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.
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