The author discusses the problem of existence and multiplicity of non-negative solutions to the problem
where is a smooth bounded domain, a smooth function, and . When the problem is asymptotically linear and the author establishes results on bifurcation from infinity when , the principal eigenvalue of the linear problem
By exploiting the relationship between the Nehari manifold and the fibering maps (maps of the form where is the Euler functional associated to ), the author studies how the Nehari manifold changes as varies. The bifurcation is then described in terms of the sign of the quantity where is the positive eigenfunction of the above linear problem corresponding to .