Weighted Trudinger-Moser inequalities in the subcritical Sobolev spaces and their applications

TL;DR

This paper investigates weighted Trudinger-Moser inequalities within subcritical Sobolev spaces, correcting previous errors, establishing convergence to the classical case, and applying results to elliptic problems with exponential nonlinearities.

Contribution

It revises an error in prior work, proves convergence of inequalities to the classical form as p approaches N, and applies these inequalities to elliptic PDEs with exponential nonlinearities.

Findings

Corrected a previous theorem in the literature.

Proved convergence of inequalities to the classical Trudinger-Moser inequality as p approaches N.

Applied inequalities to elliptic problems with exponential nonlinearities.

Abstract

We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces $\dot{W}^{1,p}_{0, \text{rad}}(B_R^N)\,(p<N)$. Our results give a revision of an error in \cite[Theorem C]{HL}. Also, our inequality converges to the original Trudinger-Moser inequality as $p \nearrow N$ including optimal exponent and concentration limit. Finally, we consider an application of our inequality to elliptic problems with exponential nonlinearity.