On the uniqueness of sign changing bound state solutions of a semilinear equation

Abstract

We establish the uniqueness of the higher radial bound state solutions of

We assume that the nonlinearity $f\in C(-\infty ,\infty )$ is an odd function satisfying some convexity and growth conditions, and has one zero at $b>0$, is nonpositive and not-identically 0 in $(0,b)$, positive in $[b,\infty )$, and is differentiable in $(0,\infty )$.