Positive eigenfunctions of a Schrodinger operator

The paper considers the eigenvalue problem -Delta u-alpha u+lambda g(x)u = 0 with u is an element of H-1(R-N), u not equal 0, where alpha, lambda is an element of R and g(x) equivalent to 0 on (Omega) over bar, g(x) is an element of (0, 1) on R-N and lim (vertical bar x vertical bar ->+infinity g) g(x) = 1 for some bounded open set Omega is an element of R-N. Given a > 0, does there exist a value of A > 0 for which the problem has a positive solution? It is shown that this occurs if and only if a lies in a certain interval (Gamma, xi(1)) and that in this case the value of lambda is unique, lambda = Lambda (alpha). The properties of the function Lambda(alpha) are also discussed.