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We consider the following semilinear elliptic equation with critical exponent: Δ  u==    K ( x )u    (   n  +  2   )   /   (   n  -  2   )u^{(n+2)/(n-2)}   ,  u&gt;&gt;    0 inℝ  n\mathbb{R}^{n}   , wheren  ≥  3{n\geq 3}   ,K  &gt;  0{K&gt;0}    is periodic in (x  1x_{1}   ,…,x  kx_{k}   ) with 1≤\leq    k&lt;&lt;    ( 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 inℝ  k{\mathbb{R}^{k}}   , including infinite lattices. We also show that for  k≥\geq    ( n -2)/2, no such solutions exist.

Multi-bump solutions of -Δ n = K ( x ) u ( n +2)/( n -2) on lattices in ℝ n