On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

Abstract

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation M λ , Λ + ( D 2 u ) − u + u p = 0 in R N , lim r → ∞ u ( r ) = 0 , where M λ , Λ + ( D 2 u ) denotes the Pucci's extremal operator with parameters 0 < λ ⩽ Λ and p > 1 . It is known that all positive solutions of this equation are radially symmetric with respect to a point in R N , so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian ( λ = Λ ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving M λ , Λ − is also considered.