Compactness of positive radial Solution Sets and corresponding $L^2$-Mass sets for "Zero Mass" Quasi-linear Schr\"odinger Equations

Abstract

We study the compactness of two sets of positive radial solutions to ``zero mass'' quasi-linear Schr\"odinger equation \[ -\Delta u-u\Delta(|u|^2)=g(u) \quad\text{in }\mathbb R^N,\quad N\ge5. \] For nonlinearities that are either strictly subcritical or asymptotically critical at infinity, we prove that the set of least energy positive radial solutions is nonempty and compact in the natural space. Moreover, in the strictly subcritical case and under additional assumptions, we show that the set of all finite $L^2$-mass positive radial solutions is compact in \(L^2(\mathbb R^N)\). The proof relies on uniform decay estimates for the corresponding positive radial solutions of the transformed semilinear equation, which yield the required uniform \(L^2\)-tail control.