Semilinear fractional elliptic equations with measures in unbounded domain

2014/03/06 by Chen, Huyuan, Yang, Jianfu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.1530

Abstract

In this paper, we study the existence of nonnegative weak solutions to (E) (-Δ)^αu+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=kδ₀ and h(s)=s^p, where k>0, p∈(0,(N)/(N-2α)) and δ₀ denotes Dirac mass at the origin. Finally, we show for p∈(0,1+(2α)/(N)] that u_k→∞ in ℝ^N as k→∞, and for p∈(1+(2α)/(N),(N)/(N-2α)) that lim_k→∞u_k(x)=c|x|^-(2α)/(p-1) with c>0, which is a classical solution of (-Δ)^αu+u^p=0 in ℝ^N∖\0\.

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