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In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of R n (n ≥ 2). Secondly, we study the geometric structure about the critical set K of solutions u for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of R n (n ≥ 3), and deduce that K exists (respectively does not exist) a rotationally symmetric critical closed surface S. In fact, in an eccentric spherical annulus domain, K is made up of finitely many isolated critical points (p 1 , p 2 , · · · , p l ) on an axis and finitely many rotationally symmetric critical Jordan curves (C 1 , C 2 , · · · , C k ) with respect to an axis.

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We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.