On non-local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

A non-local elliptic equation of Kirchhoff-type -(a integral I-omega & nabla;uI(2)dx + 1) delta u= lambda f(x)u + g(x)IuI(gamma-2)u in omega for a, lambda > 0 with Dirichlet boundary conditions is investigated for the cases where 1 < (gamma) < 2. It is well known that with the non-local effect removed and f equivalent to 1, a branch of positive solutions bifurcates from infinity at lambda = lambda(1) and no positive solution exists whenever lambda > lambda for some lambda >= lambda(1) (see K. J. Brown, Calc. Var. 22, 483-494, 2005),where lambda(1) is the principal eigenvalue of the linear problem -delta u = lambda u. 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 lambda > 0. Moreover, regions with three positive solutions are found for small value of a. 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.