Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary

Abstract

<p style='text-indent:20px;'>In this paper, we establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals on a compact Riemann surface <inline-formula><tex-math id="M1">\begin{document}$ ( \Sigma,g) $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Sigma $\end{document}</tex-math></inline-formula>. To be exact, let <inline-formula><tex-math id="M3">\begin{document}$ \lambda_1(\partial\Sigma) $\end{document}</tex-math></inline-formula> denotes the first eigenvalue of the Laplace-Beltrami operator <inline-formula><tex-math id="M4">\begin{document}$ \Delta _ { g} $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M5">\begin{document}$ \partial \Sigma $\end{document}</tex-math></inline-formula>. Moreover, for any <inline-formula><tex-math id="M6">\begin{document}$ 0\leq\alpha&lt;\lambda_1(\partial\Sigma) $\end{document}</tex-math></inline-formula>, we set <inline-formula><tex-math id="M7">\begin{document}$ \mathcal { H } = \{ u \in W^{1,2} ( \Sigma, g) : \left(\int _{\Sigma} |\nabla_g u|^2 dv_g -\alpha \int _{\partial\Sigma} {u^2}ds_g \right)^{1/2}\leq 1 \ \, \mathrm{and}\, \int _{\partial\Sigma} {u}\,ds_g = 0 \} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M8">\begin{document}$ W^{1,2}(\Sigma, g) $\end{document}</tex-math></inline-formula> is the usual Sobolev space. By the method of blow-up analysis, we first prove the supremum <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \sup\limits_{ u \in \mathcal { H } }\int _ { \partial\Sigma } e ^ {\pi u^ 2} ds_g \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>is attained by some function <inline-formula><tex-math id="M9">\begin{document}$ u_\alpha \in \mathcal{H}\cap C^{\infty} \left(\overline{ \Sigma}\right) $\end{document}</tex-math></inline-formula>. Further, we extend the result to the case of higher order eigenvalues. The results generalize those of Li-Liu [<xref ref-type="bibr" rid="b9">9</xref>] and Yang [<xref ref-type="bibr" rid="b19">19</xref>, <xref ref-type="bibr" rid="b20">20</xref>].