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

Abstract

Given $B_1(0)$ the unit ball of ${\mathbb{R}}^n$ ($n\ge 3$), 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 ${\mathrm{liminf}}_{x\to 0}|x{|}^{\frac{n-2}{2}}u(x)=0$ and ${\mathrm{limsup}}_{x\to 0}|x{|}^{\frac{n-2}{2}}u(x)\in (0,+\infty )$. The particular case $q=(n+2)/(n-2)$ requires a separate analysis which we also perform.

References in corpus (1)

• Refined asymptotics for constant scalar curvature metrics with isolated singularities

Cited by in corpus (2)

• Sharp existence and classification results for nonlinear elliptic equations in ${\mathbb{R}}^N\setminus \{0\}$ with Hardy potential
• Examples of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity