On the existence of solutions to the Orlicz–Minkowski problem for torsional rigidity

In [J Diff Equ, 269: 8549–8572, 2020], Li and Zhu studied the Orlicz–Minkowski problem for torsional rigidity, and among other things, they proved the existence of solutions to the problem regarding a continuous function φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} satisfying limx→0+φ(x)=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{x\rightarrow 0^+}\varphi (x)=\infty $$\end{document}. In this paper, with the motivation of complementing their results, we prove a new existence of solutions to the problem regarding a strictly increasing, continuously differentiable function φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} satisfying limx→0+φ(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{x\rightarrow 0^+}\varphi (x)=0$$\end{document}.