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Spectral problems related to the critical exponent in the Sobolev embedding theorem. (English) Zbl 0831.35116

Eigenvalue estimates for the Dirichlet and Neumann boundary value problems for the equation \[ \lambda \bigl( (- \Delta)^\ell u + \gamma u \bigr) = Vu \] in a bounded region \(\Omega \subset \mathbb{R}^d\) are obtained. It is supposed that \(d\) is even and \(2 \ell = d\). The main goal is to find a widest class of nonnegative weight-functions \(V\), which ensures the classical behaviour of the corresponding eigenvalue distribution function: \(N (\lambda) = O (\lambda^{-1})\) as \(\lambda \to 0\). It is shown that this class is \(L \log L\), and the estimate obtained is \[ N(\lambda) \leq C |V |\lambda^{-1}. \tag{*} \] where \(C = C (\Omega)\) and \(|\cdot |\) is the norm in the corresponding Orlicz space. (*) is an improvement of a result obtained by M. S. Birman and the author [Funkts. Anal. Prilozh. 4, 1-13 (1970; Zbl 0225.35077)]. This improvement is based upon a recent result on piecewise-polynomial approximations see the author [Isr. J. Math. 86, No. 1-3, 253-275 (1994; Zbl 0803.47045)]. The case \(\Omega=\mathbb{R}^d\) is also discussed, and applications to the spectral theory of operators \((- \Delta)^\ell u - \alpha Vu\), with the coupling parameter \(\alpha >0\), are given. The general scheme used in the paper, and also related results for \(2 \ell > d\) and \(2 \ell < d\), are presented in detail in M. Birman and the author [Am. Math. Soc. Transl., Ser. 2, 114 (1980; Zbl 0426.46020)] and in M. Birman and the author [Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Adv. Sov. Math. 7, 1-55 (1991; Zbl 0749.35026)].

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
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