2026/07/22 by Jesus A. León Tordecilla, Duván Cardona
#math.AP
We prove the existence of positive weak solutions for a quasilinear equation with exponential nonlinearity on arbitrary stratified Lie groups for horizontal q-Laplacians, also called q-sub-Laplacians on these groups. The nonlinearity combines a concave term with an exponential growth that can be subcritical, critical, or supercritical with respect to the Trudinger--Moser inequality for Lorentz spaces on stratified Lie groups. We prove a version of this inequality in this paper. Since the problem is not variational in the natural energy space, classical minimisation or critical-point techniques do not apply directly. Positive solutions to the resulting semilinear equation are then obtained via a carefully designed Galerkin method applied to this setting. Our main theorem recovers previous known results on ℝn in the complete range q∈ (1,∞), and the previous known result on the Heisenberg group ℍn for q=2n+2, and is extended here in the full range q∈ (1,∞). Other particular and fundamental cases, including the Engel group, are discussed.