C ¹-Fredholm Maps and Bifurcation for Quasilinear Elliptic Equations on ℝ^N

Summary

We discuss a broad class of second order quasilinear elliptic operators on ${\mathbb{R}^N}$ acting from the Sobolev space ${W^{2,p}}({\mathbb{R}^N})$ into ${L^p}({\mathbb{R}^N})$ for p ∈ (N∞). Conditions are given which ensure that such operators are C¹—Fredholm maps of index zero. Then we give additional assumptions which imply that they are proper on the closed bounded subsets of ${W^{2,p}}({\mathbb{R}^N})$. For operators with these properties the topological degrees developed by Fitzpatrick, Pejsachowicz and Rabier are available. We illustrate their use by deriving results about the bifurcation of connected components of solutions of quasilinear elliptic equations on ${\mathbb{R}^N}$.