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We consider fully nonlinear uniformly elliptic equations with quadraticgrowth in the gradient, such as $$ -F(x,u,Du,D^2u) =\lambda c(x)u+\langle M(x)Du, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundarycondition, 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 $\lambda c\le0$, whereuniqueness of solutions can be expected, and it was conjectured that thesolution set is more complex for noncoercive equations. This conjecture wasverified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations indivergence form, by exploiting the integral formulation of the problem. Here weshow that similar phenomena occur for general, even fully nonlinear, equationsin nondivergence form. We use different techniques based on the maximumprinciple.  We develop a new method to obtain the crucial uniform a priori bounds, whichpermit to us to use degree theory. This method is based on basic regularityestimates such as half-Harnack inequalities, and on a V\'azquez type strongmaximum principle for our kind of equations.

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient