An exterior nonlinear elliptic problem with a dynamical boundary condition

Several results on existence, nonexistence and large-time behavior of small positive solutions u= u(x, t) were obtained before for the equation - Δu= u^p, x∈R+N, t> 0 , with a linear dynamical boundary condition. Here Δ is the N-dimensional Laplacian (in x). We study the effects of the change of the domain from the half-space to the exterior of the unit ball when N≥ 3. We show that the critical exponent for the existence of positive solutions and the decay rate of small solutions are different. More precisely, for the half-space problem the critical exponent is p= (N+ 1) / (N- 1) and the decay rate is t^-(N-1), while for the exterior problem we obtain the exponent p= N/ (N- 2) and the exponential rate e^-(N-2)t.