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

TL;DR

This paper proves the uniqueness of higher radial bound state solutions for a semilinear elliptic equation with specific nonlinearities, under certain convexity and growth conditions, extending understanding of solution structure.

Contribution

It establishes the uniqueness of higher radial bound state solutions for a class of semilinear equations with odd nonlinearities under specified conditions.

Findings

Uniqueness of higher radial bound states proved

Conditions on nonlinearity include convexity and growth constraints

Results extend previous understanding of solution multiplicity

Abstract

We establish the uniqueness of the higher radial bound state solutions of $$ \Delta u +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth conditions, and either has one zero at $b>0$, is non positive and not identically 0 in $(0,b)$, and is differentiable and positive $[b,\infty)$, or is positive and differentiable in $[0,\infty)$.