Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball

[摘要] We establish a new Pohozaev-type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div(\del u\(m-2)del u) + f(u)=0 in B, and u = 0 on partial derivative B, where B is a finite ball in R-n, n greater than or equal to 3 and 1 < m less than or equal to n. Applying this main uniqueness result we can prove that the semilinear problem triangle u + mu u(p) + u(mu) = 0 in B, and u = 0 on partial derivative B, where mu > 0 and 1 less than or equal to p < q less than or equal to (n + 2)/(n - 2), has a unique positive solution when n greater than or equal to 6. This gives a complete answer to an open problem raised by Brezis and Nirenberg in 1983 in the case n greater than or equal to 6. We shall also derive some partial results to the open problem in the cases n = 3, 4, 5. (C) 1997 Academic Press.

[发布日期] 1997-08-10