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

We study the semilinear elliptic equation \begin{equation*} -\Delta u=u^\alpha |\log u|^\beta\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < \alpha < \frac{n+2}{n-2}$ 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}{\alpha-1}} \left(\log \frac{1}{|x|}\right)^{-\frac{\beta}{\alpha-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2}{\alpha-1}\right)^{1-\beta}\left(n-2-\frac{2}{\alpha-1}\right)\right]^{\frac{1}{\alpha-1}}. \end{equation*}