We study the Dirichlet problem for fully nonlinear elliptic operators: $G(D^2u, \nabla u, u, x) = f(x)$ in $\Omega$, where $\Omega$ is a bounded regular domain, and $f$ is continuous. We prove the existence, the nonexistence and the multiplicity of solutions for some particular right-hand side $f$ when $G$ has its two principal eigenvalues of different sign.
"Nonuniqueness of solutions for Dirichlet problems related to fully nonlinear singular or degenerate operators." Adv. Differential Equations 14 (11/12) 1107 - 1126, November/December 2009.