Application of elliptic regularity to bifurcation in stationary nonlinear Schrödinger equations

Abstract

We establish elliptic regularity results for linear Schrödinger operators in Sobolev spaces W 2,p=W 2,p( R N) yielding their Fredholm properties on intersections W 2, p ∩ W 2, q . In turn, these properties are used to identify a functional setting in which very general bifurcation theorems for stationary nonlinear Schrödinger equations can be proved. Instead of relying upon existing theories for bifurcation in variational problems on Hilbert space, which place stringent limitations upon the admissible nonlinearities, we show that the finite dimensional reduced problem has a natural gradient structure if suitable choices are made to perform the Lyapunov–Schmidt reduction. Bifurcation is then ensured by translating available criteria for a change of Morse index.