Hardy Sobolev Research Articles

Multiple solutions for Kirchhoff–Schrödinger problems of fractional p-Laplacian involving Sobolev–Hardy critical exponent

ABSTRACT This paper is devoted to multiple solutions to a Kirchhoff–Schrödinger type problem of fractional p-Laplacian involving the Sobolev–Hardy critical exponent and a parameter 0 $ ]]> λ > 0 . With some suitable assumptions on the potential V ( x ) and the nonlinearity f ( x , u ) , the Krasnoselskii's genus argument is exploited to show the existence of infinitely many solutions if λ is sufficiently large. Furthermore, we employ a fractional version of the concentration-compactness to prove that there are m-pairs solutions of the problem provided that λ is small enough and the nonlinear force f ( x , ⋅ ) is odd.

The ground states of Hénon equations for [formula omitted]-Laplacian in [formula omitted] involving upper weighted critical exponents

In this paper, we investigate the existence of nontrivial ground state solutions of the Hénon equations with p-Laplacian in RN involving weighted upper critical exponents including Sobolev, Hardy–Sobolev and Hénon–Sobolev critical exponents. The Nehari manifold method and the mountain pass theorem are applied to get the nontrivial ground states for the problems under a subcritical weighted perturbation. Regularity and non-existence of the nontrivial solutions are also discussed.

Infinitely Many Solutions for a Nonlinear Elliptic PDE with Multiple Hardy–Sobolev Critical Exponents

Infinitely Many Solutions for a Nonlinear Elliptic PDE with Multiple Hardy–Sobolev Critical Exponents

Ground state solutions for Schrödinger–Poisson systems on ℝ3$$ {\mathbb{R}}^3 $$ with a weighted critical exponent

In this paper, we are interested in the existence of ground state solutions for the weighted critical Schrödinger–Poisson systems. The weighted critical exponent may be the Hénon–Sobolev, the Sobolev or the Hardy–Sobolev critical exponent. Moreover, the potentials are power type, which may be singular at origin or unbounded. Due to the presence of Possion term, we can not directly obtain the boundedness of the Palais–Smale sequences of the associated functional for some cases. The Pohozaev identity, the minimax principle and some techniques are used to overcome the difficulties.

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

On the existence and multiplicity of solutions for a class of sub-Laplacian problems involving critical Sobolev–Hardy exponents on Carnot groups

ABSTRACT In this work, we study the following sub-elliptic equations on Carnot group with Hardy-type singularity and critical Sobolev–Hardy exponents where stands for the sub-Laplacian operator on Carnot group , , and is the critical Sobolev–Hardy exponent, Q is the homogeneous dimension with respect to the dilation naturally associated with , d is the natural gauge associated with the fundamental solution of on , and is the horizontal gradient associated with . Through variational methods combined with the theory of genus, we prove that our problems admit infinitely many solutions.

Infinitely many sign-changing solutions for an elliptic equation involving double critical Hardy–Sobolev–Maz’ya terms

In this paper, we consider the existence of infinitely many sign-changing solutions for an elliptic equation involving double critical Hardy–Sobolev–Maz’ya terms. By using a compactness result obtained in [C.H. Wang, J. Yang, Infinitely many solutions for an elliptic problem with double Hardy–Sobolev–Maz’ya terms, Discrete Contin. Dyn. Syst., 36(3):1603–1628, 2016], we prove the existence of these solutions by a combination of invariant sets method and Ljusternik–Schnirelman-type minimax method.

Totally null sets and capacity in Dirichlet type spaces

In the context of Dirichlet type spaces on the unit ball of C d $\mathbb {C}^d$ , also known as Hardy–Sobolev or Besov–Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthening of boundary interpolation theorems of Peller and Khrushchëv and of Cohn and Verbitsky.

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.

The semilinear elliptic equations with double weighted critical exponents

In this paper, we focus on the equation with double weighted critical exponents, including Hardy–Sobolev, Sobolev, and Hénon–Sobolev critical exponents. Applying the Nehari manifold and the mountain pass theorem, we prove the existence of the ground state solutions for the equation in RN. Based on this fact, the positive solutions of the equation on the unit ball in RN are established.

Fractional Leibniz Rules Associated to Bilinear Hermite Pseudo-Multipliers

Abstract We obtain a fractional Leibniz rule associated to bilinear Hermite pseudo-multipliers in the context of Hermite Besov and Hermite Triebel–Lizorkin spaces. As a byproduct, we show that the classes of bounded functions in these spaces (which include Hermite Sobolev and Hermite Hardy–Sobolev spaces) are algebras under pointwise multiplication. To obtain these results we develop appropriate decompositions for bilinear pseudo-multipliers and molecular estimates for certain families of functions in the Hermite setting.

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

On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces

<abstract><p>In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.</p></abstract>

A sharp Hardy–Sobolev inequality with boundary term and applications

A sharp Hardy–Sobolev inequality with boundary term and applications

Apriori decay estimates for Hardy–Sobolev–Maz’ya equations

Apriori decay estimates for Hardy–Sobolev–Maz’ya equations

Hardy–Sobolev inequalities in the half-space for double phase functionals

Hardy–Sobolev inequalities in the half-space for double phase functionals

Weighted Hardy–Sobolev inequality and global existence result of thermoelastic system on manifolds with corner‐edge singularities

This article concerns with the thermoelastic corner‐edge‐type system with singular potential function on a wedge manifold with corner singularities. First, we introduce weighted p‐Sobolev spaces on manifolds with corner‐edge singularities. Then, we prove the corner‐edge‐type Sobolev inequality, Poincar inequality and Hardy inequality and obtain some results about the compactness of embedding maps on the weighted corner‐edge Sobolev spaces. Finally, as an application of these results, we apply the potential well theory and the Faedo–Galerkin approximations to obtain the global weak solutions for the thermoelastic corner‐edge‐type system 1.1.

Well-posedness and global dynamics for the critical Hardy–Sobolev parabolic equation

We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy–Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle.

Sums of Dual Toeplitz Products on the Orthogonal Complements of the Hardy–Sobolev Spaces

Sums of Dual Toeplitz Products on the Orthogonal Complements of the Hardy–Sobolev Spaces