A topological approach to nonlocal elliptic partial differential equations on an annulus

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 .