Pointwise Estimates Near Singular Sets for Quasilinear Elliptic Equations

Abstract

In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $\Omega$ 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, $F(\cdot, \boldsymbol{\nabla} \rho_1) \leq 1$ and $F(\cdot, \boldsymbol{\nabla} \rho_2) \leq 1$ a.e. in $\mathcal{M}$. The singular set is defined by $\Gamma=\rho_1^{-1}(\{0\})$. The model problem is $-\Delta_{p(x)} u+|u|^{q-1} u=0$ 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 $\rho_1(x)=|(x_{d+1}, \ldots, x_n)|$ and $\rho_2(x)=|(x_1, \ldots, x_d)|$. The main tool in our analysis is the estimate $$ |u(x)| \leq \mathbf{C} \rho_1(x)^{-\tau} $$ near $\Gamma$ for weak solutions $u \in W_{loc}^{1, p(x)}(\bar{\Omega} \backslash(\Gamma \cup \Sigma) ; \vartheta) \cap L_{loc}^{\infty}(\bar{\Omega} \backslash(\Gamma \cup \Sigma))$, where the constants $\mathbf{C}>0$ and $\tau>0$ converge to positive values as $p^{+} \rightarrow 1$. This estimate is a key ingredient in proving that the singularity at $\Gamma$ is removable. Moreover, in a bounded domain $\Omega$, using this estimate and assuming that, for every variable exponent satisfying $1<p^{-} \leq p^{+}<\min \{2, q+1\}$, there exists a weak solution $u_p \in W_{loc}^{1, p(x)}(\Omega ; \vartheta) \cap L_{loc}^{\infty}(\Omega)$ of $$ -\operatorname{div}\left(|\boldsymbol{\nabla} u_p|_F^{p-2} \boldsymbol{\nabla} u_p\right)+|u_p|^{q-1} u_p=0 \quad \text { in } \Omega, $$ we prove that, for every $U \Subset \Omega$, there exists a subsequence $\{u_{p_m}\}$, with $p_m^{+} \rightarrow 1$, that converges to a solution $u \in B V(U ; \vartheta) \cap L^{q+1}(U ; \vartheta)$ of $$ -\Delta_1 u+|u|^{q-1} u=0 \quad \text { in } U . $$