Properties of unique positive solution for a class of nonlocal semilinear elliptic equation

Abstract

We study a class of nonlocal elliptic equationswith the Dirichlet boundary conditions in bounded domain. Under suitable assumptions on M and the nonlinear term f , the existence and new properties of a unique positive solutions are obtained via a monotone operator method and a mixed monotone operator method.

Nonzero positive solutions of nonlocal elliptic systems with functional BCs

“…has been studied, for example, by Corrêa and co-authors [11,12], Jiang and Zhai [26] and by Yan and co-authors [43,44,45].…”

“…A widely studied case is the one of Kirchoff-type equations, see for example the review by Ma [31]. Under Dirichlet boundary conditions (BCs) the equation (1.1) − a Ω |u| p dx ∆u = λf (x, u), x ∈ Ω, has been studied, for example, by Corrêa and co-authors [11,12], Jiang and Zhai [26] and by Yan and co-authors [43,44,45]. Here Ω ⊂ R m , m ≥ 1, is a domain with sufficiently smooth boundary, a is a positive function, p ∈ R.…”