Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities
Fessel achhoud, Hichem Khelifi
TL;DR
The paper tackles existence and regularity for anisotropic elliptic equations with unbounded coefficients and a singular right-hand side, modeled by $-\sum_j \partial_j([b(x)+u^q]|\partial_j u|^{p_j-2}\partial_j u) = f/u^\gamma$ in $\mathcal{D}$ with $f\in L^1(\mathcal{D})$, $f\ge0$. It employs a regularization/truncation scheme and a Schauder fixed-point argument to construct approximations $u_n$, derives uniform a priori estimates across different regimes of $\gamma$ and $q$, and passes to the limit to obtain a nonnegative weak solution with detailed anisotropic Sobolev regularity. The main contribution is a comprehensive regularity taxonomy depending on $\gamma$ and $q$, including $W_0^{1,(p_j)}(\mathcal{D})$ or weaker anisotropic spaces and corresponding $L^r$ integrability, thereby extending known isotropic results to the anisotropic, low-data setting $f\in L^1(\mathcal{D})$. The work also clarifies how the lower-order, gradient terms regularize the singular problem and discusses potential generalizations to broader data classes, connecting with prior results in the isotropic and anisotropic literature.
Abstract
In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose simplest model is \begin{equation*} -\sum_{j=1}^N\partial_{j}\left([1+u^{q}]\vert \partial_{j} u \vert^{p_{j}-2} \partial_{j} u\right)= \frac{f}{u^γ}\text{ in $\mathcal{D},$}\quad u>0 \text{ in $\mathcal{D},$} \quad u=0 \hbox{ on}\;\; \partial\mathcal{D}, \end{equation*} where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{N}$ with $N>2$, $ γ\geq0$, $q >0 $, $p_{j}>2$ for all $j=1,...,N$ and the source term $f$ belongs to $L^1(\mathcal{D})$, with $f \geq 0$ and $f \not\equiv 0$.