Quasilinear elliptic equations with critical potentials

Abstract

AbstractWe study Liouville theorems for problems of the form$\operatorname{div}_{L}({\mathscr{A}}(x,u,\nabla_{L}u))+V(x)|u|^{p-2}u=a(x)|u|^% {q-1}u\quad\text{on }{{\mathbb{R}}}^{N}$in the framework of Carnot groups. Here ${{\mathscr{A}}}$ is a vector-valued function satisfying Carathéodory condition and ${\nabla_{L}}$ 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.