Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials

TL;DR

This paper proves a new fractional Trudinger-Moser inequality involving a logarithmic convolution potential, explores extremal functions, and establishes symmetry properties of solutions in bounded intervals and the entire space.

Contribution

It introduces a novel fractional inequality with a logarithmic convolution potential and analyzes the existence and symmetry of extremal functions and solutions.

Findings

Established a fractional Trudinger-Moser inequality with logarithmic convolution.

Proved existence of extremal functions for the inequality.

Demonstrated radial symmetry and decreasing properties of solutions.

Abstract

We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential $$ \sup_{u\in W^{\frac{1}{2},2}_0(I),\|u\|_{W_0^{\frac{1}{2},2}}\leq1}\int_{I} \int_{I} \log \frac{1}{|x-y|} G(u(x))G(u(y)) \, dx \, dy<+\infty,$$ where $G(s)\leq C\frac{e^{\pi s^{2}}}{(1 + |s|)^{\gamma}}~ \forall s\in\mathbb{R}$ with some constant $C>0,\gamma\geq1$, the domain $I\subset\mathbb{R}$ is a bounded interval. This type of inequality in the entire space $\mathbb{R}$ 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.