3B.	The function given by
	(17.10)	H (t, x) = 2V (t, x) − x V (t, x) · x
	satisfies
	(17.11)	H (t, x) ≤ W (t) L1(I ),	|x| ≥ C, t I, x Rn ,
	and
	(17.12)	H (t, x) → −∞, |x| → ∞, t I, x Rn .

Theorems 17.1–17.4 show the existence of solutions, which conceivably could be constants. The following theorems provide the existence of non-constant solutions.

Theorem 17.5. If we replace hypothesis 4 in Theorem 17.1 with

4A. There are constants β > 2π 2/ T 2 and C such that

V (t, x) ≥ β|x|2, |x| > C, t I, x Rn,

then system (17.1) has a nonconstant solution.

As a variant of Theorem 17.5, we have

Theorem 17.6. The conclusion in Theorem 17.5 is the same if we replace hypothesis 2 with hypothesis 2A.

We also have

Theorem 17.7. The conclusion of Theorem 17.5 holds if we replace hypothesis 1 with hypothesis 1A and hypothesis 3 with hypothesis 3B.

We shall prove Theorems 17.1–17.7 in the next section. We use the linking method of Chapter 2.

17.2 Proofs of the theorems

We now give the proof of Theorem 17.1.

Proof. Let X be the set of vector functions x(t) described above. It is a Hilbert space with norm satisfying

		n
	x 2X =	x j 2H1 .
	We also write	j =1
		n
	x 2 =
		x j 2,
		j =1
	where · is the L2(I ) norm.

216	17.	Second-Order Periodic Systems
Let	N = {x(t) X : x j (t) ≡ constant,	1 ≤ j ≤ n},

and M = N . The dimension of N is n, and X = M N. The following is known (cf., e.g., Proposition 1.3 of [95]).

Lemma 17.8. If x M, then

x 2 ≤ T x˙ 2

∞ 12

and

x ≤ T x˙ .

2π

Proof. It suffices to prove the lemma for continuously differential periodic functions. First, consider the case T = 2π. Using Fourier series, we have

∞

(17.13)

x =

αk ϕk ,

k=−∞

where

(17.14)

αk

= (x, ϕ¯k ),

k = 0, ±1, ±2, . . . ,

and

(17.15)

ϕk (x) =

√

1

eikx ,

k = 0, ±1, ±2, . . .

2π

Thus,

(17.16)

x 2 =

∞

|αk |2.

k=−∞

If x M, then α0 = 0, and

x 2 ≤

∞

|kαk |2 = x˙ 2.

k=−∞

Moreover,

∞

2

∞

2 ∞ k−2

1

x

2

1

αk

1

kαk

2

x 2

π 2/3.

∞ ≤

k=−∞

|

| ≤ 2π k=−∞ |

|

=

2π

k=1

2π

˙

This proves the lemma for the case T = 2π. Otherwise, we let y(t) = x(T t/2π ). Then

x 2 =	T	y 2,	x˙ 2 =	2π	y˙ 2,	x ∞ = y ∞.
	2π			T

17.2. Proofs of the theorems

Thus,

x 2 =

T

2

T

y˙ 2 =

T

2

y

≤

2π

2π

2π

and

π

π T

x ∞ = y ∞ ≤

y˙ 2

=

x˙ 2 =

6

6 2π

Note that it follows that

x ∞ ≤ C x X , x X.

We define

217

x˙ 2

T x˙ 2.

12

(17.17) G(x) = x˙ 2 − 2 V (t, x(t)) dt, x X.

I

For each x X, write x = v + w, where v N, w M. For convenience, we shall use the following equivalent norm for X:

x 2X = w˙ 2 + v 2.

If x M and

x˙ 2 = ρ2 = 12 m2,

T

then Lemma 17.8 implies that x ∞ ≤ m, and we have by hypothesis 2 that V (t, x) ≤ α. Hence,

(17.18)	G(x)	≥	x˙ 2 − 2	α dt
			|x|<m
		≥	ρ2 − 2αT ≥ 0.
We also note that hypothesis 1 implies
(17.19)		G(v) ≤ 0, v N.
Take
	A =	∂ Bρ ∩ M, ρ2 =		12	m2,
				T
	B =	N,

where

Bσ = {x X : x X < σ }.

By Example 2 of Section 3.4, A links B. Moreover, if R is sufficiently large,

(17.20) sup[−G] ≤ 0 ≤ inf[−G].

A B

218 17. Second-Order Periodic Systems

Hence, we may apply Theorem 3.4 to conclude that there is a sequence {x(k)} X such that

(17.21)

G(x(k)) = x˙ (k) 2 − 2

I

V (t, x(k)(t)) dt → c ≥ 0,

(17.22)

(G

(x(k)), z)/2

=

(x(k), z)

−

x V (t, x(k))

·

z(t) dt

→

0,

z

X,

˙

˙

I

and

(17.23)

(G (x(k)), x(k))/2

x(k)

2

I

x V (t, x(k))

·

x(k) dt

→

0.

= ˙

−

If

ρk = x(k) X ≤ C,

then there is a renamed subsequence such that x(k)

converges to a limit x X weakly

in X and uniformly on I. From (17.22), we see that

(G (x), z)/2

=

(x, z)

− I

x V (t, x(t))

·

z(t) dt

=

0,

z

X,

˙ ˙

from which we conclude easily that x is a solution of (17.1).

If

ρk = x(k) X → ∞,

let x˜ (k)

= x(k)/ρk . Then x˜ (k) X = 1. Let x˜ (k) = w˜ (k) + v˜(k) , where w˜ (k)

M and

˜

2

2

˜

→

r and

˜

→

τ,

v(k)

N. There is a renamed subsequence such that

[x(k)]·

x

(k)

where r

+ τ

= 1. From (17.21) and (17.23), we obtain

and

Thus,

(17.24)

and

(17.25)

Hence,

(17.26)

[x˜ (k)]· 2 − 2 V (t, x(k)(t)) dt/ρk2 → 0

				I
[x	(k)]· 2		x	V (t, x	(k))	·	x	(k) dt/ρ2	→	0.
˜	−	I						k

2 V (t, x(k)(t)) dt/ρk2 → r 2

I

x V (t, x(k)) · x(k) dt/ρk2 → r 2.

I

	Hμ(t, x(k)(t)) dt/ρk2 →	μ		− 1 r 2.
I		2

17.2. Proofs of the theorems

219

Note that

|x˜ (k)(t)| ≤ C x˜ (k) X = C.

If

|x(k)(t)| → ∞,

then, by hypothesis 3,

lim sup

Hμ(t, x(k)(t))

≤ lim sup

Hμ(t, x(k)(t))

|x˜ (k)(t)|2 ≤ 0.

ρk2

|x(k)(t)|2

If

|x(k)(t)| ≤ C,

then

Hμ(t, x(k)(t))

→ 0.

ρk2

Hence,

lim sup

I

Hμ(t, x(k)(t)) dt/ρk2 ≤ 0.

Thus, by (17.26),

μ

− 1

r 2 ≤ 0.

2

If r

=

0

,

this contradicts the fact that μ

>

2

.

If r

=

0

,

then

w˜

(k)

→

0 uniformly

(k) 2

(k) 2

in I by Lemma 17.8. Moreover, T

|v˜

|

=

2 v˜

→

1

.

Thus, there is a renamed

subsequence such that

v˜

(k)

(k)

→ v˜ uniformly in

I. Consequently, |x

(k)

→ v˜ in N with |v˜|

= 1/ T. Hence, x˜

| → ∞ uniformly in I. Thus, by hypothesis 4,

lim inf

V (t, x(k)(t)) dt/ρk2 ≥

e lim inf

V (t, x(k)(t))

|x˜ (k)(t)|2 dt > 0.

I

|x(k)(t)|2

This contradicts (17.24). Hence, the ρk are bounded, and the proof is complete.

The proof of Theorem 17.2 is similar to that of Theorem 17.1 with the exception of inequality (17.18) resulting from hypothesis 2. In its place we reason as follows: If x M, we have, by hypothesis 2A,

G(x) ≥ x˙ 2 − 2

|x|<m

α|x(t)|2 dt − C

|x|>m

(|x|q + 1) dt

x

2

−

2α x 2

C(1

+

m2−q

+

m−q )

|

x

q dt

≥ ˙

−

|x|>m

|

x

2

(1

−

[2αT 2

/4π 2])

−

C

|

x

q dt

≥ ˙

|

|x|>m

≥

(1

−

[αT 2/2π 2]) x 2

−

C

x q

dt

X

X

I

220							17.	Second-Order Periodic Systems
	≥	(1	−	[αT 2/2π 2])			x 2	C		x q
							X −			X
	=	1 − [αT 2/2π 2] − C x qX−2 x 2X
by Lemma 17.8. Hence, we have
Lemma 17.9.
(17.27)	G(x) ≥ ε x 2X ,				x X ≤ ρ, x M,

for ρ > 0 sufficiently small, where ε < 1 − [αT 2/2π 2].

The remainder of the proof is essentially the same.

In proving Theorem 17.3, we follow the proof of Theorem 17.1 until we reach (17.26). Then we reason as follows. If

|x(k)(t)| → ∞,

then

Hμ(t, x(k)(t))

Hμ(t, x(k)(t))

lim inf

≥ lim inf

|x˜ (k)(t)|2 ≥ 0.

If

ρk2

|x(k)(t)|2

|x(k)(t)| ≤ C,

then

Hμ(t, x(k)(t))

→ 0.

ρk2

Hence,

lim inf

I

Hμ(t, x(k)(t)) dt/ρk2 ≥ 0.

Thus, by (17.26),

μ

− 1

r 2 ≥ 0.

=

2

w˜

→

If r

(k) 2

(k) 2 =

0

,

then

(k)

0 uniformly

0, this contradicts the fact that μ <

2.

If r

in I by Lemma 17.8.

Moreover, T

|v˜

|

= 2v˜

→

1

.

Hence, there is a renamed

(k)

(k)

that

v˜ → v˜ in N with |v˜|

= 1/ T. Hence, x˜

→ v˜ uniformly in

subsequence such

(k)

I. Consequently, |x

(t)| → ∞ uniformly in I. Thus, by Hypothesis 4,

lim inf

V (t, x(k)(t)) dt/ρk2 ≥

e lim inf

V (t, x(k)(t))

|x˜ (k)(t)|2 dt > 0.

I

|x(k)(t)|2

This contradicts (17.24). Hence, the ρk are bounded, and the proof is complete.

In proving Theorem 17.4, we follow the proof of Theorem 17.1 until (17.26). Assume first that r > 0. Note that (17.21) and (17.23) imply that

(17.28) H (t, x(k)(t)) dt → −c.

I