  We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a
smooth bounded domain $\Omega$, under the assumption the nonlinearity $F$ is
uniformly elliptic and positively homogeneous. Recently, it has been shown that
such operators have two principal "half" eigenvalues, and that the
corresponding Dirichlet problem possesses solutions, if both of the principal
eigenvalues are positive. In this paper, we prove the existence of solutions of
the Dirichlet problem if both principal eigenvalues are negative, provided the
"second" eigenvalue is positive, and generalize the anti-maximum principle of
Cl\'{e}ment and Peletier to homogeneous, fully nonlinear operators.
