ROTATIONALLY INVARIANT PERIODIC SOLUTIONS OF SEMILINEAR WAVE EQUATIONS

Under suitable conditions we are able to solve the semilinear wave equation in any dimension. We are also able to compute the essential spectrum of the linear wave operator for the rotationally invariant periodic case.


Introduction
In this paper we continue the work of Smiley [SM] and Ben-Naoum and Mawhin [BNM1] concerning radially symmetric solutions for the problem (1.1) Our basic assumption is that (This shows that the spectrum has at most one limit point.)We can then consider the nonlinear case (1.8) f (t, r, s) = µs + p(t, r, s), where µ is a point in the resolvent set, r = |x|, and (1.9) |p(t, r, s)| ≤ (|s| θ + 1), s ∈ R for some number θ < 1.Our main theorem is Theorem 1.1.If (1.5) holds, then (1.1) -( 1.3) has a weak rotationally invariant solution.If (1.6) holds and λ 0 < µ, assume in addition that p(t, r, s) is nondecreasing in s.If µ < λ 0 , assume that p(t, r, s) is nonincreasing in s.
The case T= 2π, 2R = π was considered in detail in [BNM1].They proved the existence of a weak solution for n even and n = 1, 3. They consider more general situations than (1.8), (1.9).However, our methods can be adjusted to cover their case as well.Uniqueness theorems were also treated in [BNM1].They also considered odd n > 5 when the spectrum of the linear problem is not dense.However, they do not establish when this is the case.
A main consideration in our approach is the following theorem concerning infinite dimensional linking.It is of interest in its own right and has several other applications.
Theorem 1.2.Let N be a closed separable subspace of a Hilbert space E. Let G be a continuously differentiable functional on E such that where P is the projection of E onto N .Let Q be a bounded open convex subset of N , and let F be a continuous map of E onto N such that

Assume
(1.12) Theorem 1.2 will be proved in Section 4. It generalizes theorems in [KS,S2,3,Wi].Theorem 1.1 will be proved in Section 3 after the essential spectrum of the linear operator is determined in Section 2.

The Spectrum of the linear operator
In proving Theorem 1.1 we shall need to calculate the spectrum of the linear operator ✷ applied to periodic rotationally symmetric functions.Specifically, we shall need Theorem 2.1.Let L 0 be the operator where ρ), where ρ = r n−1 .Assume that 8R/T = a/b, where a, b are relatively prime integers (i.e., (a, b) = 1).Then L 0 has a selfadjoint extension L having no essential spectrum other than the point then L has no essential spectrum.If n ≡ 3(mod(4, a)), then the essential spectrum of L is precisely the point λ 0 .
Proof.Let ν = (n − 2)/2, and let γ be a positive root of J ν (x) = 0, where J ν is the Bessel function of the first kind.Set (2.4) ϕ(r) = J ν (γr/R)/r ν . Then Let γ j be the j-th positive root of J ν (x) = 0, and set Then ψ jk (t, r) is an eigenfunction of L 0 with eigenvalue (2.7) It is easily checked that the functions ψ jk , when normalized, form a complete orthonormal sequence in L 2 (Ω, ρ).We shall show that the corresponding eigenvalues (2.7) are not dense in R. It will then follow that L 0 has a selfadjoint extension L with spectrum equal to the closure of the set {λ jk }.Now (2.8) where (2.9) (cf., e.g., [WA]).Thus Since the expression in the brackets is an integer, we see that either

Thus
(2.12) lim If n − 3 is not a multiple of (4, a), then can never vanish.To see this, note that if (b, k) = b, then ak/b is not an integer.Hence Thus in this case we always have On the other hand, if n ≡ 3 (mod(4, a)), then there is an infinite number of positive integers j, k such that Hence, the point λ 0 is a limit point of eigenvalues.Consequently, it is in σ e (L).This completes the proof.

The Nonlinear Case
We now turn to the problem of solving where L is the selfadjoint extension of the operator L 0 given in Theorem 2.1.Under the hypotheses of that theorem the spectrum of L is discrete.We assume that where µ is a point in the resolvent set of L and p(t, r, s) is a Carathéodory function on Ω × R such that for some number θ < 1.We have Proof.Since µ is in the resolvent set of L, there is a δ > 0 such that where the λ jk are given by (2.7).Each u ∈ L 2 (Ω, ρ) can be expanded in the form where the ψ jk are given by (2.6).Let N 0 be the subspace of those u ∈ L 2 (Ω, ρ) for which α jk = 0 if β j = τ k (cf. the proof of Theorem 2.1).For u ∈ N 0 where summation is taken over those j, k for which β j = τ k .Let E be the subspace of L 2 (Ω, ρ) consisting of those u for which (3.9) is finite.With this norm, E becomes a separable Hilbert space.Note that E ⊂ D(|L| 1/2 ), and the embedding of E N 0 into L 2 (Ω, ρ) is compact (we use (2.13) for this purpose).Let and the scalar product is that of L 2 (Ω, ρ).One checks readily that G is a where we write p(u) in place of p (t, r, u).This shows that u is a weak solution of (3.1) iff G (u) = 0.
Let N be the subspace of E spanned by the ψ jk corresponding to those λ jk < µ and let M denote the subspace of E spanned by the rest.Thus We can now make use of Theorem , where P is the projection of E onto N , then {u k } has a renamed subsequence which converges strongly in L 2 (Ω, ρ).The reason is that {v k } has such a subsequences because the embedding of E N 0 in L 2 (Ω, ρ) is compact.Thus G (u n ) → G (u) weakly in E. Hence all of the hypotheses of Theorem 1.2 are satisfied, and we can conclude that there is a sequence {u k } satisfying (1.13).Write and consequently in view of (3.3) and (3.9).Similarly If N 0 = {0}, then if follows from (3.14) and (3.15) that u k E is bounded, and consequently there is a renamed subsequence which converges weakly in E and strongly in L 2 (Ω, ρ) to a function u.Thus G (u k ) → G (u) weakly.But G (u k ) → 0. Consequently G (u) = 0, and the proof for this case is complete.If N 0 = {0}, we note that where y k → y weakly in E and L 2 (Ω, ρ) and u k → u weakly in E and strongly in L 2 (Ω, ρ).By hypothesis This shows that y k → y in E, and the proof proceeds as before.If λ 0 < µ, we apply Theorem 1.2 to −G(u) and come to the same conclusion.In this case, the inequality in (3.17) is reversed.This completes the proof.

Weak Linking
We now give a proof of Theorem 1.2.It is similar to those of [KS,S2,3,Wi].Assume that there is no sequence satisfying (1.13).Then there is a positive number δ such that (4.1) G (u) ≥ 2δ whenever u belongs to the set (4.2) Since N is separable, we can norm it with a norm |v| w satisfying and such that the topology induced by this norm is equivalent to the weak topology of N on bounded subsets of N (cf., e.g., [DS,p.426]).For u ∈ E, we write u = v + w, where v ∈ N, w ∈ M = N ⊥ , and take (4, 4) and convergence of a bounded sequence u n = v n + w n with respect to this norm means that v n converges weakly in N and w n converges strongly in M .We denote E equipped with this norm by E w .For u ∈ E 1 , let q(u) = G (u)/ G (u) .Then by (4.1) For otherwise there would be a sequence {h k } ⊂ B such that Since B is bounded in E, P h k → P u weakly in N and (I − P )h k → (I − P )u strongly in M .Hence, by hypothesis, Then Y (u) is locally Lipschitz continuous with respect to both norms.Moreover, (4.10) For u ∈ Q ∩ E 1 , let σ(t)u be the solution of (4.12) Note that σ(t)u will exist as long as σ(t)u is in B.Moreover, it is continuous in (u, t) with respect to both topologies.Next we note that if u Each u 0 ∈ B has a neighborhood W (u 0 ) in E w and a finite dimensional subspace S(u 0 ) such that Y (u) ⊂ S(u 0 ) for u ∈ W (u 0 ) ∩ B. Since σ 1 (t)v is continuous is (v, t), for each (v 0 , t 0 ) in K there are a neighborhood W (v 0 , t 0 ) ⊂ N × [0, T ] and a finite dimensional subspace S(v 0 , t 0 ) ⊂ N such that F z 1t (v) ⊂ S(v 0 , t 0 ) for (v, t) ∈ W (v 0 , t 0 ), where Since K is compact, there is a finite number of points (v j , t j ) ⊂ K such that K ⊂ W = ∪W (v j , t j ).Let S be a finite dimensional subspace of N containing p and all the S(v j , t j ).We note that ϕ (v, t) For if ϕ(v, t) = p, then σ 1 (t)v ∈ F −1 (p) = B.This implies G(σ 1 (t)v) ≥ b 0 by (1.12).But (4.13) and (1.12) imply that G(σ 1 (t)v) < b 0 for t > 0.
2δ in view of (4.6).This contradicts (4.8).Let B w be the set B with the inherited topology of E w .It is a metric space, and W (u) ∩ B is an open set in this space.Thus {W (u) ∩ B}, u ∈ B, is an open covering of the paracompact space B w .Consequently, there is a locally finite refinement {W τ } of this cover.For each τ there is an element u τ such that W τ ⊂ W (u τ ).Let {ψ τ } be a partition of unity subordinate to this covering.Each ψ τ is locally Lipschitz continuous with respect to the norm |u| w and consequently with respect to the norm of E.