Uniqueness of Positive Solutions of Δ u + f ( u )=0 in ℝ N , N ≦3

Abstract

We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation $$\Delta u+f(u)=0 \eqno{(*)}$$ in ℝ N . We assume that the function f has two zeros, the origin and u 0>0. Above u 0 the function f is positive, is locally Lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature. Below u 0, f is assumed to be non-positive, non-identically zero and merely continuous. Our results are obtained through a careful analysis of the solutions of an associated initial‐value problem, and the use of a monotone separation theorem. It is known that, for a large class of functions f, the ground states of (*) are radially symmetric. In these cases our result implies that (*) possesses at most one ground state.