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