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

TL;DR

This paper establishes the existence of two normalized standing wave solutions for a biharmonic Schrödinger equation with positive dispersion and Sobolev critical exponent, using a novel direct minimization approach without radial symmetry assumptions.

Contribution

It introduces a new energy inequality and a direct minimization method to find solutions without relying on radial symmetry or Palais-Smale sequences.

Findings

Existence of two normalized solutions: a ground state and a higher-energy state.

Analysis of the relationship between ground states and least action solutions.

Investigation of asymptotic properties, orbital stability, and global existence of solutions.

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.