Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$-\mathrm{\Delta }u-k^{2}u=Q(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,u\in W^{2,p}({\mathbb{R}}^N)$$ with $k>0,$ $N\ge 3$, $p \in \left[\left.\frac{2(N+1)}{N-1},\frac{2N}{N-2}\right)\right.$and$Q \in L^{\infty}($\mathbb{R}$^{N})$\mathrm{.}Duetothesign-changesof$Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.