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We present two methods to prove the strict monotonicity of the global branchof solutions of some PDEs in the L2 norm, and the uniqueness of thecorresponding normalized ground states. We will first discuss the key ideas andingredients of each method. Then, we will apply them to various classes ofPDEs. Our approach is applicable to other operators, domains and nonlinearitiesprovided that some hypotheses are satisfied. Our general methods are novel andself-contained, they provide two approaches that address the challenginguniqueness and strict monotonicity issues.

Strict Monotonicity of the global branch of solutions in the L2 norm and  Uniqueness of the normalized ground states for various classes of PDEs: Two  general Methods with some examples