An exterior nonlinear elliptic problem with a dynamical boundary condition

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


Several results on existence, nonexistence and large-time behavior of small positive solutions u= u(x, t) were obtained before for the equation - Δu= up, x∈R+N, t> 0 , with a linear dynamical boundary condition. Here Δ is the N-dimensional Laplacian (in x). We study the effects of the change of the domain from the half-space to the exterior of the unit ball when N≥ 3. We show that the critical exponent for the existence of positive solutions and the decay rate of small solutions are different. More precisely, for the half-space problem the critical exponent is p= (N+ 1) / (N- 1) and the decay rate is t-(N-1), while for the exterior problem we obtain the exponent p= N/ (N- 2) and the exponential rate e-(N-2)t.

Original languageEnglish
Pages (from-to)281-312
Number of pages32
JournalRevista Matematica Complutense
Issue number2
Publication statusPublished - 2017 May 1


  • Dynamical boundary condition
  • Exterior domain
  • Semilinear elliptic equation

ASJC Scopus subject areas

  • Mathematics(all)


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