Nonnegative Radial Solutions To A Class Of Elliptic Equations Involving The Pucci's Operator

In this article we are interested in the study of radially symmetric solutions to thenonlinear elliptic boundary value problemwhere Mλ±,Λdenotes the Pucci's extremal operators with parameters 0 <λ≤Λand BRis the ball of radius R in RN, N≥3.In the case of a radially symmetric function, we can give more explicit definition ofthe Pucci's operators Mλ±,Λ. Define the functionsWe see thatConsequently, y(x) = u(r) is a C2 radial solution of the problem (1.1) if and only if u isa C2 solution to the boundary value problem of the ordinary di?erential equationWe note that the problem (1.2) means two problems, one with the function h+ and theother with h?.Related to the critical exponent and eigenvalue problems of Pucci's extremal oper-ators, the following two results had been proved.Theorem 1.1 Let N≥3, then for the problem there exists a number p?+ > 1 (p?? > 1) such that there is no nontrivial solution when1 < p < p?+ (1 < p < p?? ).Theorem 1.2 The eigenvalue problemTheorem 1.2 The eigenvalue problemhas a unique eigenvalueμ±, which has a positive eigenfunction.Our main result is Theorem 1.3 as following.Theorem 1.3 Assume that N≥3 and f satisfiesThen the problem (1.2) has at least one nontrivial and nonnegative solution.For the complete reason, we will give a new and simple proof of Theorem 1.2 inAppendix of this paper.