Asymptotic Behavior of Least Energy Solutions for a Fractional Laplacian Eigenvalue Problem on Double-struck capital RN

We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P) (-.)su + V (x)u = mu u + am(x)|u| 4s N u, RN |u|2 dx = 1, u. H s (RN), where s. (0, 1), mu. R, a > 0, V (x) and m(x) are L8 (RN) functions with N = 2. We prove that there is a threshold a* s > 0 such that problem (P) has a least energy solution ua(x) for each a. (0, a* s) and ua blows up, as a a* s, at some point x0. RN, which makes V (x0) be the minimum and m(x0) be the maximum. Moreover, the precise blowup rates for ua are obtained under suitable conditions on V (x) and m(