A NOTE ON POSITIVE SOLUTIONS OF LICHNEROWICZ EQUATIONS INVOLVING THE Δλ-LAPLACIAN

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), v > 0; in R-N x R, where p > 0 and Delta(lambda) is a sub-elliptic operator of the form Delta(lambda) = Sigma(N)(i=1) partial derivative x(i) (lambda(2)(i)partial derivative x(i) ). Under some general assumptions of lambda(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 Delta(lambda)-Laplacian -Delta(lambda)u = u(-p-2) - u(p), u > 0, in R-N.