(untitled)
Let W^1,n (R^n) be the standard Sobolev space. For any τ>0 and p>n>2, we denote λ_n,p=_uW^1,n(R^n), u ot0_R^n(|∇u|^n+τ|u|^n)dx(_R^n|u|^p dx)^np . Define a norm in W^1,n (R^n) by ||u||_n,p=(_R^n(|∇u|^n+τ|u|^n)dx-α(_R^n|u|^pdx)^np )^1n , where 0 ≤ α < λ_n,p. Using a rearrangement argument and blow-up analysis, we will prove _uW^1,n(R^n), ||u||_n,p1_R^n(e^α_n |u|^nn-1 -^n-1_j=0α_n^j|u|^jnn-1 j! )dx can be attained by some function u_0 ∈ W^1,n (R^n )∩C^1 (R^n) with ∥u_0∥_n,p= 1, here α_n = nω ^1 n−1 _n−1 and ω_n−1 is the measure of the unit sphere in R^n . , number=4, journal=Journal of Partial Differential Equations, year=2023, month=Nov., pages=414–434