Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition

In this paper we prove that the only nonnegative solution to the Laplace equation .6.u=Ointhehalfspace{(xl,"' ,xn); Xl >O}subjecttotheboundarycondition ~~=uP on {x1 = 0} is the trivial solution u = 0 when pis subcritical, namely, 1 :=:; p < n~2 • This result is then used to obtain the blowup rate of a heat equation with the boundary condition au-uP an-•