Critical points of \(A\)-solutions of quasilinear elliptic equations (Q2705871)

The paper deals with the study of qualitative properties of solutions to the quasilinear degenerate equation \(\text{div} A(x,\nabla f)=0\) on a smooth \(n\)-dimensional (\(n\geq 2\)) Riemannian manifold. The main result of the paper is a counterpart of the celebrated Sard theorem. More precisely, assume that \(f\) is a solution (\(A\)-solution) of the above equation. The planar case \(n=2\) is easier, in the sense that \(f\) is a pseudoharmonic function. If \(n\geq 3\) the authors prove that the set of points at which \(f\) behaves regularly and at which \(f\) branches many times are sent by \(f\) into a set of small Hausdorff dimension.