In this article we obtain point-wise asymptotic estimates for solutions to \( \left\{ \begin{aligned} -\textrm{div}\,(w|\nabla u|^{p-2}\nabla u)&=w|u|^{q-2}u & \text {in }\Omega ,\\ u\in D^{1,p}&(\Omega ;wdx), \end{aligned} \right. \) for an open unbounded set \(\Omega \subseteq \mathbb {R}^N\) and a critical exponent \(q>p\) in the sense of Sobolev. To do so, we firstly study the quasi-linear equation \( \left\{ \begin{aligned} \textrm{div}\, \mathcal {A}(x,u,\nabla u)&=\mathcal {B}(x,u,\nabla u) & \text {in }\Omega ,\\ u\in H^{1,p}_{loc}&(\Omega ;wdx), \end{aligned} \right. \) where \(\mathcal {A}\) and \(\mathcal {B}\) are suitable Carathéodory functions which are structurally similar to \(w|\nabla u|^{p-2}\nabla u\) and \(w|u|^{p-2}u\) respectively for \(p>1\) and a \(p\) -admissible weight function \(w\) . We fill a gap in the literature and we establish interior regularity results of weak solutions to this kind of quasi-linear equations.
