Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials

2025/07/26 by Luo, Huxiao, Wang, Shiying
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.20069

Abstract

We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential sup_u∈ W^(1)/(2),2₀(I),‖u‖_W0^(1)/(2),2≤1∫_I ∫_I log (1)/(|x-y|) G(u(x))G(u(y)) dx dylt;+∞, where G(s)≤ C\frace^πs²(1 + |s|)^γ~ ∀ s∈ℝ with some constant C>0,γ≥1, the domain I⊂ℝ is a bounded interval. This type of inequality in the entire space ℝ is also considered. Moreover, we study the existence of corresponding extremal functions. In addition, by the moving plane method, we obtain the radial symmetry and radial decreasing property of positive solutions to the corresponding Euler-Lagrange equation.

Citations

• A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N
• Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains
• Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications
• Extremal function for Moser–Trudinger type inequality with logarithmic weight
• Moser functions and fractional Moser-Trudinger type inequalities
• Trudinger–Moser type inequalities with logarithmic weights in dimension N
• Hitchhiker's guide to the fractional Sobolev spaces
• Hitchhikerʼs guide to the fractional Sobolev spaces

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