A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N

In the article, we investigate Trudinger-Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of extremal functions or blow-up, where the domain is the ball or the entire space $\mathbb{R}<^>N$. We also show that the extremal functions satisfy suitable Euler-Lagrange equations. When the domain is the entire space, such equations can be derived by a N-Laplacian Schr & ouml;dinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends [16] to any dimension $N \geq 2$.