A remark on the uniqueness of positive solutions to semilinear elliptic equations with double-power nonlinearities

Abstract

We consider the uniqueness of positive solutions to \begin{equation*} \begin{cases} \triangle u +f(u)=0 & \text{in $\mathbb{R}^n$},\\ \displaystyle \lim_{\lvert x \rvert \to \infty} u(x) =0, \end{cases} \end{equation*} where $f(u)=- \omega u+u^p-u^{2p-1},$ with $\omega>0$ and $p>1$. It is known that for fixed $p>1$, a positive solution exists if and only if $\omega < \omega_p:=\dfrac{p}{(p+1)^2}$. We deduce the uniqueness in the case where $\omega$ is close to $\omega_p$, from the argument in the classical paper by Peletier and Serrin [9], thereby recovering a part of the uniqueness result of Ouyang and Shi [8] for all $\omega \in (0, \omega_p)$. In the appendix we consider the more general nonlinearity \begin{equation*} f(u)=-\omega u + u^p - u^q, \qquad \omega>0, ~~q>p>1 \end{equation*} and discuss the existence and uniqueness conditions. There we prove the fact that $f$ having positive part is equivalent to $\tilde{f}$ remaining negative, where $ \tilde{f}(u):= (uf'(u))'f(u)-uf'(u)^2 . $