I am reading the paper "Area minimizing hypersurfaces with isolated singularities" by Hardt and Simon (https://eudml.org/doc/152770) and I get stuck on equation 1.9 on page 106.
The statement is as follows: Let $C$ be a cone in $\mathbb R^{n+1}$ with $\Sigma=C\cap S^n$ smooth $n-1$ dimensional submanifold of $S^n$. Let $L_C$ be the stability operator on $C$ i.e. $$L_Cu=\frac{\partial^2u}{\partial r^2}+\frac {n-1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}(\Delta_\Sigma u+|A_\Sigma|^2u)$$ where $u=u(r,\omega)$ is a function on $C$ in polar coordinate $x=r\omega$, $\Delta_\Sigma$ is the Laplacian on $\Sigma$, and $|A_\Sigma|^2$ is the norm squared of the second fundamental form of $\Sigma$.
If $v$ has bounded $C^1$ norm and is a positive solution to $$L_Cv=a_1\cdot D^2v+r^{-1}a_2\cdot Dv+r^{-2}a_3v\quad \text{on }r\ge 1$$ where $|a_j(r,\omega)|\le Cr^{-\alpha}$ for some $\alpha>0$ and all $j=1,2,3$ then we have $$(1.9)\quad \begin{aligned} \text{either } &v=(c_1+c_2\log r)r^{-\gamma_-}\varphi_1+O(r^{-\gamma_--\alpha})\\ \text{or }&v=c_1r^{-\gamma_+}\varphi_1+O(r^{-\gamma_+-\alpha}) \quad \text{as }r\to\infty \end{aligned} $$
Here $\gamma_\pm=\frac{n-2}{2}\pm\sqrt{(\frac{n-2}{2})^2+\lambda_1}$ where $\lambda_1$ is the first eigenvalue of $\Delta_\Sigma+|A_\Sigma|^2$ and $\varphi_1$ is a corresponding positive first eigenfuntion on $\Sigma$.
The paper indicate that we need to apply Harnack inequality and (1.8) which says if $v$ is a positive solution to $L_Cv=f$ with $|f|\le Cr^{-2-\gamma_--\alpha}$ then
$$(1.8)\quad
\begin{aligned}
\text{either } &v=(c_1+c_2\log r)r^{-\gamma_-}\varphi_1+O(r^{-\gamma_--\alpha})\\
\text{or }&v=c_1r^{-\gamma_+}\varphi_1+O(r^{-\gamma_+-\alpha}) \quad \text{as }r\to\infty
\end{aligned}
$$
I think I have no problem with these two but I just don't know how to get 1.9 from these.