Improved Moser–Trudinger Inequalities with Singular Weights and Nonlinear Perturbations for Zero-Mean Functions

We study Moser–Trudinger inequalities with singular weights and nonlinear perturbations for zero-mean functions on a smooth bounded domain Ω⊂R2 with 0∈∂Ω. We consider supu∈H1(Ω), ∥∇u∥22=1∫Ωudx=0∫Ωe2π(1−β)u2q(∥u∥22)|x|2βdx, where 0<β<1 and q(t)=1+a1t+⋯+aktk has nonnegative coefficients. Under the cascading conditions 0≤a1<λ and 0≤aj≤λaj−1 (j=2,…,k), where λ is the first nonzero Neumann eigenvalue, we prove that the supremum is finite. Our results unify and extend the improved Moser–Trudinger inequality of Lu–Yang for zero-mean functions and the singular Moser–Trudinger inequality of Zhu.