Cantor families of periodic solutions for wave equations via by Berti M., Bolle P.

By Berti M., Bolle P.

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2. 17). 16). 21) with s∗ = 1 possesses a nontrivial “Mountain pass” critical set K∞ ⊂ SR∞ := { v H 1 = R∞ }. Next we fix the dimension N ∈ N of the finite dimensional subspace V1 in the orthogonal decomposition V = V1 ⊕ V2 , N depending only on ap (x). 7. 3 and its proof). 1, there are q ≤ q1 ≤ . . 1 holds. 4). 10) with ω(δ) = 1 + 2δ p−1 . 21) holds, with q = qM − p . e. 5). 2. If there is v ∈ V such that G(v) < 0, we may choose s∗ = −1. 1) with frequencies ω < 1. 1 Clearly m∞ < +∞ because |ap (x)v p+1 | ≤ C|v|p+1 ∞ ≤C v |G(v)| ≤ Ω p+1 H1 , ∀v ∈ V .

8) σ/4,s for ε small enough because M ≥ ln2 1 → +∞ as 6L0 ε ε→0 and LM := L0 2M >> χM . 8), Lω rM ε exp (−χi ) ≤ Kε exp (−χM ) . 9) Lω w − εΠW Γ(δ, λ, v1 , w) ≤ K ε exp (−χM ) σ/4,s and since χM ≥ χ− ln2 (6L0 ε) = χ−(ln2 χ) lnχ˜ (6L0 ε) = (6L0 ε)− ln2 χ and, setting α := ln2 χ ∈ (0, 1), Lω w − εΠW Γ(λ, δ, v1 , w) ≤ Cε exp − σ/4,s for 0 < δ ≤ δ0 (γ, τ ) small enough. 41 1 (6L0 ε)α ≤ C ε exp (− C ) δα References [1] A. Ambrosetti, P. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications, Journ.

Let η have minimal period 2π. 3) T then h = 0. Proof. Step 1: For any y ∈ T and α < β h(s) ds = 0 . 3), for any real polynomial P (X) := k≥q ak X k divisible by X q , P η(y + s) − η(s) h(s) ds = 0 . 5) holds for any real polynomial ak X k . 6). By the Stone-Weierstrass theorem, the set A is dense in C([−M, M ], R) because it is a subalgebra with unity and A separates the points of [−M, M ] (take any X q with q odd). As a consequence for any continuous function g ∈ C(R) g η(y + s) − η(s) h(s) ds = 0 .

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