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The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE. (English) Zbl 0988.58013
The Paneitz operator P, a conformally invariant differential operator of order four on a compact four-dimensional Riemannian manifold is studied. In terms of conformal invariants of the manifold a sufficient criterion is given under which the principal eigenvalue λ 1 (P)=0· If the Yamabe invariant is positive, there is an upper bound for another conformal invariant κ g , in the case of equality the manifold is conformally equivalent to the standard sphere.

MSC:
58J50Spectral problems; spectral geometry; scattering theory
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
58J05Elliptic equations on manifolds, general theory
58J60Relations of PDE with special manifold structures