Nonuniqueness for the dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti–Prodi phenomenon

Abstract

We study the uniformly elliptic fully nonlinear PDE F(D2u,Du,u,x)=f(x) in Ω where F is a convex positively 1-homogeneous operator and Ω⊂RN is a regular bounded domain. We prove non-existence and multiplicity results for the Dirichlet problem, when the two principal eigenvalues of F are of different sign. Our results extend to more general cases, for instance, when F is not convex, and explain in a new light the classical results of Ambrosetti–Prodi Type in elliptic PDE.