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Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents

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Abstract

We consider the following problem

$$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$

where \({ \mu \ge 0, 0 < s < 2, 0 \in \partial \Omega}\) and Ω is a bounded domain in R N. We prove that if \({N \ge 7, a(0) > 0}\) and all the principle curvatures of ∂Ω at 0 are negative, then the above problem has infinitely many solutions.

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Authors and Affiliations

  1. Department of Mathematics, The University of New England, Armidale, NSW, 2351, Australia

    Shusen Yan

  2. Department of Mathematics, Jiangxi Normal University, Nanchang, 330022, People’s Republic of China

    Jianfu Yang

Authors
  1. Shusen Yan
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  2. Jianfu Yang
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Correspondence to Shusen Yan.

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Communicated by C.S. Lin.

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Yan, S., Yang, J. Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents. Calc. Var. 48, 587–610 (2013). https://doi.org/10.1007/s00526-012-0563-7

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  • DOI: https://doi.org/10.1007/s00526-012-0563-7

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