  In this paper, we study the existence of traveling waves for a fourth order
Schr\" odinger equations with mixed dispersion, that is, solutions to
$$\Delta^2 u +\beta \Delta u +i V \nabla u +\alpha u =|u|^{p-2} u,\ in\ \R^N ,\
N\geq 2.$$ 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.
