A Legendre-Fenchel identity for the nonlinear Schr\"odinger equations on Rd×Tm\mathbb{R}^d\times\mathbb{T}^m: theory and applications

The present paper is inspired by a previous work \cite{Luo_Waveguide_MassCritical} of the author, where the large data scattering problem for the focusing cubic nonlinear Schr\"odinger equation (NLS) on $\mathbb{R}^2\times\mathbb{T}$ was studied. Nevertheless, the results from \cite{Luo_Waveguide_MassCritical} are by no means sharp, as we could not even prove the existence of ground state solutions on the formulated threshold. By making use of the variational tools introduced by the author \cite{Luo_inter}, we establish in this paper the sharpened scattering results. Yet due to the mass-critical nature of the model, we encounter the major challenge that the standard scaling arguments fail to perturb the energy functionals. We overcome this difficulty by proving a crucial Legendre-Fenchel identity for the variational problems with prescribed mass and frequency. More precisely, we build up a general framework based on the Legendre-Fenchel identity and show that the much harder or even unsolvable variational problem with prescribed mass, can in fact be equivalently solved by considering the much easier variational problem with prescribed frequency. As an application showing how the geometry of the domain affects the existence of the ground state solutions, we also prove that while all mass-critical ground states on $\mathbb{R}^d$ must possess the fixed mass $\widehat M(Q)$, the existence of mass-critical ground states on $\mathbb{R}^d\times\mathbb{T}$ is ensured for a sequence of mass numbers approaching zero.Comment: A gap in the proof of Thm 1.5 fixed. Thm. 1.5 reformulate

Scattering threshold for radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in d≥5d\geq 5

We extend the scattering result for the radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in $d\leq 4$ given by Cheng et al. to the case $d\geq 5$. The main ingredient is a suitable long time perturbation theory which is applicable for $d\geq 5$. The paper will therefore give a full characterization on the scattering threshold for the radial defocusing-focusing mass-energy double critical nonlinear Schr\"odinger equation in all dimensions $d\geq 3$

Normalized ground states and threshold scattering for focusing NLS on Rd×T\mathbb{R}^d\times\mathbb{T} via semivirial-free geometry

We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_t u+\Delta_{x,y} u=-|u|^\alpha u\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime $\alpha\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the \eqref{nls_abstract} is independent of $y$, it reduces to the focusing intercritical NLS on $\mathbb{R}^d$, which is known to have standing wave and finite time blow-up solutions. Naturally, we ask whether these special solutions with non-trivial $y$-dependence exist. In this paper we give an affirmative answer to this question. To that end, we introduce the concept of \textit{semivirial} functional and consider a minimization problem $m_c$ on the semivirial-vanishing manifold with prescribed mass $c$. We prove that for any $c\in(0,\infty)$ the variational problem $m_c$ has a ground state optimizer $u_c$ which also solves the standing wave equation $$-\Delta_{x,y}u_c+\beta_c u_c=|u|^\alpha u$$ with some $\beta_c>0$. Moreover, we prove the existence of a critical number $c_*\in(0,\infty)$ such that \begin{itemize} \item For $c\in(0,c_*)$, any optimizer $u_c$ of $m_c$ must satisfy \pt_y u_c\neq 0. \item For $c\in(c_*,\infty)$, any optimizer $u_c$ of $m_c$ must satisfy \pt_y u_c=0. \end{itemize} Finally, we prove that the previously constructed ground states characterize a sharp threshold for the bifurcation of scattering and finite time blow-up solutions in dependence of the sign of the semivirial