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

Abstract

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