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 >Osubjecttotheboundarycondition ~~=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-•