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

  • Nils AckermannEmail author
  • Alfredo Cano
  • Eric Hernández-Martínez


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.


Weak Solution Homogeneous Boundary Condition Cylindrical Domain Smooth Bounded Domain Critical Sobolev Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank the anonymous referee for carefully reading the manuscript and for suggesting several improvements.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Nils Ackermann
    • 1
    Email author
  • 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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