Global bifurcation for quasilinear elliptic equations on R-N

In this paper we discuss the global behaviour of some connected sets of sol utions (lambda, u) of a broad class of second order quasilinear elliptic eq uations - Sigma (N)(alpha,beta =1) a(alpha beta)(x, u(x), delu(x))partial derivativ e (alpha)partial derivative (beta)u(x) + b(x, u(x), delu(x), lambda) = 0 (1 ) for x is an element of R-N where lambda is a real parameter and the functio n u is required to satisfy the condition lim(\x\--> infinity) u(x) = 0. 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 pr oblem must be expressed in the form F : J x X --> Y where J is an interval, X and Y are Banach spaces and F is a C-1 map which is Fredholm and proper on closed bounded subsets. We use the usual spaces X = W-2,W-p(R-N) and Y = L-p(R-N). Then the main difficulty involves finding general conditions on a(alpha beta) and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that a(alpha beta) and b are asymptotically periodic in x as /x/ --> infinity, we have obtain ed simple conditions which are necessary and sufficient for F(lambda,(.)) : X --> Y to be Fredholm and proper on closed bounded subsets of X. In parti cular, 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 ca ses are also discussed. Even for semilinear equations of the form -Deltau(x) + f(x, u(x)) = lambdau(x), our results cover situations outside the scope of other methods in the lite rature.