In the paper 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 $\R^N$. 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\"odinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends \cite{CiWe2} to any dimension $N \geq 2$.
A sharp threshold for Trudinger–Moser type inequalities with logarithmic kernels in dimension N
Alessandro Cannone;Silvia Cingolani
2025-01-01
Abstract
In the paper 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 $\R^N$. 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\"odinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends \cite{CiWe2} to any dimension $N \geq 2$.File in questo prodotto:
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