Abstract In this paper, we study the existence of traveling waves for a fourth order Schrödinger equations with mixed dispersion, that is, solutions to ² u + u - Q (-i V) u + u =|u|^p-2 u, \ in \ RN, \ N 2, Δ 2 u + β Δ u - Q (- i V ∇) u + α u = | u | p - 2 u, in R N, N ≥ 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 (p₁, 2N/ (N-4) _+) p ∈ (p 1, 2 N / (N - 4) +), where the real number p₁ 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 Lᵖ - Lq L p - L q resolvent-type estimate which can be of independent interest and can be extended to other operators.
Casteras et al. (Sun,) studied this question.