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

TL;DR

This paper establishes a priori bounds and demonstrates the existence and multiplicity of solutions for an indefinite elliptic boundary value problem with critical gradient growth, using a new boundary weak Harnack inequality applicable to the p-Laplacian.

Contribution

It introduces a novel boundary weak Harnack inequality for the p-Laplacian and applies it to obtain bounds and multiplicity results for a class of indefinite elliptic problems with critical gradient growth.

Findings

Established an a priori upper bound for solutions without sign restrictions.

Proved existence of multiple solutions under general conditions.

Developed a new boundary weak Harnack inequality of independent interest.

Abstract

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.