Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

In this paper, we study the existence and non-existence of maximizers for the Moser–Trudinger type inequalities in R ^N of the form DN,α(a,b):=supu∈W1,N(RN),‖∇u‖LN(RN)a+‖u‖LN(RN)b=1∫RNΦN(α|u|N′)dx.Here N≥2,N′=NN-1,a,b>0,α∈(0,αN] and ΦN(t):=et-∑j=0N-2tjj! where αN:=NωN-11/(N-1) and ω _{N - 1} denotes the surface area of the unit ball in R ^N . We show the existence of the threshold α _∗ = α _∗ (a, b, N) ∈ [0 , α _N ] such that D _{N , α} (a, b) is not attained if α∈ (0 , α _∗ ) and is attained if α∈ (α _∗ , α _N ). We also provide the conditions on (a, b) in order that the inequality α _∗ < α _N holds.