Critical exponents for the Pucci's extremal operators

Abstract

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation (∗) M λ, Λ +(D 2u)+u p=0,u⩾0 in R N. Here N⩾3, p>1 and M λ, Λ + denotes the Pucci's extremal operators with parameters 0< λ⩽Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents 1<p s +<p ∗ +<p p + , that satisfy: (i) If 1<p<p ∗ + then there is no nontrivial solution of ( ∗) . (ii) If p=p ∗ + then there is a unique fast decaying solution of ( ∗) . (iii) If p ∗<p⩽p p + then there is a unique pseudo-slow decaying solution to ( ∗) . (iv) If p p +< p then there is a unique slow decaying solution to ( ∗) . Similar results are obtained for the operator M λ, Λ − . To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.