No abstract

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such asin a bounded domain with a Dirichlet boundary condition; here λ ∈ R, c, h ∈ L p (Ω), p > n ≥ 1, c 0 and the matrix M satisfies 0 < µ 1 I ≤ M ≤ µ 2 I. Recently this problem was studied in the "coercive" case λc ≤ 0, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations. * gabrielle@mat.puc-rio.br, supported by Capes PROEX/PDSE grant 88881.134627/2016-01. † bsirakov@mat.puc-rio.br

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M À : ARTICLE HISTORY

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M − .