Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

Abstract

We study the semilinear elliptic equation \begin{equation*} -Δu=u^α|\log u|^β\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset {\mathbb{R}}^n$ with $n\ge 3$, $\frac{n}{n-2}<\alpha <\frac{n+2}{n-2}$ and $-\infty <\beta <\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus \{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{α-1}} \left(\log \frac{1}{|x|}\right)^{-\fracβ{α-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2}{α-1}\right)^{1-β}\left(n-2-\frac{2}{α-1}\right)\right]^{\frac{1}{α-1}}. \end{equation*}

to appear in Adv. Nonlinear Anal