Existence and non-existence of the first eigenvalue of the perturbed Hardy–Sobolev operator

Abstract

In this paper we study the existence, non-existence and simplicity of the¯rst eigenvalue of the perturbed Hardy{Sobolev operator ¡¢ ¡ 1 4 (n ¡ 2) 2 (q=jxj 2 ) under various assumptions on the perturbation q. We study the asymptotic behaviour of the¯rst eigenfunction near the origin when the perturbation q is q = s, 0 < s < 1. We will also establish the best constant in a Hardy{Sobolev inequality proved by Adimurthi e t a l .

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

“…Remarkably, Hardy inequality in W 1,N 0 (B) is readily available in Adimurthi and Sandeep [4], that contains the Hardy term Q p | p=N , the best constant, and further correction terms.…”

A Leray–Trudinger inequality

“…The constant is well known to be the optimal one. For α = n in Lemma 2.1 we obtain (see also [3], [5] and [29,Lemma 17.4])…”

“…More generally, in analogy with versions of Hardy's inequality for p = n, it has been extended to p = n ≥ 2 by ( [3], [5] & [7]), and can be stated as follows: If Ω is a bounded domain in R n ; n ≥ 2, then 5) with the best possible constant in case 0 ∈ Ω. If we define the Leray difference…”

“…If 0 ∈ Ω, then the constant is known to be the optimal one (see [5] and [7]). Thus, (2.3) may be considered as a "substitute" of the Hardy inequality (2.2) in case p = n, which is valid in bounded domains containing the origin.…”

Weighted Hardy inequality with higher dimensional singularity on the boundary

“…It amounts to obtain a uniform control of a specific minimizing sequence for µ λ * (Ω, Σ k ) near Σ k via the H 1 -super-solution constructed. We mention that the existence and non-existence of extremals for (1) and related problems were studied in [1,6,7,8,11,12,13,17,18,19] and some references therein. We would like to mention that some of the results in this paper might of interest in the study of semilinear equations with a Hardy potential singular at a submanifold of the boundary.…”