LINEAR AND SEMILINEAR PROBLEMS INVOLVING Δλ-LAPLACIANS

In recent years a growing attention has been devoted to Delta(lambda)-Laplacians, linear second-order degenerate elliptic PDO's contained in the general class introduced by Franchi and Lanconelli in some papers dated 1983-84 [12, 13, 14]. Here we present a survey on several results appeared in literature in the previous decades, mainly regarding: (i) Geometric and functional analysis frameworks for the Delta(lambda)'s; (ii) regularity and pointwise estimates for the solutions to Delta(lambda)u = 0; (iii) Liouville theorems for entire solutions; (iv) Pohozaev identities for semilinear equations involving Delta(lambda)-Laplacians; (v) Hardy inequalities; (vi) global attractors for the parabolic and damped hyperbolic counterparts of the Delta(lambda)'s. We also show several typical examples of Delta(lambda)-Laplacians, stressing that their class contains, as very particular examples, the celebrated Baouendi-Grushin operators as well as the L-alpha,L-beta and P-alpha,P-beta operators respectively introduced by Thuy and Tri in 2002 [36] and by Thuy and Tri in 2012 [37].