Hardy–Sobolev inequalities involving mixed radially and cylindrically symmetric weights

We study minimization problems on Hardy–Sobolev type inequality. We consider the case where singularity is in interior of bounded domain \(\Omega \subset \mathbb {R}^N\). The attainability of best constants for Hardy–Sobolev type inequalities with boundary singularities have been studied so far, for example Ghoussoub and Kang (Ann Inst Henri Poincare Anal Non Lineaire 21(6):767–793, 2004), Ghoussoub and Robert (IMRP 21867:1–85, 2006), Ghoussoub and Robert (Trans Am Math Soc 361(9):4843–4870, 2009) etc.... According to their results, the mean curvature of \(\partial \Omega \) at singularity affects the attainability of the best constants. In contrast with boundary singularity case, in interior singularity case it is well known that the best Hardy–Sobolev constant $$\begin{aligned} \mu _s(\Omega ):=\left\{ \int _\Omega |\nabla u|^2 dx \Bigg | u \in H_0^1(\Omega ),\ \int _\Omega \frac{|u|^{2^*(s)}}{|x|^s}dx = 1 \right\} \end{aligned}$$is never achieved for all bounded domain \(\Omega \). We can see that the position of singularity on domain is related to the existence of minimizer. In this paper, we consider the attainability of the best constant for the embedding \(H^1(\Omega ) \hookrightarrow L^{2^*(s)}(\Omega ,|x|^{-s}dx)\) for bounded domain \(\Omega \) with \(0 \in \Omega \). In this problem, scaling invariance doesn’t hold and we can not obtain information of singularity like mean curvature.

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1<p\leq q<\infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

Let n ≥ 3, Ω ⊂ Rn be a domain with 0 ∈ Ω, then, for all the Hardy–Sobolev inequality says that and equality holds if and only if u = 0 and ((n − 2)/2)2 is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy–Sobolev type inequalities on manifolds and also on the Heisenberg group.

Minimization problem associated with an improved Hardy–Sobolev type inequality

Hardy–Sobolev inequalities in unbounded domains and heat kernel estimates

In this paper we study a class of Hardy–Sobolev type systems defined in RN and coupled by a singular critical Hardy–Sobolev term. The main novelty of this work is that the orders of the singularities are independent and contained in a wide range. By means of variational techniques, we will prove the existence of positive bound and ground states for such a system. In particular, we find solutions as minimizers or Mountain–Pass critical points of the energy functional on the underlying Nehari manifold.

Hardy–Sobolev equations on compact Riemannian manifolds

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

Optimal Hardy–Sobolev inequalities on compact Riemannian manifolds

A critical problem on the Hardy–Sobolev inequality in boundary singularity case

We first review improvements of (first-order) Sobolev and Hardy inequalities by the addition of suitable lower-order terms (these improved inequalities have been pioneered by Brezis and Nirenberg (1983) and Brezis and Vazquez (1997)). Recently, corresponding results concerning first-order Hardy–Sobolev and higher-order Sobolev and Hardy inequalities have been proved.

Qualitative properties of solutions for an integral system related to the Hardy–Sobolev inequality

We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot–Caratheodory spaces associated with a system of vector fields of Hormander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type.

Positive solutions to the weighted critical quasilinear problems

Sharp weighted Hardy type inequalities and Hardy–Sobolev type inequalities on polarizable Carnot groups