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 $N\ge 1$ 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.