The effect of Q-condition in elliptic equations involving Hardy potential and singular convection term

TL;DR

This paper proves existence and uniqueness of weak solutions for certain elliptic equations with Hardy potential and singular convection, also establishing regularity results through a contradiction approach.

Contribution

It introduces a novel contradiction method to establish existence, uniqueness, and regularity of solutions for elliptic equations with Hardy potential and singular convection terms.

Findings

Proved existence and uniqueness of weak solutions.

Established regularity of solutions under specific conditions.

Analyzed the interaction between zero order terms and data for regularization.

Abstract

Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -\Delta u=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+\lambda \frac{u}{\vert x\vert^2}+f(x),} \end{equation*} where \(\mathcal{O}\) is a bounded open set in \(\mathbb{R}^N\), $\left(\mathcal{A},\lambda\right) \in \left(0, \infty\right)^2$ and \(f\in W^{-1,2}(\mathcal{O})\). Additionally, by taking advantage of the regularizing effect of the interaction between the coefficient of the zero order term and the datum, we establish the existence, uniqueness and regularity of a weak solution to a quasi-linear boundary value problem whose simplest example is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -\Delta u +a(x)\vert u\vert^{p-2}u=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+\lambda \frac{u}{\vert x\vert^2}+f(x),} \end{equation*} under suitable assumptions on $a$ and $f$.