Existence and Uniqueness of Positive Solutions for Degenerate Logistic Neumann Problems in a Half Space

In this paper, we consider the existence and uniqueness of positive solutions of the following degenerate logitic Neumann problem in half space -Delta u = a(x)u - b(x)u(q) x is an element of T, partial derivative u/partial derivative n = 0, on partial derivative T, Where T = {x =(x(1), x(2),...x(N)):x(N) > 0}, (N >= 2), q is a constant greater than 1, a(x) and b(x) are continuous functions with b(x) nonnegative on R-N and n is outward pointing unit normal vector of partial derivative T. Under suitable conditions, we show that this problem has only one positive solution.