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A topological approach to nonlocal elliptic partial differential equations on an annulus
Author(s) -
Goodrich Christopher S.
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201900204
Subject(s) - mathematics , elliptic partial differential equation , mathematical analysis , dirichlet boundary condition , partial differential equation , annulus (botany) , boundary value problem , dirichlet problem , ordinary differential equation , differential equation , cone (formal languages) , first order partial differential equation , class (philosophy) , botany , biology , algorithm , artificial intelligence , computer science
Abstract For q ≥ 1 we consider the nonlocal ordinary differential equation− a ∫ 0 1| y | qd s y ′ ′( t ) = λ f ( t , y ( t ) ) ,0 < t < 1 , subject to the Dirichlet boundary conditions y ( 0 ) = 0 = y ( 1 ) . Due to the term a ∫ 0 1| y | qdsappearing in the equation, this is a class of nonlocal differential equations. By using a novel order cone we are able to establish existence of a positive solution to this problem by means of topological fixed point theory. The preceding problem is really a special case of a more general problem that we consider – namely, the existence of a positive radially symmetric solution to the nonlocal elliptic partial differential equation− a ∫ Ω| u | qd s Δ u ( x ) = λ g ( u ( x ) ) ,x ∈ Ω , subject to u ( x ) ≡ 0 , for x ∈ ∂ Ω , where Ω is an annular region when n ≥ 3 .