Multiplicity result for a mass supercritical NLS with a partial confinement

TL;DR

This paper proves the existence of multiple solutions for a mass supercritical nonlinear Schrödinger equation with partial confinement in three-dimensional space, using a limit process from bounded domain solutions and symmetry properties.

Contribution

It provides a non-perturbative proof of a second solution at mountain pass energy level, addressing an open problem from prior research.

Findings

Existence of multiple solutions established.

Solutions obtained as limits from bounded domain problems.

Symmetry plays a key role in convergence analysis.

Abstract

We consider an NLS equation in $\mathbb{R}^3$ with partial confinement and mass supercritical nonlinearity. In Bellazzini, Boussaid, Jeanjean and Visciglia (Comm. Math. Phys. 353, 2017, 229-251) for such a problem, a solution with a prescribed $L^2$ norm was obtained, as a local minima, and the existence of a second solution, at a mountain pass energy level, was proposed as an open problem. We give here a positive, non-perturbative, answer to this problem. Our solution is obtained as a limit of a sequence of solutions of corresponding problems on bounded domains of $\mathbb{R}^3$. The symmetry of solutions on bounded domains is used centrally in the convergence process.