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

TL;DR

This paper classifies the asymptotic behavior of positive singular solutions to a nonlinear elliptic equation with a sign-changing nonlinearity near the singularity, revealing new profiles beyond classical results.

Contribution

It extends the understanding of singular solutions by identifying and describing new asymptotic profiles when the nonlinearity changes sign.

Findings

Identifies two new singular profiles for solutions with sign-changing nonlinearities.

Provides a complete classification of all possible singular behaviors near the origin.

Analyzes special cases including the critical exponent case.

Abstract

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.