On a fourth order nonlinear Helmholtz equation

Abstract

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $Δ^2 u -βΔu + αu= Γ|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $Γ$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties.