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Maximal and minimal eigenvalues and their associated nonlinear equations. (English) Zbl 0628.47032

The spectral theory of uniformly elliptic operators A under perturbations V giving rise to operators of the form \(H_ V=A+V(x)\) on a bounded or unbounded region, such as Schrödinger operators, are considered. Suppose that \(\| V\|_ p\) is constrained, but V is otherwise unspecified. The theory of the potentials V that maximize or minimize the eigenvalues of \(H_ V\) is presented. The optimizing potentials are typically determined by equations of the form \(-\Delta u+W(x)u=\pm cu^{\alpha}+\Lambda u\). The optimization of eigenvalues also turns out to be related to the determination of the best constants in Sobolev’s inequality, and, in its one-dimensional simplification, to a classical oscillator problem with “instanton” properties.

MSC:

47F05 General theory of partial differential operators
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
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