2017/02/26 by Van Hoang Nguyen, Nguyen, Van Hoang
Mathematics · #26D10 #46E35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1702.07970
openalex publication_date 2017/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the existence and nonexistence of maximizers for variational problem concerning to the Moser--Trudinger inequality of Adimurthi--Druet type in W1,N(\mathbb RN) MT(N,β, α) =supu∈ W1,N(\mathbb RN), ‖∇ u‖NN + ‖u‖NN≤ 1 ∫\mathbb RN ΦN(β(1+α‖u‖NN)^\frac1N-1 |u|^\frac NN-1) dx, where ΦN(t) =et -∑k=0N-2 (tk)/(k!), 0≤ α< 1 both in the subcritical case β< βN and critical case β=βN with βN = N ωN-1^\frac1N-1 and ωN-1 denotes the surface area of the unit sphere in \mathbb RN. We will show that MT(N,β,α) is attained in the subcritical case if N≥ 3 or N=2 and β∈ ((2(1+2α))/((1+α)2 B2),β2) with B2 is the best constant in a Gagliardo--Nirenberg inequality in W1,2(\mathbb R2). We also show that MT(2,β,α) is not attained for β small which is different from the context of bounded domains. In the critical case, we prove that MT(N,βN,α) is attained for α≥ 0 small enough. To prove our results, we first establish a lower bound for MT(N,β,α) which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT(N,β,α) in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together the scaling argument, we show that MT(N,βN,1) =∞. Our results settle the questions left open in \citedoO2015,doO2016.