Non-local investigation of bifurcations of solutions of non-linear elliptic equations

We justify the projective fibration procedure for functionals defined on Banach spaces. Using this procedure and a dynamical approach to the study with respect to parameters, we prove that there are branches of positive solutions of non-linear elliptic equations with indefinite non-linearities. We investigate the asymptotic behaviour of these branches at bifurcation points. In the general case of equations with $p$-Laplacian we prove that there are upper bounds of branches of positive solutions with respect to the parameter.