Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators
Résumé
We study uniformly elliptic fully nonlinear equations of the type $F(D^2u, Du, u, x)=f(x)$ in $\Omega$, where $F$ is a convex positively 1-homogeneous operator and $\Omega$ is a regular bounded domain in $\rn$. 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 Ambrosetti-Prodi phenomenon in elliptic theory.