ON A CLASS OF GRAPHS WITH PRESCRIBED MEAN-CURVATURE

We study a class of quasilinear elliptic equations on the unit ball of R(n) in the divergence form SIGMA(j=1)n D(j){G(\x\2, \Du\2)D(j)u} = H (\x\) and get estimates on the boundary by using a modified barrier-fu nction technique of Bernstein. We establish a maximum principle for th e gradients of solutions and get a global gradient estimate. We prove that solutions with constant boundary condition must be radial. Finall y, we apply these results to graphs {(x,u(x)) : x is-an-element-of H(n )} where u : H(n) --> R is a smooth map of the n-hyperbolic space H(n) = B(0, 1) with the metric g = 4dx2/(1-\x\2)2 to get the existence of graphs with radial prescribed mean curvature.