Hardy Type Potentials Research Articles

On a p(x)-biharmonic singular problem with changing sign weight and with no-flux boundary condition

In the present paper, we study $p(x)-$biharmonic problem involving $q(x)-$Hardy type potential with no-flux boundary condition. By using the mountain pass type theorem and Ekeland variatoinal principle, we obtain at least two nontrivial weak solutions.

On the fractional powers of a Schrödinger operator with a Hardy-type potential

AbstractStrong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

On fractional parabolic equations with Hardy-type potentials

A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren–Poon monotonicity formula combined with a blow-up analysis.

Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality|div(A(x,t)∇u)−ut|≤M|x|2|u|, where the coefficient matrix A is Lipschitz continuous in x and t. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends an earlier result of one of us with Garofalo and Manna for the heat operator.

Self-similar solution for Hardy operator

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.

On criticality coupled sub-Laplacian systems with Hardy type potentials on Stratified Lie groups

<abstract><p>In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{aligned} &-\Delta_{\mathbb{G}}u = \frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{d(z)^{\alpha}}+ \frac{p_{1}}{2^*(\gamma)}\frac{\psi^{\gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{d(z, z_{0})^{\gamma}} +\lambda h(z)\frac{\psi^{\sigma}|u|^{q-2}u}{d(z)^{\sigma}} \, \, & \text{in } \, \, \Omega, \\ &-\Delta_{\mathbb{G}}v = \frac{\psi^{\beta}|v|^{2^*(\beta)-2}v}{d(z)^{\beta}}+ \frac{p_{2}}{2^*(\gamma)}\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{d(z, z_{0})^{\gamma}} +\lambda h(z)\frac{\psi^{\sigma}|v|^{q-2}v}{d(z)^{\sigma}}\, \, &\text{in } \, \, \Omega, \\ &\quad u = v = 0\, \, &\text{on } \, \, \partial\Omega, \end{aligned}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ -\Delta_{\mathbb{G}} $ is a sub-Laplacian on Carnot group $ \mathbb{G} $, $ \alpha, \beta, \gamma, \sigma\in [0, 2) $, $ d $ is the $ \Delta_{\mathbb{G}} $-natural gauge, $ \psi = |\nabla_{\mathbb{G}}d| $ and $ \nabla_{\mathbb{G}} $ is the horizontal gradient associated to $ \Delta_{\mathbb{G}} $. The positive parameters $ \lambda $, $ q $ satisfy $ 0 < \lambda < \infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(\gamma) $, here $ 2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2} $, $ 2^*(\beta): = \frac{2(Q-\beta)}{Q-2} $ and $ 2^*(\gamma) = \frac{2(Q-\gamma)}{Q-2} $ are the critical Hardy-Sobolev exponents, $ Q $ is the homogeneous dimension of the space $ \mathbb{G} $. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.</p></abstract>

The Hardy–Schrödinger operator on the Poincaré ball: Compactness, multiplicity, and stability of the Pohozaev obstruction

Let Ω be a smooth relatively compact domain containing zero in the Poincaré ball model of the Hyperbolic space Bn (n≥3) and let −ΔBn be the Laplace-Beltrami operator on Bn. We consider problems of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem,(1){−ΔBnu−γV2u−λu=V2⋆(s)|u|2⋆(s)−2u in Ωu=0 on ∂Ω, where γ<(n−2)24, 0<s<2, 2⋆(s):=2(n−s)n−2 being the corresponding critical Sobolev exponent, while V2 (resp., V2⋆(s)) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like 1r2 (resp. 1rs) at the origin. The bulk of this paper is a sharp blow-up analysis that we perform on certain approximate solutions of (1) with bounded but arbitrary high energies. When these approximate solutions are positive, our analysis leads to improvements of results in [5] regarding positive ground state solutions for (1), as we show that they exist whenever n≥4, 0≤γ≤(n−2)24−1 and λ>0. The latter result also holds true for n≥3 and γ>(n−2)24−1 provided the domain has a positive “hyperbolic mass”. On the other hand, the same analysis yields that if γ>(n−2)24−1 and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to (1), we show that there are infinitely many of them provided n≥5, γ<(n−2)24−4 and λ>n−2n−4(n(n−4)4−γ).

Probabilistic approach to the heat equation with a dynamic Hardy-type potential

We consider the heat equation with a dynamic potential ∂∂tu=12Δu+V(x,t)u,x∈RN,where N>2. Here the potential V is given by a Hardy-type function V(x,t)=λ|x−ξ(t)|−μ with constants λ,μ>0, and the singular point ξ(t) is a path of the N-dimensional fractional Brownian motion with the Hurst exponent 0<H<1/2. By using the Feynman–Kac formula, it is shown that the initial value problem is solvable if and only if μ<(1/H)∧N. We also study the heat equation with a singular nonhomogeneous term.

Intersection properties for singular radial solutions of quasilinear elliptic equations with Hardy type potentials

We are interested in singular positive solutions of a quasilinear elliptic equation with a singular coefficient where , , and . The differential operator includes the standard Laplace, m-Laplace and ℓ-Hessian operators. In the case , a solution of the problem gives a radial solution of . This problem has a one-parameter family of singular positive solutions and one exact singular solution. We obtain an intersection number of two singular solutions in the critical and supercritical cases. The main technical tool is a phase plane analysis.

On the evolution equation with a dynamic Hardy-type potential

Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

Multiple Solutions for Double Phase Problems with Hardy Type Potential

In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f. Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness.

Nonexistence of critical fractional Sobolev–Hardy elliptic problems

In this paper, we study the nonexistence of solutions for a fractional elliptic problem with critical Sobolev–Hardy exponents and Hardy-type potentials by using the Pohozaev identity.

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

&lt;p style='text-indent:20px;'&gt;In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [&lt;xref ref-type="bibr" rid="b10"&gt;10&lt;/xref&gt;] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (ⅰ) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ⅱ) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.&lt;/p&gt;

A Strong Unique Continuation Property for the Heat Operator with Hardy Type Potential

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality |Δu-ut|≤M|x|2|u|,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} |\\Delta u - u_t| \\le \\frac{M}{|x|^2} |u|, \\end{aligned}$$\\end{document}with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.

On the strong unique continuation property of a degenerate elliptic operator with Hardy-type potential

In this paper, we prove the strong unique continuation property for the following degenerate elliptic equation 0.1$$\begin{aligned} \Delta _zu +|z|^2\partial _t^2u = Vu,\quad (z,t) \in {\mathbb {R}}^N \times {\mathbb {R}} \end{aligned}$$where the potential V satisfies either of the following growth assumptions 0.2$$\begin{aligned}&\left| V(z,t) \right| \le \frac{f(\rho (z,t))}{\rho (z,t)^2}, \text {where} \rho \text { is as in }(2.1)\text { and }\nonumber \\&\quad f\text { satisfies the Dini integrability condition as in }(1.3). \end{aligned}$$or when $$\begin{aligned}&\left| V(z,t) \right| \le C \frac{\psi (z,t)^{\epsilon }}{\rho (z,t)^2}, \text {for some }\epsilon >0\text { with }\psi \text { as in }(2.6)\text { and } N\text { even.} \end{aligned}$$This extends some of the previous results obtained in [18] for this subfamily of Baouendi–Grushin operators. As corollaries, we obtain new unique continuation properties for solutions u to $$\begin{aligned} \Delta _{{\mathbb {H}}} u = Vu \end{aligned}$$with certain symmetries as expressed in (1.6) where $$\Delta _{{\mathbb {H}}}$$ corresponds to the sub-Laplacian on the Heisenberg group $${\mathbb {H}}^n$$.

Strong unique continuation for the higher order fractional Laplacian

In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrodinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderon type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates.

The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities

We look for solutions to a fractional Schrödinger equation of the following form (−Δ)α∕2u+V(x)−μ|x|αu=f(x,u)−K(x)|u|q−2uonRN∖{0},where V is bounded and close-to-periodic potential and −μ|x|α is a Hardy-type potential. We assume that V is positive and f has the subcritical growth but not higher than |u|q−2u. If μ is positive and small enough we find a ground state solution, i.e. a critical point of the energy being minimizer on the Nehari manifold. If μ is negative we show that there is no ground state solutions. We are also interested in an asymptotic behaviour of solutions as μ→0+ andK→0.

On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case

We start by pointing out an important ambiguity in the mathematical treatment of the study of bound state solutions of the Schrodinger equation for infinite well type potentials (studied for the first time in a pioneering article of 1928 by G. Gamow). An alternative to get a “localizing effect” for the wave packet solution of time dependent Schrodinger equation with potentials becoming singular on the boundary of a compact region \(\overline{\Omega }\) is here offered in terms of “ Hardy type potentials” in which the potential behaves like the distance to the boundary to the power \(\alpha =-2.\) We show that in this case the probability to find the particle outside \(\Omega \) is zero once we assume that at \(t=0\) the particle is located in \(\Omega \). The paper extends to the N-dimensional and evolution cases some previous results by the author.

On Nonhomogeneous Elliptic Equations with Critical Sobolev Exponent and Prescribed Singularities

In this paper we consider a class of nonhomogeneous elliptic equations involving multi-polar Hardy type potentials and a Sobolev critical nonlinearity in an open domain of $\mathbb{R}^{N}$, $N \geq 3$. By Ekeland's Variational Principle and the Mountain Pass Lemma, we prove the existence of multiple solutions under sufficient conditions on the data and the considered parameters.

On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via flat solutions: The one-dimensional case

An ambiguity in the mathematical treatment of the study of bound state solutions of the Schrödinger equation for infinite well type potentials (studied for the first time in a pioneering article of 1928 by G. Gamow) is pointed out. An alternative to prove a similar “localizing effect” is here offered “in terms Hardy type potentials” with the distance to the boundary as a variable. The existence of flat solutions (with zero normal derivative at the boundary) and solutions with compact support is here obtained by first time in the literature for elliptic problems for this kind of linear equations