Semilinear fractional elliptic equations with measures in unbounded domain

In this paper, we study the existence of nonnegative weak solutions to (E)(-Delta)(alpha)u+h(u) = nu in a general regular domain Omega, which vanish in R-N \ Omega, where (-Delta)(alpha) denotes the fractional Laplacian with alpha is an element of (0,1), nu is a nonnegative Radon measure and h : R+ -> R+ is a continuous nondecreasing function satisfying a subcritical integrability condition. Furthermore, we analyze properties of weak solution u(k) to (E) with Omega = R-N, nu = k delta(0) and h(s) = s(P), where k > 0, p is an element of (0, N/N-2 alpha) and delta(0) denotes Dirac mass at the origin. Finally, we show for p is an element of (0,1 + 2 alpha/N] that u(k) -> infinity in R-N as k -> infinity, and for p E (1 + 2 alpha/N, N/N-2 alpha) that lim(k ->infinity) u(k)(x) = c vertical bar x vertical bar(-2 alpha/p-1) with c > 0, which is a classical solution of (-Delta)(alpha)u + u(P) = 0 in R-N \ {0}. (C) 2016 Elsevier Ltd. All rights reserved.