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

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 $f$.