Sharp anisotropic singular Trudinger-Moser inequalities in the entire space. (English) Zbl 07828201
Summary: In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm \(\left(\int_{\mathbb{R}^N}F^N(\nabla u)\;\mathrm{d}x\right)^{\frac{1}{N}}\) in Sobolev-type space \(D^{N, q}(\mathbb{R}^N) \), \(N \geq 2\), \(q \geq 1\). Here \(F: \mathbb{R}^N\rightarrow[0, +\infty)\) is a convex function of class \(C^2(\mathbb{R}^N\setminus\{0\})\), which is even and positively homogeneous of degree 1. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger-Moser inequalities and discuss their sharpness under different situations, including the case \(\|F(\nabla u)\|_N \leq 1\), the case \(\|F(\nabla u)\|_N^a + \|u\|_q^b \leq 1\), and whether they are associated with exact growth.
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 35J70 | Degenerate elliptic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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