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Standing waves with prescribed mass for biharmonic NLS with positive dispersion and Sobolev critical exponent

Juntao Sun, Shuai Yao, He Zhang

TL;DR

The paper studies standing waves with prescribed mass for a biharmonic NLS with positive dispersion in the Sobolev critical regime. It develops a direct minimization framework on the mass constraint, supported by sharp energy inequalities and a biharmonic Lions lemma, to obtain two normalized solutions: a ground state with negative energy and a high-energy state with positive energy, without relying on radial symmetry or Palais–Smale sequences. It further clarifies the link between ground states and least-action solutions, derives asymptotics as the second-order dispersion parameter $\mu$ tends to zero, and establishes dynamical properties including orbital stability and global well-posedness for relevant initial data. The results provide a nonradial, variational approach at critical Sobolev exponents and yield insights into the long-time behavior of the associated Cauchy problem.

Abstract

We investigate standing waves with prescribed mass for a class of biharmonic Schrodinger equations with positive Laplacian dispersion in the Sobolev critical regime. By establishing novel energy inequalities and developing a direct minimization approach, we prove the existence of two normalized solutions for the corresponding stationary problem. The first one is a ground state with negative level, and the second one is a higher-energy solution with positive level. It is worth noting that we do not work in the space of radial functions, and do not use Palais-Smale sequences so as to avoid applying the relatively complex mini-max approach based on a strong topological argument. Finally, we explore the relationship between the ground states and the least action solutions, some asymptotic properties and dynamical behavior of solutions, such as the orbital stability and the global existence.

Standing waves with prescribed mass for biharmonic NLS with positive dispersion and Sobolev critical exponent

TL;DR

The paper studies standing waves with prescribed mass for a biharmonic NLS with positive dispersion in the Sobolev critical regime. It develops a direct minimization framework on the mass constraint, supported by sharp energy inequalities and a biharmonic Lions lemma, to obtain two normalized solutions: a ground state with negative energy and a high-energy state with positive energy, without relying on radial symmetry or Palais–Smale sequences. It further clarifies the link between ground states and least-action solutions, derives asymptotics as the second-order dispersion parameter tends to zero, and establishes dynamical properties including orbital stability and global well-posedness for relevant initial data. The results provide a nonradial, variational approach at critical Sobolev exponents and yield insights into the long-time behavior of the associated Cauchy problem.

Abstract

We investigate standing waves with prescribed mass for a class of biharmonic Schrodinger equations with positive Laplacian dispersion in the Sobolev critical regime. By establishing novel energy inequalities and developing a direct minimization approach, we prove the existence of two normalized solutions for the corresponding stationary problem. The first one is a ground state with negative level, and the second one is a higher-energy solution with positive level. It is worth noting that we do not work in the space of radial functions, and do not use Palais-Smale sequences so as to avoid applying the relatively complex mini-max approach based on a strong topological argument. Finally, we explore the relationship between the ground states and the least action solutions, some asymptotic properties and dynamical behavior of solutions, such as the orbital stability and the global existence.
Paper Structure (7 sections, 33 theorems, 230 equations, 1 figure)

This paper contains 7 sections, 33 theorems, 230 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminary results
  4. The existence of ground states
  5. The existence of high-energy solution
  6. Least action characterization
  7. The dynamical behavior

Key Result

Theorem 1.2

Let $\overline{p}<p<4$ for $5\leq N\leq 8$, or $\overline{p}<p\leq 4^{\ast }$ for $N\geq 9.$ Assume that condition (E4) holds. Then the following statements are true. $(i)$$\Psi _{\mu ,p}$ restricted to $S_{a}$ has a ground state $u_{\mu }^{+}$. This ground sate is a local minimizer of $\Psi _{\mu , Moreover, any ground state for $\Psi _{\mu ,p}$ on $S_{a}$ is a local minimizer of $\Psi _{\mu ,p}$

Figures (1)

  • Figure 1: Possible forms of fibering map

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 27 more