Abstract: In this paper, we study the following eigenvalue problem involving limiting nonlinearity
-△u=λf(x,u),x∈R^N u∈D^1,2(R^N),λ∈R¹ by a new way of using the concentration-compactness principle. Even in the nonlimiting situation, we improve the known result in the positive mass case in the sense of seeking a nontrivial solution of the above eigenvalue problem.