| Abstract: |
| We devote this talk to discussing separation property of positive radial solutions to Matukuma type equations. For the Matukuma equation
\begin{align*}
-\Delta u=\frac{1}{1+|x|^{2}}u^{p}\quad\text{in}\quad\mathbb{R}^{N},\tag{M}
\end{align*}
it is already known that if the exponent $p$ is greater than or equal to the Joseph-Lundgren exponent $p_{JL}$, then the family of positive radial solutions has separation property, i.e., any two positive radial solutions do not intersect each other. In this talk, we prove that $p_{JL}$ is critical with respect to separation property of positive radial solutions to (M), i.e., if $p\in (p_{S},p_{JL})$, then there exist two positive radial solutions intersecting each other, where $p_{S}$ denotes the Sobolev exponent. Moreover, we extend the result to general Matukuma type equations. |
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