The Hardy potential and eigenvalue problems

TL;DR: In this article, the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential was established, and the principal eigenfunctions for the LPO involving weighted Hardy potentials were examined.

TL;DR: The article studies Dirichlet problems involving the Hardy-Leray operator with multiple polars. It focuses on the properties of solutions to such problems subject to zero Dirichlet boundary conditions and obtains increasing Dirichlet eigenvalues and global weighted estimates for weak solutions. Additionally, it builds a weighted distributional framework to show the existence of weak solutions and obtain a sharp assumption for the existence of isolated singular solutions.

TL;DR: For the Dirichlet Hardy-Leray operator, Li-Yau and Karachalio as mentioned in this paper showed that the inverse-square potential does not play an essential role for the asymptotic behavior of the spectral of the problem considered.

TL;DR: In this article, the authors present some suitable distributional identities of the solutions for nonhomogeneous elliptic equations involving the Hardy-Leray potentials and study qualitative properties of the solution to the corresponding non-homogeneous problems.

TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.

TL;DR: In this paper, the equivalence between the compactness of all minimizing sequences and some strict sub-additivity conditions was shown based on a compactness lemma obtained with the help of the notion of concentration function of a measure.

TL;DR: In this paper, the authors show how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.

TL;DR: In this article, a constant positive C telle que l'inegalite suivante est vraie pour tout u∈C 0 ∞ (R n ): ||X| G u| L r≤C||x|α|Du|| L p 2 ||x|βu| L q 1−a si et seulement si on a 1/r+γ/n=a(1/p+(α−1)/n)+(1−a)(1/q+β/u), 0≤α−σ