We focus on the existence of ground state solutions to the following class of elliptic equations

\begin{equation*}-\Delta_{\mathbb{B}^N} u+V(x)u=f(u) \quad \hbox{in} \quad \mathbb{B}^N,\end{equation*}

where $\mathbb{B}^N$ is the disc model of the Hyperbolic space and $\Delta_{\mathbb{B}^N}$ denotes the Laplace–Beltrami operator with $N \geq 2$, $V:\mathbb{B}^N \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ are continuous functions that satisfy some technical conditions. With different types of the potential V, by introducing some new tricks handling the hurdle that the Hyperbolic space is not a compact manifold, we are able to obtain at least a positive ground state solution using variational methods.

As some applications for the methods adopted above, we derive the existence of normalized solutions to the elliptic problems

\begin{equation*}\left\{\begin{aligned}&-\Delta_{\mathbb{B}^N} u+V(x)u=\mu u+f(u) \quad \hbox{in} \quad \mathbb{B}^N,\\&\int_{\mathbb{B}^N}|u|^{2}\,dV_{{\mathbb{B}^N}}=a^{2},\end{aligned}\right.\end{equation*}

where a > 0, $\mu\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and f is a continuous function that fulfils the L2-subcritical or L2-supercritical growth. We do believe that it seems the first results to deal with normalized solutions for the Schrödinger equations in the Hyperbolic space.

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region $\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω0 centred at origin, a region Ω1 outside a ball, and an n-dimensional annulus Ω2. To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when $\Omega = \Omega_1$, the ibvp for the NLS equation is locally well-posed in $ C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in $H^s(\Omega_1)$ and boundary data is in $ H^{\frac{2s+1}{4}}(0, T)$ with $s \geq 0$. This is the optimal regularity for the boundary data and cannot be improved. When $\Omega = \Omega_2$, the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in $ H^s(\Omega_2)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with $s \geq 0$. In this case, the boundary data requires $1/4$ more derivative compared to the case when $\Omega = \Omega_1$. When $\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in $ H^s(\Omega_0)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or $s \gt n/2$). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with $ 0 \leq s \leq 1$, the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω0 and Ω2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation

$$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$

$\Omega =\mathbb {R}^N$

$\mathbb {R}^N\setminus \Omega$

$\rho >0$

$V\ge 0$

$V\equiv 0$

$p\in (2,2+\frac 4 N)$

$\bar u$

$V$

$\|V\|_{L^q}$

$q\ge \frac N2$

$q>1$

$N=2$

$V$

$\mathbb {R}^N\setminus \Omega$

$\bar u$

$V\equiv 0$

$\Omega =\mathbb {R}^N$

This paper is concerned with numerical method for a two-dimensional time-dependent cubic nonlinear Schrödinger equation. The approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time, respectively. We prove optimal L 2 error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

We study the propagation of wave packets for a one-dimensional system of two coupled Schrödinger equations with a cubic nonlinearity, in the semiclassical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an avoided crossing: at one given point, the gap between them reduces as the semiclassical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point the nonlinearity’s role is negligible at leading order, and the transition probabilities can be computed with the linear Landau–Zener formula.

In this paper, we review some of our recent results in the study of the dynamics ofinteracting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on thewell-posedness of the associated Cauchy problem for the infinite particle system describedby the GP hierarchy.

In this work we consider the magnetic NLS equation $$ ( \frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox{ in } \mathbb{R}^N\qquad\qquad(0.1)$$ where $N \geq 3$ , $A \colon \mathbb{R}^N \to \mathbb{R}^N$ is a magnetic potential,possibly unbounded, $V \colon \mathbb{R}^N \to \mathbb{R}$ is a multi-well electricpotential, which can vanish somewhere, f is a subcriticalnonlinear term. We prove the existence of a semiclassical multi-peaksolution $u\colon \mathbb{R}^N \to \mathbb{C}$ to (0.1), under conditionson the nonlinearity which are nearly optimal.