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Zbl 1008.49034
Marcus, Moshe; Itai, Shafrir
An eigenvalue problem related to Hardy's $L^p$ inequality.
(English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29, No.3, 581-604 (2000). ISSN 0391-173X

The paper establishes the relation between the best constant for the Hardy $L^p$ inequality and the existence of positive eigenfunctions in an appropriate Sobolev space for a corresponding singular eigenvalue problem. \par Let $\Omega$ be a smooth bounded domain in $\bbfR^n$ and $p\in (1,\infty)$. Denote by $\mu_p$ the best constant for the $L^p$ Hardy inequality, that is, $$\mu_p=\inf\left\{ u\in W_0^{1,p}(\Omega);\ \int_\Omega |\nabla u(x)|^pdx, \ \int_\Omega (|u(x)|/\delta (x))^pdx=1\right\},$$ where $\delta (x)=\text{dist} (x,\partial\Omega)$. A classical result asserts that $\mu_p\leq c_p$, where $c_p=(1-1/p)^p$ denotes the value of the best constant in Hardy's inequality for $n=1$. The main results proved in this paper are the following: (i) $\mu_p=c_p$ if and only if the Euler-Lagrange equation corresponding to the above minimization problem has no positive solution; (ii) if $\mu_p<c_p$ then $\mu_p$ is a simple eigenvalue. Moreover, if $u$ is an arbitrary positive eigenfunction then the asymptotic behaviour of $u$ is given by $C^{-1}[\delta (x)]^\alpha \leq|u(x)|\leq C[\delta (x)]^\alpha$, for any $x\in\Omega$, where $C$ is a positive constant and $\alpha$ is the unique root of the equation $\alpha^{p-1}(1-\alpha)(p-1)=\mu_p$. \par The proofs use refined techniques based on {\it a priori} estimates, comparison principles for singular quasilinear equations or the construction of sub and super solutions.
[V.Rădulescu (Craiova)]
MSC 2000:
*49R50 Variational methods for eigenvalues of operators
35J70 Elliptic equations of degenerate type
35B50 Maximum principles (PDE)
35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B45 A priori estimates
35P30 Nonlinear eigenvalue problems for PD operators

Keywords: eigenvalue problem; singular elliptic equation; Hardy's inequality; comparison principle

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