TY - JOUR

T1 - An exterior nonlinear elliptic problem with a dynamical boundary condition

AU - Fila, Marek

AU - Ishige, Kazuhiro

AU - Kawakami, Tatsuki

N1 - Funding Information:
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".

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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.

KW - Dynamical boundary condition

KW - Exterior domain

KW - Semilinear elliptic equation

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U2 - 10.1007/s13163-017-0225-6

DO - 10.1007/s13163-017-0225-6

M3 - Article

AN - SCOPUS:85012925874

VL - 30

SP - 281

EP - 312

JO - Revista Matematica Complutense

JF - Revista Matematica Complutense

SN - 1139-1138

IS - 2

ER -