Regularity and symmetry results for nonlinear degenerate elliptic equations

Abstract

In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form − div(A(|∇u|)∇u) + B (|∇u|) = f (u); in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.

The moving plane method for doubly singular elliptic equations involving a first order term

“…Moreover, we take into account the variant developed in the seminal papers of Gidas-Ni-Nirenberg [31] and Berestycki-Nirenberg [4]. We want to stress that this method is very powerful and can be adapted to several problems, hence, for this reason, we refer the reader to [2,3,11,17,19,20,21,22,23,28,30,33,34,38] for some related works regarding such a technique in bounded domains. This procedure can be performed in general domains providing partial monotonicity results near the boundary and symmetry when the domain is convex and symmetric.…”