Hardy-Sobolev Inequality Research Articles

Improved Hardy-Sobolev Inequality under Moment Constraints

This paper is inspired by Aubin's 1979 result, which established that the best constant in the Sobolev inequality on the n-sphere, S^n, can be improved under the condition of vanishing first-order moments. Recent advancements by Hang and Wang (2021) showed that Aubin's improvement can be generalized to arbitrary higher-order moments. We further extend Hang and Wang's results to the Hardy-Sobolev inequality on S^n by deriving an associated concentration-compactness principle and imposing similar moment constraints. Finally, we briefly outline a framework for extending these results to higher-order Sobolev spaces.

Attainability of the best constant of Hardy–Sobolev inequality with full boundary singularities

Abstract We consider a type of Hardy–Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a conformally invariant one using the conformal radius of the domain. The best constant of such inequality on a smooth bounded domain is achieved if and only if the domain is non‐convex. In higher dimensions, the best constant is achieved if the domain has negative mean curvature somewhere. If the mean curvature vanishes but is non‐umbilic somewhere, we also establish the attainability for some special cases. In the other direction, we also show that the best constant is not achieved if the domain is sufficiently close to a ball in sense.

Weighted Rellich and Hardy inequalities in L p(·) spaces

Abstract In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

Traveling wave phenomena of inhomogeneous half-wave equation

Traveling wave phenomena of inhomogeneous half-wave equation

On a singular epitaxial thin‐film growth equation involving logarithmic nonlinearity

This paper is concerned with the well‐posedness and asymptotic behavior for a singular epitaxial thin‐film growth equation with logarithmic nonlinearity under the Navier boundary condition. Based on the technique of cut‐off and combining with Hardy–Sobolev inequality, the technique of Faedo–Galerkin, and multiplier, we establish the local solvability. Meantime, by virtue of the family of potential wells, we obtain the threshold between the existence and nonexistence of the global solution (including the critical case) and give the upper bound of lifespan and the estimate of blow‐up rate. Furthermore, the results of blow‐up with arbitrary initial energy and the lifespan are derived.

On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

<p>In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.</p>

Global weighted regularity for the 3D axisymmetric non-resistive MHD system

We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in $ \mathbb{R}^{3} $. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time $ T $ if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.

Regularity criteria of the axisymmetric Navier-Stokes equations and Hardy-Sobolev inequality in mixed Lorentz spaces

Regularity criteria of the axisymmetric Navier-Stokes equations and Hardy-Sobolev inequality in mixed Lorentz spaces

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.

Hardy-Sobolev inequalities with distance to the boundary weight functions

Hardy-Sobolev inequalities with distance to the boundary weight functions

Best constant and Mountain-Pass solutions for a supercritical Hardy-Sobolev problem in the presence of symmetries

Best constant and Mountain-Pass solutions for a supercritical Hardy-Sobolev problem in the presence of symmetries

Hardy-Sobolev Inequalities with Dunkl Weights

Hardy-Sobolev Inequalities with Dunkl Weights

Hardy-Sobolev inequalities and weighted capacities in metric spaces

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy–Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $\Omega$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy–Sobolev inequalities.

On a fractional Hardy–Sobolev inequality with two-variables

On a fractional Hardy–Sobolev inequality with two-variables

Hardy-Sobolev inequalities and boundary growth of Sobolev functions for double phase functionals on the half space

Hardy-Sobolev inequalities and boundary growth of Sobolev functions for double phase functionals on the half space

Hardy-Sobolev inequality in Herz spaces

Our aim in this paper is to establish Hardy-Sobolev inequality in the settings of Herz spaces. As an application, we show Sobolev-type integral representation for a $C^1$-function on ${\mathbb R}^N \setminus \{0\}$ which vanishes outside the unit ball.

The second best constant for the Hardy–Sobolev inequality on manifolds

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.

Solutions for nonhomogeneous fractional (p,q)-Laplacian systems with critical nonlinearities

Abstract In this article, we aimed to study a class of nonhomogeneous fractional (p,q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities inRN{{\mathbb{R}}}^{N}. By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p,q)-Laplacian systems in the case ofN=sp=lqN=sp=lq. It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p,q)-Laplacian systems is the main novelty of this article.

P-Bessel Pairs, Hardy’s Identities and Inequalities and Hardy–Sobolev Inequalities with Monomial Weights

p-Bessel Pairs, Hardy’s Identities and Inequalities and Hardy–Sobolev Inequalities with Monomial Weights

A fractional magnetic Hardy-Sobolev inequality with two variables

A fractional magnetic Hardy-Sobolev inequality with two variables