$\nu_t$ is given in (\[2.1.2\]) and $\{g_n\}$ are a sequence of $C^1$ nondecreasing functions defined on ${\mathbb{R}}$ such that $g_n(0)=g(0)\ge 0$, $$\label{06-08-0}
 | g_n|\le g,\quad \sup_{s\in{\mathbb{R}}}|g_n(s)|=n\quad{\rm and}\quad \lim_{n\to\infty}{\|g_n-g\|}_{L^\infty_{loc}({\mathbb{R}})}=0.$$

The existence and uniqueness of weak solution to (\[2.0.6\]) is stated as follows.

\[pr 1\] Assume that $k>0$, $\alpha\in(0,1)$, $g_n$ is a $C^1$ nondecreasing function satisfying $g_n(0)\ge 0$ and (\[06-08-0\]). Then for $t\in(0,\sigma_0)$, problem (\[2.0.6\]) admits a unique weak solution $u_{n,k\nu_t}$ such that $$|u_{n,k\nu_t}|\le k\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\quad{\rm a.e.\ in}\quad \Omega$$ and $$\label{12-08-0}
{\|g_n(u_{n,k\nu_t})\|}_{L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le ck{\|\mathbb{G}_\alpha[t^{-\alpha}|\nu_t|]\|}_{L^1(\Omega)},$$ where $c>0$ independent of $t$, $k$ and $n$.

Furthermore, for any fixed $n\in{\mathbb{N}}$, $t\in(0,\sigma_0)$ and $k>0$, the mapping $\nu\mapsto u_{n,k\nu_t}$ is increasing.

[**Proof.**]{} For any $t>0$, we observe that $kt^{-\alpha}\nu_t$ is a bounded Radon measure in $\Omega$ and $g_n$ is bounded, then it follows from [@CV1 Theorem 1.1] that problem (\[2.0.6\]) admits a unique weak solution $u_{n,k\nu_t}$. Moreover, $kt^{-\alpha}\nu_t$ is increasing with respect to $\nu_t$ and $\nu_t$ is increasing with respect to $\nu$ by the definition of $\nu_t$, then applying [@CV1 Theorem 1.1], we have that for any fixed $t\in(0,\sigma_0)$ and $k>0$, the mapping $\nu\mapsto u_{n,k\nu_t}$ is increasing. $\Box$

To simplify the notation, we always write $u_{n,k\nu_t}$ by $u_{n,t}$ in this section. In order to consider the limit of 