On the uniqueness of the second bound state solution of a semilinear equation

Abstract

We establish the uniqueness of the second radial bound state solution of

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