Trudinger-Moser type inequalities for the Hessian equation with logarithmic weights

Abstract

We establish sharp Trudinger-Moser inequalities with logarithmic weights for the $k$-Hessian equation and investigate the existence of maximizers. Our analysis extends the classical results of Tian and Wang to $k$-admissible function spaces with logarithmic weights, providing a natural complement to the work of Calanchi and Ruf. Our approach relies on transforming the problem into a one-dimensional weighted Sobolev space, where we solve it using various techniques, including some radial lemmas and certain Hardy-type inequalities, which we establish in this paper, as well as a theorem due to Leckband.

References in corpus (5)

• Extremal function for Moser-Trudinger type Inequality with Logarithmic weight
• Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications
• Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions
• Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces
• On a supercritical k-Hessian inequality of Trudinger-Moser type and extremal functions