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Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case

Nicola Soave

TL;DR

This work develops a constrained variational framework for normalized ground states of the Sobolev-critical nonlinear Schrödinger equation with combined nonlinearities on $\mathbb{R}^N$, under a prescribed mass $a^2$. It introduces the Pohozaev manifold $\mathcal{P}_{a,\mu}$ as a natural constraint and analyzes the fiber maps $\Psi_u^\mu$ to study energy geometry under $L^2$-preserving dilations, obtaining existence results across $L^2$-subcritical, critical, and supercritical perturbations. A key contribution is the precise threshold condition $\mu a^{(1-\gamma_q)q}<\alpha(N,q)$, with explicit $\alpha$ depending on $N$ and $q$, which yields ground states that are real, positive, and radially symmetric with negative or positive energy according to the regime; the analysis covers both compactness of PS sequences and a Brezis–Nirenberg–type dichotomy in the critical setting. The paper also proves nonexistence results for defocusing perturbations and studies the asymptotic behavior of ground states as $\mu\to0^+$, connecting normalized solutions to the homogeneous Sobolev-critical problem and highlighting discontinuities in ground-state levels. Dynamical implications are discussed, including stability/instability and blow-up criteria, illustrating how the constrained critical points govern long-time behavior in the Sobolev-critical regime.

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities \[ -Δu= λu + μ|u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $μ|u|^{q-2} u$ we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space $\mathbb{R}^N$.

Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case

TL;DR

This work develops a constrained variational framework for normalized ground states of the Sobolev-critical nonlinear Schrödinger equation with combined nonlinearities on , under a prescribed mass . It introduces the Pohozaev manifold as a natural constraint and analyzes the fiber maps to study energy geometry under -preserving dilations, obtaining existence results across -subcritical, critical, and supercritical perturbations. A key contribution is the precise threshold condition , with explicit depending on and , which yields ground states that are real, positive, and radially symmetric with negative or positive energy according to the regime; the analysis covers both compactness of PS sequences and a Brezis–Nirenberg–type dichotomy in the critical setting. The paper also proves nonexistence results for defocusing perturbations and studies the asymptotic behavior of ground states as , connecting normalized solutions to the homogeneous Sobolev-critical problem and highlighting discontinuities in ground-state levels. Dynamical implications are discussed, including stability/instability and blow-up criteria, illustrating how the constrained critical points govern long-time behavior in the Sobolev-critical regime.

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities having prescribed mass in the \emph{Sobolev critical case}. For a -subcritical, -critical, of -supercritical perturbation we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space .

Paper Structure

This paper contains 12 sections, 26 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Compactness of Palais-Smale sequences
  5. $L^2$-subcritical perturbation
  6. $L^2$-critical perturbation
  7. $L^2$-supercritical perturbation
  8. Non-existence in the defocusing case $\mu<0$
  9. Behavior of ground states when $\mu \to 0^+$
  10. Dynamical properties
  11. Some useful estimates
  12. Boundedness of solutions

Key Result

Theorem 1.1

Let $N \ge 3$, $2<q<2^*$, and let $a,\mu>0$. There exists a constant $\alpha=\alpha(N,q)>0$ such that, if then $E_\mu|_{S_a}$ has a ground state $\tilde{u}$ with the following properties: $\tilde{u}$ is a real valued, positive, radially symmetric function, and solves stat com for some $\tilde{\lambda}<0$. Moreover, denoting by $m(a,\mu) = E_\mu(\tilde{u})$, we have that:

Theorems & Definitions (57)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Remark 1.3
  • ...and 47 more