A note on the uniqueness and the non-degeneracy of positive radial   solutions for semilinear elliptic problems and its application
  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. 
Shinji Adachi, Masataka Shibata, Tatsuya Watanabe
math.AP