  In this paper, we consider the existence and nonexistence of positive
solution for a Neumann problem with double critical exponents and fast
increasing weighted in half-space. This problem is closely related to the study
of self-similar solutions for nonlinear heat equation. By applying the Mountain
Pass Theorem without (PS) condition and the delicate estimates for the Mountain
Pass level, we obtain the existence of a positive solution under different
assumptions. Meanwhile, some nonexistence results for this problem is also
obtained by an improved Pohozaev identity and Hardy inequality according to the
value of the parameters. Particularly, we give the best lower bound of the
parameter for the existence of a positive solution of this problem if dimension
N=4.
