Multi-bump solutions for logarithmic Schr\"odinger equations
  We study spatially periodic logarithmic Schr\"odinger equations:   \begin{equation}\tag{LS}   -\Delta u + V(x)u=Q(x)u\log u^2, \quad u>0\quad \text{in}\ \mathbb{R}^N,   \end{equation} where $N\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\mathbb{Z}^N$-action. 
Kazunaga Tanaka and Chengxiang Zhang
math.AP