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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Multiplicity of positive solutions to the boundary value problems for fractional {Laplacians}},
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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/

[1] J. Byeon, “Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli”, J. Diff. Eqs., 136:1 (1997), 136–165 | DOI | MR | Zbl

[2] L. Caffarelli, L. Silvestre, “An extension problem related to the fractional Laplacian”, Comm. Part. Diff. Eqs., 32:7–9 (2007), 1245–1260 | DOI | MR | Zbl

[3] A. Capella, J. Dávila, L. Dupaigne, Y. Sire, “Regularity of radial extremal solutions for some non-local semilinear equations”, Comm. Part. Diff. Eqs., 36:8 (2011), 1353–1384 | DOI | MR | Zbl

[4] C. V. Coffman, “A non-linear boundary value problem with many positive solutions”, J. Diff. Eqs., 54:3 (1984), 429–437 | DOI | MR | Zbl

[5] A. Cotsiolis, N. K. Tavoularis, “Best constants for Sobolev inequalities for higher order fractional derivatives”, J. Math. Anal. Appl., 295:1 (2004), 225–236 | DOI | MR | Zbl

[6] F. Gazzola, H.-C. Grunau, G. Sweers, “Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions”, Ann. Mat. Pura ed Appl. (4), 189:3 (2010), 475–486 | DOI | MR | Zbl

[7] Y. Ge, “Sharp Sobolev inequalities in critical dimensions”, Michigan Math. J., 51:1 (2003), 27–45 | DOI | MR | Zbl

[8] A. Iannizzotto, S. Mosconi, M. Squassina, “$H^s$ versus $C^0$-weighted minimizers”, NoDEA Nonlinear Differential Equations Appl., 22:3 (2015), 477–497 | DOI | MR | Zbl

[9] Y. Y. Li, “Existence of many positive solutions of semilinear elliptic equations on annulus”, J. Diff. Eqs., 83:2 (1990), 348–367 | DOI | MR | Zbl

[10] R. Musina, A. I. Nazarov, “On fractional Laplacians”, Comm. Part. Diff. Eqs., 39:9 (2014), 1780–1790 | DOI | MR | Zbl

[11] R. Musina, A. I. Nazarov, “On fractional Laplacians – 3”, ESAIM: Control, Optimisation and Calculus of Variations, 22:3 (2016), 832–841 | DOI | MR | Zbl

[12] R. Musina, A. I. Nazarov, “On the Sobolev and Hardy constants for the fractional Navier Laplacian”, Nonlinear Analysis: Theory, Methods Applications, 121 (2015), 123–129 | DOI | MR | Zbl

[13] R. Musina, A. I. Nazarov, “Variational inequalities for the spectral fractional Laplacian”, Comp. Math. Math. Phys., 57:3 (2017), 373–386 | DOI | MR | Zbl

[14] R. S. Palais, “The principle of symmetric criticality”, Comm. Math. Phys., 69:1 (1979), 19–30 | DOI | MR | Zbl

[15] P. R. Stinga, J. L. Torrea, “Extension problem and Harnack's inequality for some fractional operators”, Comm. Part. Diff. Eqs., 35:11 (2010), 2092–2122 | DOI | MR | Zbl

[16] R. C. A. M. Van der Vorst, “Best constant for the embedding of the space $H^2\cap H^1_0(\Omega)$ into $L^{2N/(N-4)}$”, Differential Integral Equations, 6:2 (1993), 259–276 | MR | Zbl

[17] Dzh. N. Vatson, Teoriya besselevykh funktsii, Izdatelstvo inostrannoi literatury, 1949

[18] S. B. Kolonitskii, “Mnozhestvennost reshenii zadachi Dirikhle dlya uravneniya s $p$-laplasianom v trekhmernom sfericheskom sloe”, Algebra i analiz, 22:3 (2010), 206–221 | MR | Zbl

[19] A. Kufner, S. Fuchik, Nelineinye differentsialnye uravneniya, Nauka, 1988 | MR

[20] A. I. Nazarov, “O resheniyakh zadachi Dirikhle uravneniya, soderzhaschego $p$-laplasian, v sfericheskom sloe”, Trudy SPbMO, 10, 2004, 33–62

[21] I. M. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, 1974

[22] Kh. Tribel, Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, 1980 | MR