Uniqueness Theorems for Positive Radial Solutions of Quasilinear Elliptic Equations in a Ball

Abstract

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(|∇u|m−2emsp14;∇u)+f(u)=0 inB, andu=0 on ∂B, whereBis a finite ball in Rn,n⩾3 and 1<m⩽n. Applying this main uniqueness result we can prove that the semilinear problemΔu+μup+uq=0 inB, andu=0 on ∂B, whereμ>0 and 1⩽p<q⩽(n+2)/(n−2), has a unique positive solution whenn⩾6. This gives a complete answer to an open problem raised by Brezis and Nirenberg in 1983 in the casen⩾6. We shall also derive some partial results to the open problem in the casesn=3, 4, 5.