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

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