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

Abstract

In this paper, we study normalized solutions to the following $L^2$-supercritical nonlinear Schr\"{o}dinger equation with a periodic potential \begin{equation*} \begin{dcases} -\Delta u +V (x)u + \lambda u=\chi_{{\Omega}}(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 =\mu. \end{dcases} \end{equation*} Here $N \geq 1$, $\mu>0$ is prescribed, $\lambda \in \mathbb{R}$ is a Lagrange multiplier, $V\in C(\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \in C^1(\mathbb{R})$ is a nonlinearity having $L^2$-supercritical growth at infinity, $\Omega\subset \mathbb{R}^N$ is either a (nonempty) bounded open set with smooth boundary $\partial \Omega$ or the whole space \(\mathbb{R}^N\), and $\chi_{{\Omega}}$ is the characteristic function of ${\Omega}$. In both cases, we prove the existence of normalized solutions of mountain pass type when $\mu>0$ is small and $f$ behaves like a power function $|t|^{p-2}t$ with $2+4/N<p<2^*$ at infinity. Moreover, if $\Omega$ is bounded, $1 \leq N \leq 4$ and $f$ has $L^2$-supercritical growth near the origin, then the existence of normalized solutions is obtained for all $\mu>0$. On the other hand, when $\Omega=\mathbb{R}^N$, we prove the existence of solutions corresponding to local minimizers when $\mu>0$ is small. When $\Omega$ is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where $\Omega = \mathbb{R}^N$, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.