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On the Existence and the Profile of Nodal solutions of Elliptic Equations Involving Critical Growth

  • Thomas BartschEmail author
  • Anna Maria Micheletti
  • Angela Pistoia
Article

Abstract

We study the existence of sign changing solutions to the slightly subcritical problem
$$-\Delta u=|u|^{p-1-\epsilon} u\ {\rm in}\ \Omega,\quad u=0\ {\rm on}\ \partial \Omega,$$
where ω is a smooth bounded domain in ℝ N , N ≥ 3, p = (N + 2)/(N − 2) and ɛ > 0. We prove that, for ɛ small enough, there exist N pairs of solutions which change sign exactly once. Moreover, the nodal surface of these solutions intersects the boundary of ω, provided some suitable conditions are satisfied.

Keywords

Sign changing solutions Nodal domains Critical exponent 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Thomas Bartsch
    • 1
    Email author
  • Anna Maria Micheletti
    • 2
  • Angela Pistoia
    • 3
  1. 1.Mathematisches InstitutJustus-Liebig-Universität GiessenGiessenGermany
  2. 2.Dipartimento di Matematica Applicata“U.Dini”, Università di PisaPisaItaly
  3. 3.Dipartimento di Metodi e Modelli MatematiciUniversità di Roma “La Sapienza”RomaItaly

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