A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application

TL;DR

This paper investigates the uniqueness and non-degeneracy of positive radial solutions in semilinear elliptic equations, extending previous results to sublinear growth cases and applying findings to modified Schrödinger equations.

Contribution

It introduces new analysis techniques for sublinear nonlinearities and establishes ground state properties for modified Schrödinger equations.

Findings

Proves uniqueness of positive radial solutions in broader cases

Demonstrates non-degeneracy of solutions under new conditions

Applies results to ground states of modified Schrödinger equations

Abstract

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schr\"odinger equations.