Global structure of positive solution sets of nonlinear operator equations

Abstract

In this paper we investigate the existence of unbounded maximal subcontinua of positive solution sets of some nonlinear operator equations that bifurcates from infinity. The main feature of the paper is that the nonlinearity may not be of asymptotically linear type and may not be positone. The methods used to show our main results are different from that of Rabinowitz’ well-known global bifurcation theorems and that of some other related papers. We first obtain a sequence of unbounded subcontinua located in a pipe that bifurcates from nontrivial solutions by using a topological degree argument. Then using a result on subcontinua of superior limit in metric spaces we obtain the main results concerning unbounded maximal subcontinua of positive solution sets of the nonlinear operator equation. The main results can be applied to the semipositone problems to obtain the existence of positive solutions.