Bifurcation for some quasilinear operators

Abstract

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem $$\begin{array}{c} -\mbox{div}\, (A(x,u)\nabla u) = f(\lambda,x, u), \quad \mbox{ in } \Omega , \\u = 0, \quad \mbox{ on } \partial \Omega , \end{array} $$ where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.