Global bifurcation for asymptotically linear Schr?dinger equations

摘要

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schr?dinger equations. The method is topological, based on recent developments of degree theory. We use the inversion u → v:= u/u{double pipe}u{double pipe}~2X in an appropriate Sobolev space X = W~2,p(□~N) and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables (λ, v) ∈ □ × X. This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positiveegative solutions 'bifurcating from infinity'. We believe that, for the values of λ covered by our bifurcation approach, the existence result we obtain for positive solutions of (1) is the most general so far.