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

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