Biharmonic non-linear Schrödinger equation with an unbounded inhomogeneous term
Taif Abdullah Enaoufal, Tarek Saanouni
TL;DR
This work analyzes the focusing biharmonic inhomogeneous NLS with an unbounded weight $|x|^b$, deriving a sharp inhomogeneous Gagliardo-Nirenberg inequality with optimal constant $C_{opt}$ and a radial ground state $\zeta$ that saturates the inequality. The ground state provides a sharp energy threshold that governs a dichotomy between global existence and finite-time blow-up for energy solutions below the threshold, with radial symmetry enabling key estimates for the unbounded weight. The analysis combines variational techniques (Weinstein-style) with a localized Morawetz/variance framework to treat both energy-subcritical and mass-critical regimes, and it establishes a compact embedding crucial for the variational arguments. Overall, the paper extends the theory of inhomogeneous higher-order NLS by handling growing weights, elucidating new phenomena due to the weight and loss of translation invariance, and highlighting the role of radial symmetry in obtaining sharp long-time behavior results.
Abstract
This paper is devoted to the analysis of a focusing nonlinear biharmonic Schrödinger equation in the presence of an unbounded growing up inhomogeneous term. The first main contribution of this work is the derivation of an inhomogeneous Gagliardo-Nirenberg inequality adapted to the unbounded weight, which provides the necessary control over the nonlinear term in terms of Sobolev norms. Building on this inequality, we then investigate the long-time behavior of solutions and establish a sharp dichotomy: solutions with initial data below the ground state energy either exist globally in time or experience finite-time blow-up. A distinctive feature of our results is that the analysis of the unbounded inhomogeneous term requires the imposition of radial symmetry on the initial data, which allows us to exploit certain Strauss type Sobolev estimates that would not hold in the general non-radial case. This work complements previous studies on biharmonic Schrödinger equations with singular inhomogeneities, highlighting both the challenges and the new phenomena that arise when the nonlinearity is weighted by a growing up unbounded function, which broke the space translation invariance of the standard homogeneous associated equation.