| Abstract: |
| In this talk, we provide a full description of the asymptotic profile near zero of the positive solutions to elliptic equations of the form $-\Delta u=|x|^{-s} u^{2^*(s)-1}-\mu u^q$ in $B\setminus\{0\}$, where $B$ denotes the unit ball of $\mathbb R^n$ with $n\geq 3$, $s\in (0,2)$, $2^*(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. We show that along with a Caffarelli-Gidas-Spruck type profile, the solution may develop two new singular behaviors at zero due to the interaction between the critical Hardy-Sobolev type potential and the pure power non-linearity. We also prove the existence of all these sharp singular profiles. The results are based on works with F. Robert (University of Lorraine) and J. V\`{e}tois (McGill University). |
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