Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results

Abstract

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation left(−Δ)αu=upleft in 1emnormalΩ∖false{0false},leftfalse(−normalΔfalse)αu=0left in 1emRN∖normalΩ,where p>1, α∈(0,1), normalΩ is a bounded C2 domain in RN containing the origin, N⩾2α and the fractional Laplacian false(−normalΔfalse)α is defined in the principle value sense. We prove that any classical solution u of is a very weak solution of left(−Δ)αu=up+kδ0left in 1emnormalΩ,leftfalse(−normalΔfalse)αu=0left in 1emRN∖normalΩfor some k⩾0, where δ0 is the Dirac mass at the origin. In particular, when p⩾NN−2α, we have that k=0; when p∈(1,NN−2α), u has removable singularity at the origin if k=0 and if k>0, u satisfies that trueprefixlimx→0ufalse(xfalse)|x|N−2α=cN,αk,where cN,α>0. Furthermore, when p∈(1,NN−2α), we show that there exists k∗>0 such that problem has at least two solutions for k∈(0,k∗), a unique solution for k=k∗ and no solution for k>k∗.

Harnack‐type inequality for fractional elliptic equations with critical exponent

“…and  3R (x) is the ball in R N+1 with radius 3R and its center at the x;  + 3R =  3R ∩ R N+1 + is the upper half ball; and ′  + 3R is the flat part of  + 3R , which is the ball B 3R in R N . For other results of fractional Laplacian equations, please see some works [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and reference therein.…”

On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

“…We also refer [12,25] for further results in this directions. Nonlocal equations with measure data have been investigated in [2,7,8,10,22] and the references therein. More precisely, fractional elliptic equations with interior measure data were studied in [10,8], while the equations with measure boundary data were carried out in [22] (for absorption nonlinearity) and in [2] (for source nonlinearity).…”

Liouville theorem for the fractional Lane–Emden equation in an unbounded domain

“…By Proposition 2.2 in [14], we have that u p ∈ L q 0 (B 2r 0 (x)) with q 0 = 1 2 (1 + 1 p N N −2α ) > 1 and then…”