which is positive since both a and b are less than » +1 . This completes the proof of the

theorem.

11.4 Notes and remarks

The presentation given here is from [124]. Further work was done in [102].

Chapter 12

Rotationally Invariant Solutions

12.1 Introduction

In this chapter (and the next) we study periodic solutions of the Dirichlet problem for

the semilinear wave equation. In this chapter we study radially symmetric solutions for

the problem

u := u t t ’ u = f (t, x, u), t ∈ R, x ∈ B R ,

(12.1)

u(t, x) = 0, t ∈ R, x ∈ ‚ B R ,

(12.2)

u(t + T, x) = u(t, x), t ∈ R, x ∈ BR,

(12.3)

where

B R = {x ∈ Rn : |x| < R}.

(12.4)

In this case we have

f (t, x, u) = f (t, |x|, u), x ∈ BR.

Our basic assumption is that the ratio R/T is rational. Thus, we can write

8R/T = a/b,

(12.5)

where a, b are relatively prime positive integers. We show that

n≡3

(12.6) (mod(4, a))

implies that the linear problem corresponding to (12.1)“(12.3) has no essential spec-

trum. If

n≡3

(12.7) (mod(4, a)),

then the essential spectrum of the linear operator consists of precisely one point:

»0 = ’(n ’ 3)(n ’ 1)/4R 2 .

(12.8)

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_12,

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

142 12. Rotationally Invariant Solutions

This shows that the spectrum has at most one limit point. We can then consider the

nonlinear case

f (t, r, s) = μs + p(t, r, s),

(12.9)

where μ is a point in the resolvent set, r = |x|, and

| p(t, r, s)| ¤ C(|s|θ + 1), s ∈ R,

(12.10)

for some number θ < 1. Our main theorem is

Theorem 12.1. If (12.6) holds, then (12.1)“(12.3) has a weak rationally invariant

solution. If (12.7) holds and »0 < μ, assume, in addition, that p(t, r, s) is nondecreas-

ing in s. If μ < »0 , assume that p(t, r, s) is nonincreasing in s. Then (12.1)“(12.3)

has a weak rotationally invariant solution.

Our proof of this theorem will make use of Theorem 10.2. For the de¬nition of

essential spectrum, cf., e.g., [126] and [106].

12.2 The spectrum of the linear operator

In proving Theorem 12.1, we shall need to calculate the spectrum of the linear operator

applied to periodic rotationally symmetric functions. Speci¬cally, we shall need

Theorem 12.2. Let L 0 be the operator

L 0 u = u t t ’ u rr ’ r ’1 (n ’ 1)u r

(12.11)

applied to functions u(t, r ) in C ∞ ( ¯ ) satisfying

u(T, r ) = u(0, r ), u t (T, r ) = u t (0, r ), 0 ¤ r ¤ R,

(12.12)

u(t, R) = u R (t, 0) = 0, t ∈ R,

(12.13)

= [0, T ] — [0, R]. Then L 0 is symmetric on L 2 ( , ρ), where ρ = r n’1 .

where

Assume that 8R/T = a/b, where a, b are relatively prime integers [i.e., (a, b) = 1].

Then L 0 has a self-adjoint extension L having no essential spectrum other than the

point »0 = ’(n ’ 3)(n ’ 1)/4R 2 . If n ≡ 3 (mod(4, a)), then L has no essential

spectrum. If n ≡ 3 (mod(4, a)), then the essential spectrum of L is precisely the

point »0 .

Proof. Let ν = (n ’ 2)/2 and let γ be a positive root of Jν (x) = 0, where Jν is the

Bessel function of the ¬rst kind. Set

•(r ) = Jν (γ r/R)/r ν .

(12.14)

Then

• + (n ’ 1)• /r = (x 2 Jν + x Jν ’ ν 2 Jν )/r ν+2 = ’γ 2 Jν /R 2 .

(12.15)

12.2. The spectrum of the linear operator 143

If

ψ(t, r ) = •(r )e2πikt /T ,

(12.16)

then

L 0 ψ = [(γ /R)2 ’ (2πk/T )2 ]ψ.

(12.17)

Let γ j be the j th positive root of Jν (x) = 0, and set

ψ j k (t, r ) = r ’ν Jν (γ j r/R)e2πikt /T .

(12.18)

Then ψ j k (t, r ) is an eigenfunction of L 0 with eigenvalue

» j k = (γ j /R)2 ’ (2πk/T )2 .

(12.19)

It is easily checked that the functions ψ j k , when normalized, form a complete orthonor-

mal sequence in L 2 ( , ρ). We shall show that the corresponding eigenvalues (12.19)

are not dense in R. It will then follow that L 0 has a self-adjoint extension L with

spectrum equal to the closure of the set {» j k } (cf., e.g., [126]). Now

γ j = β j ’ (μ ’ 1)/8β j + O(β ’3 ) as β j ’ ∞,

(12.20) j

where

1 1

βj = π j+ ν’ , μ = 4ν 2

(12.21)

2 4

(cf., e.g., [157]). Thus,

[β j ’ „k ’ (μ ’ 1)/8β j + O(β ’3 )]

» j k R2 = j