Geodesic
Multiplicity of positive solutions to the boundary value problems for fractional Laplacians
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Tome 459 (2017), pp. 104-126

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We establish the so-called “multiplicity effect” for the problem $(-\Delta)^su=u^{q-1}$ in the annulus $\Omega_R=B_{R+1}\setminus B_R\in\mathbb R^n$: for each $N\in\mathbb N$ there exists $R_0$ such that for all $R \geq R_0$ this problem has at least $N$ different positive solutions. $(-\Delta)^s$ in this problem stands either for Navier-type or for Dirichlet-type fractional Laplacian. Similar results were proved earlier for the equations with the usual Laplace operator and with the $p$-Laplacian operator. 
@article{ZNSL_2017_459_a6,
     author = {N. S. Ustinov},
     title = {Multiplicity of positive solutions to the boundary value problems for fractional {Laplacians}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {104--126},
     publisher = {mathdoc},
     volume = {459},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a6/}
}
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N. S. Ustinov. Multiplicity of positive solutions to the boundary value problems for fractional Laplacians. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Tome 459 (2017), pp. 104-126. http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a6/