Let B⊂R2\documentclass[12pt]{minimal}
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\begin{document}$$B\subset {\mathbb {R}}^2$$\end{document} be the unit disk , H\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}$$\end{document} be the completion of C0∞(B)\documentclass[12pt]{minimal}
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\begin{document}$$C_{0}^{\infty }(B)$$\end{document} under the norm ||u||H=(∫B|∇u|2dx-∫Bu2(1-|x|2)2dx)1/2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} ||u||_{{\mathcal {H}}}=\bigg (\int _{B}|\nabla u|^2\textrm{d}x-\int _{B}\frac{u^2}{(1-|x|^2)^2}\textrm{d}x \bigg )^{1/2}. \end{aligned}$$\end{document}In this paper, we consider a maximum problem concerning the Hardy–Trudinger–Moser inequalities containing lower order perturbation. Namely, there exists a positive constant ε0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon _{0}$$\end{document} such that if γ≤4π+ε0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \le 4\pi +\varepsilon _{0}$$\end{document}, then supu∈H,||u||H≤1∫B(e4πu2-γu2)dx\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sup \limits _{u\in {\mathcal {H}},\;|| u||_{{\mathcal {H}}}\le 1}\int _{B}(e^{4\pi u^{2}}-\gamma u^2)\textrm{d}x \end{aligned}$$\end{document}can be achieved by some functions u0∈H\documentclass[12pt]{minimal}
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\begin{document}$$u_{0}\in {\mathcal {H}}$$\end{document} with ||u0||H=1\documentclass[12pt]{minimal}
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\begin{document}$$||u_{0}||_{{\mathcal {H}}}=1$$\end{document}. This expands the results of Wang and Ye (Adv Math 230:294–320, 2012).
