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 Zap. Nauchn. Sem. POMI, 2017, Volume 459, Pages 104–126 (Mi znsl6467)

Multiplicity of positive solutions to the boundary value problems for fractional Laplacians

N. S. Ustinov

St. Petersburg State University, St. Petersburg, Russia

Abstract: 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.

Key words and phrases: fractional Laplacians, multiplicity of solutions, Navier Laplacian, Dirichlet Laplacian.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00678A

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UDC: 517

Citation: N. S. Ustinov, “Multiplicity of positive solutions to the boundary value problems for fractional Laplacians”, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Zap. Nauchn. Sem. POMI, 459, POMI, St. Petersburg, 2017, 104–126

Citation in format AMSBIB
\Bibitem{Ust17} \by N.~S.~Ustinov \paper Multiplicity of positive solutions to the boundary value problems for fractional Laplacians \inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~46 \serial Zap. Nauchn. Sem. POMI \yr 2017 \vol 459 \pages 104--126 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6467}