Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

  • Nils Ackermann
  • Alfredo Cano
  • Eric Hernández-Martínez
Article

Abstract

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Nils Ackermann
    • 1
  • Alfredo Cano
    • 2
  • Eric Hernández-Martínez
    • 3
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico, D.F.Mexico
  2. 2.Facultad de CienciasUniversidad Autónoma del Estado de MéxicoTolucaMexico
  3. 3.Universidad Autónoma de la Ciudad de MéxicoMexico, D.F.Mexico

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