Let \(L\) be a uniformly elliptic operator and \(a(x)=a^+(x)-\varepsilon a^-(x)\) be a weight of variable sign. This paper deals with positive solutions of \(Lu=\lambda u -a(x)u^2\) in a bounded domain with homogeneous boundary conditions. This type of problem has a rich structure of solutions depending on the parameters \(\lambda\) and \(\varepsilon\). A fairly complete analysis is given based on bifurcation techniques and on comparison principles. Included is a study of stability of the various solutions.