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1 The University of Sydney Sydney 2 EDP - Equations aux dérivées partielles IECL - Institut Élie Cartan de Lorraine

Abstract : Given $B 10$ the unit ball of $\mathbb{R}^n$ $n\geq 3$, we study smooth positive singular solutions $u\in C^2B 10\setminus \{0\}$ to $-\Delta u=\frac{u^{2^\stars-1}}{|x|^s}-\mu u^q$. Here $0< s1$ and $\mu> 0$. When $\mu=0$ and $s=0$, the profile at the singularity $0$ was fully described by Caffarelli-Gidas-Spruck. We prove that when $\mu>0$ and $s>0$, besides this profile, two new profiles might occur. We provide a full description of all the singular profiles. Special attention is accorded to solutions such that $\liminf {x\to 0}|x|^{\frac{n-2}{2}}ux=0$ and $\limsup {x\to 0}|x|^{\frac{n-2}{2}}ux\in 0,+\infty$. The particular case $q=n+2-n-2$ requires a separate analysis which we also perform.





Autor: Florica C. Cîrstea - Frédéric Robert -

Fuente: https://hal.archives-ouvertes.fr/



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