On Non‐local Elliptic Equations with Sublinear Nonlinearities Involving an Eigenvalue Problem

A non‐local elliptic equation of Kirchhoff‐type for with Dirichlet boundary conditions is investigated for the cases where . It is well known that with the non‐local effect removed and , a branch of positive solutions bifurcates from infinity at and no positive solution exists whenever for some (see K. J. Brown, Calc. Var. 22, 483‐494, 2005),where is the principal eigenvalue of the linear problem . As a consequence of the non‐local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for . Moreover, regions with three positive solutions are found for small value of . Comparisons are also made of the results here with those of the elliptic problem in the absence of the non‐local term under the same prescribed conditions using numerical simulations.