A General and Unified Method to prove the Uniqueness of Ground State Solutions and the Existence/Non-existence, and Multiplicity of Normalized Solutions with applications to various NLS

TL;DR

This paper introduces a comprehensive framework for establishing the uniqueness, existence, non-existence, and multiplicity of ground state and normalized solutions for a broad class of PDEs, including fractional nonlinear Schrödinger equations.

Contribution

It provides a general, self-contained method applicable to various operators and domains, extending previous results to non-autonomous and mixed nonlinearities with new stability insights.

Findings

Unified approach to ground state solution uniqueness

Extension of results to non-autonomous nonlinear Schrödinger equations

New non-degeneracy and stability results

Abstract

We first give an abstract framework to show the uniqueness of Ground State Solutions (GSS) for a large class of PDEs. To the best of our knowledge, all the existing results in the literature only addressed particular cases. Moreover, our self-contained approach offers a general framework to study the existence/non-existence and multiplicity of normalized solutions. We will exhibit concrete examples to which our method applies, and verify all the assumptions we need. Our approach is applicable to a wide range of operators and domains provided that our hypotheses are verified. Additionally, we prove new results about the non-degeneracy and uniqueness of positive GSS in a general setting. Our findings are applicable to fractional nonlinear Schrodinger equations with non-autonomous nonlinearities. In particular, we were able to extend the main results of [13, 14] to general non-autonomous and mixed nonlinearities. This does not seem possible by using the approach developed by the authors of the above breakthrough papers. The orbital stability/instability of the standing waves will be addressed thanks to the non-degeneracy.