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 $$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$ - Δ u - k 2 u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p R N with $$k>0,$$ k > 0 , $$N \ge 3$$ N ≥ 3 , $$p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. $$ p ∈ 2 ( N + 1 ) N - 1 , 2 N N - 2 and $$Q \in L^{\infty }({\mathbb {R}}^{N})$$ Q ∈ L ∞ ( R N ) . Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.