Critical Phenomena in Linear Elliptic Problems

In this article we consider the existence and nonexistence of principal eigenvalues for the linear elliptic equation−Δu=λV(x) u in RN and u∈D1,  2(RN),whereN⩾3, andV∈L1loc(RN) withV≩0 in RN. We establish that when potentialVhas strong singularities, phenomena which appear under critical exponent nonlinearities (Brezis and Nirenberg) are present in linear problems as well.