Normalized solutions for the nonlinear Schr\"odinger equation with potential: the purely Sobolev critical case

TL;DR

This paper investigates the existence and multiplicity of positive solutions with prescribed mass for the Sobolev critical nonlinear Schrödinger equation in the presence of a potential, extending known results to more general settings.

Contribution

It establishes conditions under which local minimum and mountain-pass solutions exist for the critical NLS with potential, and applies these findings to Mean Field Games systems.

Findings

Existence of local minimum solutions under certain potential conditions.

Persistence of mountain-pass solutions with potential.

Applications to multiple solutions in ergodic Mean Field Games.

Abstract

We study the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the (stationary) nonlinear Schr\"odinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this paper, we consider the same problem, in presence of a weakly attractive, possibly irregular, potential, wondering (i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and (ii) if the existence of a mountain-pass solution persists. We provide positive answers, depending on suitable assumptions on the potential and on the mass value. Moreover, by the Hopf-Cole transform, we give some applications of our results to the existence of multiple solutions to ergodic Mean Field Games systems with potential and quadratic Hamiltonian.