Multi - Bump Solutions on Lattices

We consider the following semi-linear elliptic equation with critical exponent: −∆u = K(x)u n+2 n−2 , u > 0 in R, where n ≥ 3, K > 0 is periodic in (x1, ..., xk) with 1 ≤ k < n−2 2 . Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in R, including infinite lattices. We also show that for k ≥ n−2 2 , no such solutions exist.