Summary: In this paper we study the following non-autonomous singularly perturbed Dirichlet problem: \[ \epsilon^2 \Delta u - u + K(x) f(u) = 0, \;u>0 \text{ in } \Omega, \;u = 0 \text{ on } \partial \Omega, \] for a totally degenerate potential \(K\). Here \(\epsilon >0\) is a small parameter, \(\Omega \subset \mathbb{R}^N\) is a bounded domain with a smooth boundary, and \(f\) is an appropriate superlinear subcritical function. In particular, \(f\) satisfies \(0< \liminf_{ t \rightarrow 0+} f(t)/t^q \leq \limsup_{ t \rightarrow 0+} f(t)/t^q < + \infty\) for some \(1< q < + \infty\). We show that the least energy solutions concentrate at the maximal point of the modified distance function \(D(x) = \min \{ ( q+1) d(x, \partial A), 2 d(x, \partial \Omega) \}\), where \(A = \{ x \in \bar{ \Omega } \mid K(x) = \max_{ y \in \bar{ \Omega } } K(y) \}\) is assumed to be a totally degenerate set satisfying \(A^{\circ} \neq \varnothing\).