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Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities

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Abstract

We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: \({-div(|x|^\alpha |\nabla u|^{p-2}\nabla u)=|x|^\beta u^{p(\alpha,\beta)-1}, u(x) > 0,\,x\in\Omega\subset\mathbb{R}^N} (N\geq 3), 1<p<N\) and \({\alpha, \beta\in \mathbb{R}}\) such that \({\frac{p(N+\beta)}{N-p+\alpha} > p}\) . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in rather general, possibly degenerate or singular settings.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Giessen, Arndtstr. 2, 35392, Giessen, Germany

    Thomas Bartsch

  2. School of Mathematics and Statistics, Central China Normal University, 430079, Wuhan, People’s Rebulic of China

    Shuangjie Peng

  3. Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, 100080, Beijing, People’s Rebulic of China

    Zhitao Zhang

Authors
  1. Thomas Bartsch
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  2. Shuangjie Peng
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  3. Zhitao Zhang
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Correspondence to Thomas Bartsch.

Additional information

Supported by the Alexander von Humboldt foundation and NSFC(10571069,10631030).

Supported by the Alexander von Humboldt foundation and NSFC(10471098,10671195).

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Bartsch, T., Peng, S. & Zhang, Z. Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities. Calc. Var. 30, 113–136 (2007). https://doi.org/10.1007/s00526-006-0086-1

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  • DOI: https://doi.org/10.1007/s00526-006-0086-1

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