Global bifurcation for quasilinear elliptic equations on $\mathbb{R}^{N}$

In this paper we discuss the global behaviour of some connected sets of solutions of a broad class of second order quasilinear elliptic equations for where is a real parameter and the function u is required to satisfy the condition The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier. To use this degree the problem must be expressed in the form where J is an interval, X and Y are Banach spaces and F is a map which is Fredholm and proper on closed bounded subsets. We use the usual spaces and . Then the main difficulty involves finding general conditions on and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that and b are asymptotically periodic in x as $\left| x\right| \rightarrow\infty$, we have obtained simple conditions which are necessary and sufficient for to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form our results cover situations outside the scope of other methods in the literature. Received March 30, 1999; in final form January 17, 2000 / Published online February 5, 2001