General class of optimal Sobolev inequalities and nonlinear scalar field equations

@article{Mederski2018GeneralCO,
  title={General class of optimal Sobolev inequalities and nonlinear scalar field equations},
  author={Jaroslaw Mederski},
  journal={arXiv: Analysis of PDEs},
  year={2018}
}
We find a class of optimal Sobolev inequalities $$\Big(\int_{\mathbb{R}^N}|\nabla u|^2\, dx\Big)^{\frac{N}{N-2}}\geq C_{N,G}\int_{\mathbb{R}^N}G(u)\, dx, \quad u\in\mathcal{D}^{1,2}(\mathbb{R}^N), N\geq 3,$$ where the nonlinear function $G:\mathbb{R}\to\mathbb{R}$ of class $\mathcal{C}^1$ satisfies general growth assumptions in the spirit of the fundamental works of Berestycki and Lions. We admit, however, a wider class of problems involving zero, positive and infinite mass cases as well as $G… Expand
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