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

We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces ${\dot{W}}_{0,\text{rad}}^{1,p}(B_R^N)(p<N)$. 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.