Analysis Of The Asymptotic Behavior Of A Class Of Semilinear Elliptic Equations

We are interested in the elliptic boundary value problem in Ω with on (?)Ω where Ω (?) R~2 denotes a bounded domain with smooth boundary (?)Ω.Takasi Senba and Takashi Suzuki[23] obtained the asymptotic behavior of the blow-up solutions of this elliptic equation by complex analysis, which is restricted to solve the Laplacian problem in dimension 2. This article will give another proof.The key idea of this article is to change this elliptic equation to the equation—Δw = V(x)e~w, which we have the asymptotic behavior of the blow-up solutions according to Brezis-Merle[3]. First, we use a conformal mapping to straighten the boundary, so that we can take an even extension, thereby avoid discussing the boundary condition. Then, we get the result after a careful calculation(the key tool is Pohozaev identity). Thus the asymptotic behavior of the blow-up solutions of the elliptic equation is obtained.