Hardy Sobolev Research Articles

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.

Existence of Solutions for Fractional $(p,q)$-Laplacian Problems Involving Critical Hardy–Sobolev Nonlinearities

Existence of Solutions for Fractional $(p,q)$-Laplacian Problems Involving Critical Hardy–Sobolev Nonlinearities

Hardy–Sobolev equation with negative power and sign-changing nonlinearity on closed manifolds

Hardy–Sobolev equation with negative power and sign-changing nonlinearity on closed manifolds

On fractional inequalities on metric measure spaces with polar decomposition

Abstract 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.

Existence and Multiplicity of Nontrivial Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The PSc sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized.

Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics

In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: −Δu+V(x)u=|u|2∗−2u|y|,x=(y,z)∈Rm×Rn−m,where n>4, 2≤m<n, 2∗≔2(n−1)n−2 and potential function V(x):Rn→R. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg Inequalities on General Lie Groups

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

The Calogero–Moser derivative nonlinear Schrödinger equation

AbstractWe study the Calogero–Moser derivative nonlinear Schrödinger NLS equation posed on the Hardy–Sobolev space with suitable . By using a Lax pair structure for this ‐critical equation, we prove global well‐posedness for and initial data with sub‐critical or critical ‐mass . Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi‐soliton solutions and we prove that they exhibit energy cascades in the following strong sense such that as for every .

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯γ,uγ) of the equationF(D2uγ)+λ¯γuγrγ=0inB(0,1)∖{0},uγ=0on∂B(0,1) where uγ>0 in B(0,1)∖{0} and γ>0. We prove existence of radial solutions which are continuous on B(0,1)‾ in the case γ<2, existence of unbounded solutions in the case γ=2 and a non existence result for γ>2. We also give, in the case of Pucci's operators, the explicit value of λ¯2, which generalizes the Hardy–Sobolev constant for the Laplacian.

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Multiple high‐energy solutions for an elliptic system with critical Hardy–Sobolev nonlinearity

This paper discusses the existence of multiple high‐energy solutions for a ‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in . Considering that the “double” lack of compactness in the system is caused by the unboundedness of and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to of Struwe's classical global compactness result for double ‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work.

A Positive Solution for a Weighted p-Laplace Equation with Hardy–Sobolev’s Critical Exponent

A Positive Solution for a Weighted p-Laplace Equation with Hardy–Sobolev’s Critical Exponent

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.

Existence and multiplicity results for parameter Kirchhoff double phase problem with Hardy–Sobolev exponents

The solvability of a class of parameter Kirchhoff double phase Dirichlet problems with Hardy–Sobolev terms is considered. We focus on the existence of at least one solution, two solutions, three solutions, and infinitely many solutions to the problem, as the nonlinear terms satisfy different growth conditions, respectively. Our tools are mainly based on variational methods and critical point theory. In particular, in order to establish the relationship between singular terms and the norm of the Musielak–Orlicz–Sobolev space, we extend the Sobolev–Hardy inequality from W01,p to W01,H.

Integrability for Hardy operators of double phase

We establish Hardy–Sobolev and Hardy–Trudinger inequalities in weighted Orlicz spaces on \mathbb{R}^n . As an application, we prove Hardy–Sobolev and Hardy–Trudinger inequalities in the framework of general double phase functionals given by \varphi_p(x,t) = \varphi_1(t^p) + \varphi_2((b(x)t)^p), \quad x\in \mathbb{R}^n,\, t \ge 0, where p&gt;1 , \varphi_1, \varphi_2 are positive convex functions on (0,\infty) and b is a non-negative function on [0,\infty) which is Hölder continuous of order \theta \in (0,1] .

Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane

The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.

Classification of positive solutions to [formula omitted]-Laplace equations with critical Hardy–Sobolev exponent

Let u∈D1,p(RN) be a positive solution to the critical equation −Δpu=upa∗−1|x|ain RN,where 1<p<N, 0<a<p and pa∗=(N−a)pN−p. By exploiting the method of moving planes, we show that u is radial and strictly radially decreasing about the origin. Consequently, u must have the form u(x)=(N−a)1pN−pp−1p−1pλλp+|x|p−ap−1N−pp−afor some λ>0.

Multiple solutions for a boundary singular semilinear equation with sublinear term involving Hardy potential and Hardy–Sobolev exponent

Multiple solutions for a boundary singular semilinear equation with sublinear term involving Hardy potential and Hardy–Sobolev exponent

Almost Everywhere Convergence of Bochner–Riesz Means on Hardy–Sobolev Spaces

We investigate the convergence of Bochner–Riesz means on Hardy–Sobolev spaces. The relation between the smoothness imposed on functions and the rate of almost everywhere convergence of the generalized Bochner–Riesz means is given.