Multiplicity result for a mass supercritical NLS with a partial confinement

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.