A note on positive solutions of Lichnerowicz equations involving the $\Delta_\lambda$-Laplacian

Abstract

In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ v_t-\Delta_\lambda v=v^{-p-2}-v^p,\quad v> 0, \quad \mbox{ in }\mathbb R^N\times\mathbb R, $$ where $p> 0$ and $\Delta_\lambda$ is a sub-elliptic operator of the form $$ \Delta_\lambda=\sum_{i=1}^N\partial_{x_i}\big(\lambda_i^2\partial_{x_i}\big). $$ Under some general assumptions of $\lambda_i$ introduced by A.E. Kogoj and 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 $\Delta_\lambda$-Laplacian $$ -\Delta_\lambda u=u^{-p-2}-u^p,\quad u> 0,\quad \mbox{in }\mathbb R^N. $$