\noindent The authors study existence of positive functions \(u \in H^1_0(\Omega)\) satisfying (in the distributional sense) the quasilinear elliptic equation \[ -\sum_{i,j = 1}^{N} D_j(a_{ij}(x,u)D_i u) + \frac{1}{2} \sum_{i=1}^{N} \frac{\partial a_{ij}}{\partial s}(x,u) D_i u D_j u = g(x,u) + |u|^{2^{*} - 2} u\quad \text{in }\Omega \] where \(\Omega \subset \mathbb{R}^N\) \((N \geq 3)\) is a bounded open set, \(2^{*} = \frac{2N}{N - 2}\) is the critical Sobolev exponent and \(a_{i,j}(x,s), g(x,s)\) are given functions with \(g\) having subcritical growth. The solutions are obtained as critical points of the functional \(J:H^1_0(\Omega) \to \mathbb{R}\), \[ J(u) = \frac{1}{2} \int_{\Omega} \sum_{i,j = 1}^{N} a_{ij}(x,u)D_i u D_j u - \int_{\Omega} G(x,u) - \frac{1}{2^{*}} \int_{\Omega} |u|^{2^{*}}, \] \noindent (where \(G(x,s) = \int_0^s g(x,t) dt\)), which is continuous but not even locally Lipschitz continuous due to the dependence of \(a_{ij}\) on \(s\). The authors explore the fact that \(J\) is weakly \(C_0^{\infty}\)-differentiable, apply an appropriate version of the Mountain Pass Lemma and develop a suitable variant of the technique by Brézis \& Nirenberg to overcome the lack of compactness due to the critical exponent and finally obtain existence of positive solutions.