In this paper, we investigate the quasilinear problem given by the following equation in \(\mathbb {R}^{N}\) : \(\begin{aligned}&-\text{ div }\left( |x|^{-ap}|\nabla u|^{p-2}\nabla u\right) +|x|^{-bp^{*}}[1+\mu V(z)]|u|^{p-2}u\\ &\quad = |x|^{-bp^{*}}f(u)+|x|^{-bp^{*}}\varrho |u|^{\sigma -2}u. \end{aligned}\) Here, \(1<p<N\) , \(0\le a< \frac{N-p}{p}\) , \(a<b\le a+1\) , \(p=p(a,b)=\frac{pN}{N-dp}\) and \(d=1+a-b\) . This equation exhibits singularity not only in the nonlinearity but also in the operator. By imposing suitable assumptions on the functions V and f, we establish the existence of solutions and investigate their concentration behavior as \(\mu \rightarrow +\infty \) . We consider three cases: the subcritical case, the critical case, and the supercritical case. In the subcritical case, we have \(\varrho =0\) , while in the critical case, we have \(\varrho =1\) and \(\sigma =p^{*}\) . In the supercritical case, we have \(\varrho =1\) and \(\sigma >p^{*}\) .
