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-lambda(+),(Lambda)(D(2)u) - u + u(P) = 0 in R-N, lim(r ->infinity) u(r) = 0, where M-lambda(,Lambda)+ (D(2)u) denotes the Pucci's extremal operator with parameters 0 < lambda < Lambda 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 (lambda = Lambda). 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-lambda(,Lambda)- is also considered.