Two nonlinear days in Urbino 2017. Electron. J. Diff. Eqns., Conference 25 (2018), pp. 167-178.

Linear and semilinear problems involving Delta-lambda-laplacians

Alessia E. Kogoj, Ermanno Lanconelli

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 $L_{\alpha, \beta}$ and $P_{\alpha, \beta}$ operators respectively introduced by Thuy and Tri in 2002 [36] and by Thuy and Tri in 2012 [37].

Published September 15, 2018.
Math Subject Classifications: 35J70, 35H20, 35K65.
Key Words: Degenerate elliptic PDE; semilinear subelliptic PDE; Delta-lambda-Laplacian.

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Alessia E. Kogoj
Dipartimento di Scienze Pure e Applicate (DiSPeA)
Università degli Studi di Urbino "Carlo Bo"
Piazza della Repubblica
13 - 61029 Urbino (PU), Italy
Ermanno Lanconelli
Dipartimento di Matematica
Università degli Studi di Bologna
Piazza di Porta San Donato
5 - 40126 Bologna, Italy

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