Existence of traveling waves for a fourth order Schrödinger equation with mixed dispersion in the Helmholtz regime

In this paper, we study the existence of traveling waves for a fourth order Schrödinger equations with mixed dispersion, that is, solutions to Δ ² u +β Δ u - Q(-i V ∇ ) u +α 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.