On a fourth order nonlinear Helmholtz equation

Abstract

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