Entire Solutions To A Semilinear Elliptic Equation And Its Asymptotic Behavior

In this paper, we will study the entire solution and its asymptotic behavior of a semilinear elliptic equation△U+|χ|TeU=O inRN,T>-2(0.1)Firstly, we prove that for N≥3, the set of all radial solution of (0.1) is a one parameter family{ua} with ua(0)=a, ua(0)=0, andSecondly, we prove that for35≤N<10+4t, a≠β， then the group of ua oscillates around that of uβ infinitely many times; for N≥10+4T, a<β, then ua(r)<uβ(r), no two radial solutions of (0.1) can intersect each other.Next, the equation (0.1) exists uniqueness singular solution U(x)=In[(N-2)(2+T)]-(2+T) In|x|and it is stable when N≥10+4T.Lastly, we study the behaviour of entire solution of (0.1) at infinity:for35≤N<10+4T...