A note on positive solutions of Lichnerowicz equations involving the $Δ_λ$-Laplacian - Duong, Anh Tuan; Nguyen, Thi Quynh (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $Δ_λ$-Laplacian $$ v_t-Δ_λ v=v^{-p-2}-v^p,\quad v> 0, \quad in \mathbb R^N×\mathbb R, $$ where $p> 0$ and $Δ_λ$ is a sub-elliptic operator of the form$$Δ_λ=\sum_{i=1}^N\partial_{x_i}\big(λ_i^2\partial_{x_i}\big).$$Under some general assumptions of $λ_i$ introduced by A.E. Kogojand E. Lanconelli in Nonlinear Anal. {\bf 75} (2012), no.\ 12, 4637-4649,we shall prove a uniform lower bound of positive solutions of the equation provided that $p> 0$. Moreover, in the case $p> 1$, we shall show that the equation has only the trivial solution $v=1$.As a consequence, when $v$ is independent of the time variable, we obtain the similar results for the elliptic Lichnerowicz equation involving the $Δ_λ$-Laplacian $$ -Δ_λ u=u^{-p-2}-u^p,\quad u> 0,\quad in \mathbb R^N. $$