Bifurcation theory and related problems:: Anti-maximum principle and resonance

For a smooth bounded domain Omega subset of IRN, we consider the b.v.p. -Deltau = lambdam(x)u + g(lambda, x, u) if x is an element of Omega , u(x) = 0 if x is an element of partial derivative Omega, where m is an element of L-r(Omega) for some r is an element of (max{1, N/2}, + infinity], with m(+) not equal 0 and g is a Caratheodory function. We deduce sufficient and sharp conditions to have subcritical ("to the left") or supercritical ("to the right") bifurcations (either from zero or from infinity) at an eigenvalue Xk(M) of the associated linear weighted eigenvalue problem. Furthermore, as a consequence, we also point out the bifurcation nature of some classical results like the (local) Antimaximum Principle of Clement and Peletier and the Landesman-Lazer theorem for resonant problems. In addition, we see that the bifurcation viewpoint allows to obtain also local maximum principle and more general results for some classes of strongly resonant problems. In addition, we extend the above technique to handle quasilinear b.v.p.