On Existence and Stability Results for Normalized Ground States of Mass-Subcritical Biharmonic Nonlinear Schrödinger Equation on [math]

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4415-4439, August 2024.
Abstract. We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space [math]. Following the crucial scaling arguments introduced in [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number [math] that sharply determines the [math]-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer scale-invariant and the arguments from [Terracini, Tzvetkov, and Visciglia, Anal. PDE, 7 (2014), pp. 73–96] for determining the sharp [math]-dependence of the ground states fail. The main novelty of the present paper is to address this difficult and interesting issue: Using a different scaling argument, we show that [math]-independence of ground states with small mass still holds in the case [math] and [math]. Additionally, we also prove that ground states with sufficiently large mass must possess nontrivial [math]-dependence by appealing to some novel construction of test functions. The latter particularly holds for all parameters lying in the full mass-subcritical regime.