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.

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