Fredholm and properness properties of quasilinear elliptic operators on R-N

We discuss the Fredholm and properness properties of second-order quasiline ar elliptic operators viewed as mappings from W-2,W-p(R-N) to L-p(R-N) with N < p < infinity. The unboundedness of the domain makes the standard Sobol ev embedding theorems inadequate to investigate such issues. Instead, we de velop several new tools and methods to obtain fairly simple necessary and s ufficient conditions for such operators to be Fredholm with a given index a nd to be proper on the closed bounded subsets of W-2,W-p(R-N). It is notewo rthy that the translation invariance of the domain, well-known to be respon sible for the lack of compactness in the Sobolev embedding theorems, is tak en advantage of to establish results in the opposite direction and is indee d crucial to the proof of the properness criteria. The limitation to second -order and scalar equations chosen in our exposition is relatively unimport ant, as none of the arguments involved here relies upon either of these ass umptions. Generalizations to higher order equations or to systems are thus clearly possible with a variable amount of extra work. Various applications , notably but not limited, to global bifurcation problems, are described el sewhere.