Existence of traveling waves for a fourth order Schr\" odinger equation with mixed dispersion in the Helmholtz regime

Abstract

In this paper, we study the existence of traveling waves for a fourth order Schr\" odinger equations with mixed dispersion, that is, solutions to $${\Delta }^{2}u+\beta \Delta u+iV\nabla u+\alpha u=|u{|}^{p-2}u,in{\mathbb{R}}^N,N\ge 2.$$ We consider this equation in the Helmholtz regime, when the Fourier symbol $P$ of our operator is strictly negative at some point. Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that $p\in (p_1,2N/(N-4{)}_{+})$. The real number $p_1$ depends on the number of principal curvature of $M$ staying bounded away from $0$, where $M$ is the hypersurface defined by the roots of $P$. We also obtained estimates on the Green function of our operator and a $L^p-L^q$ resolvent estimate which can be of independent interest and can be applied to other operators.