  We consider fully nonlinear uniformly elliptic equations with quadratic
growth in the gradient, such as $$ -F(x,u,Du,D^2u) =\lambda c(x)u+\langle M(x)D
u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary
condition, here $\lambda \in\mathbb{R}$, $c,\, h \in L^p(\Omega)$, $p>n\geq 1$,
$c\gneqq 0$ and the matrix $M$ satisfies $0<\mu_1 I\leq M\leq \mu_2 I$.
Recently this problem was studied in the "coercive" case $\lambda c\le0$, where
uniqueness of solutions can be expected, and it was conjectured that the
solution set is more complex for noncoercive equations. This conjecture was
verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in
divergence form, by exploiting the integral formulation of the problem. Here we
show that similar phenomena occur for general, even fully nonlinear, equations
in nondivergence form. We use different techniques based on the maximum
principle.
  We develop a new method to obtain the crucial uniform a priori bounds, which
permit to us to use degree theory. This method is based on basic regularity
estimates such as half-Harnack inequalities, and on a V\'azquez type strong
maximum principle for our kind of equations.
