Abstract We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type L u : = − r − θ ( r α | u ′ ( r ) | β u ′ ( r ) ) ′ , where θ , β ≥ 0 and α > 0 , are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the p-Laplacian and k-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szego principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.

中文翻译：

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关于 RN 中的加权 Trudinger-Moser 不等式

摘要 我们在整个空间的加权 Sobolev 空间上建立了 Trudinger-Moser 不等式，对于一类 L u 型径向形式的拟线性椭圆算子： = − r − θ ( r α | u ′ ( r ) | β u ′ ( r ) ) ′ ，其中 θ , β ≥ 0 且 α > 0 是满足一定存在条件的常数。值得强调的是，这些算子在径向情况下推广了 p-Laplacian 和 k-Hessian 算子。我们的结果涉及分数维数、新的加权 Polya-Szego 原理以及 Gagliardo-Nirenberg 型不等式中最优常数的有界值。