Existence of an interval of principal eigenvalues for radially symmetric problems on all of

We investigate the existence of principal eigenvalues, i.e., values of λ for which the corresponding eigenfunction is positive, for the radially symmetric problem - Δ u ( x ) = λ g ( x ) u ( x ) for x ∈ R N , where g is a radially symmetric smooth bounded function which changes sign on [ 0 , ∞ ) .