Weighted Hardy inequality with higher dimensional singularity on the boundary

Abstract

Abstract. Let Ω be a smooth bounded domain in R N with N ≥ 3 and let Σ k be a closed smoothIn this paper we study the weighted Hardy inequality with weight function singular on Σ k . In particular we provide necessary and sufficient conditions for existence of minimizers.

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

“…In these papers, the finiteness of the integral Γ 1 1 − q(σ) dσ was necessary and sufficient to obtain the existence of an eigenfunction in some function space corresponding to some "critical" eigenvalue. We belive that the argument in this paper and the results in [14] might be used to study problem (4.1) but with Γ ⊂ ∂Ω.…”

Schrödinger equations with singular potentials: linear and nonlinear boundary value problems

“…The paper is organized as follows. In Section 2, we establish estimates of the first eigenfunction of −L γV by adapting an argument of [11]. This result together with the estimates of the Green and Martin kernels of [14] are crucial in the study of the notion of normalized boundary trace in Section 3.…”

Global W^1,p regularity for an elliptic problem with measure source and Leray-Hardy potential,

“…The numerous underlying problems were a source of motivation for many researchers. In order to solve (1.1) in the standard case, ν is a function, many aspects have been developed : Hardy inequality, see the references [5,15,19,21,28,29], variational methods [16,20] for example, the singularities of semilinear Hardy elliptic equation by [7,9,14,22,23,24,26,30] and the references therein. When N ≥ 3, µ 0 ≤ µ < 0, Dupaigne in [15] (also see [4,5]) studied the weak solutions for the equation L µ u = u p + tf in Ω, subject to the zero Dirichlet boundary condition, with t > 0, p > 1 and f ∈ L 1 (Ω), in the distributional sense that u ∈ L 1 (Ω), u p ∈ L 1 (Ω, ρdx) and…”