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The nonexistence of extremals for the Hardy-Trudinger-Moser inequality in the hyperbolic space. (English) Zbl 1490.35014

Summary: Let \(\mathbb{B}\) be the unit disc in \(\mathbb{R}^2\), \(\mathscr{H}\) be the completion of \(C^\infty_0(\mathbb{B})\) under the norm \[ \|u\|_{\mathscr{H}} = \left(\int_{\mathbb{B}}|\nabla u|^2\mathrm{d}x-\int_{\mathbb{B}}\frac{u^2}{(1-|u|^2)^2}\mathrm{d}x\right)^{\frac{1}{2}}, \quad\forall u\in C^\infty_0(\mathbb{B}). \] We prove that the supremum in the following inequality \[ \sup_{u\in\mathscr{H}, \|u\|_{\mathscr{H}}\leqslant1}\int_{\mathbb{B}}\exp\{4\pi(1+\alpha\|u\|^2_2)u^2\}\mathrm{d}x < +\infty \] can not be achieved by any functions in the function space \(\mathscr{H}\) when \(\alpha\) is sufficiently close to \(\lambda^-_1\), i.e., \(0 < \lambda_1-\alpha \ll 1\), where \[ \lambda_1(\mathbb{B}) = \inf_{u\in\mathscr{H}, u\not\equiv 1}\frac{\|u\|_{\mathscr{H}}^2}{\|u\|^2_2}. \] Evidently, this conclusion is complementary to that of our work [Acta Math. Sin., Engl. Ser. 36, No. 6, 711–722 (2020; Zbl 1447.35015), Theorem 1.1 (ii)].

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35B33 Critical exponents in context of PDEs
35B44 Blow-up in context of PDEs

Citations:

Zbl 1447.35015
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References:

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