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We study nonnegative solutions of the boundary value problem $$-\Delta u =\lambda c(x)u + \mu(x)|\nabla u|^2 + h(x),\quad  u\in H^1_0(\Omega)\cap L^\infty(\Omega),  \leqno(P_\lambda)$$ where $\Omega$ is a smooth bounded domain,  $\mu, c\in L^\infty(\Omega)$, $h\in L^r(\Omega)$ for some $r > n/2$ and$\mu,c,h > {\hskip -3.5mm} {\atop \neq} 0$. Our main motivation is to study the"noncoercive" case. Namely, unlike in previous work on the subject, we do notassume $\mu$ to be positive everywhere in $\Omega$.  In space dimensions up to $n=5$, we establish uniform a priori estimates forweak solutions of ($P_\lambda$) when $\lambda>0$ is bounded away from $0$. Thisis proved under the assumption that the supports of $\mu$ and $c$ intersect, acondition that we show to be actually necessary, and in some cases we furtherassume that $\mu$ is uniformly positive on the support of $c$ and/or some otherconditions.  As a consequence of our a priori estimates, assuming that ($P_0$) has asolution, we deduce the existence of a continuum ${\cal C}$ of solutions, suchthat the projection of ${\cal C}$ onto the $\lambda$-axis is an interval of theform $[0,a]$ for some $a>0$ and that the continuum ${\cal C}$ bifurcates frominfinity to the right of the axis $\lambda=0$. In particular, for each$\lambda>0$ small enough, problem $(P_\lambda)$ has at least two distinctsolutions.

A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient