Pointwise Estimates Near Singular Sets for Quasilinear Elliptic Equations

In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $\mathrm{\Omega }$ of an $n$-dimensional Finsler manifold ( $\mathcal{M},F,\vartheta$ ). We work with Lipschitz functions ${\rho }_1$ and ${\rho }_2$ satisfying distance-type properties; in particular, $F(\cdot ,\mathbf{\nabla }{\rho }_1)\le 1$ and $F(\cdot ,\mathbf{\nabla }{\rho }_2)\le 1$ a.e. in $\mathcal{M}$. The singular set is defined by $\mathrm{\Gamma }={\rho }_1^{-1}(\{0\})$. The model problem is $-{\mathrm{\Delta }}_{p(x)}u+\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u=0$ in domains of ${\mathbb{R}}^n\cong {\mathbb{R}}^d\times {\mathbb{R}}^{n-d}\cong {\rho }_1^{-1}(\{0\})\times {\rho }_2^{-1}(\{0\})$, where ${\rho }_1(x)=\mathrm{\mid }(x_{d+1},\dots ,x_n)\mathrm{\mid }$ and ${\rho }_2(x)=\mathrm{\mid }(x_1,\dots ,x_d)\mathrm{\mid }$. The main tool in our analysis is the estimate $$\mathrm{\mid }u(x)\mathrm{\mid }\le \mathbf{C}{\rho }_1(x{)}^{-\tau }$$ near $\mathrm{\Gamma }$ for weak solutions $u\in W_{loc}^{1,p(x)}(\bar{\mathrm{\Omega }}\mathrm{\backslash }(\mathrm{\Gamma }\cup \mathrm{\Sigma });\vartheta )\cap L_{loc}^{\mathrm{\infty }}(\bar{\mathrm{\Omega }}\mathrm{\backslash }(\mathrm{\Gamma }\cup \mathrm{\Sigma }))$, where the constants $\mathbf{C}>0$ and $\tau >0$ converge to positive values as $p^{+}\to 1$. This estimate is a key ingredient in proving that the singularity at $\mathrm{\Gamma }$ is removable. Moreover, in a bounded domain $\mathrm{\Omega }$, using this estimate and assuming that, for every variable exponent satisfying $1<p^{-}\le p^{+}<\min \{2,q+1\}$, there exists a weak solution $u_p\in W_{loc}^{1,p(x)}(\mathrm{\Omega };\vartheta )\cap L_{loc}^{\mathrm{\infty }}(\mathrm{\Omega })$ of $$-\mathrm{div}\left(\mathrm{\mid }\mathbf{\nabla }{u}_p{\mathrm{\mid }}_F^{p-2}\mathbf{\nabla }{u}_p\right)+\mathrm{\mid }{u}_p{\mathrm{\mid }}^{q-1}{u}_p=0\text{ in }\mathrm{\Omega },$$ we prove that, for every $U⋐\mathrm{\Omega }$, there exists a subsequence $\{u_{p_m}\}$, with $p_m^{+}\to 1$, that converges to a solution $u\in BV(U;\vartheta )\cap L^{q+1}(U;\vartheta )$ of $$-{\mathrm{\Delta }}_{1}u+\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u=0\text{ in }U\mathrm{.}$$