CRITICAL EXPONENTS FOR UNIFORMLY ELLIPTIC EXTREMAL OPERATORS by Patricio L. FELMER a and Alexander QUAAS b a Departamento de Ingenier´ ıa Matem´ atica, and Centro de Modelamiento Matem´ atico, UMR2071 CNRS-UChile Universidad de Chile, Casilla 170 Correo 3, Santiago, CHILE. b Departamento de Matem´ atica, Universidad Santa Mar´ ıa, Casilla: V-110, Avda. Espa˜ na 1680, Valpara´ ıso, CHILE. 1 Introduction A cornerstone in the study of nonlinear elliptic partial differential equations is Δu + u p =0, u> 0 in IR N , (1.1) for which a complete description of the solutions depending on the exponent p is known. The main result is the existence of a number p ∗ N =(N + 2)/(N − 2), known as critical exponent, such that when 1 <p<p ∗ N no solution to equation (1.1) exists, when p = p ∗ N then, up to scaling, equation (1.1) possesses exactly one solution whose behavior at infinity is like |x| −(N −2) and when p>p ∗ N then equation (1.1) admits radial solutions with behavior at infinity like |x| −α , for α =2/(p − 1). In the proof of these basic results various important tools has been devel- oped, such as the celebrated Pohozaev identity, energy integrals, the moving planes technique based on the maximum principle, the Kelvin transform and Harnack in- equalities. In this respect the work by Pohozaev [29], Serrin [33], Gidas, Ni and Nirenberg [18], Caffarelli, Gidas and Spruck in [5], Gidas and Spruck [20], Chen and Li [6] and, Serrin and Zou [34] have been fundamental. In the recent article [15], the authors considered a similar equation but replac- ing the Laplacian by a Pucci’s extremal operator M ± λ,Λ (D 2 u)+ u p =0, u> 0 in IR N , (1.2) 1