The Existence And Uniqueness Of The Positive Solution Of The Logistic Type Elliptic Equation In R~N

In this paper ,we consider positive solution of the non-generate logistic type elliptic equation-â–³u = a(x)u - b(x)f(u),x∈R^N, N≥2.First of all we show that under the conditiona(x), b(x) is continuous and f(u)/u increasing,there exists a minimal positive solution(?) and a maximal positive solution (?) onR^N.Forthermore ,we show that under rather general conditions ona(x),b(x) and f(u)/u for large | x |,there exists some properties of any solution of the non-generate logistic type elliptic function for all large | x | .If conditions(A₄)(A₅)(A₆) are satisfied ,there exists a unique positive solution.Then we consider the degenerate logistic type elliptic function :Ω₀ = {x : x∈R^N, b(x) = 0}, b(x) >0,x∈R^N\(?)Ω₀ is a smooth bounded nonempty domain inR^N,we show that under rather general conditions ona(x), b(x) and f(u)/u for large | x |,there exists a unique positive solution of the generate logistic type elliptic function onR^N.At last we consider the outer sphere problem with the same method. It is-â–³u = a(x)u - b(x)f(u), x∈R^N \ (?), u | (?)Ω= 0WhereΩis a bound smooth domain inR^N.Using the same methods,if we show that under rather general conditions ona(x),b(x) andf(u)/u for large | x |,there exists a unique positive solution of the outer sphere problem onR^N.Our results improve the corresponding ones in [1]and[2].