Limit problems for a Fractional p-Laplacian as p -> infinity

作者：Ferreira Raul^*; Perez Llanos Mayte^*

来源：Nodea-Nonlinear Differential Equations and Applications, 2016, 23(2): 14.

DOI：10.1007/s00030-016-0368-z

• Fractional p-Laplacian
• Viscosity solutions
• Nonlocal problems

The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain Omega subset of R-N,
{-integral(RN) vertical bar u(y) - u(x)vertical bar(p-2)(u(y) - u(x))/vertical bar x-y vertical bar(alpha p) dy = f(x, u) x is an element of Omega,
u = g(x) x is an element of R-N\Omega,
where a is an element of(0, 1) and the exponent p goes to infinity. In particular we will analyze the cases:
(i) f = f(x),
(ii) f = f(u) = vertical bar u vertical bar(theta(p)-1)u with 0 < theta(p) < p - 1 and lim(p ->infinity) theta(p)/p-1 =
Theta < 1 with g >= 0.
We show the convergence of the solutions to certain limit p -> infinity and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:
L(infinity)v(x) = L(infinity)(+)v(x) + L(infinity)(-)v(x),
where
L(infinity)(+)v(x) = sup(y is an element of RN) v(y) - v(x)/vertical bar y - x vertical bar(alpha), L(infinity)(-)v(x) = inf(y is an element of RN) v(y) - v(x)/vertical bar y - x vertical bar(alpha).

• 出版日期2016-4