I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.

The authors show that the following equation has a unique positive radial solution such that $u>0$ and $u'(r)<0$ under the assumption that $1<p<\frac{n+2}{n-2}$ when $n\geq 3.$

I was wondering if this equation or its radial solutions have been studied in the case when $p=\frac{n+2}{n-2}.$ In particular, can one expect to show the same result as Kwong in this case?