On a class of semilinear fractional elliptic equations involving outside Dirac data

The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation equation(0.1)(−Δ)αu+up=0inB1(eN),u=δ0inRN∖B1(eN), where p≥0p≥0, (−Δ)α(−Δ)α with α∈(0,1)α∈(0,1) denotes the fractional Laplacian operator in the principle value sense, B1(eN)B1(eN) is the unit ball centered at eN=(0,⋯,0,1)eN=(0,⋯,0,1) in RNRN with N≥2N≥2 and δ0δ0 is the Dirac mass concentrated at the origin. We prove that problem (Eq. (0.1)) admits a unique weak solution when p>1+2αN. Moreover, if in addition p≥N+2N−2, the weak solution vanishes as α→1−α→1−. We also show that problem (Eq. (0.1)) does not have any weak solution when p∈[0,1+2αN]. These results are very surprising since they are in total contradiction with the classical setting, i.e.−Δu+up=0inB1(eN),u=δ0inRN∖B1(eN), for which it has been proved that there are no solutions for p≥N+1N−1.