  In this paper we prove regularity results for a class of nonlinear degenerate
elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla
u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; 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.
