  Let $\Omega \subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We
consider a boundary value problem of the form $$-\Delta u = c_{\lambda}(x) u +
\mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega)\cap L^{\infty}(\Omega)$$
where $c_{\lambda}$ depends on a parameter $\lambda \in \mathbb R$, the
coefficients $c_{\lambda}$ and $h$ belong to $L^q(\Omega)$ with $q>N/2$ and
$\mu \in L^{\infty}(\Omega)$. Under suitable assumptions, but without imposing
a sign condition on any of these coefficients, we obtain an a priori upper
bound on the solutions. Our proof relies on a new boundary weak Harnack
inequality. This inequality, which is of independent interest, is established
in the general framework of the $p$-Laplacian. With this a priori bound at
hand, we show the existence and multiplicity of solutions.
