Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side

We deal with existence, non-existence and multiplicity of solutions to the model problem [GRAPHICS] where Omega subset of R-n is a smooth bounded domain, F is a 1-homogeneous fully nonlinear uniformly elliptic operator, lambda > 0, 0 < q < 1 < r < (A) over cap and (r) over cap the critical exponent ill a sense to be made precise. We set up a general framework for F in which there exists a positive threshold Lambda for existence and non-existence. Moreover a result on multiplicity is obtained for 0 < lambda < Lambda. The main difficulty comes from the viscosity setting required for this kind of operators. We also use some degree-theoretic arguments. The abstract result is applied to several examples, including Pucci extremal operators, concave (convex) operators and a class of Isaac operators. (C) 2009 Elsevier Inc. All rights reserved.