On critical exponents for the Pucci's extremal operators

In this article we study some results on the existence of radially symmetric, non-negative solutions for the nonlinear elliptic equation M-lambda,Lambda(+) + (D(2)u) +u(p) = 0 in R-N. (*) Here N greater than or equal to 3, p > 1 and M-lambda,Lambda(+) denotes the Pucci's extremal operators with parameters 0 < lambda less than or equal to Lambda. The goal is to describe the solution set in function of the parameter p. We find critical exponents 1 < p(+)* < p(+)(p) that satisfy: (i) If 1 < p < p(+)* then there is no non-trivial radial solution of (*). (ii) If p = p(+)* then there is a unique fast decaying radial solution of (*). (iii) If P+* < p less than or equal to p(+)(p) then there is a unique pseudo-slow decaying radial solution to (*). (iv) If p(+)(p) < p then there is a unique slow decaying radial solution to (*). Similar results are obtained for the operator M-lambda,Lambda(-). (C) 2003 Editions scientifiques et medicales Elsevier SAS.