In this paper, we study nonlinear Helmholtz equations (NLH)
−ΔHNu−4(N−1)2u−λ2u=Γ∣u∣p−2u in HN, N≥2 where ΔHN
denotes the Laplace-Beltrami operator in the hyperbolic space HN
and Γ∈L∞(HN) is chosen suitably. Using fixed point
and variational techniques, we find nontrivial solutions to (NLH) for all
λ>0 and p>2. The oscillatory behaviour and decay rates of radial
solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds
and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle
for the Helmholtz operator in HN. As a byproduct, we obtain simple
counterexamples to certain Strichartz estimates