A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient

Abstract

Let $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-\Delta u=c_{\lambda }(x)u+\mu (x)|\nabla u{|}^2+h(x),u\in H_0^1(\Omega )\cap L^{\infty }(\Omega )$$ 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.

This second version, which improves several results of the first version, will appear in J. Math. Pures Appl