Semilinear fractional elliptic equations with measures in unbounded domain. (November 2016) - Chen, Huyuan; Yang, Jianfu
Abstract: In this paper, we study the existence of nonnegative weak solutions to ( E ) ( − Δ ) α u + h ( u ) = ν in a general regular domain Ω, which vanish in R N ∖ Ω, where ( − Δ ) α denotes the fractional Laplacian with α ∈ ( 0, 1 ), ν 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 Ω = R N, ν = k δ 0 and h ( s ) = s p, where k > 0, p ∈ ( 0, N N − 2 α ) and δ 0 denotes Dirac mass at the origin. Finally, we show for p ∈ ( 0, 1 + 2 α N ] that u k → ∞ in R N as k → ∞, and for p ∈ ( 1 + 2 α N, N N − 2 α ) that lim k → ∞ u k ( x ) = c ∣ x ∣ − 2 α p − 1 with c > 0, which is a classical solution of ( − Δ ) α u + u p = 0 in R N ∖ { 0 } .