Bifurcation in weighted Sobolev spaces

When P(x, ∂) is a second order linear elliptic differential operator on many bifurcation problems P(x, ∂)u − λu + f(x, u) = 0 cannot be formulated as a functional equation from to irrespective of p ∊ [1, ∞], 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 |x| → ∞ or because, while well defined, f is not Fréchet differentiable. Far from being pathological, the latter may happen even when f is C^∞. 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 and where ω is an 'admissible' weight and p ∊ (1, ∞), p > N/2. Even though the admissibility of ω depends in part upon f, this still yields a bifurcation theorem in W^2,p due to the inclusion . 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 −|x|⁴Δu + (Q(x) − λ)u − g(u) = 0 is treated as an example.