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Non-uniqueness of blowing-up solutions to the Gelfand problem

  • Luca Battaglia
  • Massimo GrossiEmail author
  • Angela Pistoia
Article
  • 101 Downloads

Abstract

We consider the Gelfand problem
$$\begin{aligned} \left\{ \begin{array}{ll}-\Delta u=\rho ^2V(x)e^u&{}\quad \text {in }\Omega \\ u=0&{}\quad \text {on }\partial \Omega \end{array}\right. , \end{aligned}$$
where \(\Omega \) is a planar domain and \(\rho \) is a positive small parameter. Under some conditions on the potential \(0<V\in C^\infty \left( {\overline{\Omega }}\right) \), we provide the first examples of multiplicity for blowing-up solutions at a given point in \(\Omega \) as \(\rho \rightarrow 0.\) The argument is based on a refined Lyapunov–Schmidt reduction and the computation of the degree of a finite-dimensional map.

Mathematics Subject Classification

35J60 35J61 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Luca Battaglia
    • 1
  • Massimo Grossi
    • 2
    Email author
  • Angela Pistoia
    • 3
  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomeItaly
  2. 2.Dipartimento di MatematicaSapienza Università di RomaRomeItaly
  3. 3.Dipartimento di Scienze di Base e ApplicateSapienza Università di RomaRomeItaly

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