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,v> 0, in R^N×R, where p> 0 and Δ_λ is a sub-elliptic operator of the form Δ_λ=_i=1^N_x_i(λ_i^2_x_i). Under some general assumptions of λ_i introduced by A.E. Kogoj and E. Lanconelli in Nonlinear Anal. 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,u> 0,in R^N.