Introduction:

Variational methods are the key ingredients to produce several types of solutions to elliptic and also parabolic equations which are relevant to various fields of mathematics and mathematical physics.
On the other hand, new types of functional inequalities which describe the subtle relationship between function spaces have been found and the research of these inequalities has been a subject in the field of real analysis.
For example, recent study about Adams type higherorder TrudingerMoser inequality and the related elliptic equations is an example of the fruitful collaborations of the above two fields.
The main purpose of this Special Session is to encourage vital discussions between the groups of researches from the real analysis and PDE, and then to pursue the possible applications of such new functional inequalities to variational methods.
Critical phenomena occurring from the noncompactness of embeddings, limiting profiles, concentration phenomena, and new types of scaleinvariant inequalities will be subjects in this Session. 
