Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

We establish weighted $$L^p$$ L p -Fourier extension estimates for $$O(N-k) \times O(k)$$ O ( N - k ) × O ( k ) -invariant functions defined on the unit sphere $${\mathbb {S}}^{N-1}$$ S N - 1 , allowing for exponents p below the Stein–Tomas critical exponent $$\frac{2(N+1)}{N-1}$$ 2 ( N + 1 ) N - 1 . Moreover, in the more general setting of an arbitrary closed subgroup $$G \subset O(N)$$ G ⊂ O ( N ) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation $$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$ - Δ u - u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p ( R N ) , where Q is a nonnegative bounded and G-invariant weight function.

Extended duality methods for nonlinear Helmholtz equations

We provide extensions of the dual variational method for the nonlinear Helmholtz equation from Evéquoz and Weth. In particular we prove the existence of dual ground state solutions in the Sobolev critical case, extend the dual method beyond the standard Stein Tomas and Kenig Ruiz Sogge range and generalize the method for sign changing nonlinearities.

On Helmholtz Equations and Counterexamples to Strichartz Estimates in Hyperbolic Space

In this paper, we study nonlinear Helmholtz equations (NLH)$$\begin{equation} -\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u \quad\textrm{in}\ \mathbb{H}^N, \;N\geq 2, \end{equation}$$where $\Delta _{\mathbb{H}^N}$ denotes the Laplace–Beltrami operator in the hyperbolic space $\mathbb{H}^N$ and $\Gamma \in L^\infty (\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $\lambda>0$ and $p>2$. The oscillatory behaviour and decay rate of radial solutions is analyzed, with extensions to Cartan–Hadamard manifolds and Damek–Ricci spaces. Our results rely on a new limiting absorption principle for the Helmholtz operator in $\mathbb{H}^N$. As a byproduct, we obtain simple counterexamples to certain Strichartz estimates.