# 4月26日 唐仲伟教授学术报告（数学与统计学院）

In this talk, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:

$$(-\Delta)^{s}u=K(x)u^{\frac{N+2s}{N-2s}}, ~u> 0 ~\textmd{in}~ {\BbbR}^{N},$$

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 ${\Bbb R}^{k},$ including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the  bumps  locate in lattices in ${\Bbb R}^{k},$ 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.