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For selfadjoint extensions tilde-A of a symmetric densely defined positiveoperator A_min, the lower boundedness problem is the question of whethertilde-A is lower bounded {\it if and only if} an associated operator T inabstract boundary spaces is lower bounded. It holds when the Friedrichsextension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets1976); this applies to elliptic operators A on bounded domains.  For exterior domains, A_gamma ^{-1} is not compact, and whereas the lowerbounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from Tto tilde-A has only been known when m(T)>-m(A_gamma). We now show it forgeneral T.  The operator A_a corresponding to T=aI, generalizing the Krein-von Neumannextension A_0, appears here; its possible lower boundedness for all real a isdecisive. We study this Krein-like extension, showing for bounded domains thatthe discrete eigenvalues satisfyN_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .

Krein-like extensions and the lower boundedness problem for elliptic operators