In this paper, we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in ℝn, where n ≥ 2. More precisely, we show that for any given α > 0 and 0 < t < n, then the following two inequalities hold for ∀ u ∈ W0,1, (B), $$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {\left( {{\alpha _{n,t}} + {{\left| x \right|}^\alpha }} \right){{\left| u \right|}^{{n \over {n - 1}}}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty $$ and $$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {{\alpha _{n,t}} + {{\left| u \right|}^{^{{n \over {n - 1}}} + {{\left| x \right|}^\alpha }}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty .$$ We also consider the problem of the sharpness of the constant αn,t. Furthermore, by employing the method of estimating the lower bound and using the concentration-compactness principle, we establish the existence of extremals. These results extend the known results when t = 0 to the singular version for 0 < t < n.

中文翻译：

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奇异超临界 Trudinger-Moser 不等式和极值的存在

在本文中，我们在 ℝn 中给出了单位球 B 上的奇异超临界 Trudinger-Moser 不等式，其中 n ≥ 2。更准确地说，我们证明对于任何给定的 α > 0 和 0 < t < n，则有以下两个不等式成立 ∀ u ∈ W0,1, (B), $$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {\left( {{\alpha _{n,t}} + {{\left | x \right|}^\alpha }} \right){{\left| u \right|}^{{n \over {n - 1}}}}} \right)} \over {{{\left | x \right|}^t}}}} dx < \infty $$ 和 $$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {{\alpha _{n,t}} + {{\left| u \right |}^{^{{n \over {n - 1}}} + {{\left| x \right|}^\alpha }}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty .$$ 我们还考虑了常数αn,t 的锐度问题。此外，通过使用估计下界的方法和使用浓度-紧凑性原理，我们建立了极值的存在性。这些结果将 t = 0 时的已知结果扩展到 0 < t < n 的奇异版本。