On a fourth order nonlinear Helmholtz equation

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation Δ²u − βΔu + αu = Γ|u|$^{p-2}$u in R$^{N}$, for positive, bounded and Z$^{N}$-periodic functions Γ in the following three cases: (a) α 0, β < −2√α or (c) α = 0, β < 0. Using the dual method of Evéquoz and Weth, we find solutions to this equation and establish some of their qualitative properties