Hardy Sobolev Research Articles

Asymptotic properties of positive solutions of the Hardy–Sobolev type equations

In this paper, we are concerned with the Lane–Emden type 2m-order PDE with weight(−Δ)mu(x)=|x|σup(x),u>0 in Rn, where n⩾3, p>1, m∈[1,n/2), σ∈(−2m,0], and the more general Hardy–Sobolev type integral equationu(x)=∫Rn|y|σup(y)dy|x−y|n−α, where α∈(0,n), σ∈(−α,0]. If 0<p⩽n+σn−α, then there is not any positive solution to such an integral equation. Under the assumption of p>n+σn−α, we obtain that the integrable solution u of the integral equation (i.e. u∈Ln(p−1)α+σ(Rn)) is bounded and decays fast with rate n−α. On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one α+σp−1. In addition, the classical solution u of the 2m-order PDE satisfies the integral equation with α=2m. Therefore, for the 2m-order PDE, all the decay properties above are still true.

On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents

We consider a quasilinear elliptic system involving the critical Hardy–Sobolev exponent and the Sobolev exponent. We use variational methods and analytic techniques to establish the existence of positive solutions of the system.

INFINITELY MANY SOLUTIONS FOR HARDY–SOBOLEV–MAZ'YA EQUATION INVOLVING CRITICAL GROWTH

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy–Sobolev–Maz'ya equation with critical growth: [Formula: see text] provided N &gt; 6 + t, where [Formula: see text] when k &gt; 2, λ = 0 when k = 2, [Formula: see text] and Ω is an open bounded domain in ℝN, which contains some points x0= (0, z0).

Besov–Lipschitz and Mean Besov–Lipschitz Spaces of Holomorphic Functions on the Unit Ball

We give several characterizations of holomorphic mean Besov–Lipschitz spaces on the unit ball in \({\mathbb C^N} \) and appropriate Besov–Lipschitz spaces and prove the equivalences between them. Equivalent norms on the mean Besov–Lipschitz spaces involve different types of L p -moduli of continuity, while in characterizations of Hardy–Sobolev spaces we use not only the radial derivative but also the gradient. The characterization in terms of the best approximation by polynomials is also given.

Minimal support results for Schrödinger equations

Abstract We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is -Δu = V u for u ∈ W 0 1,2(D) with D a bounded domain in ℝ n . We use the Sobolev embedding theorem, and in some cases the Moser–Trudinger inequality and the Hardy–Sobolev inequality, to derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V, the measure of D, and a sharp Sobolev constant. In most cases, these inequalities are best possible.

Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents

We consider the following problem $$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$ where \({ \mu \ge 0, 0 0}\) and all the principle curvatures of ∂Ω at 0 are negative, then the above problem has infinitely many solutions.

Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents

AbstractIn this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy–Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Holomorphic mean Lipschitz spaces and Hardy–Sobolev spaces on the unit ball

We study two classes of holomorphic functions in the unit ball 𝔹 n of ℂ n : mean Lipschitz spaces and Hardy–Sobolev spaces. Main results include new characterizations in terms of fractional radial differential operators and various comparisons between these two classes.

New Hardy inequalities and behaviour of linear elliptic equations

In this study, we want to emphasize the role of some Hardy inequalities in the blow-up phenomena of the very weak solution of a linear equation in the sense of Brezis. Thus we present here some new Hardy inequalities related to some extended Sobolev spaces such that Sobolev–Hardy spaces, Sobolev–Zygmund spaces, or other non-standard weighted spaces. Firstly we apply those results then provide two applications of these inequalities. Secondly we improve recent results by showing that the blow-up phenomena of the gradient can also occur in Hardy spaces. The Hardy inequalities for Sobolev–Zygmund spaces are obtained via an integral formula estimating the oscillation in a ball of radius r of a general function u in the usual Sobolev space. This formula involves the notion of relative rearrangement. We shall give a pointwise estimate for the solution u of linear equation −Δu=−div(F) for a bounded function F, using the distance function δ.

Existence of symmetric solutions for singular semilinear elliptic systems with critical Hardy–Sobolev exponents

This paper deals with the singular semilinear elliptic system −Δu=μu|x|2+2αQ(x)(α+β)|x|s|u|α−2u|v|β+λ|u|q−2uin Ω,−Δv=μv|x|2+2βQ(x)(α+β)|x|s|u|α|v|β−2v+δ|v|q−2vinΩ,u=v=0on∂Ω, where Ω⊂RN(N≥3) is a smooth bounded domain, 0∈Ω and Ω is G-symmetric with respect to a subgroup G of O(N), 0≤μ<μ¯ with μ¯=(N−22)2, λ,δ≥0, 0≤s<2 and α,β>1 satisfy α+β=2∗(s)=2(N−s)N−2, 2<q<2∗=2NN−2, Q(x) is continuous and G-symmetric on Ω¯. Based upon the symmetric criticality principle of Palais and variational methods, we obtain several existence results of G-symmetric solutions under certain appropriate hypotheses on Q, q and the pair of parameters (λ,δ)∈R2.

On the Existence and Breaking Symmetry of the Ground State Solution of Hardy Sobolev Type Equations withWeighted p-LAPLACIAN

Abstract We study the existence of ground state solution, cylindrically symmetric solution, breaking cylindrical symmetry of ground state solution of the following Hardy-Sobolev type equa- tion with weighted p-Laplacian: where

Hardy–Sobolev derivatives of phase and amplitude, and their applications

In time‐frequency analysis, there are fundamental formulas expressing the mean and variance of the Fourier frequency of signals, s, originally defined in the Fourier frequency domain, in terms of integrals against the density | s(t) | 2 in the time domain. In the literature, the existing formulas are only for smooth signals, for it is the classical derivatives of the phase and amplitude of the signals that are involved. The two representations of the covariance also rely on the classical derivatives and thus are restrictive. In this fundamental study, by introducing a new type of derivatives, called Hardy–Sobolev derivatives, we extend the formulas to signals in the Sobolev space that do not usually have classical derivatives. We also investigate the corresponding formulas for periodic (infinite discrete) and finite discrete signals. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev–Hardy exponents and concave–convex nonlinearities

This paper is concerned with a singular elliptic system, which involves the Caffarelli–Kohn–Nirenberg inequality and critical Sobolev–Hardy exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL HARDY–SOBOLEV EXPONENTS AND SIGN-CHANGING WEIGHT FUNCTION

In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

Optimal logarithmic estimates in the Hardy–Sobolev space of the disk and stability results

We prove a logarithmic estimate in the Hardy–Sobolev space Hk,2, k a positive integer, of the unit disk D. This estimate extends those previously established by L. Baratchart and M. Zerner in H1,2 and by S. Chaabane and I. Feki in Hk,∞. We use it to derive logarithmic stability results for the inverse problem of identifying Robin’s coefficients in corrosion detection by electrostatic boundary measurements and for a recovery interpolation scheme in the Hardy–Sobolev space Hk,2 with interpolation points located on the boundary T of the unit disk.

Existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical Hardy–Sobolev exponents

This paper deals with the singular semilinear elliptic problem −div(|x|−2a∇u)=μu|x|2(1+a)+Q(x)|u|p−2u|x|bp+σh(x,u)inΩ,u=0on∂Ω, where Ω⊂RN(N≥3) is a smooth bounded domain, 0∈Ω and Ω is G-symmetric with respect to a subgroup G of O(N), 0≤a<N−22, σ≥0, 0≤μ<μ¯ with μ¯=(N−2−2a2)2, a≤b<a+1, p=p(a,b)=2NN−2(1+a−b), Q(x) is continuous and G-symmetric on Ω¯ and h:Ω×R↦R is a continuous nonlinearity of lower order satisfying some conditions. Based upon the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on σ, Q and h.

Trace inequalities for weighted Hardy—Sobolev spaces in the non-diagonal case

We give characterizations of the positive Borel measures μ on $\mathbb{S}^n$ so that the weighted Hardy–Sobolev spaces $H_s^p(w)$ are imbedded in Lq(dμ), for a range of s &gt; 0, 0 &lt; p, q &lt; + ∞, q ≠ p, where w is a doubling weight in the unit sphere of ℂn.

The spectrum of the [formula omitted]-Laplacian with singular weight

We use the Hardy–Sobolev inequality to characterize the first eigenvalue λ1 of the p-Laplacian with singular weight. In some cases it is shown that λ1 is positive simple, isolated and has a nonnegative corresponding eigenfunction ϕ1. Higher eigenvalues, in particular the second one, are also determined.

Fractional Hardy–Sobolev–Maz'ya inequality for domains

We prove a fractional version of the Hardy–Sobolev–Maz’ya inequality for arbitrary domains and Lp norms with p ≥ 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.

Missing terms in the weighted Hardy–Sobolev inequalities and its application

Let Ω be a bounded domain of Rn (n≥1) containing the origin. In the present paper we establish the weighted Hardy–Sobolev inequalities with sharp remainders. For example, when α=1−n/p and 1 1−n/p) and the supercritical case (α<1−n/p). As an application, we use our improved inequality to determine exactly when the first eigenvalues of the weighted eigenvalue problems for the operators represented by −div(|x|αp|∇u|p−2∇u)−μ/|x|nA1(|x|)−p|u|p−2u (the critical case) will tend to zero as μ increases to Λn,p,α. This also gives us sufficient conditions for the operators to have the positive first eigenvalue in a certain nontrivial functional framework, and we study the eigenvalue problem in the borderline case.