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We consider the Helmholtz equation $-\Delta u+V \, u - \lambda \, u = f $ on$\mathbb{R}^n$ where the potential $V:\mathbb{R}^n\to\mathbb{R}$ is constant oneach of the half-spaces $\mathbb{R}^{n-1}\times (-\infty,0)$ and$\mathbb{R}^{n-1}\times (0,\infty)$. We prove an $L^p-L^q$-Limiting AbsorptionPrinciple for frequencies $\lambda>\max \, V$ with the aid of FourierRestriction Theory and derive the existence of nontrivial solutions of linearand nonlinear Helmholtz equations.

An annulus multiplier and applications to the Limiting absorption principle for Helmholtz equations with a step potential