Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition

Article

Abstract

We consider the semilinear elliptic equation \(-\Delta u=u^p\), \(p>1\), \(u=u(x,t)\), \(x\in {\mathbb R}^N_+\), \(t>0\), with a dynamical boundary condition. We show that, for \(p<(N+1)/(N-1)\), there exist no nontrivial nonnegative local-in-time solutions. Furthermore, in the case \(p>(N+1)/(N-1)\), we determine the optimal slow decay rate at spatial infinity for initial data giving rise to global bounded positive solutions.

Mathematics Subject Classification

35J91 35B40 35J25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsComenius UniversityBratislavaSlovakia
  2. 2.Mathematical InstituteTohoku UniversityAobaJapan
  3. 3.Department of Mathematical SciencesOsaka Prefecture UniversitySakaiJapan

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