Quasilinear elliptic equations with critical potentials

We study Liouville theorems for problems of the form divL(A (x, u, ∇Lu)) + V(x)|u|p−2u = a(x)|u|q−1u on RN in the framework of Carnot groups. Here A is a vector-valued function satisfying Carathéodory condition and ∇L denotes an horizontal gradient, V is a given singular potential, a is a measurable scalar function and q > p − 1. Particular emphasis is given to the case when V is a Hardy or Gagliardo–Nirenberg potential. The results are new even in the canonical Euclidean setting.