I have been recently reading Kwong's [paper][1] on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$. 

The authors show that the above 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?


  [1]: https://link.springer.com/content/pdf/10.1007/BF00251502.pdf