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

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 $$div(\mathsf{A}\nabla v)+\mathsf{B}f(v)=0,\underset{|x|\to +\infty }{\lim }v(x)=0,x\in {\mathbb{R}}^n,$$ $n>2$, 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 $(0,c)$.

24 pages