Linear and semilinear problems involving $Delta_lambda$-laplacians

In recent years a growing attention has been devoted to ∆λ-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 ∆λ’s; (ii) regularity and pointwise estimates for the solutions to ∆λu = 0; (iii) Liouville theorems for entire solutions; (iv) Pohozaev identities for semilinear equations involving ∆λ-Laplacians; (v) Hardy inequalities; (vi) global attractors for the parabolic and damped hyperbolic counterparts of the ∆λ’s. We also show several typical examples of ∆λ-Laplacians, stressing that their class contains, as very particular examples, the celebrated Baouendi-Grushin operators as well as the Lα,β and Pα,β operators respectively introduced by Thuy and Tri in 2002 [36] and by Thuy and Tri in 2012 [37].