A uniqueness result for certain semilinear elliptic equations

Abstract

For the problem $\Delta u + f(u)=0 \ \text{ in } \ \Bbb R^n$; $u(x)\rightarrow 0, \ \text{as} \ |x| \rightarrow \infty$ we use a shooting method to prove that there is at most one positive radially symmetric solution if $u$ decays like $|x|^{-(n-2)}$ as $|x| \rightarrow \infty$, and $f$ is similar in shape to $f(u)=u^p-u^q$ with $n>2$ and $q>p>(n+2)/(n-2)$.