Biharmonic non-linear Schr\"odinger equation with an unbounded inhomogeneous term

Abstract

This paper is devoted to the analysis of a focusing nonlinear biharmonic Schr\"odinger 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 Schr\"odinger 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.