On the uniqueness of bound state solutions of a semilinear equation with weights

TL;DR

This paper proves the uniqueness of radial bound state solutions for a class of weighted elliptic equations with specific nonlinearities, extending understanding of solution behavior in weighted PDEs.

Contribution

It establishes the first rigorous proof of uniqueness for radial bound states in weighted elliptic equations with general nonlinearities under specified conditions.

Findings

Uniqueness of radial bound state solutions is proven.

Conditions on weights and nonlinearities are identified for uniqueness.

The results extend previous work on unweighted equations to weighted cases.

Abstract

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to $$ {div}\big(\mathsf A\,\nabla v\big)+\mathsf B\,f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,$$ $n>2$, where $\mathsf A$ and $\mathsf B$ are two positive, radial, smooth functions defined on $\mathbb R^n\setminus\{0\}$. We assume that the nonlinearity $f\in C(-c,c)$, $0<c\le\infty$ is an odd function satisfying some convexity and growth conditions, and has a zero at $b>0$, is non positive and not identically 0 in $(0,b)$, positive in $(b,c)$, and is differentiable in $(0,c)$.