Uniqueness of positive solutions of Δu+f(u)=0 in R<SUP>N</SUP>, N≥3
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Date
1998
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Abstract
We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation
Delta u + f(u) = 0 (*)
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 nonpositive, 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.
Delta u + f(u) = 0 (*)
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 nonpositive, 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.