Normalized solutions of $L^2$-supercritical NLS equations with periodic potentials and localized nonlinearities

In this paper, we study the existence of normalized solutions to the following $L^2$-supercritical nonlinear Schrödinger equation with a periodic potential $$\{\begin{array}{l}{\displaystyle -\mathrm{\Delta }u+V(x)u+\lambda u={\chi }_{\mathrm{\Omega }}(x)f(u)\text{in }{\mathbb{R}}^N,u>0\text{in }{\mathbb{R}}^N,{\int }_{{\mathbb{R}}^N}\text{abs}{u}^{2}dx=\mu ,}\end{array}$$ where $N\ge 1$, $\mu >0$ is prescribed, $\lambda \in \mathbb{R}$ is a Lagrange multiplier, $V\in C({\mathbb{R}}^N)$ is $1$-periodic in $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_N$, $f\in C^1(\mathbb{R})$ exhibits a general mass supercritical growth at infinity, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a (nonempty) bounded open set with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ and ${\chi }_{\mathrm{\Omega }}$ is the characteristic function of $\mathrm{\Omega }$. We prove the existence of normalized solutions for all $\mu >0$ sufficiently small. Moreover, if $f$ further has a mass-supercritical growth near the origin, then the existence result extends to every $\mu >0$. The result is obtained through a combination of the monotonicity trick, minimax principle with Morse index information for constrained functionals and blow-up analysis.