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Revista Matemática Complutense

, Volume 30, Issue 2, pp 281–312 | Cite as

An exterior nonlinear elliptic problem with a dynamical boundary condition

  • Marek FilaEmail author
  • Kazuhiro Ishige
  • Tatsuki Kawakami
Article

Abstract

Several results on existence, nonexistence and large-time behavior of small positive solutions \(u=u(x,t)\) were obtained before for the equation \(-{\varDelta }u=u^p\), \(x\in {\mathbb R}^N_+\), \(t>0\), with a linear dynamical boundary condition. Here \({\varDelta }\) 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 \(N\ge 3\). 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 \(e^{-(N-2)t}\).

Keywords

Semilinear elliptic equation Exterior domain Dynamical boundary condition 

Notes

Acknowledgements

The first author was supported by the Slovak Research and Development Agency under the contract No. APVV-14-0378 and by the VEGA Grant 1/0319/15. The second author was supported by the Grant-in-Aid for Scientific Research (A)(No. 15H02058), from Japan Society for the Promotion of Science. The third author was supported by the Grant-in-Aid for Young Scientists (B) (No. 24740107) and (No. 16K17629) from Japan Society for the Promotion of Science and by the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI”.

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Copyright information

© Universidad Complutense de Madrid 2017

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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