Fredholm and Propernes Properties of Quasilinear Elliptic Operators on RN

We discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from W^2,p( R ^N) to L^p( R ^N) with N < p < ∞. The unboundedness of the domain makes the standard Sobolev embedding theorems inadequate to investigate such issues. Instead, we develop several new tools and methods to obtain fairly simple necessary and suffcient conditions for such operators to be Fredholm with a given index and to be proper on the closed bounded subsets of W^2,p( R ^N). It is noteworthy that the translation invariance of the domain, well-known to be responsible for the lack of compactness in the Sobolev embedding theorems, is taken advantage of to establish results in the opposite direction and is indeed crucial to the proof of the properness criteria. The limitation to second-order and scalar equations chosen in our exposition is relatively unimportant, as none of the arguments involved here relies upon either of these assumptions. Generalizations to higher order equations or to systems are thus clearly possible with a variableamount of extra work. Various applications, notably but not limited, to global bifurcation problems, are described elsewhere.