TY - JOUR

T1 - Existence of positive solutions of a semilinear elliptic equation 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-0134-10 and by the VEGA grant 1/0319/15. The second author was supported by the Grant-in-Aid for for Scientific Research (B)(No. 23340035), from Japan Society for the Promotion of Science. The third author was supported by the Grant-in-Aid for Young Scientists (B) (No. 24740107) from Japan Society for the Promotion of Science.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/10/22

Y1 - 2015/10/22

N2 - We consider the semilinear elliptic equation – Δu = up, p > 1, u = u (x,t) x ∈ ℝN+, 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)$$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.

AB - We consider the semilinear elliptic equation – Δu = up, p > 1, u = u (x,t) x ∈ ℝN+, 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)$$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.

KW - 35B40

KW - 35J25

KW - 35J91

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U2 - 10.1007/s00526-015-0856-8

DO - 10.1007/s00526-015-0856-8

M3 - Article

AN - SCOPUS:84941993385

SN - 0944-2669

VL - 54

SP - 2059

EP - 2078

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

ER -