Multi-bump Solutions for a Strongly Degenerate Problem with Exponential Growth in $$\mathbb {R}^N$$

In this paper, we study a class of strongly degenerate problems with critical exponential growth in \(\mathbb {R}^N\), \(N\ge 2\). We do not assume ellipticity condition on the operator and thus the maximum principle given by Lieberman (Commun Partial Differ Equ 16:311–361, 1991) can not be accessed. Therefore, a careful and delicate analysis is necessary and some ideas can not be applied in our scenario. The arguments developed in this paper are variational and our main result completes the study made in the current literature about the subject. Moreover, when \(N=2\) or \(N=3\) the solutions model the slow steady-state flow of a fluid of Prandtl-Eyring type.