Asymptotically critical problems on higher-dimensional spheres

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10 Citations (Scopus)

Abstract

The paper is concerned with the equation -Δhu = f(u) on Sd where Δh denotes the Laplace-Beltrami operator on the standard unit sphere (Sd, h), while the continuous nonlinearity f : ℝ → ℝ oscillates either at zero or at infinity having an asymptotically critical growth in the Sobolev sense. In both cases, by using a group-theoretical argument and an appropriate variational approach, we establish the existence of [d/2] + (-1)d+1 - 1 sequences of sign-changing weak solutions in H12(Sd) whose elements in different sequences are mutually symmetrically distinct whenever f has certain symmetry and d ≥ 5. Although we are dealing with a smooth problem, we are forced to use a non-smooth version of the principle of symmetric criticality (see Kobayashi-Ôtani, J. Funct. Anal. 214 (2004), 428-449). The L - and H12-asymptotic behaviour of the sequences of solutions are also fully characterized.

Original languageEnglish
Pages (from-to)919-935
Number of pages17
JournalDiscrete and Continuous Dynamical Systems
Volume23
Issue number3
DOIs
Publication statusPublished - Mar 1 2009

Keywords

  • Asymptotically critical growth
  • Laplacian on the sphere
  • Non-smooth principle of symmetric criticality
  • Oscillatory term
  • Sign-changing solution
  • Symmetrically distinct solutions
  • Szulkin-type functional

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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