Strong barriers for weighted quasilinear equations (Q6634207)

This article is concerned with the study of the problem \(-\operatorname{div}\mathcal{A}(x, \nabla u)=f(x)\) in \(\Omega\subset \mathbf{R}^n\), subject to \(u=0\) on \(\partial \Omega\). Here \(\Omega\) is an open set with nonempty boundary while \(\operatorname{div}\mathcal{A}(x, \nabla u)\) is a nonlinear differential operator that contains the weighted \((p,w)\)-Laplacian type elliptic operator. The main goal of the article is to construct strong barriers for quasilinear operators of the above type and thus derive the existence of a weak solution. Various techniques are employed in the approach, such as: boundary Hölder estimate, a De Giorgi-Nash-Moser theory for elliptic equations, geometric Hardy inequalities.