Critical dimensions and higher order Sobolev inequalities with remainder terms

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Existence and nonexistence results for critical growth biharmonic elliptic equations (2003, methodology): “Of course, this technique fails for (4) where higher order derivatives are involved and we overcome this difficulty thanks to the decomposition method in dual cones developed in [14].”
Existence of Solutions for Singular Critical Growth Semilinear Elliptic Equations (2001, background): “Recently, an attempt was made to explain this phenomenon by means of local summability of the fundamental solutions [J, M] and with the presence of linear remainder terms in Sobolev inequalities with optimal constants [ GG1 ].”
Remainder terms in the fractional Sobolev inequality (2012, background): “Ω |u|dx ) 2 q ≥ C|Ω|− 2q |u|(2)w,q/2,Ω For fixed bounded domains Ω ⊂ R , the existence of a weak L-remainder term is due to Brezis and Lieb [6] in the case s = 2 and to Gazzola and Grunau [14] in the case of an arbitrary even positive integer s < N .”