In this paper, we study the existence of traveling waves for a fourth order Schrödinger equations with mixed dispersion, that is, solutions to \(\Delta ^2 u +\beta \Delta u - Q(-i V \nabla ) u +\alpha u =|u|^{p-2} u,\ \text {in } \ \mathbb {R}^N,\ N\ge 2,\) where the Q term represents a lower order symmetry breaking operator. We consider this equation in the Helmholtz regime, that is, when the Fourier symbol P of the operator is sign-changing and we assume that the zero set of the symbol is a manifold M. Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that \(p\in (p_1, 2N/(N-4)_+)\) , where the real number \(p_1\) depends on the number of principal curvature of M staying bounded away from 0. We also obtain estimates on the Green’s function of our operator and a \(L^p - L^q\) resolvent-type estimate which can be of independent interest and can be extended to other operators.
