A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms

TL;DR

This paper investigates the existence and regularity of solutions for a class of anisotropic elliptic equations with unbounded coefficients and singular lower order terms that depend on the gradient, considering various parameter regimes.

Contribution

It introduces new existence and regularity results for anisotropic elliptic problems with singular gradient-dependent terms and unbounded coefficients, extending previous theories.

Findings

Existence of solutions under certain parameter conditions.

Regularity results depending on the values of q and θ.

Handling of singular lower order terms in anisotropic elliptic equations.

Abstract

In this work, we study the existence and regularity results of anisotropic elliptic equations with a singular lower order term that grows naturally with respect to the gradient and unbounded coefficients. We take up the following model problem \begin{equation*} \left\{\begin{array}{ll}-\displaystyle\sum\limits_{j\in J} D_{j}\left(\left[ 1+ u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\ u>0& \hbox{in}\;\Omega, u =0 & \hbox{on}\; \partial\Omega, \end{array} \right. \end{equation*} $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $j\in J=\{1,2,\ldots,N\},$ $q>0$, $0< \theta<1$, $2\leq p_{1}\leq p_{2}\leq... \leq p_{N}$ and $f\in L^{1}(\Omega)$. Our study's conclusions will depend on the values of $q$ and $\theta$.