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 \[ \begin{dcases} -Δu +V (x)u + λu=χ_{Ω}(x)f(u)\quad \text{in }\mathbb{R}^N, u>0 \quad \text{in} \ \mathbb{R}^N, \int_{\mathbb{R}^N}\abs{u}^2\, dx =μ, \end{dcases} \] where $N \geq 1$, $μ>0$ is prescribed, $λ\in \mathbb{R}$ is a Lagrange multiplier, $V\in C(\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \in C^1(\mathbb{R})$ exhibits a general mass supercritical growth at infinity, $Ω\subset \mathbb{R}^N$ is a (nonempty) bounded open set with smooth boundary $\partial Ω$ and $χ_{Ω}$ is the characteristic function of $Ω$. We prove the existence of normalized solutions for all $μ>0$ sufficiently small. Moreover, if $f$ further has a mass-supercritical growth near the origin, then the existence result extends to every $μ>0$. The result is obtained through a combination of the monotonicity trick, minimax principle with Morse index information for constrained functionals and blow-up analysis.