  This paper considers a pair of coupled nonlinear Helmholtz equations
\begin{align*}
  -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2}
\right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*}
  -\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2}
\right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where
$\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions
in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods.
  The focus lies on necessary and sufficient conditions on the parameters
deciding whether or not both components of such solutions are nontrivial.
