Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s

¹ LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté des Sciences, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France
farina@u-picardie.fr
² Università degli Studi di Milano, Dipartimento di Matematica, via Cesare Saldini 50, 20133 Milano, Italy
enrico@math.utexas.edu

Received: 21 December 2011
Revised: 19 June 2012

Abstract

We prove pointwise gradient bounds for entire solutions of pde’s of the form

     ℒu(x) = ψ(x, u(x), ∇u(x)),

where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.