Dual variational methods for an indefinte nonlinear Helmholtz equation

We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$-\Delta u-k^2u=Q(x)|u|^{p-2}u,\quad u\in W^{2,p}(\mathbb{R}^N)$$ with $k>0, N\ge3,p\in\left[\frac{2(N+1)}{N-1},\frac{2N}{N-2}\right]$ and $Q\in L^\infty(\mathbb{R}^N)$. Due to sign-changes of $Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.