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In this paper, we study the existence of traveling waves for a fourth orderSchr\" 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 Fouriersymbol $P$ of our operator is strictly negative at some point. Under suitableassumptions, we prove the existence of solution using the dual method ofEvequoz 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 alsoobtained 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 toother operators.

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