On the Lane-Emden equations with fully nonlinear operators

On the Lane-Emden equations with fully nonlinear operators. In this Note we consider nonnegative solutions for the nonlinear equation M-lambda, Lambda(+)(D(2)u) + \x\(alpha)u(rho) = 0 in R-N, where M-lambda, Lambda(+)(D(2)u) is the so called Pucci operator M-lambda, Lambda(+)(M) = lambdaSigma(ei<0)e(i) + Lambda Sigma(ei>0)e(i), and the e(i) are the eigenvalues of M et Lambda greater than or equal to lambda > 0. We prove that if u satisfies the decreasing estimate lim(\x\-->+infinity)\x\(beta-1)u(x) = 0 for some beta satisfying (beta-1)(p-1) > 2 + alpha then u is radial. In a second time we prove that if p < N+2alpha+2/N-2 and u is a nonnegative radial solution of (1), u(x) = g(r), such that g" changes sign at most once, then u is zero. (C) 2003 Academie des sciences. Publie par Editions scientifiques et medicales Elsevier SAS. Tous droits reserves.