Let [Formula: see text] be a smooth compact Riemannian manifold of dimension [Formula: see text] and let [Formula: see text] to be a closed submanifold of dimension [Formula: see text]. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on [Formula: see text] within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.
The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.
We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.
Abstract. We consider a bounded domain Ω of R N , N ≥ 3, and h a continuous function on Ω. Let Γ be a closed curve contained in Ω. We study existence of positive solutions u ∈ H 1 0 (Ω) to the equation, σ ∈ (0, 2), and ρ Γ is the distance function to Γ. For N ≥ 4, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case N = 3, we obtain existence of ground-state solution provided the trace of the regular part of the Green of −∆ + h is positive at a point of the curve.
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