Weighted a priori estimates with powers of the distance function for elliptic equations

Abstract

Let u be a weak solution of the equation (-Δ)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω ⊂ ℝn. In this paper, we obtain some estimates for the Green's function associated to this problem. Moreover, under appropriate conditions on p, we prove some weighted Sobolev a priori estimates for the solution u, where the weight is a power of the distance function. This result extends previous work by R. G. Durán and M. Sanmartino ([Indiana Univ. Math. J. 57 (2008), no. 7, 3463-3478] and [Anal. Theory Appl. 26 (2010), no. 4, 339-349]).