Solutions with prescribed mass for $L^2$-supercritical NLS equations under Neumann boundary conditions
Xiaojun Chang, Vicenţiu D. Rădulescu, Yuxuan Zhang
TL;DR
This work addresses normalized solutions of the $L^2$-supercritical nonlinear Schrödinger equation with Neumann boundary conditions on a smooth bounded domain, under a prescribed mass $c$. The authors develop a novel variational framework: a uniform mountain-pass geometry for a family of functionals $J_\rho$, a dense-set existence result on the $L^2$-sphere via a Morse-index constrained min-max principle, and a meticulous Neumann blow-up analysis to ensure compactness. For small mass $c$, they prove the existence of mountain-pass type normalized solutions $(u,\lambda)$, not necessarily positive, with positivity of $\lambda$ under suitable sign conditions; the framework covers nonlinearities such as $f(u)=|u|^{p-2}u$ and various mixtures. A key contribution is the handling of Neumann boundary effects and the development of a domain-extension blow-up technique, yielding rigorous compactness and convergence to normalized standing waves with physical relevance in optics and Bose–Einstein condensates. The results advance the theory of normalized solutions on bounded domains and provide a flexible approach applicable to a broad class of nonlinearities.
Abstract
In this paper, we investigate the following nonlinear Schrödinger equation with Neumann boundary conditions: \begin{equation*} \begin{cases} -Δu+ λu= f(u) & {\rm in} \,~ Ω,\\ \displaystyle\frac{\partial u}{\partial ν}=0 \, &{\rm on}\,~\partial Ω\end{cases} \end{equation*} coupled with a constraint condition: \begin{equation*} \int_Ω|u|^2 dx=c, \end{equation*} where $Ω\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, $ν$ represents the unit outer normal vector to $\partial Ω$, $c$ is a positive constant, and $λ$ acts as a Lagrange multiplier. When the nonlinearity $f$ exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various nonlinearities, including cases such as $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2<q<2+\frac{4}{N}<p< 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.