Asymptotic Behavior of Radial Solutions to Singular $p$-Laplacian Equations as $p\to 1$

We study the singular limit $p\downarrow 1$ for positive radial entire solutions $u_p$ of $$-{\mathrm{\Delta }}_p{u}_p=F_p(\mathrm{\mid }x\mathrm{\mid },u_p,\mid \mathrm{\nabla }{u}_p\mid )u_p^{-{\beta }_p}\text{ in }{\mathbb{R}}^n,$$ where $n\ge 2,1<p<\min \{2,\sqrt{n}\}$, and $0\le {\beta }_p\le p-1$. Assuming that $$F_p\to F_1\text{ locally uniformly },\frac{{\beta }_p}{p-1}\to \mathbf{c}\in [0,1]\text{ as }p\downarrow 1,$$ we prove that, after passing to a subsequence, $u_{p_j}\to u$ in $C_{\text{loc }}\left({\mathbb{R}}^n\right)$ and $\mathrm{\nabla }{u}_{p_j}\mathrm{\nabla }u$ in $L_{\text{loc }}^{\mathrm{\infty }}({\mathbb{R}}^n;{\mathbb{R}}^n)$, where $u\in W^{1,\mathrm{\infty }}\left({\mathbb{R}}^n\right)$ is radial, and solves $$-{\mathrm{\Delta }}_{1}u=\mathtt{K}(\mathrm{\mid }x\mathrm{\mid })\text{ in }{\mathbb{R}}^n\mathrm{.}$$ On $\{x\in {\mathbb{R}}^n\mid \mathtt{G}(\mathrm{\mid }x\mathrm{\mid })<1\}$, we prove that $\mathtt{K}(\mathrm{\mid }x\mathrm{\mid })=w^{-\mathbf{c}}{F}_1(\mathrm{\mid }x\mathrm{\mid },u(x),0)$ and that $\mathrm{\nabla }u=0$ a.e., where $u_{p_i}^{p_i-1}\to w$. On the other hand, we show that if $\rho \mapsto F_1(r,t,\rho )$ is affine for every $r$ and $t$, then $\mathtt{K}(\mathrm{\mid }x\mathrm{\mid })=$ $w^{-\mathbf{c}}{F}_1(\mathrm{\mid }x\mathrm{\mid },u(x),\mathrm{\mid }\mathrm{\nabla }u(x)\mathrm{\mid })$ a.e. in ${\mathbb{R}}^n$. For $n=1$ and $1<p<2$, we study $$\frac{\mathrm{d}}{\mathrm{d}x}\left({\mid u_p^{\mathrm{\prime }}\mid }^{p-2}{u}_p^{\mathrm{\prime }}\right)={\mathtt{F}}_p(\mathrm{\mid }x\mathrm{\mid },u_p,\mid u_p^{\mathrm{\prime }}\mid )u_p^{-{\beta }_p}\text{ in }\mathbb{R}$$ and obtain analogous results as $p\downarrow 1$.