Ground states to bi-harmonic nonlinear Schrödinger equations with competing dispersion

Abstract

We study ground-state solutions of the 4th order nonlinear Schrödinger equation with mixed dispersion via the elliptic profile equation ${\Delta }^{2}Q+2a\Delta Q+bQ-|Q{|}^{\alpha }Q=0$, focusing on symmetry breaking of least-action ground states in 2D in the regime $a=1$, $b=1+\epsilon$, $0<\epsilon \ll 1$, where nonradiality was proved by Lenzmann and Weth [26]. We establish a quotient-to-mass relation converting small-$\epsilon$ asymptotics of the Weinstein quotient into mass asymptotics, in any dimension and symmetry class. In 1D we prove the sharp small-$\epsilon$ rate for the quotient for every $\alpha >0$ and deduce the rates for the mass, action and potential norm. In two and higher dimensions the same relation, combined with the quotient asymptotics of Lenzmann-Weth [26] and Mandel-Oliveira e Silva [27], gives the mass rates for the Knapp-type nonradial and the radial ground states. The resulting threshold ${\alpha }_K(d)=8/(d+1)$ coincides for $d\le 3$ with the existence of normalized minimizers threshold of Fernández-Jeanjean-Mandel-Mariş [16], connecting the least-action and mass-constrained formulations. We complement this with a detailed numerical study. For the cubic nonlinearity we construct a nonradial branch from a Knapp-type example, compare it with radial and angular-mode branches, and verify the predicted rates. For the quartic nonlinearity, turning points and branching are present in the mass-energy diagrams in radial or two- and four-peak nonradial solutions. Here, although least-action ground states are nonradial for small $\epsilon$, the normalized ground states of the same mass appear radial, a consequence of branching. For $\alpha \ge 4$ we find only radial ground states, consistent with the conjectured sharp form of the Stein-Tomas inequality on ${\mathbb{S}}^1$, equivalent to the radial and nonradial masses having the same leading coefficient.