Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity

Florica C. Cîrstea (1) , Frédéric Robert (2)

1 The University of Sydney
2 IECL - Institut Élie Cartan de Lorraine

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ 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 $\liminf_{x\to 0}|x|^{\frac{n-2}{2}}u(x)=0$ and $\limsup_{x\to 0}|x|^{\frac{n-2}{2}}u(x)\in (0,+\infty)$. The particular case $q=(n+2)/(n-2)$ requires a separate analysis which we also perform.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Florica C. Cîrstea, Frédéric Robert. Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity. Proceedings of the London Mathematical Society, 2016, ⟨10.1112/plms.12003⟩. ⟨hal-01259766⟩