## 唐仲伟: Solutions for conformally invariant fractional Laplacian equations with multi-bumps centered in lattices

：唐仲伟 教授

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.

（罗肖/文）