We study the mixed dispersion fourth order nonlinear Schrödinger equationwhere γ, σ > 0 and β ∈ R. We focus on standing wave solutions, namely solutions of the form ψ(x, t) = e iαt u(x), for some α ∈ R. This ansatz yields the fourth-order elliptic equationWe consider two associated constrained minimization problems: one with a constraint on the L 2 -norm and the other on the L 2σ+2 -norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.

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We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely we prove that the best constant is achieved by a non-constant solution of the associated fourth-order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.