Hardy-Sobolev Type Inequalities Research Articles

Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : inf { | u | 2 s ∗ | x | s ∫ Ω | ∇ u | 2 d x − λ 1 ∫ Ω u 2 | x − P 1 | 2 d x − λ 2 ∫ Ω u 2 | x − P 2 | 2 d x | u ∈ H 0 1 ( Ω ) , ∫ Ω | u | 2 s ∗ | x | s d x = 1 } where N ≥ 3 , Ω is a smooth domain, λ 1 , λ 2 ∈ R , 0 , P 1 , P 2 ∈ Ω , s ∈ ( 0 , 2 ) and 2 s ∗ = 2 ( N − s ) N − 2 . Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems.

Remarks on a Sobolev embedding

In this paper, we prove a Hardy-Sobolev type inequality which is posteriorly used to give a characterization of an important class of Sobolev space. The abstract setting is applied to obtain the existence of solution for a class of elliptic equations with Neumann/Robin boundary conditions.

Sharp Morrey–Sobolev inequalities and eigenvalue problems on Riemannian–Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on [Formula: see text]-dimensional Finsler manifolds having nonnegative [Formula: see text]-Ricci curvature. A byproduct of this method is a Hardy–Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler–Laplace operator. Our results are also new in the Riemannian setting.

On a Hardy–Sobolev-type inequality and applications

In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition [Formula: see text] where [Formula: see text] is an exterior domain such that [Formula: see text].

The Brezis–Nirenberg problem for fractional systems with Hardy potentials

In this work, we study the existence of positive solutions to the following fractional elliptic systems with Hardy‐type singular potentials and coupled by critical homogeneous nonlinearities where (− Δ)s denotes the fractional Laplace operator, is a smooth bounded domain such that 0 ∈ Ω, μ1, μ2 ∈ [0, ΛN, s) with ΛN, s the sharp constant of the fractional Hardy inequality, and is the fractional critical Sobolev exponent. In order to prove the main result, we study the related fractional Hardy–Sobolev type inequalities and then establish the existence of positive solutions to the systems through variational methods.

Hardy type inequalities in generalized grand Lebesgue spaces

In the present paper, we discuss generalized grand Lebesgue spaces on homogeneous Lie groups. We prove Hardy type inequalities in the context of the considered spaces. We also present Hardy–Rellich and Hardy–Sobolev type inequalities extending some recently obtained results.

Extremals for Hardy-Sobolev type inequalities with monomial weights

In this article we study the existence and non-existence of extremals for the following family of Hardy-Sobolev inequalities(∫(R+)k×RN−k|xBu|q)1q≤C(∫(R+)k×RN−k|xA∇u|p)1p, which holds for suitable values of A,B∈RN, q>p>1. Here the quantity xA (respectively xB) denotes the monomial weight defined asxA=|x1|a1⋅…⋅|xk|ak(respectively xB=|x1|b1⋅…⋅|xk|bk).

A minimization problem involving a fractional Hardy–Sobolev type inequality

In this work, we obtain existence of nontrivial solutions to a minimization problem involving a fractional Hardy–Sobolev type inequality in the case of inner singularity. Precisely, for λ > 0 , we analyze the attainability of the optimal constant μ α , λ ( Ω ) : = inf { [ u ] s , Ω 2 + λ ∫ Ω | u | 2 d x : u ∈ H s ( Ω ) , ∫ Ω | u ( x ) | 2 s , α | x | α d x = 1 } , where 0 < s < 1 , n > 4 s , 0 ≤ α < 2 s , 2 s , α = 2 ( n − α ) n − 2 s , and Ω ⊂ R n is a bounded domain such that 0 ∈ Ω .

Minimization problem associated with an improved Hardy–Sobolev type inequality

We consider the existence or non-existence of a minimizer of the following minimization problems associated with an improved Hardy–Sobolev type inequality introduced by Ioku (2019): Ia≔infu∈W01,p(BR)∖{0}∫BR|∇u|pdx∫BR|u|p∗(s)Va(x)dxpp∗(s),whereVa(x)=1|x|s1−a|x|RN−pp−1β≥1|x|s, where 1<p<N and 0≤s≤p. The minimization problem Ia is equivalent to the minimization problem associated with the classical Hardy–Sobolev inequality on RN via a transformation if we restrict ourselves to radial functions. In contrast to the classical results for a=0, we show the existence of non-radial minimizers for the Hardy–Sobolev critical exponent p∗(s)=p(N−s)N−p on bounded domains. Furthermore, as an application of the transformation, we derive an infinite-dimensional form of the classical Sobolev inequality in some sense.

Critical Hardy inequalities

We prove a range of critical Hardy inequalities and uncertainty type principles on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. Moreover, we establish a new type of critical Hardy inequality and prove Hardy-Sobolev type inequalities. Most of the obtained estimates are new already for the case of $\mathbb R^{n}$. For example, for any $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$ our results imply the range of critical Hardy inequalities of the form $$\qquad \underset{R>0}{\sup}\left\|\frac{f-f_{R}}{|x|^{\frac{n}{p}}{\log}\frac{R}{|x|}}\right\|_{L^{p}(\mathbb{R}^{n})}\leq \frac{p}{p-1}\left\| \frac{1}{|x|^{\frac{n}{p}-1}} \nabla f\right\|_{L^{p}(\mathbb{R}^{n})},\quad 1<p<\infty,$$ where $f_{R}=f(R\frac{x}{|x|})$, with sharp constant $\frac{p}{p-1}$, recovering the known cases of $p=n$ and $p=2$. Moreover, our results also imply a new type of a critical Hardy inequality of the form $$\left\|\frac{f}{|x|}\right\|_{L^{n}(\mathbb{R}^{n})}\leq n\left\|(\log|x|)\nabla f\right\|_{L^{n}\mathbb{R}^{n})}, $$ for all $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\}),$ where the constant $n$ is sharp. However, homogeneous groups provide a perfect degree of generality to talk about such estimates without using specific properties of $\mathbb R^n$ or of the Euclidean distance.

Hardy-Sobolev type inequality and supercritical extremal problem

This paper deals with Hardy-Sobolev type inequalities involving variable exponents. Our approach also enables us to prove existence results for a wide class of quasilinear elliptic equations with supercritical power-type nonlinearity with variable exponent.

Fractional Hardy–Sobolev type inequalities for half spaces and John domains

As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces.

Hardy–Sobolev–Maz’ya and related inequalities in the half-space

Starting from the Hardy–Sobolev–Maz’ya inequality, we present all known Hardy–Sobolev-type inequalities involving the distance to the boundary of a half space. We give the simpler proofs known in this particular case. Related inequalities are discussed and two open questions are stated.

Minimization problem on the Hardy–Sobolev inequality

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.

Weighted Caffarelli-Kohn-Nirenberg type inequality on the Heisenberg group

This work is devoted to establish a class of Caffarelli-Kohn-Nirenberg type inequalities on the Heisenberg group. Inspired by the idea of Chern and Lin, a function transformation is introduced. Combining some elementary inequalities and some accurate estimates, we establish a class of weighted Hardy-Sobolev type inequalities and then obtain our main result, namely Caffarelli-Kohn-Nirenberg type inequalities on the Heisenberg group.

Integrability and boundedness of extremal functions of a Hardy-Sobolev type inequality

Integrability and boundedness of extremal functions of a Hardy-Sobolev type inequality

Existence and non-existence of solutions for p-Laplacian equations with decaying cylindrical potentials

In this paper we deal with the problem−Δpu+|u|p⁎(s)−2u|y|s=|u|m−2u,u∈D1,p(RN;R) where 1<p<N, x=(y,z)∈Rk×RN−k, Δp is p-Laplacian operator, p⁎(s)=p(N−s)N−p and p⁎=pNN−p. Combining a version of the concentration compactness result by Solimini, Hardy–Sobolev type inequality with the Mountain Pass Theorem, existence of non-trivial solutions is obtained. Decay properties of these solutions are showed by applying Vassilev results. Pohozaev type identities are established in order to get non-existence results.

Existence of solution of sub-elliptic equations on the Heisenberg group with critical growth and double singularities

For a class of sub-elliptic equations on Heisenberg group \(\mathbb{H}^N\) with Hardy type singularity and critical nonlinear growth, we prove the existence of least energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].

Hardy–Sobolev inequalities in unbounded domains and heat kernel estimates

We deal with domains with infinite inner radius. More precisely, we introduce a new geometric assumption on an exterior domain Ω⊂Rn; n⩾3 (i.e. complement of smooth compact domain not containing the origin). Under this assumption, we prove the Hardy inequality with optimal constant involving the distance to the boundary. In addition, in the case n⩾4, we improve this inequality by adding a critical Sobolev norm. Furthermore, we investigate the singular case n=3 and we show that, under some additional geometric assumption on Ω, the Hardy inequality can be improved by adding a Sobolev type term with critical exponent. Also, we prove some Hardy–Sobolev type inequalities without any geometric assumptions on Ω, which are of independent interest. Finally, we prove Harnack inequality up to the boundary for the positive solutions of the problem ut=Δu+14udist2(x,∂Ω) and we prove heat kernel estimates for small times.

Minimizing problems for the Hardy-Sobolev type inequality with the singularity on the boundary

In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert (9, 10) proved that the Hardy- Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary. We generalize their results by using the idea of Brezis and Nirenberg, and as a consequence, we shall prove the existence of pos- itive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.