Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation \[ γ Δ 2 u − Δ u + α u = | u | 2 σ u , u ∈ H 2 ( R N ) , \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma } u, \qquad u \in H^2({\mathbb {R}}^N), \] under the constraint \[ ∫ R N | u | 2 d x = c > 0. \int _{{\mathbb {R}}^N}|u|^2 \, dx =c>0. \] We assume that γ > 0 , N ≥ 1 , 4 ≤ σ N > 4 N ( N − 4 ) + \gamma >0, N \geq 1, 4 \leq \sigma N > \frac {4N}{(N-4)^+} , whereas the parameter α ∈ R \alpha \in {\mathbb {R}} will appear as a Lagrange multiplier. Given c ∈ R + c \in {\mathbb {R}}^+ , we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.