In this talk, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:
$$Delta)^{s}u=K(x)u^{\frac{N+2s}{N-2s}},
u>0 in R^N$$
where $s\in (0,1)$ and $N>2+2s,$ $K>0$ is periodic in $(x_{1},\ldots, x_{k})$ with $1\leq k<\frac{N-2s}{2}$. 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 $R^k$ including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in $R^k$ the restriction on $1\leq k<\frac{N-2s}{2}$ is in some sense optimal, since we can show that for $ k\geq\frac{N-2s}{2},$ no such solutions exist. This is a joint work with Dr. Miaomiao Niu and Dr.Lushun Wang.
