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Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case

Gianmaria Verzini, Junwei Yu

TL;DR

We address normalized solutions of the Sobolev-critical nonlinear Schrödinger equation with a weakly attractive potential on $\mathbb{R}^N$, $N\ge3$, under a mass constraint. The authors develop a constrained variational framework on the $L^2$-sphere and prove the existence of both a local minimizer (ground state) and a mountain-pass solution, with the mountain-pass result confined to $3\le N\le5$ and depending on smallness properties of the potential, the mass, and the energy landscape. Compactness is achieved through concentration-compactness arguments and a Jeanjean-type minimax construction, with a precise upper bound on the mountain-pass level that prevents dichotomy. The work also leverages a Hopf-Cole transform to obtain multiplicity results for ergodic Mean Field Games systems with potential and quadratic Hamiltonian, connecting PDE variational methods to game-theoretic models. Overall, the paper extends Sobolev-critical normalized-solution theory to nonhomogeneous potentials under low regularity and nonradial settings, delivering existence and multiplicity results under sharp, dimension-dependent conditions.

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 Schrödinger 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.

Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case

TL;DR

We address normalized solutions of the Sobolev-critical nonlinear Schrödinger equation with a weakly attractive potential on , , under a mass constraint. The authors develop a constrained variational framework on the -sphere and prove the existence of both a local minimizer (ground state) and a mountain-pass solution, with the mountain-pass result confined to and depending on smallness properties of the potential, the mass, and the energy landscape. Compactness is achieved through concentration-compactness arguments and a Jeanjean-type minimax construction, with a precise upper bound on the mountain-pass level that prevents dichotomy. The work also leverages a Hopf-Cole transform to obtain multiplicity results for ergodic Mean Field Games systems with potential and quadratic Hamiltonian, connecting PDE variational methods to game-theoretic models. Overall, the paper extends Sobolev-critical normalized-solution theory to nonhomogeneous potentials under low regularity and nonradial settings, delivering existence and multiplicity results under sharp, dimension-dependent conditions.

Abstract

We study the existence and multiplicity of positive solutions in , , with prescribed -norm, for the (stationary) nonlinear Schrödinger 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 -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.
Paper Structure (5 sections, 23 theorems, 191 equations)

This paper contains 5 sections, 23 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Mountain pass geometry
  4. Ground state solution (proof of Theorem \ref{['Thm_neg']})
  5. Mountain pass solution (proof of Theorem \ref{['thm:mountain_pass']})

Key Result

Theorem 1.3

Let $N\ge3$ and assumptions def_pot, Assump_Neg hold true. There exists a positive constant $C_{0}$ such that, if then eq:main has a solution, which corresponds to a local minimizer of $E$ on ${\mathcal{M}}_{\rho}$ with negative energy, negative multiplier, and that can be chosen to be strictly positive almost everywhere in $\mathbb{R}^N$. If in addition for appropriate positive constants $C_{1}

Theorems & Definitions (52)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 42 more