Asymptotic Behavior of Radial Solutions to Singular $p$-Laplacian Equations as $p\to1$

RESUMO

We study the singular limit $p \downarrow 1$ for positive radial entire solutions $u_p$ of $$ -\Delta_p u_p=F_p\left(|x|, u_p,\left|\nabla u_p\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R}^n, $$ where $n \geq 2,1<p<\min \{2, \sqrt{n}\}$, and $0 \leq \beta_p \leq p-1$. Assuming that $$ F_p \rightarrow F_1 \quad \text { locally uniformly }, \quad \frac{\beta_p}{p-1} \rightarrow \mathbf{c} \in[0,1] \quad \text { as } p \downarrow 1, $$ we prove that, after passing to a subsequence, $u_{p_j} \rightarrow u$ in $C_{\text {loc }}\left(\mathbb{R}^n\right)$ and $\nabla u_{p_j} \stackrel{*}{\rightharpoonup} \nabla u$ in $L_{\text {loc }}^{\infty}\left(\mathbb{R}^n ; \mathbb{R}^n\right)$, where $u \in W^{1, \infty}\left(\mathbb{R}^n\right)$ is radial, and solves $$ -\Delta_1 u=\mathtt{K}(|x|) \quad \text { in } \mathbb{R}^n . $$ On $\left\{x \in \mathbb{R}^n \mid \mathtt{G}(|x|)<1\right\}$, we prove that $\mathtt{K}(|x|)=w^{-\mathbf{c}} F_1(|x|, u(x), 0)$ and that $\nabla u=0$ a.e., where $u_{p_i}^{p_i-1} \rightarrow w$. On the other hand, we show that if $\rho \mapsto F_1(r, t, \rho)$ is affine for every $r$ and $t$, then $\mathtt{K}(|x|)=$ $w^{-\mathbf{c}} F_1(|x|, u(x),|\nabla u(x)|)$ a.e. in $\mathbb{R}^n$. For $n=1$ and $1<p<2$, we study $$ \frac{\mathrm{d}}{\mathrm{~d} x}\left(\left|u_p^{\prime}\right|^{p-2} u_p^{\prime}\right)=\mathtt{F}_p\left(|x|, u_p,\left|u_p^{\prime}\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R} $$ and obtain analogous results as $p \downarrow 1$.