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Schrödinger operators with negative potentials and Lane-Emden densities. (English) Zbl 1388.35135

The authors consider the smallest eigenvalue of the Schrödinger operator \(-\Delta+V\) subject to Dirichlet boundary conditions in an open set \(\Omega \subset \mathbb{R^N}\). The potential \(V\) is assumed to be negative. A condition is given that nevertheless the spectrum is positive in form of an explicit bound on \(V\). The bound is obtained with the aid of a Hardy-type inequality which uses a weight-function depending on the solution of the so-called Lane-Emden equation \(-\Delta u = u^{q-1}\) subject to Dirichlet boundary conditions, where \(1 \leq q <2\). Applications are given to special cases of \(\Omega\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
49S05 Variational principles of physics
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