Bifurcation in weighted Sobolev spaces

When P(x, partial derivative) is a second order linear elliptic differential operator on R(N), many bifurcation problems P(x, partial derivative)u - lambda u + f (x, u) = 0 cannot be formulated as a functional equation from W(2,p) := W(2,p)(R(N)) to L(p) := L(p)(R(N)) irrespective of p is an element of[1, infinity], either because the Nemystskii operator f(u) := f (x, u) does not map W(2,p) to L(p) due to the growth of f as vertical bar x vertical bar -> infinity or because, while well defined, f is not Frechet differentiable. Far from being pathological, the latter may happen even when f is C(infinity). In this paper, we show that all these difficulties may often be circumvented by replacing the spaces W(2,p) and L(p) by weighted spaces W(2,p). and L(omega)(p) where omega is an 'admissible' weight and p is an element of (1, infinity), p > N/2. Even though the admissibility of omega depends in part upon f, this still yields a bifurcation theorem in W(2,p) due to the inclusion W(omega)(2,p) hooked right arrow W(2,p). In addition, this approach can be fine tuned to discuss bifurcation in some degenerate elliptic problems after a suitable change of the variables x and u. The problem -vertical bar x vertical bar(4) Delta u + (Q(x)- lambda)u - g(u) = 0 is treated as an example.